THE MANIN CONJECTURE IN DIMENSION 2
T.D. BROWNING
Contents
1. Introduction 2
1.1. Notation 5
1.2. The Manin conjectures 6
1.3. Degree 3 surfaces 10
1.4. Degree 4 surfaces 13
1.5. Degree > 5 surfaces 17
1.6. Universal torsors 19
2. Further heuristics 20
2.1. The lines on a cubic surface 21
2.2. Cubic characters and Jacobi sums 23
2.3. The Hardy–Littlewood circle method 25
3. The A1 del Pezzo surface of degree 6 34
3.1. Elementary considerations 35
3.2. The asymptotic formula 38
4. The D4 del Pezzo surface of degree 3 44
4.1. Elementary considerations 45
4.2. A crude upper bound 48
4.3. A better upper bound 50
References 55
Date: November 12, 2018.
2000 Mathematics Subject Classification. 11G35 (14G05, 14G10).
http://arxiv.org/abs/0704.1217v1
2 T.D. BROWNING
1. Introduction
The study of integer solutions to Diophantine equations is a topic that
is almost as old as mathematics itself. Since its inception at the hands of
Diophantus of Alexandria in 250 A.D., it has been found to relate to virtu-
ally every mathematical field. The purpose of these lecture notes is to focus
attention upon an aspect of Diophantine equations that has only crystallised
within the last few decades, and which exhibits a fascinating interplay be-
tween the subjects of analytic number theory and algebraic geometry.
Suppose that we are given a polynomial f ∈ Z[x1, . . . , xn], and write
Sf := {x = (x1, . . . , xn) ∈ Zn \ {0} : f(x) = 0}
for the corresponding locus of non-zero solutions. There are a number of
basic questions that can be asked about the set Sf . When is Sf non-empty?
How large is Sf when it is non-empty? When Sf is infinite can we describe
the set in some way? A lot of the work to date has been driven by trying
to understand the situation for equations in only n = 2 or 3 variables. The
last 50 years in particular has delivered a remarkable level of understanding
concerning the arithmetic of curves. In stark contrast to this, the situation
for equations in 4 or more variables remains a relatively untamed frontier,
with only a scattering of results available.
We will restrict attention to the study of Diophantine equations f = 0
for which the corresponding zero set Sf is infinite. The description that we
will aim for is quantative in nature, the main goal being to understand how
the counting function
N(f ;B) := #{x ∈ Sf : |x| 6 B} (1.1)
behaves, as B → ∞. Here, as throughout these lecture notes, |z| denotes the
norm max16i6n |zi| for any z ∈ Rn. Aside from being intrinsically interesting
in their own right, as we will see shortly, the study of functions likeN(f ;B) is
often an effective means of determining whether or not the equation f = 0
has any non-trivial integer solutions at all. In many applications of the
Hardy–Littlewood circle method, for example, one is able to prove that Sf
is non-empty by showing that N(f ;B) > 0 for large enough values of B. In
fact the method usually carries with it a proof of the fact that Sf is infinite.
In the context of the circle method at least, it is useful to have a general
idea of which polynomials f might have an infinite zero locus Sf .
Suppose that f ∈ Z[x1, . . . , xn] has degree d > 1. Then for the vectors
x ∈ Zn counted by N(f ;B), the values of f(x) will all be of order Bd.
In fact a positive proportion of them will have exact order Bd. Thus the
probability that a randomly chosen value of f(x) should vanish might be
expected to be of order 1/Bd. Since the number of x to be considered has
order Bn, this leads us to the following general expectation.
Heuristic. When n > d we have
Bn−d ≪ N(f ;B) ≪ Bn−d. (1.2)
As a crude first approximation, therefore, this heuristic tells us that we
might expect polynomials whose degree does not exceed the number of vari-
ables to have infinitely many solutions. Unfortunately there are a number
THE MANIN CONJECTURE IN DIMENSION 2 3
of things that can conspire to upset this heuristic expectation. First and
foremost, local conditions will often provide a reason for N(f ;B) to be iden-
tically zero no matter the values of d and n. By local obstructions we mean
that the obvious necessary conditions for Sf to be non-empty fail. These
are the conditions that the equation f(x) = 0 should have a real solution
x ∈ Rn, and secondly, that the congruence
f(x) ≡ 0 (mod pk)
should be soluble for every prime power pk. When f is homogeneous we
must take care to ignore the trivial solution x = (0, . . . , 0) in both cases.
It is quite easy to construct examples that illustrate the failure of these
local conditions. For example, when d is even, the equation
x2d1 + · · ·+ x2dn = 0
doesn’t have any integer solutions, since it patently doesn’t have any real
solutions. Let us now exhibit an example, due to Mordell [50], of a poly-
nomial equation that fails to have integer solutions because it fails to have
solutions as a congruence modulo a prime p. Let K be an algebraic number
field of degree d, with ring of integers OK . Write
N(y1, . . . , yd) := NK/Q(y1ω1 + · · · + ydωd)
for the corresponding norm form, where ω1, . . . , ωd is a basis for K over Q.
It is clear that N is a homogeneous polynomial of degree d, with coefficients
in Z.
Exercise 1. Let y ∈ Zn and let p be a rational prime such that the ideal
(p) ⊂ OK is prime. Show that p | N(y) if and only if p | y.
We define the homogeneous polynomial
f1 := N(x1, . . . , xd) + pN(xd+1, . . . , x2d) + · · ·+ pd−1N(xd2−d+1, . . . , xd2),
(1.3)
which has degree d and d2 variables. We claim that the only integer solution
to the equation f1(x) = 0 is the trivial solution x = 0. To see this we argue
by contradiction. Thus we suppose there to be a vector x ∈ Zd2 such that
f1(x) = 0, with gcd(x1, . . . , xd2) = 1. Viewed modulo p we deduce that
p | N(x1, . . . , xd), whence p | x1, . . . xd by Exercise 1. Writing xi = pyi for
1 6 i 6 d, and substituting into the equation f1 = 0, we deduce that
pd−1N(y1, . . . , yd) +N(xd+1, . . . , x2d) + · · ·+ pd−2N(xd2−d+1, . . . , xd2) = 0.
But then we deduce in a similar fashion that p | xd+1, . . . , x2d. We may
clearly continue in this fashion, ultimately concluding that p | x1, . . . , xd2 ,
which is a contradiction. This polynomial illustrates that for any d it is
possible to construct examples of homogeneous polynomials in d2 variables
that have no non-zero integer solutions. This fits with the facts rather well:
when d = 2 we know from Meyer’s theorem that an indefinite quadratic form
always has non-trivial solutions as soon as its rank is at least 5. Similarly,
it is conjectured that 10 variables are always enough to ensure the solubility
in integers of an arbitrary homogeneous cubic equation.
So far we have only seen examples of polynomials f for which the zero
locus Sf is empty. In this case the corresponding counting function N(f ;B)
4 T.D. BROWNING
is particularly easy to estimate! There are also examples which show that
N(f ;B) may grow in quite unexpected ways, even when n > d. An equation
that illustrates excessive growth is provided by the polynomial
f2 := x
1 − x2(xd−13 + · · · + x
Here there are “trivial” solutions of the type (0, 0, a3, . . . , an) which already
contribute ≫ Bn−2 to the counting function N(f ;B), whereas (1.2) predicts
that we should have exponent n− d.
It is also possible to construct examples of varieties which demonstrate
inferior growth. Let n > d2 and choose any d2 linear forms L1, . . . , Ld2 ∈
Z[x1, . . . , xn] that are linearly independent over Q. Consider the form
f3 := f1(L1(x1, . . . , xn), . . . , Ld2(x1, . . . , xn)),
where f1 is given by (1.3). Then it is clear that N(f3;B) has the same order
of magnitude as the counting function associated to the system of linear
forms L1 = · · · = Ld2 = 0. Since these forms are linearly independent we
deduce that N(f3;B) has order of magnitude B
n−d2 , whereas (1.2) led us
to expect an exponent n− d.
We have seen lots of reasons why (1.2) might fail — how about some
evidence supporting it? One of the most outstanding achievements in this
direction is the following very general result due to Birch [4].
Theorem 1.1. Suppose f ∈ Z[x1, . . . , xn] is a non-singular homogeneous
polynomial of degree d in n > (d−1)2d variables. Assume that f(x) = 0 has
non-trivial solutions in R and each p-adic field Qp. Then there is a constant
cf > 0 such that
N(f ;B) ∼ cfBn−d,
as B → ∞.
Birch’s result doesn’t apply to either of the polynomials f2, f3 that we con-
sidered above, since both of these actually have a rather large singular locus.
Since generic homogeneous polynomials are non-singular, Birch’s result an-
swers our initial questions completely for typical forms with n > (d− 1)2d.
It would be of considerable interest to reduce the lower bound for n, but ex-
cept for d 6 4 this has not been done. Theorem 1.1 is established using the
circle method, and exhibits a common feature of all Diophantine problems
successfully tackled via this machinery: the number of variables involved
needs to be large compared to the degree. In particular, there is an obvious
disparity between the range for n in Birch’s result and the range for n in
(1.2). The main aim of these lecture notes is to discuss the situation when
n is comparable in size with d.
It turns out that phrasing things in terms of single polynomial equations
is far too restrictive. It is much more satisfactory to work with projective
algebraic varieties V ⊆ Pn−1. All of the varieties that we will work with
are assumed to be cut out by a finite system of homogeneous equations,
all of which are defined over Q. In line with the above, our main interest
lies with those varieties for which we expect the set V (Q) = V ∩ Pn−1(Q)
to be infinite. Let x = [x] ∈ Pn−1(Q) be a projective rational point, with
x = (x1, . . . , xn) ∈ Zn chosen so that gcd(x1, . . . , xn) = 1. Then we define
THE MANIN CONJECTURE IN DIMENSION 2 5
the height of x to be H(x) := |x|, where as usual |z| denotes the norm
max16i6n |zi|. Given any subset U ⊆ V , we may then define the counting
function
NU (B) := #{x ∈ U(Q) : H(x) 6 B}, (1.4)
for each B > 1. The main difference between this counting function and the
quantity introduced in (1.1) is that we are now only interested in primitive
integer solutions, by which we mean that the components of the vector
x ∈ Zn should share no common prime factors. When the polynomial in
(1.1) is homogeneous, this formulation has the advantage of treating all
scalar multiples of a given non-zero integer solution as a single point.
Recall the definition of the Möbius function µ : N → {0, 1}, which is given
µ(n) =
0, if p2 | n for some prime p,
1, if n = 1,
(−1)r, if n = p1 · · · pr for distinct primes p1, . . . , pr.
The Möbius function is a multiplicative arithmetic function, and will play a
very useful rôle in our work.
Exercise 2. Let S ⊆ Zn be an arbitrary set. Show that
#{x ∈ S : gcd(x1, . . . , xn) = 1} =
µ(k)#{x ∈ S : k | xi, (1 6 i 6 n)}.
We now have the tools with which to relate the counting function (1.4) to
our earlier counting function N(f ;B) in (1.1), when U = V and V ⊂ Pn−1 is
a hypersurface with underlying homogeneous polynomial f ∈ Z[x1, . . . , xn].
On noting that x and −x represent the same point in Pn−1, it follows from
Exercise 2 that
NV (B) =
µ(k)N(f ;B/k). (1.5)
When f is non-singular of degree d, with n > (d − 1)2d, it can be deduced
from Theorem 1.1 that NV (B) ∼ c̃fBn−d, where c̃f = 12ζ(n− d)
−1cf .
Returning to the counting function (1.4), it is easy to check that NU (B)
is bounded for each B, no matter what the choice of U and V . This fol-
lows on combining the fact that NV (B) 6 NPn−1(B) with the self-evident
inequalities NPn−1(B) 6 #{x ∈ Zn : |x| 6 B} 6 (2B + 1)n. In fact it is not
so hard to establish an asymptotic formula for NPn−1(B).
Exercise 3. Let n > 2. Use Exercise 2 to show that
NPn−1(B) =
Bn +On
Bn−1(logB)bn
where b2 = 1 and bn = 0 for n > 2.
1.1. Notation. Before embarking on the main thrust of these lecture notes,
we take a moment to summarise some of the key pieces of notation that we
will make use of.
6 T.D. BROWNING
• A(x) = O(B(x)) means that there exists a constant c > 0 and x0 ∈ R
such that |A(x)| 6 cB(x) for all x > x0. Throughout our work we
will follow the convention that the implied constant is absolute unless
explicitly indicated otherwise by an appropriate subscript. We will
often use the alternative notation A(x) ≪ B(x) or B(x) ≫ A(x).
• A(x) ≍ B(x) means A(x) ≪ B(x) ≪ A(x).
• A(x) = o(B(x)) means limx→∞A(x)/B(x) = 0.
• A(x) ∼ B(x) means limx→∞A(x)/B(x) = 1.
• N = {1, 2, 3, ...} will denote the set of natural numbers.
• Zn will denote the set of primitive vectors in Zn, and Zn∗ will denote
the set of v ∈ Zn such that v1 · · · vn 6= 0.
• |z| := max16i6n |zi|, for any vector z ∈ Rn.
1.2. The Manin conjectures. Around 1989 Manin initiated a program
to relate the asymptotic behaviour of counting functions to the intrinsic
geometry of the underlying variety, for suitable families of algebraic varieties.
It is precisely this rich interplay between arithmetic and geometry that this
set of lecture notes aims to communicate.
Several of the varieties that we have looked at so far have many rational
points, in the sense that NV (B) grows like a power of B. For such varieties
it is natural to look at the quantity
βV := lim
logNV (B)
assuming that this limit exists. In general we may consider βU for any
Zariski open subset U ⊆ V . It is clear that βU gives a measure of “how
large” the set U(Q) is, since we will have
BβU−ε ≪ NU (B) ≪ BβU+ε
for sufficiently large values of B and any ε > 0. The insight of Manin was
to try and relate βU to the geometry of V via the introduction of a certain
quantity α(V ). Before defining this quantity we will need some facts from
algebraic geometry. The facts that we will need are summarised in more
detail in the book of Hindry and Silverman [44, §A].
Assume that V ⊂ Pn−1 is non-singular and let Div(V ) be the free abelian
group generated by finite formal sums of the shape D =
nY Y , with nY ∈
Z and Y running over geometrically irreducible codimension 1 subvarieties
of V . A divisor D ∈ Div(V ) is effective if nY > 0 for all Y , and D is
said to be principal if D =
Y ordY (f)Y = Df , say, for some rational
function f ∈ C(V ). The intuitive idea behind the definition of the ordY
function for a codimension 1 subvariety Y is that ordY (f) = k if f has a
zero of order k along Y , while ordY (f) = −k if f has a pole of order k along
Y . If f has neither a zero nor a pole along Y , then ordY (f) = 0. Since
Df + Dg = Dfg and D1/f = −Df , the principal divisors form a subgroup
PDiv(V ) of Div(V ). We define the geometric Picard group associated to V
to be
(V ) := Div(V )/PDiv(V ).
A divisor class [D] ∈ PicQ(V ) is effective if there exists an effective di-
visor in the class. One may also construct the geometric Néron–Severi
THE MANIN CONJECTURE IN DIMENSION 2 7
group NSQ(V ), which is Div(V ) modulo a further equivalence relation called
“algebraic equivalence”. When V is covered by curves of genus zero, as
in all the cases of interest to us in these lecture notes, it turns out that
NSQ(V ) = PicQ(V ). We illustrate the definition of PicQ(V ) by calculating
it in the simplest possible case V = Pn−1.
Lemma 1.1. We have Pic
(Pn−1) = Z.
Proof. An irreducible divisor on Pn−1 has the form Y = {F = 0} for some
absolutely irreducible form F ∈ C[x1, . . . , xn]. For such a divisor, define the
degree of Y to be degY = degF . Extend the definition of degree additively,
so that
nY deg Y.
The map deg : Div(Pn−1) → Z is clearly a homomorphism, and to establish
the lemma it will suffice to show that the kernel of this map is precisely the
subgroup PDiv(Pn−1). To see this, we note that degDf = 0 for any rational
function f = F1/F2. Indeed, the sum of the positive degree terms will be
degF1, whereas the sum of the negative degree terms will be degF2, and this
two degrees must coincide in order to have a well-defined rational function.
Conversely, if D = n1Y1 + · · · + nkYk has degree zero, with Yi = {Fi = 0}
for 1 6 i 6 k, then f = F
1 · · ·F
k is a well-defined rational function on
Pn−1 with Df = D. This completes the proof of the lemma. �
Returning to the setting of arbitrary non-singular varieties V ∈ Pn−1, let
H ∈ Div(V ) be a divisor corresponding to a hyperplane section. Further-
more, let KV ∈ Div(V ) be the canonical divisor. This is a common abuse
of notation: really KV refers to the class of Dω in PicQ(V ) for any differen-
tial (dimV )-form ω of V . It would take us too far afield to include precise
definitions of these objects here. We may now define the real number
α(V ) := inf{r ∈ R : r[H] + [KV ] ∈ Λeff(V )},
where
Λeff(V ) := {c1[D1] + · · ·+ ck[Dk] : ci ∈ R>0, [Di] ∈ NSQ(V ) effective}
is the so-called effective cone of divisors. It does not matter too much if this
definition is currently meaningless: the main thing is that α(V ) depends in
an explicit way on the geometry of V over C. We now have the following
basic conjecture due to Batyrev and Manin [1, Conjecture A].
Conjecture 1.1. For all ε > 0 there exists a Zariski open subset U ⊆ V
such that βU 6 α(V ) + ε.
A non-singular variety V ⊂ Pn−1 is said to be Fano if KV does not lie in
the closure of the effective cone Λeff(V ) ⊂ NSQ(V )⊗Z R. This is equivalent
to −KV being ample, and implies in particular that V is covered by rational
curves. As an example, suppose that V is a complete intersection, with
V = W1 ∩ · · · ∩ Wt for hypersurfaces Wi ⊂ Pn−1 of degree di. Then V is
Fano if and only if d1+ · · ·+dt < n. With this in mind we have the following
supplementary prediction.
8 T.D. BROWNING
Conjecture 1.2. Assume that V is Fano and V (Q) is Zariski dense in V .
Then there exists a Zariski open subset U ⊆ V such that βU = α(V ).
We have α(V ) = n − d1 − · · · − dt when V is a non-singular complete
intersection as above. In particular, when V is a hypersurface of degree
d we may deduce from Theorem 1.1 that Conjecture 1.2 holds when n is
sufficiently large in terms of d. It also holds for n > 3 when d = 2 (see
Heath-Brown [37], for example). Finally we remark that Conjecture 1.2
holds for projective space. This follows from Exercise 3 and the fact that
[KPn−1 ] = [−nH] in PicQ(Pn−1), whence α(Pn−1) = n.
The title of these lecture notes suggests that we will focus our attention
on the situation for varieties of dimension 2. Before doing so, let us consider
the situation for curves briefly. For simplicity we will discuss only projective
plane curves V ⊂ P2 of degree d. There is a natural trichotomy among
such curves, according to the genus g of the curve. For curves with g = 0,
otherwise known as rational curves, it is possible to show that NV (B) ∼
2/d. This is in complete accordance with the Manin conjecture. It is
an amusing exercise to check that such an asymptotic formula holds with
d = 2 when V is given by the equation x21 + x
2 = x
3, for example. When
g = 1 and V (Q) 6= ∅, the curve is elliptic and it has been shown by Néron
[44, Theorem B.6.3] that
NV (B) ∼ cV (logB)rV /2,
where rV denotes the rank of V . Thus although there can be infinitely many
points in V (Q), we see that the corresponding counting function grows much
more slowly than for rational curves. Elliptic curves are not Fano, and
so this is not covered by the Manin conjecture. However it does confirm
Conjecture 1.1, since α(V ) = 0. When g > 2 the work of Faltings [28] shows
that V (Q) is always finite, and so it does not make sense to study NV (B).
Let us now concern ourselves with Fano varieties of dimension 2. We
begin with some simple-minded numerics. Suppose that we are given a Fano
variety V of dimension 2 and degree d, which is a non-singular complete
intersection in Pn−1. Thus V = W1 ∩ · · · ∩Wt for hypersurfaces Wi ⊂ Pn−1
of degree di, and we assume that the intersection is transversal at a generic
point of V . We are not interested in hyperplane sections of V , and so we
will assume without loss of generality that di > 2 for each 1 6 i 6 t. Then
the following inequalities must be satisfied:
(1) d1 + · · ·+ dt < n, [Fano]
(2) n− 1− t = 2, [complete intersection of dimension 2]
(3) d = d1 · · · dt, [Bézout]
(4) dt > · · · > d1 > 2.
It follows that the only possibilities are
(d; d1, . . . , dt;n; t) ∈
(2; 2; 4; 1), (3; 3; 4; 1), (4; 2, 2; 5; 2)
These surfaces correspond to a quadric in P3, a cubic surface in P3, and an
intersection of 2 quadrics in P4, respectively. We have already observed that
the Manin conjecture holds for quadrics. Hence one would like to examine
the latter two surfaces. In fact these are the most familiar examples of
“del Pezzo surfaces”. We will see in §1.5 that not all del Pezzo surfaces are
THE MANIN CONJECTURE IN DIMENSION 2 9
complete intersections, and so we have missed out on several surfaces in this
analysis. Nonetheless, a substantial portion of these lecture notes will focus
on cubic surfaces in P3 and intersections of 2 quadrics in P4.
It is now time to give a formal definition of a del Pezzo surface. Let us
begin with a discussion of non-singular del Pezzo surfaces. Let d > 3. Then
a del Pezzo surface of degree d is a non-singular surface S ⊂ Pd of degree
d, with very ample anticanonical divisor −KS. This latter condition is
equivalent to the equality [−KS ] = [H] in PicQ(S), for a hyperplane section
H ∈ Div(S). The facts that we will recall here are all established in the
book of Manin [49], for example. It is well-known that del Pezzo surfaces
S ⊂ Pd arise either as the quadratic Veronese embedding of a quadric in P3,
which is a del Pezzo surface of degree 8 in P8 (isomorphic to P1 × P1), or
as the blow-up of P2 along 9 − d points in general position, in which case
the degree of S satisfies 3 6 d 6 9. We will meet the notion of “general
position” when d = 3 in §2.1. Since [−KS ] = [H] in PicQ(S), we see that
α(S) = 1 for non-singular del Pezzo surfaces of degree d.
The geometry of del Pezzo surfaces is very beautiful and well-worth study-
ing. However, to avoid straying from the main focus of these lecture notes,
we will content ourselves with simply quoting the facts that are needed.
One of the remarkable features of del Pezzo surfaces of small degree is that
each such surface contains finitely many lines. The precise number of lines
is recorded in Table 1.
d number of lines
Table 1. Lines on non-singular del Pezzo surfaces of degree d
It turns out that dealing with del Pezzo surfaces of degree d gets easier
as the degree increases. In these lecture notes we will focus our attention
on the del Pezzo surfaces of degree d ∈ {3, 4, 5, 6}. It turns out that for del
Pezzo surfaces of degree d, the geometric Picard group PicQ(S) is a finitely
generated free Z-module, with
PicQ(S)
∼= Z10−d. (1.6)
This is established in Manin [49], where an explicit basis for the group is
also provided (see §2.1 for a concrete example). Let K be a splitting field
for the finitely many lines contained in S. The final invariant that we will
need to introduce is the Picard group
Pic(S) := Pic
(S)Gal(K/Q) (1.7)
of the surface. This is just the set of elements in PicQ(S) that are fixed by
the action of the Galois group. Write ρS for the rank of Pic(S). Let U ⊂ S
be the Zariski open subset formed by deleting the finitely many lines from
S. Then we have the following [1, Conjecture C ′].
10 T.D. BROWNING
Conjecture 1.3. Suppose that S ⊂ Pd is a non-singular del Pezzo surface
of degree d. Then there exists a non-negative constant cS,H such that
NU (B) = cS,HB(logB)
1 + o(1)
. (1.8)
In these lecture notes this is what will commonly be termed as “the Manin
conjecture”. Note that the exponent of B agrees with Conjecture 1.2, since
α(S) = 1. Moreover the exponent of logB is at most 9 − d, since the
geometric Picard group has rank 10 − d. We will develop some heuristics
to support this power of logB in §2. The value of the constant cS,H has
also received a conjectural interpretation at the hands of Peyre [51], an
interpretation that has been extended by Batyrev and Tschinkel [2], and by
Salberger [55].
There are a number of refinements to Conjecture 1.3 that are currently
emerging, which we will not have space to discuss here. Some of these are
discussed in more details in the author’s survey [14, §2], for example. One
such refinement is that there should exist a polynomial P ∈ R[x] of degree
ρS − 1, and a real number δ > 0, such that
NU (B) = BP (logB) +O(B
1−δ). (1.9)
One obviously expects the leading coefficient of P to agree with Peyre’s
prediction, but there has so far been rather little investigation of the lower
order terms. All of the del Pezzo surfaces that we have discussed so far have
been non-singular. In the following section we will meet some singular ones.
1.3. Degree 3 surfaces. The del Pezzo surfaces S ⊂ P3 of degree 3 are the
geometrically integral cubic surfaces in P3, which are not ruled by lines. In
particular, this definition covers both singular and non-singular del Pezzo
surfaces of degree 3. Given such a surface S defined over Q, we may always
find an absolutely irreducible cubic form C ∈ Z[x1, x2, x3, x4] such that S
is defined by the equation C = 0. In this section we will discuss the Manin
conjecture in the context of cubic surfaces. Let us begin by considering the
situation for non-singular cubic surfaces, for which one takes U ⊂ S to be
the open subset formed by deleting the famous 27 lines. Peyre and Tschinkel
[53, 54] have provided ample numerical evidence for the validity of the Manin
conjecture for diagonal cubic surfaces. However we are still rather far away
from proving it for any single example. The best upper bound available is
NU (B) = Oε,S(B
4/3+ε), (1.10)
due to Heath-Brown [38]. This applies when the surface S contains 3 copla-
nar lines defined over Q, and in particular to the Fermat cubic surface
x31 + x
2 = x
3 + x
Heath-Brown [40] has extended the bound (1.10) to all non-singular cubic
surfaces, subject to a natural conjecture concerning the size of the rank of
elliptic curves over Q.
The problem of proving lower bounds is somewhat easier. Under the as-
sumption that S contains a pair of skew lines defined over Q, Slater and
Swinnerton-Dyer [59] have shown that NU (B) ≫S B(logB)ρS−1, as pre-
dicted by the Manin conjecture. This does not apply to the Fermat cubic
THE MANIN CONJECTURE IN DIMENSION 2 11
surface, however, since the only skew lines contained in this surface are
defined over Q(
It turns out that much more can be said if one permits S to contain
isolated singularities. For the remainder of this section let S ⊂ P3 be a
geometrically integral cubic surface, which has only isolated singularities
and is not a cone. Then there exists a unique “minimal desingularisation”
π : S̃ → S of the surface, which is just a sequence of blow-up maps, and
furthermore, that the asymptotic formula (1.8) is still expected to hold, with
ρS now taken to be the rank of the Picard group of S̃. As usual U ⊂ S is
obtained by deleting all of the lines from S. The classification of singular
cubic surfaces S is a well-established subject, and can be traced back to the
work of Cayley [18] and Schläfli [56] over a century ago. A contemporary
classification of singular cubic surfaces has since been given by Bruce and
Wall [16], over Q. Of course, if one is interested in a classification over the
ground field Q, then many more singularity types can occur (see Lipman
[48], for example). In Table 2 we have provided a classification table of the
20 singularity types over Q, including the number of lines that each surface
contains. We will presently meet some explicit examples of cubic forms
C ∈ Z[x1, x2, x3, x4] that typify some surface types.
type # lines singularity
i 21 A1
ii 16 2A1
iii 15 A2
iv 12 3A1
v 11 A1 +A2
vi 10 A3
vii 9 4A1
viii 8 2A1 +A2
ix 7 A1 +A3
x 7 2A2
xi 6 A4
xii 6 D4
xiii 5 2A1 +A3
xiv 5 A1 + 2A2
xv 4 A1 +A4
xvi 3 A5
xvii 3 D5
xviii 3 3A2
ix 2 A1 +A5
xx 1 E6
Table 2. Classification (over Q) of singular del Pezzo sur-
faces of degree 3 in P3
The labelling of each singularity type corresponds to the “Dynkin dia-
gram” that describes the intersection behaviour of the exceptional divisors
obtained by resolving the singularities in the surface. For example, consider
12 T.D. BROWNING
the cubic surface
S1 = {x21x3 + x2x23 + x34 = 0}. (1.11)
Up to isomorphism over Q this is the unique cubic surface of type xx in
the table, and is discussed further in [34]. The process of resolving the
singularity gives 6 exceptional divisors E1, . . . , E6 and produces the minimal
desingularisation S̃1 of the surface S1. If L denotes the strict transform of
the unique line on S1, then L,E1, . . . , E6 satisfy the intersection behaviour
encoded in the Dynkin diagram
E1 E3 E6 E5 E4 L
There is a line connecting two divisors in this diagram if and only if they
meet in S̃1. In what follows the reader can simply think of these Dynkin
diagrams as a convenient way to label the surface type.
It turns out, as discussed in [16], that some types of surfaces do not have
a single normal form, but an infinite family. This happens precisely for the
surfaces of type i, ii, iii, iv, v, vi and ix. By [16, Lemma 4] the type xii
surface, with a D4 singularity, is the only surface that has more than one
normal form, but not a family. In fact it has precisely two normal forms,
given by
S2 = {x1x2(x1 + x2) + x4(x1 + x2 + x3)2 = 0} (1.12)
S3 = {x1x2x3 + x4(x1 + x2 + x3)2 = 0}. (1.13)
That these equations actually define distinct surfaces can be seen by calcu-
lating the corresponding Hessians in each case.
Let S̃ denote the minimal desingularisation of any surface S from Table 2,
and assume that all of its singularities and lines are defined over Q. In this
case the surface is said to be split, and it follows that the Picard group
of S̃ has maximal rank 7 by (1.6), since Pic(S̃) = Pic
(S̃). For example,
[L], [E1], . . . , [E6] provide a basis for Pic(S̃1). One would like to try and
establish (1.8) for each such surface S, with ρS = 7. Several del Pezzo
surfaces are actually special cases of varieties for which the Manin conjecture
is already known to hold. Recall that a variety of dimension D is said
to be toric if it contains the algebraic group variety GDm as a dense open
subset, whose natural action on itself extends to all of the variety. The
Manin conjecture has been established for all toric varieties by Batyrev and
Tschinkel [3]. It can be checked that the surface representing type xviii is
toric. In fact this particular surface has been studied by numerous authors,
including la Bretèche [5], la Bretèche and Swinnerton-Dyer [11], Fouvry
[29], Heath-Brown and Moroz [43], and Salberger [55]. Of the unconditional
asymptotic formulae obtained, the most impressive is the first. This consists
of an estimate like (1.9) for any δ ∈ (0, 1/8), with degP = 6.
The next surface to have received serious attention is the Cayley cubic
surface
S4 = {x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 = 0},
THE MANIN CONJECTURE IN DIMENSION 2 13
which is the type vii surface in the table. Heath-Brown [42] has shown that
there exist absolute constants A1, A2 > 0 such that
A1B(logB)
6 NU (B) 6 A2B(logB)
An estimate of precisely the same form has been obtained by the author [13]
for the D4 surface S3 in (1.13). In both cases the lines in the surface are
all defined over Q, so that the surfaces are split. Thus the corresponding
Picard groups have rank 7 and the exponents of B and logB agree with
Manin’s prediction. In this set of lecture notes we will establish an upper
bound for the remaining D4 cubic surface S2 in (1.12). This will be carried
out in §4 in two basic attacks. First we will give a completely self-contained
account of the upper bound NU (B) = Oε(B
1+ε), for any ε > 0. Next, by
making use of the work in [13], we will establish the following finer result.
Theorem 1.2. Let S2 be given by (1.12). We have NU (B) ≪ B(logB)6.
The cubic surface S2 contains the unique singular point [0, 0, 0, 1], together
with the 6 lines
xi = x4 = 0, x1 + x2 = xj = 0, xi = x1 + x2 + x3 = 0, (1.14)
for distinct indices i ∈ {1, 2} and j ∈ {3, 4}. Thus the surface is split and it
follows that Theorem 1.2 agrees with the Manin conjecture.
Exercise 4. Check that (1.14) are all of the lines contained in S2.
The final surface to have been studied extensively is the E6 cubic surface
S1 that we discussed above. The figure below, which was constructed by Der-
enthal, shows all the rational points of height 6 1000 on this surface. Recent
joint work of the author
with la Bretèche and Der-
enthal [12] has succeeded
in establishing the Manin
conjecture for this surface.
In fact an asymptotic for-
mula of the shape (1.9) is
achieved, with P of degree
6 and any δ ∈ (0, 1/11).
It should be remarked that
Dr. Michael Joyce has
also established the Manin
conjecture for S1 in his
doctoral thesis, albeit with
a weaker error term of O(B(logB)5).
1.4. Degree 4 surfaces. A quartic del Pezzo surface S ⊂ P4, that is defined
over Q, can be recognised as the zero locus of a suitable pair of quadratic
forms Q1, Q2 ∈ Z[x1, . . . , x5]. Again we do not stipulate that the surface
should be non-singular. As usual let U ⊂ S denote the open subset formed
by deleting all of the lines from S. Let us begin by discussing the situation
for non-singular surfaces, where there are 16 lines to delete. The best result
available is the estimate NU (B) = Oε,S(B
1+ε), valid for any quartic non-
singular del Pezzo surface S ⊂ P4 containing a conic defined over Q. This
14 T.D. BROWNING
result was established in an unpublished note due to Salberger in 2001. It
would be interesting to see whether one could adapt the methods of [40] to
show that NU (B) = Oε,S(B
5/4+ε) for any non-singular del Pezzo surface of
degree 4, assuming the same hypothesis on the ranks of elliptic curves.
As previously, it emerges that much more can be said if one permits S to
contain isolated singularities. For the remainder of this section let S ⊂ P4
be a geometrically integral intersection of two quadric hypersurfaces, which
has only isolated singularities and is not a cone, and let S̃ be the minimal
desingularisation of S. Then the asymptotic formula (1.8) is still expected
to hold, with ρS now taken to be the rank of the Picard group of S̃, and
U ⊂ S obtained by deleting all of the lines from S. In particular, when S is
split one always has ρS = 6. The classification of singular quartic del Pezzo
surfaces can be extracted from the work of Hodge and Pedoe [45, Book IV,
§XIII.11], where it is phrased in terms of the so-called Segre symbol. The
Segre symbol of a matrix M ∈ M5(C) is defined as follows. If the Jordan
form of M has Jordan blocks of sizes a1, . . . , an, with a1+ · · ·+an = 5, then
the Segre symbol is the symbol
(a1, . . . , an)
with extra parentheses around the Jordan blocks with equal eigenvalues.
Suppose that our quartic del Pezzo surface S is defined by a pair of quadric
hypersurfaces, with underlying symmetric matrices A,B ∈ M5(Q). Then
the Segre symbol of S is defined to be the Segre symbol associated to A−1B.
A crucial property of the Segre symbol is that it does not depend on the
choice ofA andB in the pencil of quadrics defining S. Since we are assuming
that S is not a cone, one may always suppose that A,B are chosen so that
A has full rank.
To illustrate the calculation of the Segre symbol, let us consider the sur-
face S defined by the pair of equations
x1x2 + x3x4 = 0, x1x4 + x2x3 + x3x5 + x4x5 = 0. (1.15)
Let A,B ∈ M5(Q) denote the underlying matrices of the first and second
equations, respectively. Then A has rank 4, and so we replace it with
A + 2B, which has full rank. A simple calculation reveals that the matrix
(A+ 2B)−1B has Jordan form

0 0 0 0 0
0 1 0 0 0
0 0 1
0 0 0 1
0 0 0 0 1

This matrix has 4 Jordan blocks, one of size 2 and the rest of size 1. The
eigenvalues associated to the different Jordan blocks are all different, and so
it follows that the surface (1.15) has Segre symbol (2, 1, 1, 1).
Exercise 5. Find any matrices A,B ∈ M5(Q) so that the corresponding
surface xtAx = xtBx = 0 is non-singular. Show that the surface has Segre
symbol (1, 1, 1, 1, 1).
THE MANIN CONJECTURE IN DIMENSION 2 15
So far we have given a very easy way to check the isomorphism type of
a given singular del Pezzo surface of degree 4. How do we match this up
with a classification according to the singularity type, as in our discussion
of cubic surfaces in Table 2? It turns out that up to isomorphism over Q,
there are 15 possible singularity types for S. Over Q, Coray and Tsfasman
[23, Proposition 6.1] have calculated the extended Dynkin diagrams for all
of the 15 types, and Knörrer [47] has determined the precise correspondence
between the singularity type and the Segre symbol. Table 3 is extracted
from this body of work, and matches each possible singularity type with the
Segre symbol, and the number of lines that the surfaces contains.
type Segre symbol # lines singularity
i (2,1,1,1) 12 A1
ii (2,2,1) 9 2A1
iii ((1,1),1,1,1) 8 2A1
iv (3,1,1) 8 A2
v ((1,1),2,1) 6 3A1
vi (3,2) 6 A1 +A2
vii (4,1) 5 A3
viii ((2,1),1,1) 4 A3
ix ((1,1),(1,1),1) 4 4A1
x ((1,1),3) 4 2A1 +A2
xi ((2,1),2) 3 A1 +A3
xii (5) 3 A4
xiii ((3,1),1) 2 D4
xiv ((2,1),(1,1)) 2 2A1 +A3
xv ((4,1)) 1 D5
Table 3. Classification (over Q) of singular del Pezzo sur-
faces of degree 4 in P4
In general, given a particular Segre symbol, its not entirely straightfor-
ward to determine explicit equations that define a singular del Pezzo surface
of degree 4 having this symbol. Nonetheless in Table 4 we have done pre-
cisely this for each Segre symbol that occurs. In doing so we have retrieved
some of the calculations carried out by Derenthal [26]. An important feature
of the table is that the surfaces recorded are split over Q. It remains a sig-
nificant open challenge to establish the Manin conjecture for the 15 surfaces
given in Table 4. This will furnish a proof of the Manin conjecture for the
class of split singular del Pezzo surfaces of degree 4 that are defined over Q,
and is undoubtedly a key stepping stone on the way towards a resolution of
the conjecture for all del Pezzo surfaces. There is huge potential for further
work in this area, and I hope that these lecture notes succeed in showing
that analytic number theorists are well placed to make an important con-
tribution.
Exercise 6. Calculate the Segre symbol for each of the surfaces in Table 4,
and check they match up with the correct singularity type in Table 3.
16 T.D. BROWNING
type Q1(x) Q2(x)
i x1x2 − x3x4 x1x4 − x2x3 + x3x5 + x4x5
ii x1x2 − x3x4 x1x4 − x2x3 + x3x5 + x25
iii x1x2 − x23 x1x3 − x2x3 + x4x5
iv x1x2 − x3x4 (x1 + x2 + x3 + x4)x5 − x3x4
v x1x2 − x23 x2x3 + x23 + x4x5
vi x1x2 − x3x4 x1x5 + x2x3 + x4x5
vii x1x2 − x3x4 x1x4 + x2x4 + x3x5
viii x1x4 − (x2 − x3)x5 (x1 + x4)(x2 + x3) + x2x3
ix x1x2 − x23 x23 − x4x5
x x1x2 − x23 x2x3 − x4x5
xi x1x4 − x3x5 x1x2 + x2x4 + x23
xii x1x2 − x3x4 x1x5 + x2x3 + x24
xiii x1x4 − x2x5 x1x2 + x2x4 + x23
xiv x1x2 − x23 x21 − x4x5
xv x1x2 − x23 x1x5 + x2x3 + x24
Table 4. Split surfaces representing the 15 singularity types
Whereas they share the same singularity type, the surfaces of type vii
and viii differ because in the former there are 5 lines, 4 of which pass
through the singularity, whereas in the latter all 4 lines pass through the
singularity. Similarly, an important difference between the surfaces of type
ii and iii in Tables 3 and 4 is that for the surface of type ii, the line joining
the two singularities is contained in the surface, whereas for the surface of
type iii it is not. When the 2 singular points are defined over a quadratic
extension of Q, the latter surface is called an Iskovskih surface. There is
ample evidence available (see Coray and Tsfasman [23], for example) to the
effect that Iskovskih surfaces are the most arithmetically interesting surfaces
among the singular del Pezzo surfaces of degree 4. In fact they are the only
such surfaces for which the Hasse principle can fail to hold. The main
focus of these lecture notes is upon the situation for split singular del Pezzo
surfaces, and so we will say no more about Iskovskih surfaces here.
As usual, let S̃ denote the minimal desingularisation of any surface S from
Table 4. Then the Picard group of S̃ has rank ρS = 6. The goal recorded
above is to try and establish (1.8) for each S. As in the case of singular
cubic surfaces several of the surfaces are actually special cases of varieties
for which the Manin conjecture is already known to hold. Thus it can be
shown that the surfaces representing types ix, x, xiv are all toric, so that
(1.8) already holds in these cases by the work of Batyrev and Tschinkel [3].
In a very real sense these surfaces are the “easiest” to deal with in our list.
Exercise 7. Show that NU (B) = Oε(B
1+ε) for the surfaces of type ix, x
and xiv.
It has also been shown by Chambert-Loir and Tschinkel [19] that the
Manin conjecture is true for equivariant compactifications of the algebraic
group G2a. Although identifying such surfaces in the table is not entirely
routine, it transpires that the type xv surface (with a D5 singularity) is
THE MANIN CONJECTURE IN DIMENSION 2 17
covered by this work. In joint work with la Bretèche, the author [8] has
provided an independent proof of the Manin conjecture for this particular
surface. In addition to obtaining a finer asymptotic formula of the shape
given in (1.9), this work has provided a useful line of attack for several other
singular del Pezzo surfaces.
In Table 5 we have recorded a list of progress towards the final resolution
of the Manin conjecture for the split singular del Pezzo surfaces of degree 4.
We have included the relevant reference in the literature, and whether the
result attained amounts to an asymptotic formula for the counting function,
or an upper bound. We will not pay attention here to the quality of the error
term in the asymptotic formula, but each upper bound is of the correct order
of magnitude B(logB)5. There is still plenty left to do!
type type of estimate achieved
v upper bound [14]
ix asymptotic formula [3]
x asymptotic formula [3]
xiii asymptotic formula [27]
xiv asymptotic formula [3]
xv asymptotic formula [8]
Table 5. Summary of progress for the split singular del
Pezzo surfaces of degree 4
It is also interesting to try and establish the Manin conjecture for singular
del Pezzo surfaces of degree 4 that are not split over the ground field. In
further joint work of the author with la Bretèche [9], the Manin conjecture
is established for the surface
x1x2 − x23 = 0, x21 + x2x5 + x24 = 0.
This surface has a D4 singularity and is isomorphic over Q(i) to the surface
of type xiii in Table 4. The Picard group of S̃ has rank 4 in this case, and
an asymptotic formula of the shape (1.9) is obtained for any δ ∈ (0, 3/32),
with P a polynomial of degree 3.
1.5. Degree > 5 surfaces. It turns out that all del Pezzo surfaces of degree
d > 7 are toric [26, Proposition 8], and that all non-singular del Pezzo
surfaces of degree d > 6 are toric. Thus (1.8) already holds in these cases by
the work of Batyrev and Tschinkel [3]. For non-singular del Pezzo surfaces
S ⊂ P5 of degree 5, the situation is rather less satisfactory. In fact there
are very few instances for which the Manin conjecture has been established.
The most significant of these is due to la Bretèche [7], who has proved the
conjecture for the split non-singular del Pezzo surface S of degree 5, in which
the 10 lines are all defined over Q. To be precise, if U ⊂ S denotes the open
subset formed by deleting the lines from S, then la Bretèche shows that
NU (B) = c0B(logB)
log logB
for a certain constant c0 > 0. This confirms Conjecture 1.3, since we have
seen in (1.6) that Pic(S) ∼= Z5 for split non-singular del Pezzo surfaces
18 T.D. BROWNING
of degree 5. The other major achievement in the setting of quintic del
Pezzo surfaces is a result of la Bretèche and Fouvry [10], where the Manin
conjecture is established for a surface that is not split, but contains lines
defined over Q(i).
So far we have only discussed the situation for non-singular del Pezzo
surfaces of degree d > 5. Let us now turn to the singular setting. When
d = 6 it emerges that there exist such surfaces that are not toric, and so are
not covered by [3]. We will focus attention on the situation for del Pezzo
surfaces of degree 6, following the investigation of Derenthal [26], where
the degree 5 surfaces are also considered. In view of [23, Proposition 8.3],
Table 6 lists all possible types of singular del Pezzo surfaces of degree 6.
type # lines singularity
i 4 A1
ii 3 A1
iii 2 2A1
iv 2 A2
v 1 A1 +A2
Table 6. Classification (over Q) of singular del Pezzo sur-
faces of degree 6
As noted in [26, §5], the surfaces of type i, iii and v are all toric and
so do not interest us here. Any singular del Pezzo surface of degree 6 can
be realised as the intersection of 9 quadrics in P6. For example, the type iv
surface is cut out by the system of equations
x1x6 − x4x5 = x1x7 − x2x5 = x1x7 − x3x4 = x3x7 + x4x5 + x25
= x5x7 − x3x4 = x2x7 + x24 + x4x5 = x4x7 − x2x6
= x4x6 + x5x6 + x
7 = x2x3 − x1x4 + x1x5 = 0.
(1.16)
In this set of lecture notes we will establish the Manin conjecture for the
type ii surface, which has the simplest possible singularity. When S ⊂ P6
is a split surface of type ii, then there is unimodular change of variables
that takes S into the surface with equations
x21 − x2x4 = x1x5 − x3x4 = x1x3 − x2x5 = x1x6 − x3x5
= x2x6 − x23 = x4x6 − x25 = x21 + x1x4 + x5x7
= x1x2 + x
1 + x3x7 = x1x3 + x1x5 + x6x7 = 0.
(1.17)
Let S̃ denote the minimal desingularisation of S. It follows from (1.6) that
Pic(S̃) ∼= Z4 since S is split. We will establish the following result in §3.
Theorem 1.3. Let S ⊂ P6 be the A1 surface given by (1.17). Then there
exist constants c1, c2 > 0 such that
NU (B) = c1B(logB)
3 + c2B(logB)
B logB
where
THE MANIN CONJECTURE IN DIMENSION 2 19
σ∞ = 6
{u,t,v∈R: 0<u,ut2,uv2,|tv(t−v)|61}
dtdudv. (1.18)
Since Pic(S̃) has rank 4, the exponents of B and logB in this asymptotic
formula are in complete agreement with Conjecture 1.3. Although we will
not give details here, it turns out that the value of the constant c1 also
confirms the prediction of Peyre [51] in this case. It is hoped that our proof
of Theorem 1.3 will encourage other researchers to try their hand at proving
asymptotic formulae for NU (B). With this in mind Exercise 8 is more of a
research problem, and its resolution will therefore conclude the proof of the
Manin conjecture for all split (non-singular or singular) del Pezzo surfaces
of degree 6.
Exercise 8. Establish an asymptotic formula for the type iv surface in
Table 6, with underlying equations (1.16).
1.6. Universal torsors. Universal torsors were originally introduced by
Colliot-Thélène and Sansuc [20, 21] to aid in the study of the Hasse principle
and weak approximation for rational varieties. Since their inception it is
now well-recognised that they also have a central rôle to play in proofs of
the Manin conjecture for Fano varieties, and in particular, for del Pezzo
surfaces. Let S ⊂ Pd be a del Pezzo surface of degree d ∈ {3, 4, 5, 6}, and
let S̃ denote the minimal desingularisation of S if it is singular, and S̃ = S
otherwise. Let E1, . . . , E10−d ∈ Div(S̃) be generators for PicQ(S̃), and let
E×i = Ei \ {zero section}. Working over Q, a universal torsor above S̃ is
given by the action of G10−dm on the map
π : E×1 ×eS · · · ×eS E
10−d → S̃.
A proper discussion of universal torsors would take us too far afield, and the
reader may consult the survey of Peyre [52] for further details, or indeed the
construction of Hassett and Tschinkel [34]. The latter outlines an alterna-
tive approach to universal torsors via the Cox ring. Given the usual open
subset U ⊂ S, the general theory of universal torsors ensures that there
is a partition of U(Q) into a disjoint union of patches, each of which is in
bijection with a suitable set of integral points on a universal torsor above S̃.
The guiding principle behind the use of universal torsors is simply that
they ought to be arithmetically simpler than the original variety. In our work
it will suffice to think of universal torsors as “particularly nice parametri-
sations” of rational points on the surface. The universal torsors that we
encounter in these lecture notes all have embeddings as affine hypersurfaces
of high dimension. Moreover, in each case we will show how the underlying
equation of the universal torsor can be deduced in a completely elementary
fashion, without any recourse to geometry whatsoever. The torsor equations
we will meet all take the shape
A+B + C = 0,
for monomials A,B,C of various degrees in the appropriate variables. As
in many examples of counting problems for higher dimensional varieties,
one can occasionally gain leverage by fixing some of the variables at the
20 T.D. BROWNING
outset, in order to be left with a counting problem for a family of small
dimensional varieties. If one is sufficiently clever about which variables to
fix first, one is sometimes left with a quantity that we know how to estimate
— and crucially — whose error term we can control once summed over the
remaining variables.
As a concrete example, we note that Hassett and Tschinkel [34] have
calculated the universal torsor for the cubic surface (1.11). It is shown that
there is a unique universal torsor above S̃1, and that it is given by the
equation
4s5 + y
2s2 + y
1s3 = 0,
for variables y1, y2, yℓ, s1, s2, s3, sℓ, s4, s5, s6. One of the variables does not
explicitly appear in this equation, and the torsor should be thought of as
being embedded in A10. It turns out that the way to proceed here is to fix
all of the variables apart from y1, y2, yℓ. One may then view the equation as
a congruence
y22s2 ≡ −y31s21s3 (mod s3ℓs24s5),
in order to take care of the summation over yℓ. This is the approach taken
in [12], the next step being to employ very standard facts about the number
of integer solutions to polynomial congruences that are restricted to lie in
certain regions. One if left with a main term and an error term, which the
remaining variables need to be summed over. While the treatment of the
main term is relatively routine, the treatment of the error term presents a
much more serious obstacle.
The universal torsors that turn up in the proofs of Theorems 1.2 and 1.3
can also be embedded in affine space as hypersurfaces. We will see in §3 that
the approach discussed above also produces results for the del Pezzo surface
of degree 6 considered in Theorem 1.3. In the proof of Theorem 1.2 in §4
our approach will be more obviously geometric, and we will actually view
the equation as a family of projective lines, and also as a family of conics.
We will then call upon techniques from the geometry of numbers to count
the relevant solutions.
2. Further heuristics
We have seen in §1.2, and in particular in the statement of Conjecture 1.3,
that for any del Pezzo surface S ⊂ Pd one expects a growth rate like
cSB(logB)
A for the counting function NU (B). We have already given some
motivation for the exponent of B in (1.2). The focus of the present section
is to produce much more sophisticated heuristics than we have previously
met. In particular we will gain an insight into the exponent of logB that
appears in the Manin conjecture.
For ease of presentation we restrict attention to non-singular diagonal
cubic surfaces S ⊂ P3. Thus
S = {a1x31 + a2x32 + a3x33 + a4x34 = 0}, (2.1)
for a = (a1, . . . , a4) ∈ N4 such that gcd(a1, . . . , a4) = 1. Define
P := {3} ∪ {p : p | a1a2a3a4}. (2.2)
THE MANIN CONJECTURE IN DIMENSION 2 21
This is the set of primes p for which the reduction of S modulo p is singular.
Before passing to the arithmetic of diagonal cubic surfaces, we will need to
discuss some of their geometry.
2.1. The lines on a cubic surface. The facts that we will need in this
section are explained in detail in the books of Hartshorne [33] and Manin
[49]. In general, a non-singular cubic surface S ⊂ P3 is obtained by blowing
up P2 along a collection of 6 points P1, . . . , P6 in general position. By general
position we mean that no 3 of them are collinear and they do not all lie on
a conic. The 27 lines on the surface arise in the following way. There are
6 exceptional divisors Ei above Pi, for 1 6 i 6 6, and 15 strict transforms
Li,j of the lines going through precisely 2 points Pi, Pj , for 1 6 i < j 6 6.
Finally there are the 6 strict transforms Qi of the conics going through all
but one of the 6 points. If Λ is the strict transform of a line in P2 that
doesn’t go through any of the Pi, then a basis of the geometric Picard group
(S) is given by
[Λ], [E1], . . . , [E6].
The remaining divisors may be expressed in terms of these elements via the
relations
[Li,j] = [Λ]− [Ei]− [Ej ], [Qi] = 2[Λ]−
j 6=i
[Ej ]. (2.3)
The class of the anti-canonical divisor −KS is given by [−KS ] = 3[Λ] −∑6
i=1[Ej ], although we will not need this fact in our work. One can check
that the hyperplane section has class −3[Λ] +
i=1[Ej ] in PicQ(S), so that
the cubic surface has very ample anticanonical divisor, as claimed in §1.2.
When S takes the shape (2.1) it is not hard to write down the 27 lines
explicitly. The calculations that we present below are based on those carried
out by Peyre and Tschinkel [54, §2]. Fix a cubic root α (resp. α′, α′′) of
a2/a1 (resp. a3/a1, a4/a1). We will assume that α ∈ Q if a2/a1 (resp.
a3/a1, a4/a1) is a cube in Q. Put
, β′ =
, β′′ =
We denote by θ a primitive cube root of one. Let i run over elements of
Z/3Z. Then the 27 lines on the cubic surface (2.1) are given by the equations
iαx2 = 0,
iβx4 = 0,
iαx2 = 0,
i+1βx4 = 0,
L′′i :
iαx2 = 0,
i+2βx4 = 0,
iα′x3 = 0,
iβ′x2 = 0,
M ′i :
iα′x3 = 0,
i+1β′x2 = 0,
M ′′i :
iα′x3 = 0,
i+2β′x2 = 0,
iα′′x4 = 0,
iβ′′x3 = 0,
N ′i :
iα′′x4 = 0,
i+1β′′x3 = 0,
N ′′i :
iα′′x4 = 0,
i+2β′′x3 = 0.
Let K = Q(θ, α, α′, α′′). It is a Galois extension of Q, and in the generic
case has degree 54 with Galois group Gal(K/Q) ∼= (Z/3Z)3 ⋊ Z/2Z.
We need to equate these lines to the divisors Ei, Li,j , Qi that we met
earlier. There is a certain degree of freedom in doing this, as discussed in
22 T.D. BROWNING
[33, §V.4], but it turns out that the choice
E1 = L0, E2 = L1, E3 = L2,
E4 = M1, E5 = M
2, E6 = M
Q1 = L
1, Q2 = L
2, Q3 = L
Q4 = M0, Q5 = M
1, Q6 = M
L1,2 = L
1 , L2,3 = L
2, L1,3 = L
L4,5 = M
1 , L5,6 = M2, L4,6 = M
L1,4 = N0, L1,5 = N1, L1,6 = N2,
L2,4 = N
1, L2,5 = N
2, L2,6 = N
L3,4 = N
2 , L3,5 = N
0 , L3,6 = N
(2.4)
is satisfactory. In assigning lines to E1, . . . , E6, all that is required is that
they should all be mutually skew. Given any cubic surface of the shape
(2.1), we now have the tools with which to compute the Picard group (1.7).
In fact, from this point forwards the process requires little more than basic
linear algebra.
Let us illustrate the procedure by calculating the Picard group for a spe-
cial case. Consider the Fermat surface
S1 = {x31 + x32 + x33 + x34 = 0}. (2.5)
In this case α = α′ = α′′ = β = β′ = β′′ = 1, in the notation above, and
K = Q(θ) is a quadratic field extension. We wish to find elements of the
geometric Picard group PicQ(S1) that are fixed by the action of Gal(K/Q).
Thus we want vectors c = (c0, . . . , c6) ∈ Z7 such that
(c0[Λ] + c1[E1] + · · ·+ c6[E6])σ = c0[Λ] + c1[E1] + · · ·+ c6[E6], (2.6)
for every σ ∈ Gal(K/Q). Under the action of Gal(K/Q) it is not hard to
check that Λ and E1 are fixed, that E2 and E3 are swapped, and that E4
(resp. E5, E6) is taken to L5,6 (resp. L4,5, L4,6). Using (2.3) one sees that
the left hand side of (2.6) is equal to
(c0 + c4 + c5 + c6)[Λ]+c1[E1] + c2[E3] + c3[E2]
− (c5 + c6)[E4]− (c4 + c5)[E5]− (c4 + c6)[E6],
in Pic
(S1). Thus we are interested in the space of c ∈ Z7 for which
c4 + c5 + c6 = 0, c2 − c3 = 0, c4 + 2c5 = 0, c4 + 2c6 = 0.
This system of homogeneous linear equations in 7 variables has underlying
matrix of rank 3. Thus the space of solutions has rank 7− 3 = 4, and so we
may conclude that Pic(S1) ∼= Z4. In fact a little thought reveals that the 4
elements
[Λ], [E1], [E2] + [E3],−2[E4] + [E5] + [E6],
provide a basis for Pic(S1).
Next, consider the surface
S2 = {x31 + x32 + x33 + px34 = 0} (2.7)
THE MANIN CONJECTURE IN DIMENSION 2 23
for a prime p. When p = 2 or 3, the arithmetic of this surface has been
considered in some detail by Heath-Brown [36], who provides some numer-
ical evidence to the effect that the corresponding counting function NU (B)
should grow like cpB for a certain constant cp > 0.
Exercise 9. Show that Pic(S2) ∼= Z.
This calculation has also been carried out by Colliot-Thélène, Kanevsky
and Sansuc [22, p. 12]. More generally, it is known that the Picard group
of the surface (2.1) has rank 1 if and only if the ratio
aσ(1)aσ(2)
aσ(3)aσ(4)
is not a cube in Q, for each permutation σ of (1, 2, 3, 4). This result is due
to Segre [57].
2.2. Cubic characters and Jacobi sums. Throughout this section let p
be a prime. Recall that a (multiplicative) character on Fp = Z/pZ is a map
χ : F∗p → C∗ such that
χ(ab) = χ(a)χ(b)
for all a, b ∈ F∗p. The trivial character ε is defined by the relation ε(a) = 1
for all a ∈ F∗p. It is convenient to extend the domain of definition to all of
Fp by assigning χ(0) = 0 if χ 6= ε and ε(0) = 1.
We begin by collecting together a few basic facts, all of which are estab-
lished in [46, §8].
Lemma 2.1. Let p be a prime. Then the following hold:
(1) Let χ be a character on Fp and let a ∈ F∗p. Then χ(1) = 1, χ(a) is
a (p− 1)-th root of unity, and χ(a−1) = χ(a) = χ(a)−1.
(2) For any character χ on Fp we have
χ(a) =
0, if χ 6= ε,
p, if χ = ε.
(3) The set of characters on Fp forms a cyclic group of order p− 1.
It follows from part (3) of Lemma 2.1 that χp−1 = ε for any character on
Fp. We define the order of a character to be the least positive integer n such
that χn = ε. In our work we will mainly be concerned with the characters of
order 3. Let us turn briefly to the topic of generalised Jacobi sums. Given
any characters χ1, . . . , χr on Fp, a Jacobi sum is a sum of the shape
J0(χ1, . . . , χr) :=
t=(t1,...,tr)∈F
t1+···+tr≡0 mod p
χ1(t1) · · ·χr(tr).
The key fact that we will need concerning these sums is that
|J0(χ1, . . . , χr)| =
0, if χ1 · · ·χr 6= ε,
(p − 1)pr/2−1, if χ1 · · ·χr = ε.
(2.8)
This is established in [46, §8.5].
Let p be a rational prime. We proceed to consider p as an element of
the ring of integers Z[θ] associated to the quadratic field Q(θ) obtained by
24 T.D. BROWNING
adjoining a primitive cube root of unity θ. It follows from basic algebraic
number theory that p is a prime in Z[θ] if p ≡ 2 mod 3, whereas it splits
as p = ππ if p ≡ 1 mod 3, where π is a prime in Z[θ]. When p ≡ 2 mod 3
the only cubic character on Fp is the trivial character ε. On the other
hand, when p = ππ ≡ 1 mod 3 then there are precisely two non-trivial cubic
characters χπ, χπ on Fp, where
χω(·) =
is the cubic residue symbol for any prime ω in Z[θ]. All of these facts are
established in [46, §9].
It turns out that Jacobi sums can be used to give formulae for the number
of solutions to appropriate equations over finite fields. Given any q ∈ N, let
N(q) := #{x mod q : a1x31 + · · ·+ a4x34 ≡ 0 (mod q)}, (2.9)
N∗(q) := #
x mod q :
1 + · · ·+ a4x34 ≡ 0 (mod q),
gcd(q, x1, . . . , x4) = 1
. (2.10)
When p is a prime not belonging to the finite set of primes P defined in
(2.2), we can write down a very precise expression for N(p). Thus it follows
from [46, §8.7] that
N(p) = p3 +
χ1,χ2,χ3,χ4
1 )χ2(a
2 )χ3(a
3 )χ4(a
4 )J0(χ1, χ2, χ3, χ4)
where the summation is over all non-trivial cubic characters χi : F
p → C
such that χ1χ2χ3χ4 = ε.
Exercise 10. Let p 6∈ P be a prime, and let χ1, χ2, χ3, χ4 be non-trivial
cubic characters on Fp such that χ1χ2χ3χ4 = ε. Deduce from (2.8) that
J0(χ1, χ2, χ3, χ4) = p(p− 1).
It follows from Exercise 10 that
N∗(p) = N(p)− 1 = p3 + p(p− 1)δp(a)− 1. (2.11)
for any prime p 6∈ P, where
δp(a) :=
χ1,χ2,χ3,χ4
1 )χ2(a
2 )χ3(a
3 )χ4(a
Let a ∈ F∗p and suppose that p splits as ππ. Then it will be useful to observe
χπ(a) + χπ(a) =
2, if a is a cubic residue modulo π,
−1, otherwise. (2.12)
We have δp(a) = 0 when p ≡ 2 mod 3, since there are then no non-trivial
cubic characters modulo p. When p ≡ 1 mod 3, with p = ππ 6∈ P, we have
δp(a) =χπ
(a1a2
(a1a2
(a1a3
(a1a3
(a1a4
(a1a4
THE MANIN CONJECTURE IN DIMENSION 2 25
Still with this choice of prime p, let νp(a) denote the number of indices
i ∈ {2, 3, 4} for which the cubic character χπ( a1aiajak ) is equal to 1, with {i, j, k}
a permutation of {2, 3, 4}. Then we may deduce from (2.12) that
δp(a) =
0, if p ≡ 2 mod 3,
3νp(a)− 3, if p ≡ 1 mod 3,
(2.13)
when p 6∈ P.
2.3. The Hardy–Littlewood circle method. We are now ready to con-
sider the counting function NU (B) that is associated to the diagonal cubic
surface S ⊂ P3 given in (2.1). Our aim is to provide heuristic evidence in
support of Manin’s original conjecture, and we will say rather little about
the predicted value of the constant. The Hardy–Littlewood circle method
is an extremely effective means of estimating counting functions associated
to projective algebraic varieties, but it only works when the dimension of
the variety is substantially larger than the degree. We have already seen
evidence of this in the statement of Theorem 1.1, which is based on an
application of the circle method. Although it has not been made to pro-
duce asymptotic formulae for the counting functions associated to del Pezzo
surfaces, in this section we will see how the Hardy–Littlewood method can
still be used as a useful heuristic tool. The key idea is to consider only the
contribution from the major arcs.
We will simplify matters by applying the heuristic to count all of the
rational points on S, rather than restricting attention to the open subset U .
Although the details are formidable, it is in fact possible to obtain upper
bounds for NU (B) using the circle method. Thus Heath-Brown [39] has
shown that NU (B) = Oε,S(B
3/2+ε) under a certain hypothesis concerning
the Hasse–Weil L-function associated to the surface. An interesting feature
of this work is that the contribution from the rational points lying on rational
lines in the surface is successfully separated out. When the surface contains
no lines defined over Q, such as the surface given by (2.7) for example, one
obviously has
NU (B) = NS(B) +O(1).
When S contains lines defined over Q there is a general consensus among
people working on the circle method that the dominant contribution (ie. the
contribution from the points on rational lines) should come from the minor
arc integral.
In what follows let e(z) := e2πiz for any z ∈ R. As usual, Z4 denotes the
set of primitive vectors in Z4. The igniting spark in the Hardy–Littlewood
circle method is the simple identity
e(αn)dα =
1, if n = 0,
0, if n ∈ Z \ {0}.
26 T.D. BROWNING
On taking into account the fact that x and −x represent the same point in
projective space, and applying Exercise 2, we deduce that
NS(B) =
|x|6B
e(α(a1x
1 + · · ·+ a4x34))dα
|x|6B/k
e(α(a1x
1 + · · ·+ a4x34))dα
S(α)dα,
say. Let us write P = B/k and IA(P ) :=
S(α)dα, for any bounded
subset A ⊂ R. The cubic exponential sum S(α) can actually be rather large
when α is well-approximated by a rational number with small denominator.
For example, we clearly have S(0) = 24P 4 + O(P 3). The philosophy that
underpins the Hardy–Littlewood method is that one expects S(α) to be
small for values of α ∈ [0, 1] that are not well-approximated by rational
numbers with small denominator. This is notoriously difficult to prove in
general, and as indicated above, is expected to be false in the present setting!
Our heuristic will be based on analysing IM(P ) for a suitable choice of
“major arcs” M. We will not give full details here, the gaps being easily
filled by consulting the relevant techniques in Davenport [25]. Let ε > 0 be
a small parameter. Given a, q ∈ Z such that
1 6 a 6 q 6 P ε, gcd(a, q) = 1, (2.14)
we define the interval
M(a, q) :=
− P−3+ε, a
+ P−3+ε
We take as major arcs the union
q6P ε
16a6q
gcd(a,q)=1
M(a, q).
It is clear that M contains all the points in the interval [0, 1] that are well-
approximated by rational numbers with small denominator.
Exercise 11. Show that M is a disjoint union for ε < 1.
The “minor arcs” are defined to be m := [0, 1] \M, and we will proceed
under the assumption that the minor arc integral Im(P ) can be ignored.
Actually we will also ignore the contribution that this term makes once it
is summed up over values of k. In truth there will be several points in the
argument where we will simply ignore subsidiary contributions. We will
indicate all of these by an appearance of the word “error”. Thus, to begin
with, we have
NU (B) =
µ(k)IM(P ) + error .
THE MANIN CONJECTURE IN DIMENSION 2 27
Here we have made the further assumption that the contribution NS\U(B)
from the points lying on lines in S arises in the minor arc integral.
Let a, q ∈ Z such that (2.14) holds, and let α = a/q + z ∈ M(a, q). Let
C(x) denote the diagonal cubic form in (2.1). We now break the sum into
congruence classes modulo q, giving
S(a/q + z) =
r mod q
e(aC(r)/q)
x∈Z4∩[−P,P ]4
x≡r mod q
e(zC(x)). (2.15)
We would like to replace the discrete variable x by a continuous one in the
inner sum, and the summation over x by an integral. For this we will appeal
to the following general result.
Lemma 2.2. Let P > 1, let a ∈ Zn and let r ∈ N such that r 6 P . Let F
be a function on Rn all of whose first order partial derivatives exist and are
continuous on R := [−P,P ]n. Define
MF := sup
16i6n
Then we have
x∈Zn∩R
x≡a mod r
e(F (x)) =
e(F (t))dt +O
(Pn−1(1 + PMF )
Proof. Our proof of Lemma 2.2 is based on the Euler–Maclaurin summation
formula [60, §I.0]. Let Bk(x) denote the kth Bernoulli polynomial, for k ∈
Z>0, and let s ∈ Z>0. Let A,B ∈ Z, with A < B. For any function
f : R → C whose (s+1)-th derivative f (s+1) exists and is continuous on the
interval [A,B], the Euler–Maclaurin summation formula states that
A<n6B
f(n) =
f(t)dt+
(−1)k+1Bk+1(0)
(k + 1)!
f (k)(B)− f (k)(A)
(−1)s
(s+ 1)!
Bs+1(t)f
(s+1)(t)dt.
(2.16)
Let a ∈ Z and r ∈ N. We will apply this result with f0(x) = f(a+ rx) and
, B0 =
B − a
Taking s = 0 in the Euler–Maclaurin formula, we therefore deduce that
A<n6B
n≡a mod r
f(n) =
f(t)dt− f(B0)− f(A0)
(t− a
f ′(t)dt,
(2.17)
since B1(x) = x− [x]− 12 .
We are now ready to establish Lemma 2.2, which we will do by induction
on n. Write Sn for the n-dimensional sum that is to be estimated. The case
28 T.D. BROWNING
n = 1 of Lemma 2.2 follows from (2.17) with f(x) = e(F (x)). Assuming
now that n > 2, we have
y∈Z∩[−P,P ]
y≡a1 mod r
x2,...,xn
e(G(x2, . . . , xn)),
where G(x2, . . . , xn) = F (y, x2, . . . , xn), and the sum over x2, . . . , xn is over
all integers in [−P,P ] such that xi ≡ ai mod r for 2 6 i 6 n.We may employ
the induction hypothesis to estimate the inner sum in n − 1 variables. It
therefore follows that
y∈Z∩[−P,P ]
y≡a1 mod r
[−P,P ]n−1
e(G(t2, . . . , tn))dt2 · · · dtn
(P (n−2)(1 + PMG)
Now there are O(P/r) integers y in the interval [−P,P ] that are congruent
to a1 modulo r, since r 6 P by assumption. Moreover, it is clear that
MG 6 MF . Hence
y∈Z∩[−P,P ]
y≡a1 mod r
f(y) +O
(P (n−1)(1 + PMF )
where
f(y) =
[−P,P ]n−1
e(F (y, t2, . . . , tn))dt2 · · · dtn.
The statement of Lemma 2.2 is now an easy consequence of (2.17). �
It is clear from the proof of Lemma 2.2 that when F has partial derivatives
to a higher order, one may obtain a much sharper estimate by including
higher order terms in the Euler–Maclaurin summation formula. The present
bound is satisfactory for our purposes, however.
Returning to (2.15) we apply Lemma 2.2 with
F (x) = zC(x), n = 4, a = r, r = q.
In particular we have q 6 P ε 6 P , as required for the lemma. Furthermore,
MF = MzC ≪ |z|P 2 6 P−1+ε for any α = a/q+z ∈ M(a, q). It follows that
S(a/q + z) = q−4T (a, q)VP (z) +O(P
3+2ε)
on the major arcs, where
T (a, q) :=
r mod q
e(aC(r)/q), VR(z) :=
[−R,R]4
e(zC(x))dx. (2.18)
The set of major arcs has meas(M) = O(P−3+3ε). On carrying out the
integration over z and the summation over a and q one is therefore led to
THE MANIN CONJECTURE IN DIMENSION 2 29
the conclusion that
IM(P ) = P
q6P ε
16a6q
gcd(a,q)=1
q−4T (a, q)
|z|6P−3+ε
V1(zP
3)dz +O(P 5ε)
q6P ε
16a6q
gcd(a,q)=1
q−4T (a, q)
|z|6P ε
V1(z)dz +O(P
Define
I(R) :=
|z|6R
V1(z)dz =
|z|6R
[−1,1]4
e(zC(x))dxdz.
It can be shown that I(R) is a bounded function of R, and furthermore,
I(R) → I0 > 0 as R → ∞. A standard calculation reveals that the limit I0
is equal to 2σ∞, where
σ∞ :=
dx1dx2dx3
1 + a2x
2 + a3x
is the real density of solutions. Here the integral is over x1, x2, x3 ∈ [−1, 1]
such that |(a1x31 + a2x32 + a3x33)/a4| 6 1. We now define
S(R) :=
16a6q
gcd(a,q)=1
T (a, q).
On bringing everything together, our investigation has so far succeeded in
showing that
NU (B) = σ∞
µ(k)PS(P ε) + error, (2.19)
for a suitably small value of ε > 0, where P = B/k.
Our task is now to examine the sum S(R), as R → ∞. Let us write
Sq :=
16a6q
gcd(a,q)=1
T (a, q),
where T (a, q) is the complete exponential sum defined in (2.18). Then
S(R) =
q6R q
−4Sq. It turns out that Sq is a multiplicative function
of q. This can be established along the lines of [25, Lemma 5.1]. Recall the
definition (2.9) of N(q). We now come to the key relation between Sq and
N(q) at prime power values of q.
Exercise 12. Let p be a prime and let e > 1. Use Lemma 2.1 to show that
Spe = p
eN(pe)− p3+eN(pe−1).
Let us for the moment ignore considerations of convergence, and consider
the local factors
e=0 p
−4eSpe in the infinite product formula for S(∞).
Now it follows from Exercise 12 that
p−4eSpe = 1 +
p−3eN(pe)− p3−3eN(pe−1)
= p−3EN(pE),
30 T.D. BROWNING
for any E > 1. Hence, formally speaking, we have S(∞) =
p τp, where
τp := lim
p−3eN(pe).
If S(R) was convergent, which it certainly is not in general, we could then
conclude from (2.19) that
NU (B) = Bσ∞
τp + error . (2.20)
Arguing formally, we now replace the summation over k by its Euler product,
concluding that
NU (B) = Bσ∞
σp + error, (2.21)
with σp := (1− 1/p)τp. Note that
σp = lim
p−3eN∗(pe), (2.22)
in the notation of (2.10), which clearly follows from the observation that
N∗(pe) = N(pe)− 8N(pe−3),
for any e > 3. The estimate in (2.21) therefore gives a heuristic asymptotic
formula for NU (B) which is visibly a product of local densities. Among
other things, we have assumed that S(R) is convergent in formulating this
heuristic. This is expected to be true for diagonal cubic surfaces whose
Picard group has rank 1, but not in general.
One can make the transition from (2.20) to (2.21) completely rigorous
if the Hardy–Littlewood heuristic produces a leading term involving Pα
with exponent α > 1. The outcome is that to go from counting points
on the affine cone to counting projective points, one merely replaces N(pe)
by N∗(pe). For α = 1, however, the “renormalization” procedure remains
heuristic. Let
∗(R) :=
q−4S∗q ,
where
S∗q :=
16a6q
gcd(a,q)=1
r mod q
gcd(r,q)=1
e(aC(r)/q).
It is easily checked that S∗q is a multiplicative function of q, and furthermore,
that the corresponding version of Exercise 12 holds, relating S∗pe to N
∗(pe).
In fact, formally speaking, one has
p−4eS∗pe =
with σp given by (2.22). Bearing all of this in mind we will proceed under
the bold assumption that (2.19) can be replaced by
NU (B) = Bσ∞S
∗(B) + error, (2.23)
where we have taken ε = 1 in the expressions for S(Bε) and S∗(Bε).
THE MANIN CONJECTURE IN DIMENSION 2 31
We now turn to a finer analysis of S∗(B), as B → ∞. Our task is to
determine the analytic properties of the corresponding Dirichlet series
F (s) :=
(2.24)
for s = σ+it ∈ C. Armed with this analysis we will ultimately apply Perron’s
formula to obtain an estimate forS∗(R). Using the multiplicativity of q−sS∗q
we deduce that
F (s) =
σp(s), σp(s) :=
p−esS∗pe . (2.25)
In examining F (s) it clearly suffices to ignore the value of the factors σp(s)
at any finite collection of primes p. With this in mind we will try and
determine σp(s) for p 6∈ P, where P is given by (2.2). Recall the definition
(2.10) of N∗(q).
Exercise 13. Let e > 1 and let p 6∈ P be a prime. Use Hensel’s lemma to
show that N∗(pe) = p3e−3N∗(p).
It therefore follows from Exercise 13 that
σp(s) = 1 +
peN∗(pe)− p3+eN∗(pe−1)
= 1− 1
N∗(p)
for any p 6∈ P. Hence (2.11) yields
σp(s) = 1 +
δp(a)
− δp(a)
where δp(a) is given by (2.13).
We now pursue our analysis in the special case a = (1, 1, 1, 1) of the
Fermat cubic surface (2.5). Now it is clear from (2.13) that δp(1, 1, 1, 1) = 0
if p ≡ 2 mod 3 and
δp(1, 1, 1, 1) = 3νp(1, 1, 1, 1) − 3 = 6
if p ≡ 1 mod 3. Let λ : Z → C be the real Dirichlet character of order 2
defined by
λ(n) :=
), if 3 ∤ n,
0, otherwise,
where (n
) is the Legendre symbol. Then we may write
σp(s) = 1 +
3(1 + λ(p))
− 3(1 + λ(p))
1− λ(p)
pmin{σ−2,2σ−2}
for any p 6∈ P. Let L(s, λ) denote the usual Dirichlet L-function associated
to λ. When a = (1, 1, 1, 1) we have therefore succeeded in showing that
F (s) = ζ(s− 3)3L(s− 3, λ)3G(s), (2.26)
32 T.D. BROWNING
where G(s) is a function that is holomorphic and bounded on the half-plane
σ > 7/2+ δ, for any δ > 0. For future reference we note that G(4) has local
factors
Gp(4) =
(1− 1
)7(1 + 7
), if p ≡ 1 mod 3,
(1− 1
)4(1 + 1
)3(1 + 1
), if p ≡ 2 mod 3. (2.27)
Although we will not prove it here, it can be deduced from (2.22) and
Hensel’s lemma that
G3(4) =
3−3eN∗(3e) =
. (2.28)
We are now ready for our application of Perron’s formula, which we will
apply in the following form.
Lemma 2.3. Let F (s) =
n=1 ann
−s be a Dirichlet series with abscissa of
absolute convergence σa. Suppose that x 6∈ Z and let c > σa. Then we have
∫ c+iT
F (s)
|an|n−c
| log(x/n)|
for any T > 1.
Proof. Let c > 0. The lemma follows from the identity
∫ c+iT
1 +O(xc(T | log x|)−1), if x > 1,
+O(cT−1), if x = 1,
O(xc(T | log x|)−1), if 0 < x < 1,
which is a straightforward exercise in contour integration. �
In our case we have aq = S
q and we are interested in the Dirichlet series
F (s + 4), in the notation of (2.24). In order to apply Lemma 2.3 we will
need an upper bound for this quantity. For our purposes the trivial upper
bound aq ≪ q5 is sufficient. Thus the Dirichlet series F (s+ 4) is absolutely
convergent for σ > 2. Taking c = 2 + ε for any ε > 0, we may deduce from
Lemma 2.3 that
∗(B) =
∫ c+iT
F (s+ 4)
n1+ε| log(B/n)|
for any T > 1 and any B 6∈ Z. It is not hard to see that the error term here
is ≪ T−1Bc. We now apply Cauchy’s residue theorem to the rectangular
contour C joining c′ − iT , c′ + iT , c+ iT and c− iT , where c′ = −1/2 + ε.
The relation (2.26) implies that in this region F (s + 4)Bs/s has a unique
pole at s = 0, and it is a pole of order 4. It has residue
Ress=0
F (s+ 4)Bs
L(1, λ)3G(4)P (logB)
where P ∈ R[x] is a monic polynomial of degree 3. Putting all of this
together we have therefore shown that
∗(B) =
L(1, λ)3G(4)P (logB)
+O(E(B)), (2.29)
THE MANIN CONJECTURE IN DIMENSION 2 33
where
E(B) =
( ∫ c′+iT
c′−iT
∫ c−iT
c′−iT
∫ c′+iT
)∣∣∣H(s)3
∣∣∣ds,
for any T > 1, and where H(s) = ζ(s + 1)L(s + 1, λ). Here we have used
the fact that G(s+ 4) is bounded on the half-plane ℜe(s) > c′.
To make the analysis simpler, it will be convenient to proceed under the
assumption that the Lindelöf hypothesis holds for ζ(s), and also for the
Dirichlet L-function L(s, λ). This could be avoided at the cost of extra
effort, but there seems no harm in supposing it here. Thus we may assume
the bounds
ζ(σ + it) ≪ε |t|ε, L(σ + it, λ) ≪ε |t|ε,
for any σ ∈ [1/2, 1] and any |t| > 1. It therefore follows that
H(σ + it) ≪ε
|t|ε, if −1/2 6 σ 6 0,
1, if σ > 0,
for any |t| > 1, which gives
∫ c−iT
c′−iT
∣∣∣H(s)3
∣∣∣ds ≪ε
BσT−1+3εdσ ≪ε BcT−1+3ε.
One obtains the same estimate for the contribution from the remaining
horizontal contour. Turning to vertical integral, we find that
∫ c′+iT
c′−iT
∣∣∣H(s)3
∣∣∣ds ≪ Bc
|H(1/2 + ε+ it)|3
1 + |t| dt
≪ε Bc
(1 + |t|)3ε−1dt
≪ε Bc
T 3ε,
under the assumption of Lindelöf hypothesis. This shows that
E(B) ≪ε BεT 3ε
for any T > 1. Taking T sufficiently large, we therefore conclude from (2.29)
∗(B) =
L(1, λ)3G(4)P (logB)
+O(B−∆),
for some ∆ > 0.
We are now ready to return to the Hardy–Littlewood major arc analysis
which led us to (2.23). Substituting in our estimate for S∗(B), we conclude
NU1(B) ∼ c1B(logB)3 (2.30)
where U1 ⊂ S1 is the usual open subset of the Fermat surface (2.5), and
σ∞L(1, λ)
3G(4)
Now it follows from the class number formula that L(1, λ) = π
3/9. Hence,
on combining this with (2.27) and (2.28), our heuristic argument has led us
34 T.D. BROWNING
to the expectation that (2.30) holds, with
p≡1 mod 3
p≡2 mod 3
The exponents of B and logB in (2.30) agree with the Manin conjecture,
since we have already seen in §2.1 that the Picard group of S1 has rank 4.
It is interesting to compare our analysis with the work of Peyre and
Tschinkel [54], who calculate the leading constant cPeyre in Peyre’s refine-
ment [51] of the conjectured asymptotic formula for NU1(B). It turns out
cPeyre = γ(S1)c1,
with γ(S1) = 7/3. For a general non-singular cubic surface S ⊂ P3, the
constant γ(S) is defined to be the volume
γ(S) :=
e−〈−KS ,t〉dt.
Thus, in general terms, γ(S) measures the volume of the polytope obtained
by intersecting the dual of Λeff(S) with a certain affine hyperplane. In
particular γ(S) ∈ Q for any non-singular cubic surface S, and γ(S) = 1 if
and only if the corresponding Picard group has rank 1.
Exercise 14. Let S2 denote the surface (2.7). Using a similar argument,
show that one expects an asymptotic formula of the shape NS2(B) ∼ c2B
for some constant c2 > 0. Check your answer with the heuristic formula
obtained by Heath-Brown [36].
3. The A1 del Pezzo surface of degree 6
In this section we will establish Theorem 1.3. Any line in P6 is defined by
the intersection of 5 hyperplanes. It is not hard to see that the equations
x1 = x2 = x3 = x5 = x6 = 0, x1 = x3 = x4 = x5 = x6 = 0,
x3 = x5 = x6 = x1 + x4 = x1 + x2 = 0,
(3.1)
all define lines contained in the singular del Pezzo surface S given by (1.17).
Table 6 ensures that these are the only lines contained in S. By definition
U is the open subset of S on which none of these equations hold. We begin
by establishing the following result.
Lemma 3.1. We have
NU (B) = 2M(B) +O(B),
where M(B) denotes the number of x ∈ Z7 such that
x21 − x2x4 = x1x5 − x3x4 = x1x3 − x2x5 = x1x6 − x3x5
= x2x6 − x23 = x4x6 − x25 = x21 − x1x4 + x5x7
= x21 − x1x2 − x3x7 = x1x3 − x1x5 + x6x7 = 0,
(3.2)
with gcd(x1, . . . , x7) = 1, 0 < |x1|, x2, x3, x4, |x5|, x6 6 B and |x7| 6 B.
THE MANIN CONJECTURE IN DIMENSION 2 35
Proof. In view of the fact that x and −x represent the same point in P6, we
NU (B) =
#{x ∈ Z7 : |x| 6 B, (1.17) holds, but (3.1) does not},
where Z7 denotes the set of primitive vectors in Z7. We need to consider the
contribution to the right hand side from points such that xi = 0, for some
1 6 i 6 7. Let us begin by considering the contribution from vectors x ∈ Z7
for which x1 = 0. But then the equations in (1.17) imply that x2x4 = 0. If
x2 = 0, it is straightforward to check that either x satisfies the first system
of equations in (3.1), or else
x0 = x2 = x3 = x7 = 0, x4x6 = x
Such points are therefore confined to a plane conic. We therefore obtain
O(B) points overall with x1 = x2 = 0. If on the other hand x1 = x4 = 0,
then a similar analysis shows that there are O(B) points in this case too. In
view of the first equation in (1.17), the contribution from vectors x such that
x2x4 = 0 is also O(B). Let us now consider the contribution from vectors
x such that x3 = 0 and x1x2x4 6= 0. It is easily checked that the only such
vectors have x5 = x6 = 0 and x1 + x4 = x1 + x2 = 0, and so must lie on a
line contained in S. Finally, arguing in a similar fashion, we see that there
are no points contained in S with x5x6 = 0 and x1x2x3x4 6= 0. We have
therefore shown that
NU (B) =
#{x ∈ Z7 : x1 · · · x6 6= 0, |x| 6 B, (3.2) holds}+O(B).
Here we have noted that there is an obvious unimodular transformation that
takes the set of equations in (1.17) into (3.2).
We would now like to restrict our attention to positive values of x1, . . . , x6.
The equations for S imply that x2, x4, x6 all share the same sign. On ab-
sorbing the minus sign into x1 there is a clear bijection between solutions
to (3.2) with x2, x4, x6 < 0 and solutions with x2, x4, x6 > 0. We choose to
count the former. Arguing similarly, by absorbing the minus signs into x1
and x7, we see that there is a bijection between the solutions to (3.2) with
x3 < 0 and x2, x4, x6 > 0, and the solutions with x2, x3, x4, x6 > 0. Fixing
our attention on the latter set of points, we therefore complete the proof of
Lemma 3.1. �
Let S̃ denote the minimal desingularisation of the surface S. By determin-
ing the Cox ring associated to S̃, Derenthal [26] has calculated the universal
torsor above S̃. In this setting it is defined by a single equation
s1y1 − s2y2 + s3y3 = 0, (3.3)
embedded in A7. In particular one of the variables does not appear explicitly
in the equation.
3.1. Elementary considerations. As promised in §1.6, we proceed to
show how NU (B) can be related to a count of the integer points on the
corresponding universal torsor, which in this case is given by (3.3). Our
deduction of this fact is completely elementary, and is based on an analysis
of the integer solutions to the system of equations (3.2). It is still somewhat
36 T.D. BROWNING
mysterious as to how or why this rather low-brow process should ultimately
lead to the same outcome! Typical of the facts that we will employ is the
following.
Exercise 15. Show that the general solution of the equation xy = z2 is
x = a2c, y = b2c, z = abc,
with |µ(c)| = 1.
Given s0 ∈ R and s = (s1, s2, s3),y = (y1, y2, y3) ∈ R3, define
Ψ(s0, s,y) := max
|s30s21s22s23|, |y1y2y3|, |s0s21y21|, |s0s22y22|
. (3.4)
We are now ready to record our translation of the problem to the universal
torsor.
Lemma 3.2. We have
NU (B) = 2#
(s0, s,y) ∈ Z7 :
Ψ(s0, s,y) 6 B, (3.3) holds,
s0, s1, s2, s3, y1 > 0,
gcd(yi, s0sjsk) = 1,
gcd(si, sj) = 1
+O(B),
with i, j, k a permutation of 1, 2, 3 in the coprimality conditions.
Proof. Let x ∈ Z7 be a primitive vector counted by M(B), as defined in
the statement of Lemma 3.1. Combining the first equation in (3.2) with
Exercise 15 we see that
x1 = a1a2a4, x2 = a
2a1, x4 = a
for integers a1, a2, a4 such that a1, a2 > 0 and
|µ(a1)| = gcd(a1 gcd(a2, a4)2, x3, x5, x6, x7) = 1.
Inserting this into the equation x2x6 = x
3 we deduce that a1a2 | x3, whence
x3 = a1a2a3, x6 = a1a
for a positive integer a3 such that
|µ(a1)| = gcd(a1 gcd(a2, a4)2, a1a3 gcd(a2, a3), x5, x7) = 1.
Substituting this into the equation x4x6 = x
5, we deduce that
x5 = a1a3a4,
|µ(a1)| = gcd(a1, x7) = gcd(a2, a3, a4, x7) = 1. (3.5)
Note that the second equation in (3.2) implies that x1, x5 must share the
same sign, which here is the sign of a4. The equations x1x5 = x3x4, x1x3 =
x2x5 and x1x6 = x3x5 reveal no new information. Turning instead to the
equation x21 = x1x4 − x5x7, we obtain
2a4 = a1a2a
4 − a3x7. (3.6)
The coprimality conditions imply that a1 | a3. Moreover, we deduce from
this equation that a2a4 | a3x7/a1. We may therefore write
a2 = a23a27, a4 = a43a47,
THE MANIN CONJECTURE IN DIMENSION 2 37
for integers a2i, a4i, with i = 3, 7, such that a2i, a43, |a47| > 0, and
a1a23a43 | a3, a27a47 | x7.
Thus there exist further integers b3, a7 with b3 > 0, such that
a3 = a1a23a43b3, x7 = a27a47a7,
with (3.5) and (3.6) becoming
|µ(a1)| = gcd(a1, a27a47a7) = gcd(a23a27, a23a43b3, a43a47, a27a47a7) = 1,
a23a27 = a43a47 − b3a7,
respectively. The final two equations are redundant. Let us write d for the
highest common factor of a23, a43, b3. Thus
a23 = da
23, a43 = da
43, b3 = db
for positive integers d, a′23, a
43, b
3. On making these substitutions the equa-
tion remains the same, but with appropriate accents added, whereas the
coprimality conditions become
|µ(a1)| = gcd(da1, a27a47a7) = gcd(a′23a27, a′23a′43b′3, a′43a47, a27a47a7)
= gcd(a′23, a
43, b
Now any n ∈ N can be written uniquely in the form n = ab2 for a, b ∈ N
such that |µ(a)| = 1. We may therefore make the change of variables
(s0; s1, s2, s3; y1, y2, y3) = (a1d
2; a′23, a
43, b
3; a27, a47, a7).
Bringing everything together, we have therefore established the existence of
(s0, s,y) ∈ Z7 such that (3.3) holds, with
s0, s1, s2, s3, y1 > 0, (3.7)
gcd(s0, y1y2y3) = gcd(s1, s2, s3) = gcd(s1y1, s2y2, s1s2s3, y1y2y3) = 1.
Note that y2 is automatically non-zero for s0, s,y satisfying the remaining
conditions. Once combined with (3.3), it is easy to check that the latter
coprimality conditions are equivalent to the conditions
gcd(y1, s0s2s3) = gcd(y2, s0s1s3) = gcd(y3, s0s1s2) = 1,
gcd(s1, s2) = gcd(s1, s3) = gcd(s2, s3) = 1,
(3.8)
that appear in the statement of the lemma.
At this point we may summarise our argument as follows. Let T ⊂ Z7
denote the set of (s0, s,y) ∈ Z7 such that (3.3), (3.7) and (3.8) hold. Then
for any primitive vector x counted by M(B), we have shown that there
38 T.D. BROWNING
exists (s0, s,y) ∈ T such that


x1 = s0s1s2y1y2,
x2 = s0s
x3 = s
1s2s3y1,
x4 = s0s
x5 = s
2s3y2,
x6 = s
x7 = y1y2y3.
Conversely, we leave it as an exercise to check that any (s0, s,y) ∈ T pro-
duces a primitive point x ∈ Z7 such that (3.2) holds, with
|x1|, x2, x3, x4, |x5|, x6 > 0.
We may now conclude that M(B) is equal to the number of (s0, s,y) ∈ T
such that
i=1,2
|s0s1s2y1y2|, |s0s2i y2i |, |s20s1s2s3siyi|, |s30s21s22s23|, |y1y2y3|
In view of the fact that |s0s1s2y1y2| =
|s0s21y21|
|s0s22y22 |, and furthermore,
|s20s1s2s3siyi| =
|s30s21s22s23|
|s0s2i y2i |, it follows that this height condition
is equivalent to Ψ(s0, s,y) 6 B for any (s0, s,y) ∈ T , where Ψ is given
by (3.4). In summary we have therefore shown that M(B) is equal to the
number of (s0, s,y) ∈ T such that Ψ(s0, s,y) 6 B. Once inserted into
Lemma 3.1, this completes the proof of Lemma 3.2. �
At first glance it might seem a little odd that the height restriction
|s0s23y23 | 6 B doesn’t explicitly appear in the lemma. However, (3.3) implies
that 0 < s1y1 = s2y2 − s3y3 for any (s0, s,y) ∈ T , whence the restriction
Ψ(s0, s,y) 6 B is plainly equivalent to max{|s0s23y23|,Ψ(s0, s,y)} 6 B. We
have preferred not to include it explicitly in the statement of Lemma 3.2
however.
3.2. The asymptotic formula. Our starting point is Lemma 3.2. Let
T (B) denote the quantity on the right hand side that is to be estimated.
Once taken together with (3.3), the height condition Ψ(s0, s,y) 6 B is
equivalent to
i=1,2
|s30s21s22s23|, |s0s2i y2i |, |y1y2(s1y1 − s2y2)/s3|
Define
X0 :=
(s30s21s22s23
, Xi :=
(s1s2s3B
for i = 1, 2. Then the height conditions above can be rewritten as
|X0| 6 1, |f1(y1)| 6 1, |f2(y2)| 6 1, |g(y1, y2)| 6 1,
where
fi(y) := X0
, g(y1, y2) :=
for i = 1, 2. In order to count solutions to the equation (3.3), our plan will
be to view the equation as a congruence
s1y1 − s2y2 ≡ 0 (mod s3),
THE MANIN CONJECTURE IN DIMENSION 2 39
which has the effect of automatically taking care of the summation over
y3. In order to make this approach viable we will need to first extract the
coprimality conditions on the y3 variable.
Define the set
S := {(s0, s) ∈ N4 : gcd(si, sj) = 1, X0 6 1}, (3.9)
with i, j generic indices from the set {1, 2, 3}. We now apply Möbius in-
version, as in Exercise 2, in order to remove the coprimality condition
gcd(y3, s0s1s2) = 1. Thus we find that
T (B) =
(s0,s)∈S
k3|s0s1s2
µ(k3)#


y ∈ Z3 :
gcd(y1, s0s2s3) = 1,
gcd(y2, s0s1s3) = 1,
y1 > 0,
s1y1 − s2y2 + k3s3y3 = 0,
|fi(yi)| 6 1, |g(y1, y2)| 6 1


Now it is clear that the summand vanishes unless gcd(k3, s1s2) = 1. Hence
T (B) =
(s0,s)∈S
k3|s0
gcd(k3,s1s2)=1
µ(k3)Sk3(B), (3.10)
where
Sk3(B) := #


y1, y2 ∈ Z :
gcd(y1, s0s2s3) = 1,
gcd(y2, s0s1s3) = 1,
y1 > 0,
s1y1 ≡ s2y2 mod k3s3,
|fi(yi)| 6 1, |g(y1, y2)| 6 1


Clearly Sk3(B) depends on the parameters s0 and s, in addition to k3 and B.
We now turn to the estimation of Sk3(B), for which we need the following
basic result.
Exercise 16. Let b > a and q > 0. Show that
#{n ∈ Z ∩ (a, b] : n ≡ n0 mod q} =
+O(1).
We will fix y2 and apply Exercise 16 to handle the summation over y1. Be-
fore this we must use Möbius inversion to remove the coprimality condition
gcd(y1, s0s2s3) = 1 from the summand. Thus we find that
Sk3(B) =
k1|s0s2s3
µ(k1)#
y1, y2 ∈ Z :
gcd(y2, s0s1s3) = 1,
k1s1y1 ≡ s2y2 mod k3s3,
|f1(k1y1)| 6 1, |f2(y2)| 6 1,
|g(k1y1, y2)| 6 1, y1 > 0
In view of the other coprimality conditions, the summand plainly vanishes
unless gcd(k1, k3s3) = 1. We may therefore write ρ ∈ Z for the (unique)
inverse of k1s1 modulo k3s3, whence
Sk3(B) =
k1|s0s2
gcd(k1,k3s3)=1
µ(k1)Sk1,k3(B), (3.11)
40 T.D. BROWNING
Sk1,k3(B) :=
y2∈Z: |f2(y2)|61
gcd(y2,s0s1s3)=1
y1 ∈ N :
y1 ≡ ρs2y2 mod k3s3,
|f1(k1y1)| 6 1,
|g(k1y1, y2)| 6 1
An application of Exercise 16 now reveals that
Sk1,k3(B) =
y2∈Z: |f2(y2)|61
gcd(y2,s0s1s3)=1
(X1F1(X0, y2/X2)
k1k3s3
+O(1)
, (3.12)
where
F1(u, v) :=
{t∈R>0: |ut
2|,|tv(t−v)|61}
We close this section by showing that once summed over all (s0, s, y2) ∈ N5,
the error term in (3.12) makes a satisfactory overall contribution to the error
term in Theorem 1.3. Using the fact that
k|n |µ(k)| = 2ω(n), we find that
this contribution is
(s0,s)∈S
4ω(s0)2ω(s2)X2
= B1/2
(s0,s)∈S
4ω(s0)2ω(s2)
s0,s1,s2∈N
4ω(s0)2ω(s2)
s20s1s
≪ B logB.
This is satisfactory for Theorem 1.3, and so we may henceforth ignore the
error term in the above estimate for Sk1,k3(B).
Define the arithmetic function
φ∗(n) :=
where as is common convention the product is over distinct prime divisors
of n. It will be useful to note that
φ∗(mn) =
φ∗(m)φ∗(n)
φ∗(gcd(m,n))
, (3.13)
for any m,n ∈ N. We must now sum over the variable y2, for which we will
employ the following basic result.
Exercise 17. Let I ⊂ R be an interval, let a ∈ N and let f : R → R>0 be a
function that is continuously differentiable on I. Use (2.16) to show that
n∈Z∩I
gcd(n,a)=1
f(n) = φ∗(a)
f(t)dt+O
2ω(a) sup
|f(t)|
We may now return to (3.12). Using Exercise 17 we deduce that
Sk1,k3(B) =
φ∗(s0s1s3)X1X2F2(X0)
k1k3s3
(2ω(s0s1s3)X1
k1k3s3
, (3.14)
THE MANIN CONJECTURE IN DIMENSION 2 41
where
F2(u) :=
{t,v∈R: t>0, |ut2|,|uv2|,|tv(t−v)|61}
dtdv.
We must now estimate the overall contribution to NU (B) from the error
term in this estimate, once summed up over the remaining variables. This
gives
(s0,s)∈S
4ω(s0)2ω(s2)2ω(s0s1s3)X1
≪ B1/3
s0,s1,s2,s3∈N
8ω(s0)2ω(s1s2s3)s
≪ B logB,
by summing over s2 6
B/(s30s
3). This is satisfactory for Theorem 1.3,
and so we may henceforth ignore the error term in (3.14). As pointed out
to the author by Régis de la Bretèche, it is easy to sharpen this error term
to O(B) using the fact that φ∗ has constant average order.
Now it is trivial to check that
gcd(d,a)=1
φ∗(n)
φ∗(gcd(a, n))
for any a, n ∈ N. Bringing together (3.10), (3.11) and (3.14), we conclude
T (B) =
(s0,s)∈S
k3|s0
gcd(k3,s1s2)=1
µ(k3)
φ∗(s0s2)φ
∗(s0s1s3)
φ∗(gcd(k3s3, s0s2))
X1X2F2(X0)
where S is given by (3.9). It is clear that gcd(k3s3, s0s2) = gcd(k3s3, s0).
Let us define the arithmetic function
ϑ(s0, s) =
φ∗(s0s2)φ
∗(s0s1s3)
φ∗(gcd(s0, s3))
k3|s0
gcd(k3,s1s2)=1
µ(k3)
φ∗(gcd(k3, s0, s3))
φ∗(gcd(k3, s0))
when gcd(si, sj) = 1 for 1 6 i < j 6 3, and ϑ(s0, s) = 0 otherwise. It follows
from (3.13) that
ϑ(s0, s) =
φ∗(s0s2)φ
∗(s0s1s3)
φ∗(gcd(s0, s3))
p|gcd(s0,s3)
p∤s1s2
p∤s1s2s3
(1− 2
= φ∗(s0s2)φ
∗(s0s1s3)
p∤s1s2s3
(1− 2
= φ∗(s0)φ
∗(s1s2s3)
p∤s1s2s3
42 T.D. BROWNING
when gcd(si, sj) = 1 for 1 6 i < j 6 3. On recalling the definitions of
X1,X2, we deduce that
T (B) = B2/3
∆(n)F2
(n/B)1/3
, (3.15)
where
∆(n) :=
ϑ(s0, s)
(s1s2s3)1/3
, (3.16)
for any n ∈ N.
We will use Perron’s formula to estimate
n6B ∆(n), before combining
it with partial summation to estimate (3.15). Consider the Dirichlet series
D(s) :=
n=1 ∆(n)n
−s. We have
D(s+ 1/3) =
s0,s1,s2,s3=1
ϑ(s0, s)
s3s+10 s
and it is straightforward to check that D(s+ 1/3) =
p ap(s), with
ap(s) = 1 +
3(1− 1/p)
p2s+1(1− 1/p2s+1) +
(1− 1/p)(1 − 2/p)
p3s+1(1− 1/p3s+1)
3(1 − 1/p)2
p5s+2(1− 1/p2s+1)(1 − 1/p3s+1) .
Hence D(s+ 1/3) = E1(s)E2(s), where E1(s) = ζ(2s+ 1)
3ζ(3s + 1) and
E2(s) =
D(s+ 1/3)
ζ(2s+ 1)3ζ(3s+ 1)
p4σ+2
(3.17)
on the half-plane ℜe(s) > −1/2. In particular, E1(s) has a meromorphic
continuation to all of C with a pole of order 4 at s = 0, and E2(s) is
holomorphic and bounded on the half-plane ℜe(s) > −1/4.
Let c = 1/3 + ε for any ε > 0, and let T > 1. Then an application of
Lemma 2.3 reveals that
∆(n) =
∫ c+iT
E1(s − 1/3)E2(s− 1/3)
ds+Oε
(B1/3+ε
provided that B 6∈ Z. We apply Cauchy’s residue theorem to the rectangular
contour C joining the points 1/6− iT , 1/6 + iT , c+ iT and c− iT . Now the
residue of E1(s− 1/3)E2(s− 1/3)Bs/s at s = 1/3 is clearly
Ress=1/3
E1(s− 1/3)E2(s− 1/3)
E2(0)
B1/3P (logB),
for some monic polynomial P ∈ R[x] of degree 3. Define the difference
E(B) =
∆(n)− E2(0)
B1/3P (logB).
Then it follows that
E(B) ≪ε
B1/3+ε
(∫ 1/6+iT
1/6−iT
∫ c−iT
1/6−iT
∫ 1/6+iT
)∣∣∣E1(s − 1/3)
∣∣∣ds,
since E2(s−1/3) is holomorphic and bounded on the half-plane ℜe(s) > 1/6.
THE MANIN CONJECTURE IN DIMENSION 2 43
We proceed to estimate the contribution from the horizontal contours.
Recall the well-known convexity bounds
ζ(σ + it) ≪ε
|t|(1−σ)/3+ε, if σ ∈ [1/2, 1],
|t|(3−4σ)/6+ε, if σ ∈ [0, 1/2],
for any |t| > 1. A proof of these can be found in [60, §II.3.4], for example.
It therefore follows that
E1(σ − 1/3 + it) ≪ε |t|1−3σ+ε (3.18)
for any σ ∈ [1/6, 1/3) and any |t| > 1. We may now deduce that
∫ c−iT
1/6−iT
∣∣∣E1(s− 1/3)
∣∣∣ds ≪ε
BσT−3σ+εdσ
B1/3+εT ε
B1/6T ε
T 1/2
(3.19)
One obtains the same estimate for the contribution from the remaining
horizontal contour. Turning to the vertical contour, (3.18) gives
∫ 1/6+iT
1/6−iT
∣∣∣E1(s− 1/3)
∣∣∣ds ≪ B1/6
|E1(−1/6 + it)|
1 + |t| dt
≪ B1/6
|t|1/2+ε
1 + |t| dt
≪ B1/6T 1/2+ε.
Once combined with (3.19), we conclude that
E(B) ≪ε B1/3+εT−1+ε +B1/6T 1/2+ε,
for any T > 1. Taking T = B1/9 we obtain
∆(n) =
E2(0)
B1/3P (logB) +Oε(B
2/9+ε),
for any ε > 0.
We are now ready to complete the proof of Theorem 1.3. For this it
suffices to combine the latter estimate with partial summation in (3.15),
and then apply Lemma 3.2. In this way we deduce that
NU (B) = 2T (B) +O(B logB)
σ∞E2(0)
BQ(logB) +Oε(B
8/9+ε) +O(B logB),
for a further cubic monic polynomial Q ∈ R[x]. Here σ∞ = 6
F2(u)du is
given by (1.18), and it follows from (3.17) that
E2(0) =
This therefore completes the proof of Theorem 1.3.
44 T.D. BROWNING
4. The D4 del Pezzo surface of degree 3
In this section we consider Manin’s conjecture for the cubic surface
S2 = {[x1, x2, x3, x4] ∈ P3 : x1x2(x1 + x2) + x4(x1 + x2 + x3)2 = 0},
considered in (1.12) Let U2 ⊂ S2 be the open subset formed by deleting the
lines (1.14) from S2. Our task is to estimate NU2(B). In doing so it will
clearly suffice to establish the estimate for any surface that is obtained from
S2 via a unimodular transformation. In view of this we will make the change
of variables
t1 = x1, t2 = x2, t3 = x1 + x2 + x3, t4 = −x4,
which brings S2 into the shape
t1t2(t1 + t2) = t
3t4, (4.1)
and which we henceforth denote by S. The 6 lines on the surface (4.1) take
the shape
ti = tj = 0, tj = t1 + t2 = 0,
where i denotes a generic element of the set {1, 2}, and j an element of
{3, 4}. If U ⊂ S denotes the open subset formed by deleting these lines
from the surface, then we have t3t4 = 0 for any [t] 6∈ U . It now follows that
NU (B) =
#{t ∈ Z4 : (4.1) holds, |t| 6 B, t3t4 6= 0}.
As in the argument of Lemma 3.1, the factor 1
reflects the fact that t and −t
represent the same point in P3. There is a clear symmetry between solutions
such that t3 is positive and negative. Similarly, (4.1) is invariant under the
transformation t1 = −z1, t2 = −z2, t3 = z3 and t4 = −z4. Thus we have
NU (B) = 2#{t ∈ Z4 : (4.1) holds, |t| 6 B, t3, t4 > 1}, (4.2)
for any B > 1.
In the following section we will explicate the relation between (4.2) and
counting integral points on the corresponding universal torsor. When it
comes to the latter task, we will be led to consider the counting problem for
rational points on plane curves of degree 1 and 2. The estimates that we
require will need to be completely uniform in the coefficients of the equations
defining the curves.
Given any plane curve C ⊂ P2 of degree d > 1, that is defined over Q, let
NC(B) := #{x ∈ C(Q) : H(x) 6 B}.
As usual we write Zn for the set of primitive vectors in Zn, and Zn∗ for the set
of vectors in Zn with no components equal to zero. We will restrict our at-
tention to curves that are defined by diagonal ternary forms. We clearly have
NC(B) =
Md(a;B,B,B) for certain non-zero integers a1, a2, a3, where
Md(a;B) := #{x ∈ Z3 : a1xd1 + a2xd2 + a3xd3, |xi| 6 Bi}, (4.3)
and B = (B1, B2, B3). Let us begin with the situation for projective lines.
The following result is due to Heath-Brown [35, Lemma 3].
THE MANIN CONJECTURE IN DIMENSION 2 45
Lemma 4.1. Let a ∈ Z3∗ and let Bi > 0. Then we have
M1(a;B) ≪ 1 +
B1B2B3
max |ai|Bi
Lemma 4.1 shows that there are only O(1) rational points on lines of
sufficiently large height. If one has a line L ⊂ P2 given by the equation
a.x = 0, for a ∈ Z3∗ , then the height of L is simply defined to beH(L) := |a|.
When L is an arbitrary line in Pn, which is defined over Q, there is still a
very natural way of defining its height. The height of L is just the height
of the rational point in the Grassmannian G(1, n) that corresponds to the
line. We will not need this fact in our work. Let L ⊂ P2 be an arbitrary line
defined over Q. Then it follows from Lemma 4.1 that
NL(B) ≪ 1 +
≪ B2.
This is essentially best possible, as can be seen by taking n = 2 in Exercise 3.
Turning to curves of higher degree, we have the following result, which is
a special case of a result due to the author and Heath-Brown [15, Corollary
Lemma 4.2. Let a ∈ Z3∗ such that gcd(ai, aj) = 1, and let Bi > 0. Then
we have
M2(a;B) ≪
B1B2B3
|a1a2a3|
τ(a1a2a3),
where τ(n) :=
d|n 1 denotes the usual divisor function.
In keeping with our discussion of lines, let us consider to what extent our
estimate reflects the true growth rate ofNC(B), for a quadratic curve C ⊂ P2
that is defined by a diagonal equation with pairwise coprime coefficients.
Recall the estimate τ(n) = Oε(n
ε), that holds for any ε > 0. Writing ‖C‖
for the maximum modulus of the coefficients defining C, we deduce from
Lemma 4.2 that
NC(B) ≪ε ‖C‖εB.
This should be compared with the work of Heath-Brown [41, Theorem 3]
that shows NC(B) ≪d,ε B2/d+ε, for any irreducible plane curve C ⊂ P2 of
degree d.
Both Lemma 4.1 and Lemma 4.2 are established using the geometry of
numbers.
4.1. Elementary considerations. We proceed to show how NU (B) can
be related to a count of the integer points on the corresponding universal
torsor. Our argument is in complete analogy to that presented in §3.1,
although the individual steps differ somewhat. If S̃ denotes the minimal
desingularisation of the surface S, then Derenthal [26] has calculated the
universal torsor over S̃, it being embedded in A10 by a single equation
s1u1y
1 + s2u2y
2 + s3u3y
3 = 0. (4.4)
Note that one of the variables does not appear explicitly in the equation.
We will need the following basic fact.
46 T.D. BROWNING
Exercise 18. Let a, b ∈ N. Show that a | b2 if and only if a = uv2 for
u, v ∈ N such that u is square-free and uv | b.
Given v ∈ R and s,u,y ∈ R3, define
Ψ(v, s,u,y) := max
{ |s1s2s3|, |u21u22u23v3y1y2y3|
|s1u21u2u3v2y21|, |s2u1u22u3v2y22 |
. (4.5)
We are now ready to record our translation of the problem to the universal
torsor.
Lemma 4.3. We have
NU (B) = 2#


(v, s,u,y) ∈ N4 × Z3 × N3 :
u3 > 0, Ψ(v, s,u,y) 6 B
(4.4) holds,
|µ(u1u2u3)| = 1,
gcd(s1s2s3, u1u2u3v) = 1,
gcd(yi, yj) = 1,
gcd(yi, sj , sk) = 1,


where i, j, k denote distinct elements from the set {1, 2, 3}.
Proof. Let t ∈ Z4 be a vector such that (4.1) holds, with t3, t4 > 1. Write
η14 = gcd(t1, t4), η24 = gcd(t2, t4/η14), η12 = gcd(t1/η14, t2/η24).
Then η12, η14, η24 ∈ N and there exists z4 ∈ N and z1, z2 ∈ Z such that
t1 = η12η14z1, t2 = η12η24z2, t4 = η14η24z4.
Moreover, it is not hard to deduce that
gcd(η12z1, η24z4) = gcd(η12z2, z4) = gcd(z1, z2) = 1,
gcd(t3, η14, η12η24z2) = 1.
Under this substitution the equation (4.1) becomes
η312z1z2(η14z1 + η24z2) = t
It follows that η312 | t23 in any given integer solution. Exercise 18 therefore
implies that there exist u, v, z3 ∈ N such that |µ(u)| = 1 and
η12 = uv
2, t3 = u
2v3z3,
z1z2(η14z1 + η24z2) = uz
We proceed to consider the effect of the divisibility condition z1z2 | uz23 that
this equation entails.
Recall that gcd(z1, z2) = gcd(z1, z4) = gcd(z2, z4) = 1. Since z1z2 | uz23 ,
there must exist u1, u2, u3, w1, w2, w3 ∈ Z such that w1, w2, w3, u3 > 0 and
u = u1u2u3, z1 = u1w
1, z2 = u2w
2, z3 = w1w2w3.
Here we have used the fact that if p is a prime such that p ∤ u and p | z1z2,
then p must divide z1 or z2 to even order. Under these substitutions our
equation becomes
η14u1w
1 + η24u2w
2 = u3w
THE MANIN CONJECTURE IN DIMENSION 2 47
Moreover, we will have the corresponding coprimality conditions
gcd(u1u2u3vw1, η24z4) = gcd(u1u2u3vw2, z4) = gcd(u1w1, u2w2) = 1,
(4.6)
|µ(u1u2u3)| = 1, gcd(u21u22u23v3w1w2w3, η14, η24u1u22u3v2w22) = 1. (4.7)
We now set s = (η14, η24, z4) and y = w, and replace (u1, u2, u3) by
(−u1,−u2, u3). Tracing through our argument, one sees that we have made
the transformation 
t1 = −s1u21u2u3v2y21,
t2 = −s2u1u22u3v2y22,
t3 = u
3y1y2y3,
t4 = s1s2s3.
In particular, it is clear that the height condition |x| 6 B is equivalent to
Ψ(v, s,u,y) 6 B, in the notation of (4.5). We now observe that under
this transformation the equation (4.1) becomes (4.4), and the coprimality
relations (4.6) and (4.7) can be rewritten
gcd(s2s3, u1u2u3vy1) = gcd(s3, u1u2u3vy2) = gcd(u1y1, u2y2) = 1,
|µ(u1u2u3)| = 1, gcd(s1, u1u2u3vy2 gcd(y3, s2)) = 1.
We can combine these relations with (4.4) to simplify them still further. In
fact, once combined with (4.4), we claim that they are equivalent to the
conditions
|µ(u1u2u3)| = 1, gcd(s1s2s3, u1u2u3v) = gcd(yi, yj) = gcd(yi, sj , sk) = 1,
appearing in the statement of the lemma. To establish the forward impli-
cation, it suffices to show that gcd(y1, y3) = gcd(y2, y3) = 1, the remaining
conditions being immediate. But these two conditions follow on combining
(4.4) with the fact that gcd(y1, s2u2y2) = 1. To see the reverse implication,
the conditions are all immediate apart from
gcd(y1, s2s3) = gcd(y2, s1s3) = gcd(u1, y2) = gcd(u2, y1) = 1.
But each of these is an easy consequence of the assumed coprimality rela-
tions, and (4.4). Finally, we leave it as an exercise to the reader to check
that each (v, s,u,y) counted in the right hand side of Lemma 4.3 produces
a primitive solution of (4.1) with t3, t4 > 1. This completes the proof of
Lemma 4.3. �
In what follows let us write i for a generic element of the set {1, 2, 3}. Fix
a choice of v ∈ N and Si, Ui, Yi > 0, and write
N = Nv(S;U;Y) (4.8)
for the total contribution to NU (B) in Lemma 4.3 from s,u,y contained in
the intervals
Si/2 < si 6 Si, Ui/2 < |ui| 6 Ui, Yi/2 < yi 6 Yi. (4.9)
Write
S = S1S2S3, U = U1U2U3, Y = Y1Y2Y3.
48 T.D. BROWNING
If N = 0 there is nothing to prove, and so we assume henceforth that the
dyadic ranges in (4.9) produce a non-zero value of N . In particular we must
S ≪ B U2Y ≪ B/v3, SiUUiY 2i ≪ B/v2. (4.10)
In this set of lecture notes we will provide two upper bounds for NU (B).
The object of our first bound is to merely establish linear growth, without
worrying about the factor involving logB that we expect to see. By ignoring
some of the technical machinery needed to get better bounds it is hoped that
the overall methodology will be brought into focus. Later we will indicate
how the expected upper bound can be retrieved with a little more work.
4.2. A crude upper bound. We begin by establishing linear growth for
NU (B). Note that (4.10) forces the inequalities Si, Ui, Yi ≪ B. We proceed
to establish the following upper bound.
Lemma 4.4. We have
NU (B) ≪ (logB)9
v6B1/3
Si,Ui,Yi>0
Nv(S;U;Y),
where the maximum is over Si, Ui, Yi > 0 such that (4.10) holds.
Proof. Our starting point is Lemma 4.3. It follows from (4.5) that v 6 B1/3
for any v, s,u,y that contributes to the right hand side. Let us fix a choice
of v ∈ N such that v 6 B1/3, and cover the ranges for s,u,y with dyadic
intervals. Thus for fixed integers σi, νi, ηi > 0, we write
Si = 2
σi , Ui = 2
νi , Yi = 2
and consider the contribution from s,u,y in the range (4.9). But this is
just N = Nv(S;U;Y). Now we have already seen that N = 0 unless (4.10)
holds. Finally, since each Si, Ui, Yi is O(B), it follows that the number of
dyadic intervals needed is O((logB)9). This completes the proof of the
lemma. �
We may now restrict our attention to bounding Nv(S;U;Y) for fixed
values of Si, Ui, Yi > 0 such that (4.10) holds, and fixed v 6 B
1/3. In the
arguments that follow it will be necessary to focus attention on primitive
vectors s ∈ N3. To enable this we draw out possible common factors between
s1, s2, s3, obtaining
Nv(S;U;Y) =
N ∗v (k−1S;U;Y). (4.11)
Here N ∗v (S;U;Y) is defined as for Nv(S;U;Y) but with the extra condition
that gcd(s1, s2, s3) = 1. Let us write S
i = Si/k and S
′ = k−1S.
Recall the equation (4.4) that we must count solutions to, which it will be
convenient to denote by T , and which we will think of as defining a variety
in P2 ×P2 ×P2, with homogeneous coordinates s,u,y. The key idea will be
to count points on the fibres of projections π : T → P2 × P2. This amounts
to fixing six of the variables and estimating the number of points on the
resulting plane curve. Since this family of curves will vary with B, so it is
vital to obtain bounds that are completely uniform in the coefficients of the
defining equation.
THE MANIN CONJECTURE IN DIMENSION 2 49
Let us begin by fixing the variables u,y, and estimating the corresponding
number of vectors s. Now it follows from the coprimality conditions in
Lemma 4.3 that
gcd(u1y
1 , u2y
2, u3y
3) = 1.
For fixed u,y, (4.4) defines a line in P2. We clearly have
N ∗v (S′;U;Y) 6
M1(a;S
in the notation of (4.3), with ai = uiy
i . Since a is primitive, it therefore
follows from Lemma 4.1 that
N ∗v (S′;U;Y) ≪
k2 maxSiUiY
≪ UY + k−2S2/3U2/3Y 1/3.
Here we have used the trivial lower bound max{a, b, c} > (abc)1/3, valid for
any a, b, c > 0. Using (4.10) we conclude that
N ∗v (S′;U;Y) ≪ UY +
. (4.12)
The second term here will be satisfactory from our point of view, but the
first is disastrous, since we will run into trouble when it comes to summing
over k in (4.11).
It turns out that an altogether different bound is required to handle the
contribution from really small values of S′. For this we will fix values of s,u
in (4.4), and count points on the resulting family of conics. First we need
to record the coprimality relation
gcd(siui, sjuj) = 1,
which we claim holds for any of the vectors s,u,y in which we are interested.
But this follows on noting that gcd(si, sj) = 1 for any primitive vector s ∈ Z3
such that (4.4) holds and gcd(si, u1u2u3) = gcd(yi, sj, sk) = 1. We now have
N ∗v (S′;U;Y) 6
M2(a;Y),
in the notation of (4.3), with ai = siui. In particular ai is non-zero and
gcd(ai, aj) = 1 in the statement of Lemma 4.2, whence
N ∗v (S′;U;Y) ≪
k3/2Y 1/2
S1/2U1/2
2ω(s1s2s3u1u2u3).
In view of the bounds S,U ≪ B, we clearly have
2ω(s1s2s3u1u2u3) ≪ε (s1s2s3u1u2u3)ε ≪ε (SU)ε ≪ε B2ε.
Once inserted into our bound for N ∗v (S′;U;Y), and combined with (4.10),
we deduce that
N ∗v (S′;U;Y) ≪ε B2ε
S1/2U1/2Y 1/2
SUB2ε
B1+2ε
k3/2v3/2
(4.13)
Here the second term will provide a satisfactory contribution, and we will
balance the first term with our earlier estimate (4.12).
50 T.D. BROWNING
Note that
k3/2v3/2
by (4.10). It therefore follows from (4.13) and (4.12) that
N ∗v (S′;U;Y) ≪ε
B1+2ε
k3/2v
Once inserted into (4.11), and then into the statement of Lemma 4.4, we
may conclude that
NU (B) ≪ε (logB)9
v6B1/3
B1+2ε
k3/2v
≪ε B1+2ε(logB)10
≪ε B1+3ε.
Recall that NU2(B) 6 NU (B), where U ⊂ S is the open subset associated to
the surface (4.1), and NU2(B) is the counting function associated to (1.12).
On redefining the choice of parameter ε > 0, we have therefore established
the following result.
Theorem 4.1. We have NU2(B) ≪ε B1+ε, for any ε > 0.
The reader will note that there many places in our argument where we
have been wasteful. The most damaging has been in our use of the trivial
bound 2ω(n) = Oε(n
ε), in the deduction of (4.13). Using the fact that 2ω(n)
has average order ζ(2)−1 log n, it is not particularly difficult to replace the
Bε in Theorem 4.1 with a large power of logB.
Exercise 19. By analysing the proof of Theorem 4.1, find an explicit value
of A > 6 such that NU2(B) ≪ B(logB)A.
In the next section we will be able to show that the value A = 6 is an
admissible exponent, as claimed in Theorem 1.2.
4.3. A better upper bound. Crucial to the proof of Theorem 4.1 was an
investigation of the density of integer solutions to the equation (4.4). It is
in our treatment of this equation that we will hope to gain some saving.
Lets put the problem on a more general footing. For any A,B,C ∈ R3>1,
let M(A,B,C) denote the number of a,b, c ∈ Z3∗ such that
a1b1c
1 + a2b2c
2 + a3b3c
3 = 0 (4.14)
|ai| 6 Ai, |bi| 6 Bi, |ci| 6 Ci,
gcd(ai, cj) = gcd(ci, cj) = 1 (4.15)
|µ(a1a2a3)| = 1, gcd(ai, bj , bk) = 1. (4.16)
Here, we recall that Z3∗ denotes the set of primitive vectors in Z
3 with all
components non-zero. It will be convenient to set
A = A1A2A3, B = B1B2B3, C = C1C2C3.
THE MANIN CONJECTURE IN DIMENSION 2 51
Arguing exactly as in the previous section, it is not difficult to deduce from
Lemmas 4.1 and 4.2 that
M(A,B,C) ≪ε Amin{C,AεB1+ε}+A2/3B2/3C1/3 +A1/2+εB1/2+εC1/2
for any ε > 0, whence
M(A,B,C) ≪ε A2/3B2/3C1/3 +A1+εB1/2+εC1/2. (4.17)
By working a little harder, we would like to replace the terms Aε, Bε with
something rather smaller.
The main problem to be faced emerges in the application of Lemma 4.2,
which gives
M(A,B,C) ≪
|a1a2a3b1b2b3|1/2
2ω(a1a2a3b1b2b3).
Rather than using the trivial bound 2ω(n) = Oε(n
ε), as above, we can try
to make use of the fact that 2ω(n) has average order ζ(2)−1 log n in order to
get some saving. Following this line of thought it is fairly straightforward
to show that AεBε can be replaced by (logA)3/2(logB)3 in (4.17). However
this would still not be enough to deduce the best possible upper bound for
NU (B) that we would like. Let us simplify matters by considering only the
contribution
S(A,B) =
2ω(a1a2a3b1b2b3),
to the above estimate for M(A,B,C). Then S(A,B) has exact order of
magnitude
(logAi)(logBi),
so how can we hope to do better than this? The crucial observation comes
in noting that we are only interested in summing over values of a,b for
which the corresponding conic (4.14) has a non-zero solution c ∈ Z3, with
gcd(ci, cj) = 1. If we denote this finer quantity by S∗(A,B), then it is
actually possible to show that
S∗(A,B) ≪ AB. (4.18)
This is established in [13, Lemma 1], and is simply a facet of the well-known
fact that a random plane conic doesn’t have a rational point. This should
be compared with the work of Serre [58]. Using the large sieve inequality,
Serre has shown
#{y ∈ Z3 : |y| 6 Y, (−y1y3,−y2y3)Q = 1} ≪
(log Y )3/2
where
(a, b)Q =
1, if ax2 + by2 = z2 has a solution (x, y, z) 6= 0 in Q3,
−1, otherwise,
denotes the Hilbert symbol. Guo [32] has established an asymptotic for-
mula for the corresponding quantity in which one counts only odd values of
y1, y2, y3 such that the product y1y2y3 is square-free.
52 T.D. BROWNING
Thus in addition to considering the density of integer solutions to diagonal
quadratic equations, as in the previous section, we also need to consider how
often such an equation has at least one non-trivial integer solution in order
to derive sufficiently sharp bounds. The outcome of this investigation is the
following result, which is established in [13, Lemma 2].
Lemma 4.5. For any ε > 0, we have
M(A,B,C) ≪ε A2/3B2/3C1/3 + στAB1/2C1/2,
where
σ = 1 +
min{A,B}ε
min{BiBj}1/16
, τ = 1 +
min{BiBj}1/16
It is clear that this constitutes a substantial sharpening over our earlier
estimate (4.17) for M(A,B,C). Nonetheless this is still not enough on
its own, and we will need an alternative estimate when B1, B2, B3 have
particularly awkward sizes. The following result is rather easy to establish.
Lemma 4.6. We have
M(A,B,C) ≪ ABiBj(Ck + CiCjA−1k )(logAC)
for any permutation {i, j, k} of the set {1, 2, 3}.
Proof. For fixed integers a, b, q, let ρ(q; a, b) denote the number of solutions
to the congruence at2 + b ≡ 0 mod q. We then have
ρ(q; a, b) 6
|µ(d)|
. (4.19)
It will clearly suffice to establish Lemma 4.6 in the case (i, j, k) = (1, 2, 3),
say. Now it follows from (4.14) that for given ai, b1, b2, c3, and each corre-
sponding solution t of the congruence
a1b1t
2 + a2b2 ≡ 0 (mod a3c23),
we must have c1 ≡ tc2 mod a3c23 in any solution to be counted. This gives
rise to an equation of the form h.w = 0, with h = (1,−t, a3c23) and w =
(c1, c2, k). Upon recalling that gcd(c1, c2) = 1 from (4.15), an application of
Lemma 4.1 therefore yields the bound
≪ ρ(a3c23; a1b2, a2b2)
|a3c23|
for the number of possible b3, c1, c2 given fixed choices of ai, b1, b2 and c3. It
now follows from (4.19) that
M(A,B,C) ≪
ai,b1,b2,c3
ρ(a3c
3; a1b2, a2b2)
|a3c23|
ai,b1,b2,c3
d|a3c3
|µ(d)|
(−a1a2b1b2
|a3c23|
ai,b1,b2,c3
τ(a3)τ(c3) + C1C2
ai,b1,b2,c3
τ(a3)τ(c3)
|a3c23|
THE MANIN CONJECTURE IN DIMENSION 2 53
A simple application of partial summation now reveals that
M(A,B,C) ≪
AB1B2C3 +A1A2B1B2C1C2
(logAC)2,
as required to complete the proof of Lemma 4.6. �
We are now ready to combine Lemmas 4.5 and 4.6 to get a sharper upper
bound for NU (B). Taking Lemma 4.3 as our starting point we need to
bound the quantity N = Nv(S;U;Y) defined in (4.8), for fixed choices of
v ∈ N and Si, Ui, Yi > 0. As previously we will need to extract common
factors from s1, s2, s3, leading to the equality (4.11). Writing S
i = Si/k and
S′ = k−1S, as before, it is a simple matter to check that we have
N ∗v (S′;U;Y) 6 M(U,S′,Y),
with (a,b, c) = (u, s,y). Indeed we plainly have
gcd(ui, yj) = gcd(yi, yj) = 1, |µ(u1u2u3)| = gcd(ui, sj , sk) = 1,
and u, s,y ∈ Z4∗ , for any vectors counted by N ∗v (S′;U;Y), as required for
M(U,S′,Y). It now follows from (4.11) and Lemma 4.5 that
Nv(S;U;Y) ≪ε
(U2/3S2/3Y 1/3
+ k2/16στ
US1/2Y 1/2
≪ε U2/3S2/3Y 1/3 + στUS1/2Y 1/2,
for any ε > 0, where
σ = 1 +
min{S,U}ε
min{SiSj}1/16
, τ = 1 +
min{SiSj}1/16
In order to obtain our final estimate for NU (B) we need to sum this bound
over all positive integers v 6 B1/3, as in Lemma 4.4, and over all possible
dyadic intervals for Si, Ui, Yi, subject to (4.10).
Suppose for the moment that we want to sum over all possible dyadic
intervals X 6 |x| < 2X, for which |x| 6 X . Then in deducing Lemma 4.4
we employed the basic bound O(logX ) for the number of possible choices for
X. In the present investigation we will be more efficient and take advantage
of the easily established estimates
Xδ ≪δ
1, if δ < 0,
X δ, if δ > 0,
where the sum is over dyadic intervals for X 6 X . We will make frequent
use of these bounds without further mention.
Returning to our estimate for Nv(S;U;Y), we may now conclude from
the bound Yi 6 B
1/2/(v2SiUUi)
1/2 in (4.10) that
NU (B) ≪ε
v6B1/3
Si,Ui,Yi
U2/3S2/3Y 1/3 + στUS1/2Y 1/2
≪ε B1/2
v6B1/3
Si,Ui
v6B1/3
Si,Ui,Yi
στUS1/2Y 1/2
≪ε B(logB)6 +
v6B1/3
Si,Ui,Yi
στUS1/2Y 1/2.
54 T.D. BROWNING
The first term on the right-hand side is clearly satisfactory, and it remains
to deal with the second term, which we denote by R for convenience. We
would like to show that R ≪ B(logB)6.
Suppose without loss of generality that S1 6 S2 6 S3, so that in particular
min{SiSj} = S1S2 in σ and τ. If there is a constant A > 0 such that
S3 6 (S1S2)
A, then it follows that
σ 6 (S1S2)
ε−1/16Sε3 6 (S1S2)
(1+A)ε−1/16 ≪ 1,
provided that ε is sufficiently small. Taking τ ≪ logB, we may then argue
as above to conclude that there is a contribution of O(B(logB)6) to R from
this case. Suppose now that there exists A′ > 0 such that U 6 (S1S2)
Then we have σ ≪ 1 and τ ≪ logB, so that there is a contribution of
O(B(logB)6) to R in this case too.
Finally it remains to consider the contribution to NU (B) from Si, Ui, Yi
such that
S1S2 6 min{S3, U}δ, (4.20)
for some small value of δ > 0, with S1 6 S2 6 S3. Let us denote this
contribution N0, say. To estimate N0 we will return to the task of estimat-
ing Nv(S;U;Y) for fixed v, Si, Ui, Yi, but this time apply Lemma 4.6 with
(i, j, k) = (1, 2, 3). This gives
Nv(S;U;Y) ≪ (logB)2
US1S2Y3 + U1U2S1S2Y1Y2
We must now sum over dyadic intervals for Si, Ui, Yi. Thus it follows from
the bound Yi 6 B
1/2/(v2SiUUi)
1/2 in (4.10) that
N0 ≪ (logB)2
v6B1/3
Si,Ui,Yi
US1S2Y3 + U1U2S1S2Y1Y2
≪ (logB)2
v6B1/3
Si,Ui
Y1,Y2
B1/2U1/2S1S2
Si,Ui,Y3
B(S1S2U1U2)
Since U2 ≪ B/(v3Y1Y2) in (4.10), and S1S2 ≪ Sδ3 by (4.20), we therefore
deduce that the overall contribution from the first inner sum is
≪ (logB)2
v6B1/3
Si,Y1,Y2
U2,U3
B3/4S1S2
v7/4S
S2,S3,Y1,Y2
U2,U3
B3/4(logB)2
1/2−δ
Turning to the contribution from the second inner sum, we deduce from a
second application of (4.20) that
N0 ≪ B + (logB)2
Si,Ui,Y3
B(S1S2U1U2)
≪ B + (logB)2
S2,S3,Ui,Y3
U (1−δ)/2
≪ B(logB)5.
THE MANIN CONJECTURE IN DIMENSION 2 55
Once combined with our earlier work, this therefore concludes the proof of
Theorem 1.2.
Exercise 20. By mimicking the argument in [13, §5], establish the lower
bound NU2(B) ≫ B(logB)6.
Acknowledgements. The author is grateful to Roger Heath-Brown for
a number of useful conversations relating to the contents of §2.3, and to
Michael Harvey for spotting several typographical errors in an earlier draft.
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School of Mathematics, University of Bristol, Bristol BS8 1TW
E-mail address: t.d.browning@bristol.ac.uk
	1. Introduction
	1.1. Notation
	1.2. The Manin conjectures
	1.3. Degree 3 surfaces
	1.4. Degree 4 surfaces
	1.5. Degree 5 surfaces
	1.6. Universal torsors
	2. Further heuristics
	2.1. The lines on a cubic surface
	2.2. Cubic characters and Jacobi sums
	2.3. The Hardy–Littlewood circle method
	3. The A1 del Pezzo surface of degree 6
	3.1. Elementary considerations
	3.2. The asymptotic formula
	4. The D4 del Pezzo surface of degree 3
	4.1. Elementary considerations
	4.2. A crude upper bound
	4.3. A better upper bound
	References

  These lecture notes describe the current state of affairs for Manin's
conjecture in the context of del Pezzo surfaces.

Introduction 2
1.1. Notation 5
1.2. The Manin conjectures 6
1.3. Degree 3 surfaces 10
1.4. Degree 4 surfaces 13
1.5. Degree > 5 surfaces 17
1.6. Universal torsors 19
2. Further heuristics 20
2.1. The lines on a cubic surface 21
2.2. Cubic characters and Jacobi sums 23
2.3. The Hardy–Littlewood circle method 25
3. The A1 del Pezzo surface of degree 6 34
3.1. Elementary considerations 35
3.2. The asymptotic formula 38
4. The D4 del Pezzo surface of degree 3 44
4.1. Elementary considerations 45
4.2. A crude upper bound 48
4.3. A better upper bound 50
References 55
Date: November 12, 2018.
2000 Mathematics Subject Classification. 11G35 (14G05, 14G10).
http://arxiv.org/abs/0704.1217v1
2 T.D. BROWNING
1. Introduction
The study of integer solutions to Diophantine equations is a topic that
is almost as old as mathematics itself. Since its inception at the hands of
Diophantus of Alexandria in 250 A.D., it has been found to relate to virtu-
ally every mathematical field. The purpose of these lecture notes is to focus
attention upon an aspect of Diophantine equations that has only crystallised
within the last few decades, and which exhibits a fascinating interplay be-
tween the subjects of analytic number theory and algebraic geometry.
Suppose that we are given a polynomial f ∈ Z[x1, . . . , xn], and write
Sf := {x = (x1, . . . , xn) ∈ Zn \ {0} : f(x) = 0}
for the corresponding locus of non-zero solutions. There are a number of
basic questions that can be asked about the set Sf . When is Sf non-empty?
How large is Sf when it is non-empty? When Sf is infinite can we describe
the set in some way? A lot of the work to date has been driven by trying
to understand the situation for equations in only n = 2 or 3 variables. The
last 50 years in particular has delivered a remarkable level of understanding
concerning the arithmetic of curves. In stark contrast to this, the situation
for equations in 4 or more variables remains a relatively untamed frontier,
with only a scattering of results available.
We will restrict attention to the study of Diophantine equations f = 0
for which the corresponding zero set Sf is infinite. The description that we
will aim for is quantative in nature, the main goal being to understand how
the counting function
N(f ;B) := #{x ∈ Sf : |x| 6 B} (1.1)
behaves, as B → ∞. Here, as throughout these lecture notes, |z| denotes the
norm max16i6n |zi| for any z ∈ Rn. Aside from being intrinsically interesting
in their own right, as we will see shortly, the study of functions likeN(f ;B) is
often an effective means of determining whether or not the equation f = 0
has any non-trivial integer solutions at all. In many applications of the
Hardy–Littlewood circle method, for example, one is able to prove that Sf
is non-empty by showing that N(f ;B) > 0 for large enough values of B. In
fact the method usually carries with it a proof of the fact that Sf is infinite.
In the context of the circle method at least, it is useful to have a general
idea of which polynomials f might have an infinite zero locus Sf .
Suppose that f ∈ Z[x1, . . . , xn] has degree d > 1. Then for the vectors
x ∈ Zn counted by N(f ;B), the values of f(x) will all be of order Bd.
In fact a positive proportion of them will have exact order Bd. Thus the
probability that a randomly chosen value of f(x) should vanish might be
expected to be of order 1/Bd. Since the number of x to be considered has
order Bn, this leads us to the following general expectation.
Heuristic. When n > d we have
Bn−d ≪ N(f ;B) ≪ Bn−d. (1.2)
As a crude first approximation, therefore, this heuristic tells us that we
might expect polynomials whose degree does not exceed the number of vari-
ables to have infinitely many solutions. Unfortunately there are a number
THE MANIN CONJECTURE IN DIMENSION 2 3
of things that can conspire to upset this heuristic expectation. First and
foremost, local conditions will often provide a reason for N(f ;B) to be iden-
tically zero no matter the values of d and n. By local obstructions we mean
that the obvious necessary conditions for Sf to be non-empty fail. These
are the conditions that the equation f(x) = 0 should have a real solution
x ∈ Rn, and secondly, that the congruence
f(x) ≡ 0 (mod pk)
should be soluble for every prime power pk. When f is homogeneous we
must take care to ignore the trivial solution x = (0, . . . , 0) in both cases.
It is quite easy to construct examples that illustrate the failure of these
local conditions. For example, when d is even, the equation
x2d1 + · · ·+ x2dn = 0
doesn’t have any integer solutions, since it patently doesn’t have any real
solutions. Let us now exhibit an example, due to Mordell [50], of a poly-
nomial equation that fails to have integer solutions because it fails to have
solutions as a congruence modulo a prime p. Let K be an algebraic number
field of degree d, with ring of integers OK . Write
N(y1, . . . , yd) := NK/Q(y1ω1 + · · · + ydωd)
for the corresponding norm form, where ω1, . . . , ωd is a basis for K over Q.
It is clear that N is a homogeneous polynomial of degree d, with coefficients
in Z.
Exercise 1. Let y ∈ Zn and let p be a rational prime such that the ideal
(p) ⊂ OK is prime. Show that p | N(y) if and only if p | y.
We define the homogeneous polynomial
f1 := N(x1, . . . , xd) + pN(xd+1, . . . , x2d) + · · ·+ pd−1N(xd2−d+1, . . . , xd2),
(1.3)
which has degree d and d2 variables. We claim that the only integer solution
to the equation f1(x) = 0 is the trivial solution x = 0. To see this we argue
by contradiction. Thus we suppose there to be a vector x ∈ Zd2 such that
f1(x) = 0, with gcd(x1, . . . , xd2) = 1. Viewed modulo p we deduce that
p | N(x1, . . . , xd), whence p | x1, . . . xd by Exercise 1. Writing xi = pyi for
1 6 i 6 d, and substituting into the equation f1 = 0, we deduce that
pd−1N(y1, . . . , yd) +N(xd+1, . . . , x2d) + · · ·+ pd−2N(xd2−d+1, . . . , xd2) = 0.
But then we deduce in a similar fashion that p | xd+1, . . . , x2d. We may
clearly continue in this fashion, ultimately concluding that p | x1, . . . , xd2 ,
which is a contradiction. This polynomial illustrates that for any d it is
possible to construct examples of homogeneous polynomials in d2 variables
that have no non-zero integer solutions. This fits with the facts rather well:
when d = 2 we know from Meyer’s theorem that an indefinite quadratic form
always has non-trivial solutions as soon as its rank is at least 5. Similarly,
it is conjectured that 10 variables are always enough to ensure the solubility
in integers of an arbitrary homogeneous cubic equation.
So far we have only seen examples of polynomials f for which the zero
locus Sf is empty. In this case the corresponding counting function N(f ;B)
4 T.D. BROWNING
is particularly easy to estimate! There are also examples which show that
N(f ;B) may grow in quite unexpected ways, even when n > d. An equation
that illustrates excessive growth is provided by the polynomial
f2 := x
1 − x2(xd−13 + · · · + x
Here there are “trivial” solutions of the type (0, 0, a3, . . . , an) which already
contribute ≫ Bn−2 to the counting function N(f ;B), whereas (1.2) predicts
that we should have exponent n− d.
It is also possible to construct examples of varieties which demonstrate
inferior growth. Let n > d2 and choose any d2 linear forms L1, . . . , Ld2 ∈
Z[x1, . . . , xn] that are linearly independent over Q. Consider the form
f3 := f1(L1(x1, . . . , xn), . . . , Ld2(x1, . . . , xn)),
where f1 is given by (1.3). Then it is clear that N(f3;B) has the same order
of magnitude as the counting function associated to the system of linear
forms L1 = · · · = Ld2 = 0. Since these forms are linearly independent we
deduce that N(f3;B) has order of magnitude B
n−d2 , whereas (1.2) led us
to expect an exponent n− d.
We have seen lots of reasons why (1.2) might fail — how about some
evidence supporting it? One of the most outstanding achievements in this
direction is the following very general result due to Birch [4].
Theorem 1.1. Suppose f ∈ Z[x1, . . . , xn] is a non-singular homogeneous
polynomial of degree d in n > (d−1)2d variables. Assume that f(x) = 0 has
non-trivial solutions in R and each p-adic field Qp. Then there is a constant
cf > 0 such that
N(f ;B) ∼ cfBn−d,
as B → ∞.
Birch’s result doesn’t apply to either of the polynomials f2, f3 that we con-
sidered above, since both of these actually have a rather large singular locus.
Since generic homogeneous polynomials are non-singular, Birch’s result an-
swers our initial questions completely for typical forms with n > (d− 1)2d.
It would be of considerable interest to reduce the lower bound for n, but ex-
cept for d 6 4 this has not been done. Theorem 1.1 is established using the
circle method, and exhibits a common feature of all Diophantine problems
successfully tackled via this machinery: the number of variables involved
needs to be large compared to the degree. In particular, there is an obvious
disparity between the range for n in Birch’s result and the range for n in
(1.2). The main aim of these lecture notes is to discuss the situation when
n is comparable in size with d.
It turns out that phrasing things in terms of single polynomial equations
is far too restrictive. It is much more satisfactory to work with projective
algebraic varieties V ⊆ Pn−1. All of the varieties that we will work with
are assumed to be cut out by a finite system of homogeneous equations,
all of which are defined over Q. In line with the above, our main interest
lies with those varieties for which we expect the set V (Q) = V ∩ Pn−1(Q)
to be infinite. Let x = [x] ∈ Pn−1(Q) be a projective rational point, with
x = (x1, . . . , xn) ∈ Zn chosen so that gcd(x1, . . . , xn) = 1. Then we define
THE MANIN CONJECTURE IN DIMENSION 2 5
the height of x to be H(x) := |x|, where as usual |z| denotes the norm
max16i6n |zi|. Given any subset U ⊆ V , we may then define the counting
function
NU (B) := #{x ∈ U(Q) : H(x) 6 B}, (1.4)
for each B > 1. The main difference between this counting function and the
quantity introduced in (1.1) is that we are now only interested in primitive
integer solutions, by which we mean that the components of the vector
x ∈ Zn should share no common prime factors. When the polynomial in
(1.1) is homogeneous, this formulation has the advantage of treating all
scalar multiples of a given non-zero integer solution as a single point.
Recall the definition of the Möbius function µ : N → {0, 1}, which is given
µ(n) =
0, if p2 | n for some prime p,
1, if n = 1,
(−1)r, if n = p1 · · · pr for distinct primes p1, . . . , pr.
The Möbius function is a multiplicative arithmetic function, and will play a
very useful rôle in our work.
Exercise 2. Let S ⊆ Zn be an arbitrary set. Show that
#{x ∈ S : gcd(x1, . . . , xn) = 1} =
µ(k)#{x ∈ S : k | xi, (1 6 i 6 n)}.
We now have the tools with which to relate the counting function (1.4) to
our earlier counting function N(f ;B) in (1.1), when U = V and V ⊂ Pn−1 is
a hypersurface with underlying homogeneous polynomial f ∈ Z[x1, . . . , xn].
On noting that x and −x represent the same point in Pn−1, it follows from
Exercise 2 that
NV (B) =
µ(k)N(f ;B/k). (1.5)
When f is non-singular of degree d, with n > (d − 1)2d, it can be deduced
from Theorem 1.1 that NV (B) ∼ c̃fBn−d, where c̃f = 12ζ(n− d)
−1cf .
Returning to the counting function (1.4), it is easy to check that NU (B)
is bounded for each B, no matter what the choice of U and V . This fol-
lows on combining the fact that NV (B) 6 NPn−1(B) with the self-evident
inequalities NPn−1(B) 6 #{x ∈ Zn : |x| 6 B} 6 (2B + 1)n. In fact it is not
so hard to establish an asymptotic formula for NPn−1(B).
Exercise 3. Let n > 2. Use Exercise 2 to show that
NPn−1(B) =
Bn +On
Bn−1(logB)bn
where b2 = 1 and bn = 0 for n > 2.
1.1. Notation. Before embarking on the main thrust of these lecture notes,
we take a moment to summarise some of the key pieces of notation that we
will make use of.
6 T.D. BROWNING
• A(x) = O(B(x)) means that there exists a constant c > 0 and x0 ∈ R
such that |A(x)| 6 cB(x) for all x > x0. Throughout our work we
will follow the convention that the implied constant is absolute unless
explicitly indicated otherwise by an appropriate subscript. We will
often use the alternative notation A(x) ≪ B(x) or B(x) ≫ A(x).
• A(x) ≍ B(x) means A(x) ≪ B(x) ≪ A(x).
• A(x) = o(B(x)) means limx→∞A(x)/B(x) = 0.
• A(x) ∼ B(x) means limx→∞A(x)/B(x) = 1.
• N = {1, 2, 3, ...} will denote the set of natural numbers.
• Zn will denote the set of primitive vectors in Zn, and Zn∗ will denote
the set of v ∈ Zn such that v1 · · · vn 6= 0.
• |z| := max16i6n |zi|, for any vector z ∈ Rn.
1.2. The Manin conjectures. Around 1989 Manin initiated a program
to relate the asymptotic behaviour of counting functions to the intrinsic
geometry of the underlying variety, for suitable families of algebraic varieties.
It is precisely this rich interplay between arithmetic and geometry that this
set of lecture notes aims to communicate.
Several of the varieties that we have looked at so far have many rational
points, in the sense that NV (B) grows like a power of B. For such varieties
it is natural to look at the quantity
βV := lim
logNV (B)
assuming that this limit exists. In general we may consider βU for any
Zariski open subset U ⊆ V . It is clear that βU gives a measure of “how
large” the set U(Q) is, since we will have
BβU−ε ≪ NU (B) ≪ BβU+ε
for sufficiently large values of B and any ε > 0. The insight of Manin was
to try and relate βU to the geometry of V via the introduction of a certain
quantity α(V ). Before defining this quantity we will need some facts from
algebraic geometry. The facts that we will need are summarised in more
detail in the book of Hindry and Silverman [44, §A].
Assume that V ⊂ Pn−1 is non-singular and let Div(V ) be the free abelian
group generated by finite formal sums of the shape D =
nY Y , with nY ∈
Z and Y running over geometrically irreducible codimension 1 subvarieties
of V . A divisor D ∈ Div(V ) is effective if nY > 0 for all Y , and D is
said to be principal if D =
Y ordY (f)Y = Df , say, for some rational
function f ∈ C(V ). The intuitive idea behind the definition of the ordY
function for a codimension 1 subvariety Y is that ordY (f) = k if f has a
zero of order k along Y , while ordY (f) = −k if f has a pole of order k along
Y . If f has neither a zero nor a pole along Y , then ordY (f) = 0. Since
Df + Dg = Dfg and D1/f = −Df , the principal divisors form a subgroup
PDiv(V ) of Div(V ). We define the geometric Picard group associated to V
to be
(V ) := Div(V )/PDiv(V ).
A divisor class [D] ∈ PicQ(V ) is effective if there exists an effective di-
visor in the class. One may also construct the geometric Néron–Severi
THE MANIN CONJECTURE IN DIMENSION 2 7
group NSQ(V ), which is Div(V ) modulo a further equivalence relation called
“algebraic equivalence”. When V is covered by curves of genus zero, as
in all the cases of interest to us in these lecture notes, it turns out that
NSQ(V ) = PicQ(V ). We illustrate the definition of PicQ(V ) by calculating
it in the simplest possible case V = Pn−1.
Lemma 1.1. We have Pic
(Pn−1) = Z.
Proof. An irreducible divisor on Pn−1 has the form Y = {F = 0} for some
absolutely irreducible form F ∈ C[x1, . . . , xn]. For such a divisor, define the
degree of Y to be degY = degF . Extend the definition of degree additively,
so that
nY deg Y.
The map deg : Div(Pn−1) → Z is clearly a homomorphism, and to establish
the lemma it will suffice to show that the kernel of this map is precisely the
subgroup PDiv(Pn−1). To see this, we note that degDf = 0 for any rational
function f = F1/F2. Indeed, the sum of the positive degree terms will be
degF1, whereas the sum of the negative degree terms will be degF2, and this
two degrees must coincide in order to have a well-defined rational function.
Conversely, if D = n1Y1 + · · · + nkYk has degree zero, with Yi = {Fi = 0}
for 1 6 i 6 k, then f = F
1 · · ·F
k is a well-defined rational function on
Pn−1 with Df = D. This completes the proof of the lemma. �
Returning to the setting of arbitrary non-singular varieties V ∈ Pn−1, let
H ∈ Div(V ) be a divisor corresponding to a hyperplane section. Further-
more, let KV ∈ Div(V ) be the canonical divisor. This is a common abuse
of notation: really KV refers to the class of Dω in PicQ(V ) for any differen-
tial (dimV )-form ω of V . It would take us too far afield to include precise
definitions of these objects here. We may now define the real number
α(V ) := inf{r ∈ R : r[H] + [KV ] ∈ Λeff(V )},
where
Λeff(V ) := {c1[D1] + · · ·+ ck[Dk] : ci ∈ R>0, [Di] ∈ NSQ(V ) effective}
is the so-called effective cone of divisors. It does not matter too much if this
definition is currently meaningless: the main thing is that α(V ) depends in
an explicit way on the geometry of V over C. We now have the following
basic conjecture due to Batyrev and Manin [1, Conjecture A].
Conjecture 1.1. For all ε > 0 there exists a Zariski open subset U ⊆ V
such that βU 6 α(V ) + ε.
A non-singular variety V ⊂ Pn−1 is said to be Fano if KV does not lie in
the closure of the effective cone Λeff(V ) ⊂ NSQ(V )⊗Z R. This is equivalent
to −KV being ample, and implies in particular that V is covered by rational
curves. As an example, suppose that V is a complete intersection, with
V = W1 ∩ · · · ∩ Wt for hypersurfaces Wi ⊂ Pn−1 of degree di. Then V is
Fano if and only if d1+ · · ·+dt < n. With this in mind we have the following
supplementary prediction.
8 T.D. BROWNING
Conjecture 1.2. Assume that V is Fano and V (Q) is Zariski dense in V .
Then there exists a Zariski open subset U ⊆ V such that βU = α(V ).
We have α(V ) = n − d1 − · · · − dt when V is a non-singular complete
intersection as above. In particular, when V is a hypersurface of degree
d we may deduce from Theorem 1.1 that Conjecture 1.2 holds when n is
sufficiently large in terms of d. It also holds for n > 3 when d = 2 (see
Heath-Brown [37], for example). Finally we remark that Conjecture 1.2
holds for projective space. This follows from Exercise 3 and the fact that
[KPn−1 ] = [−nH] in PicQ(Pn−1), whence α(Pn−1) = n.
The title of these lecture notes suggests that we will focus our attention
on the situation for varieties of dimension 2. Before doing so, let us consider
the situation for curves briefly. For simplicity we will discuss only projective
plane curves V ⊂ P2 of degree d. There is a natural trichotomy among
such curves, according to the genus g of the curve. For curves with g = 0,
otherwise known as rational curves, it is possible to show that NV (B) ∼
2/d. This is in complete accordance with the Manin conjecture. It is
an amusing exercise to check that such an asymptotic formula holds with
d = 2 when V is given by the equation x21 + x
2 = x
3, for example. When
g = 1 and V (Q) 6= ∅, the curve is elliptic and it has been shown by Néron
[44, Theorem B.6.3] that
NV (B) ∼ cV (logB)rV /2,
where rV denotes the rank of V . Thus although there can be infinitely many
points in V (Q), we see that the corresponding counting function grows much
more slowly than for rational curves. Elliptic curves are not Fano, and
so this is not covered by the Manin conjecture. However it does confirm
Conjecture 1.1, since α(V ) = 0. When g > 2 the work of Faltings [28] shows
that V (Q) is always finite, and so it does not make sense to study NV (B).
Let us now concern ourselves with Fano varieties of dimension 2. We
begin with some simple-minded numerics. Suppose that we are given a Fano
variety V of dimension 2 and degree d, which is a non-singular complete
intersection in Pn−1. Thus V = W1 ∩ · · · ∩Wt for hypersurfaces Wi ⊂ Pn−1
of degree di, and we assume that the intersection is transversal at a generic
point of V . We are not interested in hyperplane sections of V , and so we
will assume without loss of generality that di > 2 for each 1 6 i 6 t. Then
the following inequalities must be satisfied:
(1) d1 + · · ·+ dt < n, [Fano]
(2) n− 1− t = 2, [complete intersection of dimension 2]
(3) d = d1 · · · dt, [Bézout]
(4) dt > · · · > d1 > 2.
It follows that the only possibilities are
(d; d1, . . . , dt;n; t) ∈
(2; 2; 4; 1), (3; 3; 4; 1), (4; 2, 2; 5; 2)
These surfaces correspond to a quadric in P3, a cubic surface in P3, and an
intersection of 2 quadrics in P4, respectively. We have already observed that
the Manin conjecture holds for quadrics. Hence one would like to examine
the latter two surfaces. In fact these are the most familiar examples of
“del Pezzo surfaces”. We will see in §1.5 that not all del Pezzo surfaces are
THE MANIN CONJECTURE IN DIMENSION 2 9
complete intersections, and so we have missed out on several surfaces in this
analysis. Nonetheless, a substantial portion of these lecture notes will focus
on cubic surfaces in P3 and intersections of 2 quadrics in P4.
It is now time to give a formal definition of a del Pezzo surface. Let us
begin with a discussion of non-singular del Pezzo surfaces. Let d > 3. Then
a del Pezzo surface of degree d is a non-singular surface S ⊂ Pd of degree
d, with very ample anticanonical divisor −KS. This latter condition is
equivalent to the equality [−KS ] = [H] in PicQ(S), for a hyperplane section
H ∈ Div(S). The facts that we will recall here are all established in the
book of Manin [49], for example. It is well-known that del Pezzo surfaces
S ⊂ Pd arise either as the quadratic Veronese embedding of a quadric in P3,
which is a del Pezzo surface of degree 8 in P8 (isomorphic to P1 × P1), or
as the blow-up of P2 along 9 − d points in general position, in which case
the degree of S satisfies 3 6 d 6 9. We will meet the notion of “general
position” when d = 3 in §2.1. Since [−KS ] = [H] in PicQ(S), we see that
α(S) = 1 for non-singular del Pezzo surfaces of degree d.
The geometry of del Pezzo surfaces is very beautiful and well-worth study-
ing. However, to avoid straying from the main focus of these lecture notes,
we will content ourselves with simply quoting the facts that are needed.
One of the remarkable features of del Pezzo surfaces of small degree is that
each such surface contains finitely many lines. The precise number of lines
is recorded in Table 1.
d number of lines
Table 1. Lines on non-singular del Pezzo surfaces of degree d
It turns out that dealing with del Pezzo surfaces of degree d gets easier
as the degree increases. In these lecture notes we will focus our attention
on the del Pezzo surfaces of degree d ∈ {3, 4, 5, 6}. It turns out that for del
Pezzo surfaces of degree d, the geometric Picard group PicQ(S) is a finitely
generated free Z-module, with
PicQ(S)
∼= Z10−d. (1.6)
This is established in Manin [49], where an explicit basis for the group is
also provided (see §2.1 for a concrete example). Let K be a splitting field
for the finitely many lines contained in S. The final invariant that we will
need to introduce is the Picard group
Pic(S) := Pic
(S)Gal(K/Q) (1.7)
of the surface. This is just the set of elements in PicQ(S) that are fixed by
the action of the Galois group. Write ρS for the rank of Pic(S). Let U ⊂ S
be the Zariski open subset formed by deleting the finitely many lines from
S. Then we have the following [1, Conjecture C ′].
10 T.D. BROWNING
Conjecture 1.3. Suppose that S ⊂ Pd is a non-singular del Pezzo surface
of degree d. Then there exists a non-negative constant cS,H such that
NU (B) = cS,HB(logB)
1 + o(1)
. (1.8)
In these lecture notes this is what will commonly be termed as “the Manin
conjecture”. Note that the exponent of B agrees with Conjecture 1.2, since
α(S) = 1. Moreover the exponent of logB is at most 9 − d, since the
geometric Picard group has rank 10 − d. We will develop some heuristics
to support this power of logB in §2. The value of the constant cS,H has
also received a conjectural interpretation at the hands of Peyre [51], an
interpretation that has been extended by Batyrev and Tschinkel [2], and by
Salberger [55].
There are a number of refinements to Conjecture 1.3 that are currently
emerging, which we will not have space to discuss here. Some of these are
discussed in more details in the author’s survey [14, §2], for example. One
such refinement is that there should exist a polynomial P ∈ R[x] of degree
ρS − 1, and a real number δ > 0, such that
NU (B) = BP (logB) +O(B
1−δ). (1.9)
One obviously expects the leading coefficient of P to agree with Peyre’s
prediction, but there has so far been rather little investigation of the lower
order terms. All of the del Pezzo surfaces that we have discussed so far have
been non-singular. In the following section we will meet some singular ones.
1.3. Degree 3 surfaces. The del Pezzo surfaces S ⊂ P3 of degree 3 are the
geometrically integral cubic surfaces in P3, which are not ruled by lines. In
particular, this definition covers both singular and non-singular del Pezzo
surfaces of degree 3. Given such a surface S defined over Q, we may always
find an absolutely irreducible cubic form C ∈ Z[x1, x2, x3, x4] such that S
is defined by the equation C = 0. In this section we will discuss the Manin
conjecture in the context of cubic surfaces. Let us begin by considering the
situation for non-singular cubic surfaces, for which one takes U ⊂ S to be
the open subset formed by deleting the famous 27 lines. Peyre and Tschinkel
[53, 54] have provided ample numerical evidence for the validity of the Manin
conjecture for diagonal cubic surfaces. However we are still rather far away
from proving it for any single example. The best upper bound available is
NU (B) = Oε,S(B
4/3+ε), (1.10)
due to Heath-Brown [38]. This applies when the surface S contains 3 copla-
nar lines defined over Q, and in particular to the Fermat cubic surface
x31 + x
2 = x
3 + x
Heath-Brown [40] has extended the bound (1.10) to all non-singular cubic
surfaces, subject to a natural conjecture concerning the size of the rank of
elliptic curves over Q.
The problem of proving lower bounds is somewhat easier. Under the as-
sumption that S contains a pair of skew lines defined over Q, Slater and
Swinnerton-Dyer [59] have shown that NU (B) ≫S B(logB)ρS−1, as pre-
dicted by the Manin conjecture. This does not apply to the Fermat cubic
THE MANIN CONJECTURE IN DIMENSION 2 11
surface, however, since the only skew lines contained in this surface are
defined over Q(
It turns out that much more can be said if one permits S to contain
isolated singularities. For the remainder of this section let S ⊂ P3 be a
geometrically integral cubic surface, which has only isolated singularities
and is not a cone. Then there exists a unique “minimal desingularisation”
π : S̃ → S of the surface, which is just a sequence of blow-up maps, and
furthermore, that the asymptotic formula (1.8) is still expected to hold, with
ρS now taken to be the rank of the Picard group of S̃. As usual U ⊂ S is
obtained by deleting all of the lines from S. The classification of singular
cubic surfaces S is a well-established subject, and can be traced back to the
work of Cayley [18] and Schläfli [56] over a century ago. A contemporary
classification of singular cubic surfaces has since been given by Bruce and
Wall [16], over Q. Of course, if one is interested in a classification over the
ground field Q, then many more singularity types can occur (see Lipman
[48], for example). In Table 2 we have provided a classification table of the
20 singularity types over Q, including the number of lines that each surface
contains. We will presently meet some explicit examples of cubic forms
C ∈ Z[x1, x2, x3, x4] that typify some surface types.
type # lines singularity
i 21 A1
ii 16 2A1
iii 15 A2
iv 12 3A1
v 11 A1 +A2
vi 10 A3
vii 9 4A1
viii 8 2A1 +A2
ix 7 A1 +A3
x 7 2A2
xi 6 A4
xii 6 D4
xiii 5 2A1 +A3
xiv 5 A1 + 2A2
xv 4 A1 +A4
xvi 3 A5
xvii 3 D5
xviii 3 3A2
ix 2 A1 +A5
xx 1 E6
Table 2. Classification (over Q) of singular del Pezzo sur-
faces of degree 3 in P3
The labelling of each singularity type corresponds to the “Dynkin dia-
gram” that describes the intersection behaviour of the exceptional divisors
obtained by resolving the singularities in the surface. For example, consider
12 T.D. BROWNING
the cubic surface
S1 = {x21x3 + x2x23 + x34 = 0}. (1.11)
Up to isomorphism over Q this is the unique cubic surface of type xx in
the table, and is discussed further in [34]. The process of resolving the
singularity gives 6 exceptional divisors E1, . . . , E6 and produces the minimal
desingularisation S̃1 of the surface S1. If L denotes the strict transform of
the unique line on S1, then L,E1, . . . , E6 satisfy the intersection behaviour
encoded in the Dynkin diagram
E1 E3 E6 E5 E4 L
There is a line connecting two divisors in this diagram if and only if they
meet in S̃1. In what follows the reader can simply think of these Dynkin
diagrams as a convenient way to label the surface type.
It turns out, as discussed in [16], that some types of surfaces do not have
a single normal form, but an infinite family. This happens precisely for the
surfaces of type i, ii, iii, iv, v, vi and ix. By [16, Lemma 4] the type xii
surface, with a D4 singularity, is the only surface that has more than one
normal form, but not a family. In fact it has precisely two normal forms,
given by
S2 = {x1x2(x1 + x2) + x4(x1 + x2 + x3)2 = 0} (1.12)
S3 = {x1x2x3 + x4(x1 + x2 + x3)2 = 0}. (1.13)
That these equations actually define distinct surfaces can be seen by calcu-
lating the corresponding Hessians in each case.
Let S̃ denote the minimal desingularisation of any surface S from Table 2,
and assume that all of its singularities and lines are defined over Q. In this
case the surface is said to be split, and it follows that the Picard group
of S̃ has maximal rank 7 by (1.6), since Pic(S̃) = Pic
(S̃). For example,
[L], [E1], . . . , [E6] provide a basis for Pic(S̃1). One would like to try and
establish (1.8) for each such surface S, with ρS = 7. Several del Pezzo
surfaces are actually special cases of varieties for which the Manin conjecture
is already known to hold. Recall that a variety of dimension D is said
to be toric if it contains the algebraic group variety GDm as a dense open
subset, whose natural action on itself extends to all of the variety. The
Manin conjecture has been established for all toric varieties by Batyrev and
Tschinkel [3]. It can be checked that the surface representing type xviii is
toric. In fact this particular surface has been studied by numerous authors,
including la Bretèche [5], la Bretèche and Swinnerton-Dyer [11], Fouvry
[29], Heath-Brown and Moroz [43], and Salberger [55]. Of the unconditional
asymptotic formulae obtained, the most impressive is the first. This consists
of an estimate like (1.9) for any δ ∈ (0, 1/8), with degP = 6.
The next surface to have received serious attention is the Cayley cubic
surface
S4 = {x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 = 0},
THE MANIN CONJECTURE IN DIMENSION 2 13
which is the type vii surface in the table. Heath-Brown [42] has shown that
there exist absolute constants A1, A2 > 0 such that
A1B(logB)
6 NU (B) 6 A2B(logB)
An estimate of precisely the same form has been obtained by the author [13]
for the D4 surface S3 in (1.13). In both cases the lines in the surface are
all defined over Q, so that the surfaces are split. Thus the corresponding
Picard groups have rank 7 and the exponents of B and logB agree with
Manin’s prediction. In this set of lecture notes we will establish an upper
bound for the remaining D4 cubic surface S2 in (1.12). This will be carried
out in §4 in two basic attacks. First we will give a completely self-contained
account of the upper bound NU (B) = Oε(B
1+ε), for any ε > 0. Next, by
making use of the work in [13], we will establish the following finer result.
Theorem 1.2. Let S2 be given by (1.12). We have NU (B) ≪ B(logB)6.
The cubic surface S2 contains the unique singular point [0, 0, 0, 1], together
with the 6 lines
xi = x4 = 0, x1 + x2 = xj = 0, xi = x1 + x2 + x3 = 0, (1.14)
for distinct indices i ∈ {1, 2} and j ∈ {3, 4}. Thus the surface is split and it
follows that Theorem 1.2 agrees with the Manin conjecture.
Exercise 4. Check that (1.14) are all of the lines contained in S2.
The final surface to have been studied extensively is the E6 cubic surface
S1 that we discussed above. The figure below, which was constructed by Der-
enthal, shows all the rational points of height 6 1000 on this surface. Recent
joint work of the author
with la Bretèche and Der-
enthal [12] has succeeded
in establishing the Manin
conjecture for this surface.
In fact an asymptotic for-
mula of the shape (1.9) is
achieved, with P of degree
6 and any δ ∈ (0, 1/11).
It should be remarked that
Dr. Michael Joyce has
also established the Manin
conjecture for S1 in his
doctoral thesis, albeit with
a weaker error term of O(B(logB)5).
1.4. Degree 4 surfaces. A quartic del Pezzo surface S ⊂ P4, that is defined
over Q, can be recognised as the zero locus of a suitable pair of quadratic
forms Q1, Q2 ∈ Z[x1, . . . , x5]. Again we do not stipulate that the surface
should be non-singular. As usual let U ⊂ S denote the open subset formed
by deleting all of the lines from S. Let us begin by discussing the situation
for non-singular surfaces, where there are 16 lines to delete. The best result
available is the estimate NU (B) = Oε,S(B
1+ε), valid for any quartic non-
singular del Pezzo surface S ⊂ P4 containing a conic defined over Q. This
14 T.D. BROWNING
result was established in an unpublished note due to Salberger in 2001. It
would be interesting to see whether one could adapt the methods of [40] to
show that NU (B) = Oε,S(B
5/4+ε) for any non-singular del Pezzo surface of
degree 4, assuming the same hypothesis on the ranks of elliptic curves.
As previously, it emerges that much more can be said if one permits S to
contain isolated singularities. For the remainder of this section let S ⊂ P4
be a geometrically integral intersection of two quadric hypersurfaces, which
has only isolated singularities and is not a cone, and let S̃ be the minimal
desingularisation of S. Then the asymptotic formula (1.8) is still expected
to hold, with ρS now taken to be the rank of the Picard group of S̃, and
U ⊂ S obtained by deleting all of the lines from S. In particular, when S is
split one always has ρS = 6. The classification of singular quartic del Pezzo
surfaces can be extracted from the work of Hodge and Pedoe [45, Book IV,
§XIII.11], where it is phrased in terms of the so-called Segre symbol. The
Segre symbol of a matrix M ∈ M5(C) is defined as follows. If the Jordan
form of M has Jordan blocks of sizes a1, . . . , an, with a1+ · · ·+an = 5, then
the Segre symbol is the symbol
(a1, . . . , an)
with extra parentheses around the Jordan blocks with equal eigenvalues.
Suppose that our quartic del Pezzo surface S is defined by a pair of quadric
hypersurfaces, with underlying symmetric matrices A,B ∈ M5(Q). Then
the Segre symbol of S is defined to be the Segre symbol associated to A−1B.
A crucial property of the Segre symbol is that it does not depend on the
choice ofA andB in the pencil of quadrics defining S. Since we are assuming
that S is not a cone, one may always suppose that A,B are chosen so that
A has full rank.
To illustrate the calculation of the Segre symbol, let us consider the sur-
face S defined by the pair of equations
x1x2 + x3x4 = 0, x1x4 + x2x3 + x3x5 + x4x5 = 0. (1.15)
Let A,B ∈ M5(Q) denote the underlying matrices of the first and second
equations, respectively. Then A has rank 4, and so we replace it with
A + 2B, which has full rank. A simple calculation reveals that the matrix
(A+ 2B)−1B has Jordan form

0 0 0 0 0
0 1 0 0 0
0 0 1
0 0 0 1
0 0 0 0 1

This matrix has 4 Jordan blocks, one of size 2 and the rest of size 1. The
eigenvalues associated to the different Jordan blocks are all different, and so
it follows that the surface (1.15) has Segre symbol (2, 1, 1, 1).
Exercise 5. Find any matrices A,B ∈ M5(Q) so that the corresponding
surface xtAx = xtBx = 0 is non-singular. Show that the surface has Segre
symbol (1, 1, 1, 1, 1).
THE MANIN CONJECTURE IN DIMENSION 2 15
So far we have given a very easy way to check the isomorphism type of
a given singular del Pezzo surface of degree 4. How do we match this up
with a classification according to the singularity type, as in our discussion
of cubic surfaces in Table 2? It turns out that up to isomorphism over Q,
there are 15 possible singularity types for S. Over Q, Coray and Tsfasman
[23, Proposition 6.1] have calculated the extended Dynkin diagrams for all
of the 15 types, and Knörrer [47] has determined the precise correspondence
between the singularity type and the Segre symbol. Table 3 is extracted
from this body of work, and matches each possible singularity type with the
Segre symbol, and the number of lines that the surfaces contains.
type Segre symbol # lines singularity
i (2,1,1,1) 12 A1
ii (2,2,1) 9 2A1
iii ((1,1),1,1,1) 8 2A1
iv (3,1,1) 8 A2
v ((1,1),2,1) 6 3A1
vi (3,2) 6 A1 +A2
vii (4,1) 5 A3
viii ((2,1),1,1) 4 A3
ix ((1,1),(1,1),1) 4 4A1
x ((1,1),3) 4 2A1 +A2
xi ((2,1),2) 3 A1 +A3
xii (5) 3 A4
xiii ((3,1),1) 2 D4
xiv ((2,1),(1,1)) 2 2A1 +A3
xv ((4,1)) 1 D5
Table 3. Classification (over Q) of singular del Pezzo sur-
faces of degree 4 in P4
In general, given a particular Segre symbol, its not entirely straightfor-
ward to determine explicit equations that define a singular del Pezzo surface
of degree 4 having this symbol. Nonetheless in Table 4 we have done pre-
cisely this for each Segre symbol that occurs. In doing so we have retrieved
some of the calculations carried out by Derenthal [26]. An important feature
of the table is that the surfaces recorded are split over Q. It remains a sig-
nificant open challenge to establish the Manin conjecture for the 15 surfaces
given in Table 4. This will furnish a proof of the Manin conjecture for the
class of split singular del Pezzo surfaces of degree 4 that are defined over Q,
and is undoubtedly a key stepping stone on the way towards a resolution of
the conjecture for all del Pezzo surfaces. There is huge potential for further
work in this area, and I hope that these lecture notes succeed in showing
that analytic number theorists are well placed to make an important con-
tribution.
Exercise 6. Calculate the Segre symbol for each of the surfaces in Table 4,
and check they match up with the correct singularity type in Table 3.
16 T.D. BROWNING
type Q1(x) Q2(x)
i x1x2 − x3x4 x1x4 − x2x3 + x3x5 + x4x5
ii x1x2 − x3x4 x1x4 − x2x3 + x3x5 + x25
iii x1x2 − x23 x1x3 − x2x3 + x4x5
iv x1x2 − x3x4 (x1 + x2 + x3 + x4)x5 − x3x4
v x1x2 − x23 x2x3 + x23 + x4x5
vi x1x2 − x3x4 x1x5 + x2x3 + x4x5
vii x1x2 − x3x4 x1x4 + x2x4 + x3x5
viii x1x4 − (x2 − x3)x5 (x1 + x4)(x2 + x3) + x2x3
ix x1x2 − x23 x23 − x4x5
x x1x2 − x23 x2x3 − x4x5
xi x1x4 − x3x5 x1x2 + x2x4 + x23
xii x1x2 − x3x4 x1x5 + x2x3 + x24
xiii x1x4 − x2x5 x1x2 + x2x4 + x23
xiv x1x2 − x23 x21 − x4x5
xv x1x2 − x23 x1x5 + x2x3 + x24
Table 4. Split surfaces representing the 15 singularity types
Whereas they share the same singularity type, the surfaces of type vii
and viii differ because in the former there are 5 lines, 4 of which pass
through the singularity, whereas in the latter all 4 lines pass through the
singularity. Similarly, an important difference between the surfaces of type
ii and iii in Tables 3 and 4 is that for the surface of type ii, the line joining
the two singularities is contained in the surface, whereas for the surface of
type iii it is not. When the 2 singular points are defined over a quadratic
extension of Q, the latter surface is called an Iskovskih surface. There is
ample evidence available (see Coray and Tsfasman [23], for example) to the
effect that Iskovskih surfaces are the most arithmetically interesting surfaces
among the singular del Pezzo surfaces of degree 4. In fact they are the only
such surfaces for which the Hasse principle can fail to hold. The main
focus of these lecture notes is upon the situation for split singular del Pezzo
surfaces, and so we will say no more about Iskovskih surfaces here.
As usual, let S̃ denote the minimal desingularisation of any surface S from
Table 4. Then the Picard group of S̃ has rank ρS = 6. The goal recorded
above is to try and establish (1.8) for each S. As in the case of singular
cubic surfaces several of the surfaces are actually special cases of varieties
for which the Manin conjecture is already known to hold. Thus it can be
shown that the surfaces representing types ix, x, xiv are all toric, so that
(1.8) already holds in these cases by the work of Batyrev and Tschinkel [3].
In a very real sense these surfaces are the “easiest” to deal with in our list.
Exercise 7. Show that NU (B) = Oε(B
1+ε) for the surfaces of type ix, x
and xiv.
It has also been shown by Chambert-Loir and Tschinkel [19] that the
Manin conjecture is true for equivariant compactifications of the algebraic
group G2a. Although identifying such surfaces in the table is not entirely
routine, it transpires that the type xv surface (with a D5 singularity) is
THE MANIN CONJECTURE IN DIMENSION 2 17
covered by this work. In joint work with la Bretèche, the author [8] has
provided an independent proof of the Manin conjecture for this particular
surface. In addition to obtaining a finer asymptotic formula of the shape
given in (1.9), this work has provided a useful line of attack for several other
singular del Pezzo surfaces.
In Table 5 we have recorded a list of progress towards the final resolution
of the Manin conjecture for the split singular del Pezzo surfaces of degree 4.
We have included the relevant reference in the literature, and whether the
result attained amounts to an asymptotic formula for the counting function,
or an upper bound. We will not pay attention here to the quality of the error
term in the asymptotic formula, but each upper bound is of the correct order
of magnitude B(logB)5. There is still plenty left to do!
type type of estimate achieved
v upper bound [14]
ix asymptotic formula [3]
x asymptotic formula [3]
xiii asymptotic formula [27]
xiv asymptotic formula [3]
xv asymptotic formula [8]
Table 5. Summary of progress for the split singular del
Pezzo surfaces of degree 4
It is also interesting to try and establish the Manin conjecture for singular
del Pezzo surfaces of degree 4 that are not split over the ground field. In
further joint work of the author with la Bretèche [9], the Manin conjecture
is established for the surface
x1x2 − x23 = 0, x21 + x2x5 + x24 = 0.
This surface has a D4 singularity and is isomorphic over Q(i) to the surface
of type xiii in Table 4. The Picard group of S̃ has rank 4 in this case, and
an asymptotic formula of the shape (1.9) is obtained for any δ ∈ (0, 3/32),
with P a polynomial of degree 3.
1.5. Degree > 5 surfaces. It turns out that all del Pezzo surfaces of degree
d > 7 are toric [26, Proposition 8], and that all non-singular del Pezzo
surfaces of degree d > 6 are toric. Thus (1.8) already holds in these cases by
the work of Batyrev and Tschinkel [3]. For non-singular del Pezzo surfaces
S ⊂ P5 of degree 5, the situation is rather less satisfactory. In fact there
are very few instances for which the Manin conjecture has been established.
The most significant of these is due to la Bretèche [7], who has proved the
conjecture for the split non-singular del Pezzo surface S of degree 5, in which
the 10 lines are all defined over Q. To be precise, if U ⊂ S denotes the open
subset formed by deleting the lines from S, then la Bretèche shows that
NU (B) = c0B(logB)
log logB
for a certain constant c0 > 0. This confirms Conjecture 1.3, since we have
seen in (1.6) that Pic(S) ∼= Z5 for split non-singular del Pezzo surfaces
18 T.D. BROWNING
of degree 5. The other major achievement in the setting of quintic del
Pezzo surfaces is a result of la Bretèche and Fouvry [10], where the Manin
conjecture is established for a surface that is not split, but contains lines
defined over Q(i).
So far we have only discussed the situation for non-singular del Pezzo
surfaces of degree d > 5. Let us now turn to the singular setting. When
d = 6 it emerges that there exist such surfaces that are not toric, and so are
not covered by [3]. We will focus attention on the situation for del Pezzo
surfaces of degree 6, following the investigation of Derenthal [26], where
the degree 5 surfaces are also considered. In view of [23, Proposition 8.3],
Table 6 lists all possible types of singular del Pezzo surfaces of degree 6.
type # lines singularity
i 4 A1
ii 3 A1
iii 2 2A1
iv 2 A2
v 1 A1 +A2
Table 6. Classification (over Q) of singular del Pezzo sur-
faces of degree 6
As noted in [26, §5], the surfaces of type i, iii and v are all toric and
so do not interest us here. Any singular del Pezzo surface of degree 6 can
be realised as the intersection of 9 quadrics in P6. For example, the type iv
surface is cut out by the system of equations
x1x6 − x4x5 = x1x7 − x2x5 = x1x7 − x3x4 = x3x7 + x4x5 + x25
= x5x7 − x3x4 = x2x7 + x24 + x4x5 = x4x7 − x2x6
= x4x6 + x5x6 + x
7 = x2x3 − x1x4 + x1x5 = 0.
(1.16)
In this set of lecture notes we will establish the Manin conjecture for the
type ii surface, which has the simplest possible singularity. When S ⊂ P6
is a split surface of type ii, then there is unimodular change of variables
that takes S into the surface with equations
x21 − x2x4 = x1x5 − x3x4 = x1x3 − x2x5 = x1x6 − x3x5
= x2x6 − x23 = x4x6 − x25 = x21 + x1x4 + x5x7
= x1x2 + x
1 + x3x7 = x1x3 + x1x5 + x6x7 = 0.
(1.17)
Let S̃ denote the minimal desingularisation of S. It follows from (1.6) that
Pic(S̃) ∼= Z4 since S is split. We will establish the following result in §3.
Theorem 1.3. Let S ⊂ P6 be the A1 surface given by (1.17). Then there
exist constants c1, c2 > 0 such that
NU (B) = c1B(logB)
3 + c2B(logB)
B logB
where
THE MANIN CONJECTURE IN DIMENSION 2 19
σ∞ = 6
{u,t,v∈R: 0<u,ut2,uv2,|tv(t−v)|61}
dtdudv. (1.18)
Since Pic(S̃) has rank 4, the exponents of B and logB in this asymptotic
formula are in complete agreement with Conjecture 1.3. Although we will
not give details here, it turns out that the value of the constant c1 also
confirms the prediction of Peyre [51] in this case. It is hoped that our proof
of Theorem 1.3 will encourage other researchers to try their hand at proving
asymptotic formulae for NU (B). With this in mind Exercise 8 is more of a
research problem, and its resolution will therefore conclude the proof of the
Manin conjecture for all split (non-singular or singular) del Pezzo surfaces
of degree 6.
Exercise 8. Establish an asymptotic formula for the type iv surface in
Table 6, with underlying equations (1.16).
1.6. Universal torsors. Universal torsors were originally introduced by
Colliot-Thélène and Sansuc [20, 21] to aid in the study of the Hasse principle
and weak approximation for rational varieties. Since their inception it is
now well-recognised that they also have a central rôle to play in proofs of
the Manin conjecture for Fano varieties, and in particular, for del Pezzo
surfaces. Let S ⊂ Pd be a del Pezzo surface of degree d ∈ {3, 4, 5, 6}, and
let S̃ denote the minimal desingularisation of S if it is singular, and S̃ = S
otherwise. Let E1, . . . , E10−d ∈ Div(S̃) be generators for PicQ(S̃), and let
E×i = Ei \ {zero section}. Working over Q, a universal torsor above S̃ is
given by the action of G10−dm on the map
π : E×1 ×eS · · · ×eS E
10−d → S̃.
A proper discussion of universal torsors would take us too far afield, and the
reader may consult the survey of Peyre [52] for further details, or indeed the
construction of Hassett and Tschinkel [34]. The latter outlines an alterna-
tive approach to universal torsors via the Cox ring. Given the usual open
subset U ⊂ S, the general theory of universal torsors ensures that there
is a partition of U(Q) into a disjoint union of patches, each of which is in
bijection with a suitable set of integral points on a universal torsor above S̃.
The guiding principle behind the use of universal torsors is simply that
they ought to be arithmetically simpler than the original variety. In our work
it will suffice to think of universal torsors as “particularly nice parametri-
sations” of rational points on the surface. The universal torsors that we
encounter in these lecture notes all have embeddings as affine hypersurfaces
of high dimension. Moreover, in each case we will show how the underlying
equation of the universal torsor can be deduced in a completely elementary
fashion, without any recourse to geometry whatsoever. The torsor equations
we will meet all take the shape
A+B + C = 0,
for monomials A,B,C of various degrees in the appropriate variables. As
in many examples of counting problems for higher dimensional varieties,
one can occasionally gain leverage by fixing some of the variables at the
20 T.D. BROWNING
outset, in order to be left with a counting problem for a family of small
dimensional varieties. If one is sufficiently clever about which variables to
fix first, one is sometimes left with a quantity that we know how to estimate
— and crucially — whose error term we can control once summed over the
remaining variables.
As a concrete example, we note that Hassett and Tschinkel [34] have
calculated the universal torsor for the cubic surface (1.11). It is shown that
there is a unique universal torsor above S̃1, and that it is given by the
equation
4s5 + y
2s2 + y
1s3 = 0,
for variables y1, y2, yℓ, s1, s2, s3, sℓ, s4, s5, s6. One of the variables does not
explicitly appear in this equation, and the torsor should be thought of as
being embedded in A10. It turns out that the way to proceed here is to fix
all of the variables apart from y1, y2, yℓ. One may then view the equation as
a congruence
y22s2 ≡ −y31s21s3 (mod s3ℓs24s5),
in order to take care of the summation over yℓ. This is the approach taken
in [12], the next step being to employ very standard facts about the number
of integer solutions to polynomial congruences that are restricted to lie in
certain regions. One if left with a main term and an error term, which the
remaining variables need to be summed over. While the treatment of the
main term is relatively routine, the treatment of the error term presents a
much more serious obstacle.
The universal torsors that turn up in the proofs of Theorems 1.2 and 1.3
can also be embedded in affine space as hypersurfaces. We will see in §3 that
the approach discussed above also produces results for the del Pezzo surface
of degree 6 considered in Theorem 1.3. In the proof of Theorem 1.2 in §4
our approach will be more obviously geometric, and we will actually view
the equation as a family of projective lines, and also as a family of conics.
We will then call upon techniques from the geometry of numbers to count
the relevant solutions.
2. Further heuristics
We have seen in §1.2, and in particular in the statement of Conjecture 1.3,
that for any del Pezzo surface S ⊂ Pd one expects a growth rate like
cSB(logB)
A for the counting function NU (B). We have already given some
motivation for the exponent of B in (1.2). The focus of the present section
is to produce much more sophisticated heuristics than we have previously
met. In particular we will gain an insight into the exponent of logB that
appears in the Manin conjecture.
For ease of presentation we restrict attention to non-singular diagonal
cubic surfaces S ⊂ P3. Thus
S = {a1x31 + a2x32 + a3x33 + a4x34 = 0}, (2.1)
for a = (a1, . . . , a4) ∈ N4 such that gcd(a1, . . . , a4) = 1. Define
P := {3} ∪ {p : p | a1a2a3a4}. (2.2)
THE MANIN CONJECTURE IN DIMENSION 2 21
This is the set of primes p for which the reduction of S modulo p is singular.
Before passing to the arithmetic of diagonal cubic surfaces, we will need to
discuss some of their geometry.
2.1. The lines on a cubic surface. The facts that we will need in this
section are explained in detail in the books of Hartshorne [33] and Manin
[49]. In general, a non-singular cubic surface S ⊂ P3 is obtained by blowing
up P2 along a collection of 6 points P1, . . . , P6 in general position. By general
position we mean that no 3 of them are collinear and they do not all lie on
a conic. The 27 lines on the surface arise in the following way. There are
6 exceptional divisors Ei above Pi, for 1 6 i 6 6, and 15 strict transforms
Li,j of the lines going through precisely 2 points Pi, Pj , for 1 6 i < j 6 6.
Finally there are the 6 strict transforms Qi of the conics going through all
but one of the 6 points. If Λ is the strict transform of a line in P2 that
doesn’t go through any of the Pi, then a basis of the geometric Picard group
(S) is given by
[Λ], [E1], . . . , [E6].
The remaining divisors may be expressed in terms of these elements via the
relations
[Li,j] = [Λ]− [Ei]− [Ej ], [Qi] = 2[Λ]−
j 6=i
[Ej ]. (2.3)
The class of the anti-canonical divisor −KS is given by [−KS ] = 3[Λ] −∑6
i=1[Ej ], although we will not need this fact in our work. One can check
that the hyperplane section has class −3[Λ] +
i=1[Ej ] in PicQ(S), so that
the cubic surface has very ample anticanonical divisor, as claimed in §1.2.
When S takes the shape (2.1) it is not hard to write down the 27 lines
explicitly. The calculations that we present below are based on those carried
out by Peyre and Tschinkel [54, §2]. Fix a cubic root α (resp. α′, α′′) of
a2/a1 (resp. a3/a1, a4/a1). We will assume that α ∈ Q if a2/a1 (resp.
a3/a1, a4/a1) is a cube in Q. Put
, β′ =
, β′′ =
We denote by θ a primitive cube root of one. Let i run over elements of
Z/3Z. Then the 27 lines on the cubic surface (2.1) are given by the equations
iαx2 = 0,
iβx4 = 0,
iαx2 = 0,
i+1βx4 = 0,
L′′i :
iαx2 = 0,
i+2βx4 = 0,
iα′x3 = 0,
iβ′x2 = 0,
M ′i :
iα′x3 = 0,
i+1β′x2 = 0,
M ′′i :
iα′x3 = 0,
i+2β′x2 = 0,
iα′′x4 = 0,
iβ′′x3 = 0,
N ′i :
iα′′x4 = 0,
i+1β′′x3 = 0,
N ′′i :
iα′′x4 = 0,
i+2β′′x3 = 0.
Let K = Q(θ, α, α′, α′′). It is a Galois extension of Q, and in the generic
case has degree 54 with Galois group Gal(K/Q) ∼= (Z/3Z)3 ⋊ Z/2Z.
We need to equate these lines to the divisors Ei, Li,j , Qi that we met
earlier. There is a certain degree of freedom in doing this, as discussed in
22 T.D. BROWNING
[33, §V.4], but it turns out that the choice
E1 = L0, E2 = L1, E3 = L2,
E4 = M1, E5 = M
2, E6 = M
Q1 = L
1, Q2 = L
2, Q3 = L
Q4 = M0, Q5 = M
1, Q6 = M
L1,2 = L
1 , L2,3 = L
2, L1,3 = L
L4,5 = M
1 , L5,6 = M2, L4,6 = M
L1,4 = N0, L1,5 = N1, L1,6 = N2,
L2,4 = N
1, L2,5 = N
2, L2,6 = N
L3,4 = N
2 , L3,5 = N
0 , L3,6 = N
(2.4)
is satisfactory. In assigning lines to E1, . . . , E6, all that is required is that
they should all be mutually skew. Given any cubic surface of the shape
(2.1), we now have the tools with which to compute the Picard group (1.7).
In fact, from this point forwards the process requires little more than basic
linear algebra.
Let us illustrate the procedure by calculating the Picard group for a spe-
cial case. Consider the Fermat surface
S1 = {x31 + x32 + x33 + x34 = 0}. (2.5)
In this case α = α′ = α′′ = β = β′ = β′′ = 1, in the notation above, and
K = Q(θ) is a quadratic field extension. We wish to find elements of the
geometric Picard group PicQ(S1) that are fixed by the action of Gal(K/Q).
Thus we want vectors c = (c0, . . . , c6) ∈ Z7 such that
(c0[Λ] + c1[E1] + · · ·+ c6[E6])σ = c0[Λ] + c1[E1] + · · ·+ c6[E6], (2.6)
for every σ ∈ Gal(K/Q). Under the action of Gal(K/Q) it is not hard to
check that Λ and E1 are fixed, that E2 and E3 are swapped, and that E4
(resp. E5, E6) is taken to L5,6 (resp. L4,5, L4,6). Using (2.3) one sees that
the left hand side of (2.6) is equal to
(c0 + c4 + c5 + c6)[Λ]+c1[E1] + c2[E3] + c3[E2]
− (c5 + c6)[E4]− (c4 + c5)[E5]− (c4 + c6)[E6],
in Pic
(S1). Thus we are interested in the space of c ∈ Z7 for which
c4 + c5 + c6 = 0, c2 − c3 = 0, c4 + 2c5 = 0, c4 + 2c6 = 0.
This system of homogeneous linear equations in 7 variables has underlying
matrix of rank 3. Thus the space of solutions has rank 7− 3 = 4, and so we
may conclude that Pic(S1) ∼= Z4. In fact a little thought reveals that the 4
elements
[Λ], [E1], [E2] + [E3],−2[E4] + [E5] + [E6],
provide a basis for Pic(S1).
Next, consider the surface
S2 = {x31 + x32 + x33 + px34 = 0} (2.7)
THE MANIN CONJECTURE IN DIMENSION 2 23
for a prime p. When p = 2 or 3, the arithmetic of this surface has been
considered in some detail by Heath-Brown [36], who provides some numer-
ical evidence to the effect that the corresponding counting function NU (B)
should grow like cpB for a certain constant cp > 0.
Exercise 9. Show that Pic(S2) ∼= Z.
This calculation has also been carried out by Colliot-Thélène, Kanevsky
and Sansuc [22, p. 12]. More generally, it is known that the Picard group
of the surface (2.1) has rank 1 if and only if the ratio
aσ(1)aσ(2)
aσ(3)aσ(4)
is not a cube in Q, for each permutation σ of (1, 2, 3, 4). This result is due
to Segre [57].
2.2. Cubic characters and Jacobi sums. Throughout this section let p
be a prime. Recall that a (multiplicative) character on Fp = Z/pZ is a map
χ : F∗p → C∗ such that
χ(ab) = χ(a)χ(b)
for all a, b ∈ F∗p. The trivial character ε is defined by the relation ε(a) = 1
for all a ∈ F∗p. It is convenient to extend the domain of definition to all of
Fp by assigning χ(0) = 0 if χ 6= ε and ε(0) = 1.
We begin by collecting together a few basic facts, all of which are estab-
lished in [46, §8].
Lemma 2.1. Let p be a prime. Then the following hold:
(1) Let χ be a character on Fp and let a ∈ F∗p. Then χ(1) = 1, χ(a) is
a (p− 1)-th root of unity, and χ(a−1) = χ(a) = χ(a)−1.
(2) For any character χ on Fp we have
χ(a) =
0, if χ 6= ε,
p, if χ = ε.
(3) The set of characters on Fp forms a cyclic group of order p− 1.
It follows from part (3) of Lemma 2.1 that χp−1 = ε for any character on
Fp. We define the order of a character to be the least positive integer n such
that χn = ε. In our work we will mainly be concerned with the characters of
order 3. Let us turn briefly to the topic of generalised Jacobi sums. Given
any characters χ1, . . . , χr on Fp, a Jacobi sum is a sum of the shape
J0(χ1, . . . , χr) :=
t=(t1,...,tr)∈F
t1+···+tr≡0 mod p
χ1(t1) · · ·χr(tr).
The key fact that we will need concerning these sums is that
|J0(χ1, . . . , χr)| =
0, if χ1 · · ·χr 6= ε,
(p − 1)pr/2−1, if χ1 · · ·χr = ε.
(2.8)
This is established in [46, §8.5].
Let p be a rational prime. We proceed to consider p as an element of
the ring of integers Z[θ] associated to the quadratic field Q(θ) obtained by
24 T.D. BROWNING
adjoining a primitive cube root of unity θ. It follows from basic algebraic
number theory that p is a prime in Z[θ] if p ≡ 2 mod 3, whereas it splits
as p = ππ if p ≡ 1 mod 3, where π is a prime in Z[θ]. When p ≡ 2 mod 3
the only cubic character on Fp is the trivial character ε. On the other
hand, when p = ππ ≡ 1 mod 3 then there are precisely two non-trivial cubic
characters χπ, χπ on Fp, where
χω(·) =
is the cubic residue symbol for any prime ω in Z[θ]. All of these facts are
established in [46, §9].
It turns out that Jacobi sums can be used to give formulae for the number
of solutions to appropriate equations over finite fields. Given any q ∈ N, let
N(q) := #{x mod q : a1x31 + · · ·+ a4x34 ≡ 0 (mod q)}, (2.9)
N∗(q) := #
x mod q :
1 + · · ·+ a4x34 ≡ 0 (mod q),
gcd(q, x1, . . . , x4) = 1
. (2.10)
When p is a prime not belonging to the finite set of primes P defined in
(2.2), we can write down a very precise expression for N(p). Thus it follows
from [46, §8.7] that
N(p) = p3 +
χ1,χ2,χ3,χ4
1 )χ2(a
2 )χ3(a
3 )χ4(a
4 )J0(χ1, χ2, χ3, χ4)
where the summation is over all non-trivial cubic characters χi : F
p → C
such that χ1χ2χ3χ4 = ε.
Exercise 10. Let p 6∈ P be a prime, and let χ1, χ2, χ3, χ4 be non-trivial
cubic characters on Fp such that χ1χ2χ3χ4 = ε. Deduce from (2.8) that
J0(χ1, χ2, χ3, χ4) = p(p− 1).
It follows from Exercise 10 that
N∗(p) = N(p)− 1 = p3 + p(p− 1)δp(a)− 1. (2.11)
for any prime p 6∈ P, where
δp(a) :=
χ1,χ2,χ3,χ4
1 )χ2(a
2 )χ3(a
3 )χ4(a
Let a ∈ F∗p and suppose that p splits as ππ. Then it will be useful to observe
χπ(a) + χπ(a) =
2, if a is a cubic residue modulo π,
−1, otherwise. (2.12)
We have δp(a) = 0 when p ≡ 2 mod 3, since there are then no non-trivial
cubic characters modulo p. When p ≡ 1 mod 3, with p = ππ 6∈ P, we have
δp(a) =χπ
(a1a2
(a1a2
(a1a3
(a1a3
(a1a4
(a1a4
THE MANIN CONJECTURE IN DIMENSION 2 25
Still with this choice of prime p, let νp(a) denote the number of indices
i ∈ {2, 3, 4} for which the cubic character χπ( a1aiajak ) is equal to 1, with {i, j, k}
a permutation of {2, 3, 4}. Then we may deduce from (2.12) that
δp(a) =
0, if p ≡ 2 mod 3,
3νp(a)− 3, if p ≡ 1 mod 3,
(2.13)
when p 6∈ P.
2.3. The Hardy–Littlewood circle method. We are now ready to con-
sider the counting function NU (B) that is associated to the diagonal cubic
surface S ⊂ P3 given in (2.1). Our aim is to provide heuristic evidence in
support of Manin’s original conjecture, and we will say rather little about
the predicted value of the constant. The Hardy–Littlewood circle method
is an extremely effective means of estimating counting functions associated
to projective algebraic varieties, but it only works when the dimension of
the variety is substantially larger than the degree. We have already seen
evidence of this in the statement of Theorem 1.1, which is based on an
application of the circle method. Although it has not been made to pro-
duce asymptotic formulae for the counting functions associated to del Pezzo
surfaces, in this section we will see how the Hardy–Littlewood method can
still be used as a useful heuristic tool. The key idea is to consider only the
contribution from the major arcs.
We will simplify matters by applying the heuristic to count all of the
rational points on S, rather than restricting attention to the open subset U .
Although the details are formidable, it is in fact possible to obtain upper
bounds for NU (B) using the circle method. Thus Heath-Brown [39] has
shown that NU (B) = Oε,S(B
3/2+ε) under a certain hypothesis concerning
the Hasse–Weil L-function associated to the surface. An interesting feature
of this work is that the contribution from the rational points lying on rational
lines in the surface is successfully separated out. When the surface contains
no lines defined over Q, such as the surface given by (2.7) for example, one
obviously has
NU (B) = NS(B) +O(1).
When S contains lines defined over Q there is a general consensus among
people working on the circle method that the dominant contribution (ie. the
contribution from the points on rational lines) should come from the minor
arc integral.
In what follows let e(z) := e2πiz for any z ∈ R. As usual, Z4 denotes the
set of primitive vectors in Z4. The igniting spark in the Hardy–Littlewood
circle method is the simple identity
e(αn)dα =
1, if n = 0,
0, if n ∈ Z \ {0}.
26 T.D. BROWNING
On taking into account the fact that x and −x represent the same point in
projective space, and applying Exercise 2, we deduce that
NS(B) =
|x|6B
e(α(a1x
1 + · · ·+ a4x34))dα
|x|6B/k
e(α(a1x
1 + · · ·+ a4x34))dα
S(α)dα,
say. Let us write P = B/k and IA(P ) :=
S(α)dα, for any bounded
subset A ⊂ R. The cubic exponential sum S(α) can actually be rather large
when α is well-approximated by a rational number with small denominator.
For example, we clearly have S(0) = 24P 4 + O(P 3). The philosophy that
underpins the Hardy–Littlewood method is that one expects S(α) to be
small for values of α ∈ [0, 1] that are not well-approximated by rational
numbers with small denominator. This is notoriously difficult to prove in
general, and as indicated above, is expected to be false in the present setting!
Our heuristic will be based on analysing IM(P ) for a suitable choice of
“major arcs” M. We will not give full details here, the gaps being easily
filled by consulting the relevant techniques in Davenport [25]. Let ε > 0 be
a small parameter. Given a, q ∈ Z such that
1 6 a 6 q 6 P ε, gcd(a, q) = 1, (2.14)
we define the interval
M(a, q) :=
− P−3+ε, a
+ P−3+ε
We take as major arcs the union
q6P ε
16a6q
gcd(a,q)=1
M(a, q).
It is clear that M contains all the points in the interval [0, 1] that are well-
approximated by rational numbers with small denominator.
Exercise 11. Show that M is a disjoint union for ε < 1.
The “minor arcs” are defined to be m := [0, 1] \M, and we will proceed
under the assumption that the minor arc integral Im(P ) can be ignored.
Actually we will also ignore the contribution that this term makes once it
is summed up over values of k. In truth there will be several points in the
argument where we will simply ignore subsidiary contributions. We will
indicate all of these by an appearance of the word “error”. Thus, to begin
with, we have
NU (B) =
µ(k)IM(P ) + error .
THE MANIN CONJECTURE IN DIMENSION 2 27
Here we have made the further assumption that the contribution NS\U(B)
from the points lying on lines in S arises in the minor arc integral.
Let a, q ∈ Z such that (2.14) holds, and let α = a/q + z ∈ M(a, q). Let
C(x) denote the diagonal cubic form in (2.1). We now break the sum into
congruence classes modulo q, giving
S(a/q + z) =
r mod q
e(aC(r)/q)
x∈Z4∩[−P,P ]4
x≡r mod q
e(zC(x)). (2.15)
We would like to replace the discrete variable x by a continuous one in the
inner sum, and the summation over x by an integral. For this we will appeal
to the following general result.
Lemma 2.2. Let P > 1, let a ∈ Zn and let r ∈ N such that r 6 P . Let F
be a function on Rn all of whose first order partial derivatives exist and are
continuous on R := [−P,P ]n. Define
MF := sup
16i6n
Then we have
x∈Zn∩R
x≡a mod r
e(F (x)) =
e(F (t))dt +O
(Pn−1(1 + PMF )
Proof. Our proof of Lemma 2.2 is based on the Euler–Maclaurin summation
formula [60, §I.0]. Let Bk(x) denote the kth Bernoulli polynomial, for k ∈
Z>0, and let s ∈ Z>0. Let A,B ∈ Z, with A < B. For any function
f : R → C whose (s+1)-th derivative f (s+1) exists and is continuous on the
interval [A,B], the Euler–Maclaurin summation formula states that
A<n6B
f(n) =
f(t)dt+
(−1)k+1Bk+1(0)
(k + 1)!
f (k)(B)− f (k)(A)
(−1)s
(s+ 1)!
Bs+1(t)f
(s+1)(t)dt.
(2.16)
Let a ∈ Z and r ∈ N. We will apply this result with f0(x) = f(a+ rx) and
, B0 =
B − a
Taking s = 0 in the Euler–Maclaurin formula, we therefore deduce that
A<n6B
n≡a mod r
f(n) =
f(t)dt− f(B0)− f(A0)
(t− a
f ′(t)dt,
(2.17)
since B1(x) = x− [x]− 12 .
We are now ready to establish Lemma 2.2, which we will do by induction
on n. Write Sn for the n-dimensional sum that is to be estimated. The case
28 T.D. BROWNING
n = 1 of Lemma 2.2 follows from (2.17) with f(x) = e(F (x)). Assuming
now that n > 2, we have
y∈Z∩[−P,P ]
y≡a1 mod r
x2,...,xn
e(G(x2, . . . , xn)),
where G(x2, . . . , xn) = F (y, x2, . . . , xn), and the sum over x2, . . . , xn is over
all integers in [−P,P ] such that xi ≡ ai mod r for 2 6 i 6 n.We may employ
the induction hypothesis to estimate the inner sum in n − 1 variables. It
therefore follows that
y∈Z∩[−P,P ]
y≡a1 mod r
[−P,P ]n−1
e(G(t2, . . . , tn))dt2 · · · dtn
(P (n−2)(1 + PMG)
Now there are O(P/r) integers y in the interval [−P,P ] that are congruent
to a1 modulo r, since r 6 P by assumption. Moreover, it is clear that
MG 6 MF . Hence
y∈Z∩[−P,P ]
y≡a1 mod r
f(y) +O
(P (n−1)(1 + PMF )
where
f(y) =
[−P,P ]n−1
e(F (y, t2, . . . , tn))dt2 · · · dtn.
The statement of Lemma 2.2 is now an easy consequence of (2.17). �
It is clear from the proof of Lemma 2.2 that when F has partial derivatives
to a higher order, one may obtain a much sharper estimate by including
higher order terms in the Euler–Maclaurin summation formula. The present
bound is satisfactory for our purposes, however.
Returning to (2.15) we apply Lemma 2.2 with
F (x) = zC(x), n = 4, a = r, r = q.
In particular we have q 6 P ε 6 P , as required for the lemma. Furthermore,
MF = MzC ≪ |z|P 2 6 P−1+ε for any α = a/q+z ∈ M(a, q). It follows that
S(a/q + z) = q−4T (a, q)VP (z) +O(P
3+2ε)
on the major arcs, where
T (a, q) :=
r mod q
e(aC(r)/q), VR(z) :=
[−R,R]4
e(zC(x))dx. (2.18)
The set of major arcs has meas(M) = O(P−3+3ε). On carrying out the
integration over z and the summation over a and q one is therefore led to
THE MANIN CONJECTURE IN DIMENSION 2 29
the conclusion that
IM(P ) = P
q6P ε
16a6q
gcd(a,q)=1
q−4T (a, q)
|z|6P−3+ε
V1(zP
3)dz +O(P 5ε)
q6P ε
16a6q
gcd(a,q)=1
q−4T (a, q)
|z|6P ε
V1(z)dz +O(P
Define
I(R) :=
|z|6R
V1(z)dz =
|z|6R
[−1,1]4
e(zC(x))dxdz.
It can be shown that I(R) is a bounded function of R, and furthermore,
I(R) → I0 > 0 as R → ∞. A standard calculation reveals that the limit I0
is equal to 2σ∞, where
σ∞ :=
dx1dx2dx3
1 + a2x
2 + a3x
is the real density of solutions. Here the integral is over x1, x2, x3 ∈ [−1, 1]
such that |(a1x31 + a2x32 + a3x33)/a4| 6 1. We now define
S(R) :=
16a6q
gcd(a,q)=1
T (a, q).
On bringing everything together, our investigation has so far succeeded in
showing that
NU (B) = σ∞
µ(k)PS(P ε) + error, (2.19)
for a suitably small value of ε > 0, where P = B/k.
Our task is now to examine the sum S(R), as R → ∞. Let us write
Sq :=
16a6q
gcd(a,q)=1
T (a, q),
where T (a, q) is the complete exponential sum defined in (2.18). Then
S(R) =
q6R q
−4Sq. It turns out that Sq is a multiplicative function
of q. This can be established along the lines of [25, Lemma 5.1]. Recall the
definition (2.9) of N(q). We now come to the key relation between Sq and
N(q) at prime power values of q.
Exercise 12. Let p be a prime and let e > 1. Use Lemma 2.1 to show that
Spe = p
eN(pe)− p3+eN(pe−1).
Let us for the moment ignore considerations of convergence, and consider
the local factors
e=0 p
−4eSpe in the infinite product formula for S(∞).
Now it follows from Exercise 12 that
p−4eSpe = 1 +
p−3eN(pe)− p3−3eN(pe−1)
= p−3EN(pE),
30 T.D. BROWNING
for any E > 1. Hence, formally speaking, we have S(∞) =
p τp, where
τp := lim
p−3eN(pe).
If S(R) was convergent, which it certainly is not in general, we could then
conclude from (2.19) that
NU (B) = Bσ∞
τp + error . (2.20)
Arguing formally, we now replace the summation over k by its Euler product,
concluding that
NU (B) = Bσ∞
σp + error, (2.21)
with σp := (1− 1/p)τp. Note that
σp = lim
p−3eN∗(pe), (2.22)
in the notation of (2.10), which clearly follows from the observation that
N∗(pe) = N(pe)− 8N(pe−3),
for any e > 3. The estimate in (2.21) therefore gives a heuristic asymptotic
formula for NU (B) which is visibly a product of local densities. Among
other things, we have assumed that S(R) is convergent in formulating this
heuristic. This is expected to be true for diagonal cubic surfaces whose
Picard group has rank 1, but not in general.
One can make the transition from (2.20) to (2.21) completely rigorous
if the Hardy–Littlewood heuristic produces a leading term involving Pα
with exponent α > 1. The outcome is that to go from counting points
on the affine cone to counting projective points, one merely replaces N(pe)
by N∗(pe). For α = 1, however, the “renormalization” procedure remains
heuristic. Let
∗(R) :=
q−4S∗q ,
where
S∗q :=
16a6q
gcd(a,q)=1
r mod q
gcd(r,q)=1
e(aC(r)/q).
It is easily checked that S∗q is a multiplicative function of q, and furthermore,
that the corresponding version of Exercise 12 holds, relating S∗pe to N
∗(pe).
In fact, formally speaking, one has
p−4eS∗pe =
with σp given by (2.22). Bearing all of this in mind we will proceed under
the bold assumption that (2.19) can be replaced by
NU (B) = Bσ∞S
∗(B) + error, (2.23)
where we have taken ε = 1 in the expressions for S(Bε) and S∗(Bε).
THE MANIN CONJECTURE IN DIMENSION 2 31
We now turn to a finer analysis of S∗(B), as B → ∞. Our task is to
determine the analytic properties of the corresponding Dirichlet series
F (s) :=
(2.24)
for s = σ+it ∈ C. Armed with this analysis we will ultimately apply Perron’s
formula to obtain an estimate forS∗(R). Using the multiplicativity of q−sS∗q
we deduce that
F (s) =
σp(s), σp(s) :=
p−esS∗pe . (2.25)
In examining F (s) it clearly suffices to ignore the value of the factors σp(s)
at any finite collection of primes p. With this in mind we will try and
determine σp(s) for p 6∈ P, where P is given by (2.2). Recall the definition
(2.10) of N∗(q).
Exercise 13. Let e > 1 and let p 6∈ P be a prime. Use Hensel’s lemma to
show that N∗(pe) = p3e−3N∗(p).
It therefore follows from Exercise 13 that
σp(s) = 1 +
peN∗(pe)− p3+eN∗(pe−1)
= 1− 1
N∗(p)
for any p 6∈ P. Hence (2.11) yields
σp(s) = 1 +
δp(a)
− δp(a)
where δp(a) is given by (2.13).
We now pursue our analysis in the special case a = (1, 1, 1, 1) of the
Fermat cubic surface (2.5). Now it is clear from (2.13) that δp(1, 1, 1, 1) = 0
if p ≡ 2 mod 3 and
δp(1, 1, 1, 1) = 3νp(1, 1, 1, 1) − 3 = 6
if p ≡ 1 mod 3. Let λ : Z → C be the real Dirichlet character of order 2
defined by
λ(n) :=
), if 3 ∤ n,
0, otherwise,
where (n
) is the Legendre symbol. Then we may write
σp(s) = 1 +
3(1 + λ(p))
− 3(1 + λ(p))
1− λ(p)
pmin{σ−2,2σ−2}
for any p 6∈ P. Let L(s, λ) denote the usual Dirichlet L-function associated
to λ. When a = (1, 1, 1, 1) we have therefore succeeded in showing that
F (s) = ζ(s− 3)3L(s− 3, λ)3G(s), (2.26)
32 T.D. BROWNING
where G(s) is a function that is holomorphic and bounded on the half-plane
σ > 7/2+ δ, for any δ > 0. For future reference we note that G(4) has local
factors
Gp(4) =
(1− 1
)7(1 + 7
), if p ≡ 1 mod 3,
(1− 1
)4(1 + 1
)3(1 + 1
), if p ≡ 2 mod 3. (2.27)
Although we will not prove it here, it can be deduced from (2.22) and
Hensel’s lemma that
G3(4) =
3−3eN∗(3e) =
. (2.28)
We are now ready for our application of Perron’s formula, which we will
apply in the following form.
Lemma 2.3. Let F (s) =
n=1 ann
−s be a Dirichlet series with abscissa of
absolute convergence σa. Suppose that x 6∈ Z and let c > σa. Then we have
∫ c+iT
F (s)
|an|n−c
| log(x/n)|
for any T > 1.
Proof. Let c > 0. The lemma follows from the identity
∫ c+iT
1 +O(xc(T | log x|)−1), if x > 1,
+O(cT−1), if x = 1,
O(xc(T | log x|)−1), if 0 < x < 1,
which is a straightforward exercise in contour integration. �
In our case we have aq = S
q and we are interested in the Dirichlet series
F (s + 4), in the notation of (2.24). In order to apply Lemma 2.3 we will
need an upper bound for this quantity. For our purposes the trivial upper
bound aq ≪ q5 is sufficient. Thus the Dirichlet series F (s+ 4) is absolutely
convergent for σ > 2. Taking c = 2 + ε for any ε > 0, we may deduce from
Lemma 2.3 that
∗(B) =
∫ c+iT
F (s+ 4)
n1+ε| log(B/n)|
for any T > 1 and any B 6∈ Z. It is not hard to see that the error term here
is ≪ T−1Bc. We now apply Cauchy’s residue theorem to the rectangular
contour C joining c′ − iT , c′ + iT , c+ iT and c− iT , where c′ = −1/2 + ε.
The relation (2.26) implies that in this region F (s + 4)Bs/s has a unique
pole at s = 0, and it is a pole of order 4. It has residue
Ress=0
F (s+ 4)Bs
L(1, λ)3G(4)P (logB)
where P ∈ R[x] is a monic polynomial of degree 3. Putting all of this
together we have therefore shown that
∗(B) =
L(1, λ)3G(4)P (logB)
+O(E(B)), (2.29)
THE MANIN CONJECTURE IN DIMENSION 2 33
where
E(B) =
( ∫ c′+iT
c′−iT
∫ c−iT
c′−iT
∫ c′+iT
)∣∣∣H(s)3
∣∣∣ds,
for any T > 1, and where H(s) = ζ(s + 1)L(s + 1, λ). Here we have used
the fact that G(s+ 4) is bounded on the half-plane ℜe(s) > c′.
To make the analysis simpler, it will be convenient to proceed under the
assumption that the Lindelöf hypothesis holds for ζ(s), and also for the
Dirichlet L-function L(s, λ). This could be avoided at the cost of extra
effort, but there seems no harm in supposing it here. Thus we may assume
the bounds
ζ(σ + it) ≪ε |t|ε, L(σ + it, λ) ≪ε |t|ε,
for any σ ∈ [1/2, 1] and any |t| > 1. It therefore follows that
H(σ + it) ≪ε
|t|ε, if −1/2 6 σ 6 0,
1, if σ > 0,
for any |t| > 1, which gives
∫ c−iT
c′−iT
∣∣∣H(s)3
∣∣∣ds ≪ε
BσT−1+3εdσ ≪ε BcT−1+3ε.
One obtains the same estimate for the contribution from the remaining
horizontal contour. Turning to vertical integral, we find that
∫ c′+iT
c′−iT
∣∣∣H(s)3
∣∣∣ds ≪ Bc
|H(1/2 + ε+ it)|3
1 + |t| dt
≪ε Bc
(1 + |t|)3ε−1dt
≪ε Bc
T 3ε,
under the assumption of Lindelöf hypothesis. This shows that
E(B) ≪ε BεT 3ε
for any T > 1. Taking T sufficiently large, we therefore conclude from (2.29)
∗(B) =
L(1, λ)3G(4)P (logB)
+O(B−∆),
for some ∆ > 0.
We are now ready to return to the Hardy–Littlewood major arc analysis
which led us to (2.23). Substituting in our estimate for S∗(B), we conclude
NU1(B) ∼ c1B(logB)3 (2.30)
where U1 ⊂ S1 is the usual open subset of the Fermat surface (2.5), and
σ∞L(1, λ)
3G(4)
Now it follows from the class number formula that L(1, λ) = π
3/9. Hence,
on combining this with (2.27) and (2.28), our heuristic argument has led us
34 T.D. BROWNING
to the expectation that (2.30) holds, with
p≡1 mod 3
p≡2 mod 3
The exponents of B and logB in (2.30) agree with the Manin conjecture,
since we have already seen in §2.1 that the Picard group of S1 has rank 4.
It is interesting to compare our analysis with the work of Peyre and
Tschinkel [54], who calculate the leading constant cPeyre in Peyre’s refine-
ment [51] of the conjectured asymptotic formula for NU1(B). It turns out
cPeyre = γ(S1)c1,
with γ(S1) = 7/3. For a general non-singular cubic surface S ⊂ P3, the
constant γ(S) is defined to be the volume
γ(S) :=
e−〈−KS ,t〉dt.
Thus, in general terms, γ(S) measures the volume of the polytope obtained
by intersecting the dual of Λeff(S) with a certain affine hyperplane. In
particular γ(S) ∈ Q for any non-singular cubic surface S, and γ(S) = 1 if
and only if the corresponding Picard group has rank 1.
Exercise 14. Let S2 denote the surface (2.7). Using a similar argument,
show that one expects an asymptotic formula of the shape NS2(B) ∼ c2B
for some constant c2 > 0. Check your answer with the heuristic formula
obtained by Heath-Brown [36].
3. The A1 del Pezzo surface of degree 6
In this section we will establish Theorem 1.3. Any line in P6 is defined by
the intersection of 5 hyperplanes. It is not hard to see that the equations
x1 = x2 = x3 = x5 = x6 = 0, x1 = x3 = x4 = x5 = x6 = 0,
x3 = x5 = x6 = x1 + x4 = x1 + x2 = 0,
(3.1)
all define lines contained in the singular del Pezzo surface S given by (1.17).
Table 6 ensures that these are the only lines contained in S. By definition
U is the open subset of S on which none of these equations hold. We begin
by establishing the following result.
Lemma 3.1. We have
NU (B) = 2M(B) +O(B),
where M(B) denotes the number of x ∈ Z7 such that
x21 − x2x4 = x1x5 − x3x4 = x1x3 − x2x5 = x1x6 − x3x5
= x2x6 − x23 = x4x6 − x25 = x21 − x1x4 + x5x7
= x21 − x1x2 − x3x7 = x1x3 − x1x5 + x6x7 = 0,
(3.2)
with gcd(x1, . . . , x7) = 1, 0 < |x1|, x2, x3, x4, |x5|, x6 6 B and |x7| 6 B.
THE MANIN CONJECTURE IN DIMENSION 2 35
Proof. In view of the fact that x and −x represent the same point in P6, we
NU (B) =
#{x ∈ Z7 : |x| 6 B, (1.17) holds, but (3.1) does not},
where Z7 denotes the set of primitive vectors in Z7. We need to consider the
contribution to the right hand side from points such that xi = 0, for some
1 6 i 6 7. Let us begin by considering the contribution from vectors x ∈ Z7
for which x1 = 0. But then the equations in (1.17) imply that x2x4 = 0. If
x2 = 0, it is straightforward to check that either x satisfies the first system
of equations in (3.1), or else
x0 = x2 = x3 = x7 = 0, x4x6 = x
Such points are therefore confined to a plane conic. We therefore obtain
O(B) points overall with x1 = x2 = 0. If on the other hand x1 = x4 = 0,
then a similar analysis shows that there are O(B) points in this case too. In
view of the first equation in (1.17), the contribution from vectors x such that
x2x4 = 0 is also O(B). Let us now consider the contribution from vectors
x such that x3 = 0 and x1x2x4 6= 0. It is easily checked that the only such
vectors have x5 = x6 = 0 and x1 + x4 = x1 + x2 = 0, and so must lie on a
line contained in S. Finally, arguing in a similar fashion, we see that there
are no points contained in S with x5x6 = 0 and x1x2x3x4 6= 0. We have
therefore shown that
NU (B) =
#{x ∈ Z7 : x1 · · · x6 6= 0, |x| 6 B, (3.2) holds}+O(B).
Here we have noted that there is an obvious unimodular transformation that
takes the set of equations in (1.17) into (3.2).
We would now like to restrict our attention to positive values of x1, . . . , x6.
The equations for S imply that x2, x4, x6 all share the same sign. On ab-
sorbing the minus sign into x1 there is a clear bijection between solutions
to (3.2) with x2, x4, x6 < 0 and solutions with x2, x4, x6 > 0. We choose to
count the former. Arguing similarly, by absorbing the minus signs into x1
and x7, we see that there is a bijection between the solutions to (3.2) with
x3 < 0 and x2, x4, x6 > 0, and the solutions with x2, x3, x4, x6 > 0. Fixing
our attention on the latter set of points, we therefore complete the proof of
Lemma 3.1. �
Let S̃ denote the minimal desingularisation of the surface S. By determin-
ing the Cox ring associated to S̃, Derenthal [26] has calculated the universal
torsor above S̃. In this setting it is defined by a single equation
s1y1 − s2y2 + s3y3 = 0, (3.3)
embedded in A7. In particular one of the variables does not appear explicitly
in the equation.
3.1. Elementary considerations. As promised in §1.6, we proceed to
show how NU (B) can be related to a count of the integer points on the
corresponding universal torsor, which in this case is given by (3.3). Our
deduction of this fact is completely elementary, and is based on an analysis
of the integer solutions to the system of equations (3.2). It is still somewhat
36 T.D. BROWNING
mysterious as to how or why this rather low-brow process should ultimately
lead to the same outcome! Typical of the facts that we will employ is the
following.
Exercise 15. Show that the general solution of the equation xy = z2 is
x = a2c, y = b2c, z = abc,
with |µ(c)| = 1.
Given s0 ∈ R and s = (s1, s2, s3),y = (y1, y2, y3) ∈ R3, define
Ψ(s0, s,y) := max
|s30s21s22s23|, |y1y2y3|, |s0s21y21|, |s0s22y22|
. (3.4)
We are now ready to record our translation of the problem to the universal
torsor.
Lemma 3.2. We have
NU (B) = 2#
(s0, s,y) ∈ Z7 :
Ψ(s0, s,y) 6 B, (3.3) holds,
s0, s1, s2, s3, y1 > 0,
gcd(yi, s0sjsk) = 1,
gcd(si, sj) = 1
+O(B),
with i, j, k a permutation of 1, 2, 3 in the coprimality conditions.
Proof. Let x ∈ Z7 be a primitive vector counted by M(B), as defined in
the statement of Lemma 3.1. Combining the first equation in (3.2) with
Exercise 15 we see that
x1 = a1a2a4, x2 = a
2a1, x4 = a
for integers a1, a2, a4 such that a1, a2 > 0 and
|µ(a1)| = gcd(a1 gcd(a2, a4)2, x3, x5, x6, x7) = 1.
Inserting this into the equation x2x6 = x
3 we deduce that a1a2 | x3, whence
x3 = a1a2a3, x6 = a1a
for a positive integer a3 such that
|µ(a1)| = gcd(a1 gcd(a2, a4)2, a1a3 gcd(a2, a3), x5, x7) = 1.
Substituting this into the equation x4x6 = x
5, we deduce that
x5 = a1a3a4,
|µ(a1)| = gcd(a1, x7) = gcd(a2, a3, a4, x7) = 1. (3.5)
Note that the second equation in (3.2) implies that x1, x5 must share the
same sign, which here is the sign of a4. The equations x1x5 = x3x4, x1x3 =
x2x5 and x1x6 = x3x5 reveal no new information. Turning instead to the
equation x21 = x1x4 − x5x7, we obtain
2a4 = a1a2a
4 − a3x7. (3.6)
The coprimality conditions imply that a1 | a3. Moreover, we deduce from
this equation that a2a4 | a3x7/a1. We may therefore write
a2 = a23a27, a4 = a43a47,
THE MANIN CONJECTURE IN DIMENSION 2 37
for integers a2i, a4i, with i = 3, 7, such that a2i, a43, |a47| > 0, and
a1a23a43 | a3, a27a47 | x7.
Thus there exist further integers b3, a7 with b3 > 0, such that
a3 = a1a23a43b3, x7 = a27a47a7,
with (3.5) and (3.6) becoming
|µ(a1)| = gcd(a1, a27a47a7) = gcd(a23a27, a23a43b3, a43a47, a27a47a7) = 1,
a23a27 = a43a47 − b3a7,
respectively. The final two equations are redundant. Let us write d for the
highest common factor of a23, a43, b3. Thus
a23 = da
23, a43 = da
43, b3 = db
for positive integers d, a′23, a
43, b
3. On making these substitutions the equa-
tion remains the same, but with appropriate accents added, whereas the
coprimality conditions become
|µ(a1)| = gcd(da1, a27a47a7) = gcd(a′23a27, a′23a′43b′3, a′43a47, a27a47a7)
= gcd(a′23, a
43, b
Now any n ∈ N can be written uniquely in the form n = ab2 for a, b ∈ N
such that |µ(a)| = 1. We may therefore make the change of variables
(s0; s1, s2, s3; y1, y2, y3) = (a1d
2; a′23, a
43, b
3; a27, a47, a7).
Bringing everything together, we have therefore established the existence of
(s0, s,y) ∈ Z7 such that (3.3) holds, with
s0, s1, s2, s3, y1 > 0, (3.7)
gcd(s0, y1y2y3) = gcd(s1, s2, s3) = gcd(s1y1, s2y2, s1s2s3, y1y2y3) = 1.
Note that y2 is automatically non-zero for s0, s,y satisfying the remaining
conditions. Once combined with (3.3), it is easy to check that the latter
coprimality conditions are equivalent to the conditions
gcd(y1, s0s2s3) = gcd(y2, s0s1s3) = gcd(y3, s0s1s2) = 1,
gcd(s1, s2) = gcd(s1, s3) = gcd(s2, s3) = 1,
(3.8)
that appear in the statement of the lemma.
At this point we may summarise our argument as follows. Let T ⊂ Z7
denote the set of (s0, s,y) ∈ Z7 such that (3.3), (3.7) and (3.8) hold. Then
for any primitive vector x counted by M(B), we have shown that there
38 T.D. BROWNING
exists (s0, s,y) ∈ T such that


x1 = s0s1s2y1y2,
x2 = s0s
x3 = s
1s2s3y1,
x4 = s0s
x5 = s
2s3y2,
x6 = s
x7 = y1y2y3.
Conversely, we leave it as an exercise to check that any (s0, s,y) ∈ T pro-
duces a primitive point x ∈ Z7 such that (3.2) holds, with
|x1|, x2, x3, x4, |x5|, x6 > 0.
We may now conclude that M(B) is equal to the number of (s0, s,y) ∈ T
such that
i=1,2
|s0s1s2y1y2|, |s0s2i y2i |, |s20s1s2s3siyi|, |s30s21s22s23|, |y1y2y3|
In view of the fact that |s0s1s2y1y2| =
|s0s21y21|
|s0s22y22 |, and furthermore,
|s20s1s2s3siyi| =
|s30s21s22s23|
|s0s2i y2i |, it follows that this height condition
is equivalent to Ψ(s0, s,y) 6 B for any (s0, s,y) ∈ T , where Ψ is given
by (3.4). In summary we have therefore shown that M(B) is equal to the
number of (s0, s,y) ∈ T such that Ψ(s0, s,y) 6 B. Once inserted into
Lemma 3.1, this completes the proof of Lemma 3.2. �
At first glance it might seem a little odd that the height restriction
|s0s23y23 | 6 B doesn’t explicitly appear in the lemma. However, (3.3) implies
that 0 < s1y1 = s2y2 − s3y3 for any (s0, s,y) ∈ T , whence the restriction
Ψ(s0, s,y) 6 B is plainly equivalent to max{|s0s23y23|,Ψ(s0, s,y)} 6 B. We
have preferred not to include it explicitly in the statement of Lemma 3.2
however.
3.2. The asymptotic formula. Our starting point is Lemma 3.2. Let
T (B) denote the quantity on the right hand side that is to be estimated.
Once taken together with (3.3), the height condition Ψ(s0, s,y) 6 B is
equivalent to
i=1,2
|s30s21s22s23|, |s0s2i y2i |, |y1y2(s1y1 − s2y2)/s3|
Define
X0 :=
(s30s21s22s23
, Xi :=
(s1s2s3B
for i = 1, 2. Then the height conditions above can be rewritten as
|X0| 6 1, |f1(y1)| 6 1, |f2(y2)| 6 1, |g(y1, y2)| 6 1,
where
fi(y) := X0
, g(y1, y2) :=
for i = 1, 2. In order to count solutions to the equation (3.3), our plan will
be to view the equation as a congruence
s1y1 − s2y2 ≡ 0 (mod s3),
THE MANIN CONJECTURE IN DIMENSION 2 39
which has the effect of automatically taking care of the summation over
y3. In order to make this approach viable we will need to first extract the
coprimality conditions on the y3 variable.
Define the set
S := {(s0, s) ∈ N4 : gcd(si, sj) = 1, X0 6 1}, (3.9)
with i, j generic indices from the set {1, 2, 3}. We now apply Möbius in-
version, as in Exercise 2, in order to remove the coprimality condition
gcd(y3, s0s1s2) = 1. Thus we find that
T (B) =
(s0,s)∈S
k3|s0s1s2
µ(k3)#


y ∈ Z3 :
gcd(y1, s0s2s3) = 1,
gcd(y2, s0s1s3) = 1,
y1 > 0,
s1y1 − s2y2 + k3s3y3 = 0,
|fi(yi)| 6 1, |g(y1, y2)| 6 1


Now it is clear that the summand vanishes unless gcd(k3, s1s2) = 1. Hence
T (B) =
(s0,s)∈S
k3|s0
gcd(k3,s1s2)=1
µ(k3)Sk3(B), (3.10)
where
Sk3(B) := #


y1, y2 ∈ Z :
gcd(y1, s0s2s3) = 1,
gcd(y2, s0s1s3) = 1,
y1 > 0,
s1y1 ≡ s2y2 mod k3s3,
|fi(yi)| 6 1, |g(y1, y2)| 6 1


Clearly Sk3(B) depends on the parameters s0 and s, in addition to k3 and B.
We now turn to the estimation of Sk3(B), for which we need the following
basic result.
Exercise 16. Let b > a and q > 0. Show that
#{n ∈ Z ∩ (a, b] : n ≡ n0 mod q} =
+O(1).
We will fix y2 and apply Exercise 16 to handle the summation over y1. Be-
fore this we must use Möbius inversion to remove the coprimality condition
gcd(y1, s0s2s3) = 1 from the summand. Thus we find that
Sk3(B) =
k1|s0s2s3
µ(k1)#
y1, y2 ∈ Z :
gcd(y2, s0s1s3) = 1,
k1s1y1 ≡ s2y2 mod k3s3,
|f1(k1y1)| 6 1, |f2(y2)| 6 1,
|g(k1y1, y2)| 6 1, y1 > 0
In view of the other coprimality conditions, the summand plainly vanishes
unless gcd(k1, k3s3) = 1. We may therefore write ρ ∈ Z for the (unique)
inverse of k1s1 modulo k3s3, whence
Sk3(B) =
k1|s0s2
gcd(k1,k3s3)=1
µ(k1)Sk1,k3(B), (3.11)
40 T.D. BROWNING
Sk1,k3(B) :=
y2∈Z: |f2(y2)|61
gcd(y2,s0s1s3)=1
y1 ∈ N :
y1 ≡ ρs2y2 mod k3s3,
|f1(k1y1)| 6 1,
|g(k1y1, y2)| 6 1
An application of Exercise 16 now reveals that
Sk1,k3(B) =
y2∈Z: |f2(y2)|61
gcd(y2,s0s1s3)=1
(X1F1(X0, y2/X2)
k1k3s3
+O(1)
, (3.12)
where
F1(u, v) :=
{t∈R>0: |ut
2|,|tv(t−v)|61}
We close this section by showing that once summed over all (s0, s, y2) ∈ N5,
the error term in (3.12) makes a satisfactory overall contribution to the error
term in Theorem 1.3. Using the fact that
k|n |µ(k)| = 2ω(n), we find that
this contribution is
(s0,s)∈S
4ω(s0)2ω(s2)X2
= B1/2
(s0,s)∈S
4ω(s0)2ω(s2)
s0,s1,s2∈N
4ω(s0)2ω(s2)
s20s1s
≪ B logB.
This is satisfactory for Theorem 1.3, and so we may henceforth ignore the
error term in the above estimate for Sk1,k3(B).
Define the arithmetic function
φ∗(n) :=
where as is common convention the product is over distinct prime divisors
of n. It will be useful to note that
φ∗(mn) =
φ∗(m)φ∗(n)
φ∗(gcd(m,n))
, (3.13)
for any m,n ∈ N. We must now sum over the variable y2, for which we will
employ the following basic result.
Exercise 17. Let I ⊂ R be an interval, let a ∈ N and let f : R → R>0 be a
function that is continuously differentiable on I. Use (2.16) to show that
n∈Z∩I
gcd(n,a)=1
f(n) = φ∗(a)
f(t)dt+O
2ω(a) sup
|f(t)|
We may now return to (3.12). Using Exercise 17 we deduce that
Sk1,k3(B) =
φ∗(s0s1s3)X1X2F2(X0)
k1k3s3
(2ω(s0s1s3)X1
k1k3s3
, (3.14)
THE MANIN CONJECTURE IN DIMENSION 2 41
where
F2(u) :=
{t,v∈R: t>0, |ut2|,|uv2|,|tv(t−v)|61}
dtdv.
We must now estimate the overall contribution to NU (B) from the error
term in this estimate, once summed up over the remaining variables. This
gives
(s0,s)∈S
4ω(s0)2ω(s2)2ω(s0s1s3)X1
≪ B1/3
s0,s1,s2,s3∈N
8ω(s0)2ω(s1s2s3)s
≪ B logB,
by summing over s2 6
B/(s30s
3). This is satisfactory for Theorem 1.3,
and so we may henceforth ignore the error term in (3.14). As pointed out
to the author by Régis de la Bretèche, it is easy to sharpen this error term
to O(B) using the fact that φ∗ has constant average order.
Now it is trivial to check that
gcd(d,a)=1
φ∗(n)
φ∗(gcd(a, n))
for any a, n ∈ N. Bringing together (3.10), (3.11) and (3.14), we conclude
T (B) =
(s0,s)∈S
k3|s0
gcd(k3,s1s2)=1
µ(k3)
φ∗(s0s2)φ
∗(s0s1s3)
φ∗(gcd(k3s3, s0s2))
X1X2F2(X0)
where S is given by (3.9). It is clear that gcd(k3s3, s0s2) = gcd(k3s3, s0).
Let us define the arithmetic function
ϑ(s0, s) =
φ∗(s0s2)φ
∗(s0s1s3)
φ∗(gcd(s0, s3))
k3|s0
gcd(k3,s1s2)=1
µ(k3)
φ∗(gcd(k3, s0, s3))
φ∗(gcd(k3, s0))
when gcd(si, sj) = 1 for 1 6 i < j 6 3, and ϑ(s0, s) = 0 otherwise. It follows
from (3.13) that
ϑ(s0, s) =
φ∗(s0s2)φ
∗(s0s1s3)
φ∗(gcd(s0, s3))
p|gcd(s0,s3)
p∤s1s2
p∤s1s2s3
(1− 2
= φ∗(s0s2)φ
∗(s0s1s3)
p∤s1s2s3
(1− 2
= φ∗(s0)φ
∗(s1s2s3)
p∤s1s2s3
42 T.D. BROWNING
when gcd(si, sj) = 1 for 1 6 i < j 6 3. On recalling the definitions of
X1,X2, we deduce that
T (B) = B2/3
∆(n)F2
(n/B)1/3
, (3.15)
where
∆(n) :=
ϑ(s0, s)
(s1s2s3)1/3
, (3.16)
for any n ∈ N.
We will use Perron’s formula to estimate
n6B ∆(n), before combining
it with partial summation to estimate (3.15). Consider the Dirichlet series
D(s) :=
n=1 ∆(n)n
−s. We have
D(s+ 1/3) =
s0,s1,s2,s3=1
ϑ(s0, s)
s3s+10 s
and it is straightforward to check that D(s+ 1/3) =
p ap(s), with
ap(s) = 1 +
3(1− 1/p)
p2s+1(1− 1/p2s+1) +
(1− 1/p)(1 − 2/p)
p3s+1(1− 1/p3s+1)
3(1 − 1/p)2
p5s+2(1− 1/p2s+1)(1 − 1/p3s+1) .
Hence D(s+ 1/3) = E1(s)E2(s), where E1(s) = ζ(2s+ 1)
3ζ(3s + 1) and
E2(s) =
D(s+ 1/3)
ζ(2s+ 1)3ζ(3s+ 1)
p4σ+2
(3.17)
on the half-plane ℜe(s) > −1/2. In particular, E1(s) has a meromorphic
continuation to all of C with a pole of order 4 at s = 0, and E2(s) is
holomorphic and bounded on the half-plane ℜe(s) > −1/4.
Let c = 1/3 + ε for any ε > 0, and let T > 1. Then an application of
Lemma 2.3 reveals that
∆(n) =
∫ c+iT
E1(s − 1/3)E2(s− 1/3)
ds+Oε
(B1/3+ε
provided that B 6∈ Z. We apply Cauchy’s residue theorem to the rectangular
contour C joining the points 1/6− iT , 1/6 + iT , c+ iT and c− iT . Now the
residue of E1(s− 1/3)E2(s− 1/3)Bs/s at s = 1/3 is clearly
Ress=1/3
E1(s− 1/3)E2(s− 1/3)
E2(0)
B1/3P (logB),
for some monic polynomial P ∈ R[x] of degree 3. Define the difference
E(B) =
∆(n)− E2(0)
B1/3P (logB).
Then it follows that
E(B) ≪ε
B1/3+ε
(∫ 1/6+iT
1/6−iT
∫ c−iT
1/6−iT
∫ 1/6+iT
)∣∣∣E1(s − 1/3)
∣∣∣ds,
since E2(s−1/3) is holomorphic and bounded on the half-plane ℜe(s) > 1/6.
THE MANIN CONJECTURE IN DIMENSION 2 43
We proceed to estimate the contribution from the horizontal contours.
Recall the well-known convexity bounds
ζ(σ + it) ≪ε
|t|(1−σ)/3+ε, if σ ∈ [1/2, 1],
|t|(3−4σ)/6+ε, if σ ∈ [0, 1/2],
for any |t| > 1. A proof of these can be found in [60, §II.3.4], for example.
It therefore follows that
E1(σ − 1/3 + it) ≪ε |t|1−3σ+ε (3.18)
for any σ ∈ [1/6, 1/3) and any |t| > 1. We may now deduce that
∫ c−iT
1/6−iT
∣∣∣E1(s− 1/3)
∣∣∣ds ≪ε
BσT−3σ+εdσ
B1/3+εT ε
B1/6T ε
T 1/2
(3.19)
One obtains the same estimate for the contribution from the remaining
horizontal contour. Turning to the vertical contour, (3.18) gives
∫ 1/6+iT
1/6−iT
∣∣∣E1(s− 1/3)
∣∣∣ds ≪ B1/6
|E1(−1/6 + it)|
1 + |t| dt
≪ B1/6
|t|1/2+ε
1 + |t| dt
≪ B1/6T 1/2+ε.
Once combined with (3.19), we conclude that
E(B) ≪ε B1/3+εT−1+ε +B1/6T 1/2+ε,
for any T > 1. Taking T = B1/9 we obtain
∆(n) =
E2(0)
B1/3P (logB) +Oε(B
2/9+ε),
for any ε > 0.
We are now ready to complete the proof of Theorem 1.3. For this it
suffices to combine the latter estimate with partial summation in (3.15),
and then apply Lemma 3.2. In this way we deduce that
NU (B) = 2T (B) +O(B logB)
σ∞E2(0)
BQ(logB) +Oε(B
8/9+ε) +O(B logB),
for a further cubic monic polynomial Q ∈ R[x]. Here σ∞ = 6
F2(u)du is
given by (1.18), and it follows from (3.17) that
E2(0) =
This therefore completes the proof of Theorem 1.3.
44 T.D. BROWNING
4. The D4 del Pezzo surface of degree 3
In this section we consider Manin’s conjecture for the cubic surface
S2 = {[x1, x2, x3, x4] ∈ P3 : x1x2(x1 + x2) + x4(x1 + x2 + x3)2 = 0},
considered in (1.12) Let U2 ⊂ S2 be the open subset formed by deleting the
lines (1.14) from S2. Our task is to estimate NU2(B). In doing so it will
clearly suffice to establish the estimate for any surface that is obtained from
S2 via a unimodular transformation. In view of this we will make the change
of variables
t1 = x1, t2 = x2, t3 = x1 + x2 + x3, t4 = −x4,
which brings S2 into the shape
t1t2(t1 + t2) = t
3t4, (4.1)
and which we henceforth denote by S. The 6 lines on the surface (4.1) take
the shape
ti = tj = 0, tj = t1 + t2 = 0,
where i denotes a generic element of the set {1, 2}, and j an element of
{3, 4}. If U ⊂ S denotes the open subset formed by deleting these lines
from the surface, then we have t3t4 = 0 for any [t] 6∈ U . It now follows that
NU (B) =
#{t ∈ Z4 : (4.1) holds, |t| 6 B, t3t4 6= 0}.
As in the argument of Lemma 3.1, the factor 1
reflects the fact that t and −t
represent the same point in P3. There is a clear symmetry between solutions
such that t3 is positive and negative. Similarly, (4.1) is invariant under the
transformation t1 = −z1, t2 = −z2, t3 = z3 and t4 = −z4. Thus we have
NU (B) = 2#{t ∈ Z4 : (4.1) holds, |t| 6 B, t3, t4 > 1}, (4.2)
for any B > 1.
In the following section we will explicate the relation between (4.2) and
counting integral points on the corresponding universal torsor. When it
comes to the latter task, we will be led to consider the counting problem for
rational points on plane curves of degree 1 and 2. The estimates that we
require will need to be completely uniform in the coefficients of the equations
defining the curves.
Given any plane curve C ⊂ P2 of degree d > 1, that is defined over Q, let
NC(B) := #{x ∈ C(Q) : H(x) 6 B}.
As usual we write Zn for the set of primitive vectors in Zn, and Zn∗ for the set
of vectors in Zn with no components equal to zero. We will restrict our at-
tention to curves that are defined by diagonal ternary forms. We clearly have
NC(B) =
Md(a;B,B,B) for certain non-zero integers a1, a2, a3, where
Md(a;B) := #{x ∈ Z3 : a1xd1 + a2xd2 + a3xd3, |xi| 6 Bi}, (4.3)
and B = (B1, B2, B3). Let us begin with the situation for projective lines.
The following result is due to Heath-Brown [35, Lemma 3].
THE MANIN CONJECTURE IN DIMENSION 2 45
Lemma 4.1. Let a ∈ Z3∗ and let Bi > 0. Then we have
M1(a;B) ≪ 1 +
B1B2B3
max |ai|Bi
Lemma 4.1 shows that there are only O(1) rational points on lines of
sufficiently large height. If one has a line L ⊂ P2 given by the equation
a.x = 0, for a ∈ Z3∗ , then the height of L is simply defined to beH(L) := |a|.
When L is an arbitrary line in Pn, which is defined over Q, there is still a
very natural way of defining its height. The height of L is just the height
of the rational point in the Grassmannian G(1, n) that corresponds to the
line. We will not need this fact in our work. Let L ⊂ P2 be an arbitrary line
defined over Q. Then it follows from Lemma 4.1 that
NL(B) ≪ 1 +
≪ B2.
This is essentially best possible, as can be seen by taking n = 2 in Exercise 3.
Turning to curves of higher degree, we have the following result, which is
a special case of a result due to the author and Heath-Brown [15, Corollary
Lemma 4.2. Let a ∈ Z3∗ such that gcd(ai, aj) = 1, and let Bi > 0. Then
we have
M2(a;B) ≪
B1B2B3
|a1a2a3|
τ(a1a2a3),
where τ(n) :=
d|n 1 denotes the usual divisor function.
In keeping with our discussion of lines, let us consider to what extent our
estimate reflects the true growth rate ofNC(B), for a quadratic curve C ⊂ P2
that is defined by a diagonal equation with pairwise coprime coefficients.
Recall the estimate τ(n) = Oε(n
ε), that holds for any ε > 0. Writing ‖C‖
for the maximum modulus of the coefficients defining C, we deduce from
Lemma 4.2 that
NC(B) ≪ε ‖C‖εB.
This should be compared with the work of Heath-Brown [41, Theorem 3]
that shows NC(B) ≪d,ε B2/d+ε, for any irreducible plane curve C ⊂ P2 of
degree d.
Both Lemma 4.1 and Lemma 4.2 are established using the geometry of
numbers.
4.1. Elementary considerations. We proceed to show how NU (B) can
be related to a count of the integer points on the corresponding universal
torsor. Our argument is in complete analogy to that presented in §3.1,
although the individual steps differ somewhat. If S̃ denotes the minimal
desingularisation of the surface S, then Derenthal [26] has calculated the
universal torsor over S̃, it being embedded in A10 by a single equation
s1u1y
1 + s2u2y
2 + s3u3y
3 = 0. (4.4)
Note that one of the variables does not appear explicitly in the equation.
We will need the following basic fact.
46 T.D. BROWNING
Exercise 18. Let a, b ∈ N. Show that a | b2 if and only if a = uv2 for
u, v ∈ N such that u is square-free and uv | b.
Given v ∈ R and s,u,y ∈ R3, define
Ψ(v, s,u,y) := max
{ |s1s2s3|, |u21u22u23v3y1y2y3|
|s1u21u2u3v2y21|, |s2u1u22u3v2y22 |
. (4.5)
We are now ready to record our translation of the problem to the universal
torsor.
Lemma 4.3. We have
NU (B) = 2#


(v, s,u,y) ∈ N4 × Z3 × N3 :
u3 > 0, Ψ(v, s,u,y) 6 B
(4.4) holds,
|µ(u1u2u3)| = 1,
gcd(s1s2s3, u1u2u3v) = 1,
gcd(yi, yj) = 1,
gcd(yi, sj , sk) = 1,


where i, j, k denote distinct elements from the set {1, 2, 3}.
Proof. Let t ∈ Z4 be a vector such that (4.1) holds, with t3, t4 > 1. Write
η14 = gcd(t1, t4), η24 = gcd(t2, t4/η14), η12 = gcd(t1/η14, t2/η24).
Then η12, η14, η24 ∈ N and there exists z4 ∈ N and z1, z2 ∈ Z such that
t1 = η12η14z1, t2 = η12η24z2, t4 = η14η24z4.
Moreover, it is not hard to deduce that
gcd(η12z1, η24z4) = gcd(η12z2, z4) = gcd(z1, z2) = 1,
gcd(t3, η14, η12η24z2) = 1.
Under this substitution the equation (4.1) becomes
η312z1z2(η14z1 + η24z2) = t
It follows that η312 | t23 in any given integer solution. Exercise 18 therefore
implies that there exist u, v, z3 ∈ N such that |µ(u)| = 1 and
η12 = uv
2, t3 = u
2v3z3,
z1z2(η14z1 + η24z2) = uz
We proceed to consider the effect of the divisibility condition z1z2 | uz23 that
this equation entails.
Recall that gcd(z1, z2) = gcd(z1, z4) = gcd(z2, z4) = 1. Since z1z2 | uz23 ,
there must exist u1, u2, u3, w1, w2, w3 ∈ Z such that w1, w2, w3, u3 > 0 and
u = u1u2u3, z1 = u1w
1, z2 = u2w
2, z3 = w1w2w3.
Here we have used the fact that if p is a prime such that p ∤ u and p | z1z2,
then p must divide z1 or z2 to even order. Under these substitutions our
equation becomes
η14u1w
1 + η24u2w
2 = u3w
THE MANIN CONJECTURE IN DIMENSION 2 47
Moreover, we will have the corresponding coprimality conditions
gcd(u1u2u3vw1, η24z4) = gcd(u1u2u3vw2, z4) = gcd(u1w1, u2w2) = 1,
(4.6)
|µ(u1u2u3)| = 1, gcd(u21u22u23v3w1w2w3, η14, η24u1u22u3v2w22) = 1. (4.7)
We now set s = (η14, η24, z4) and y = w, and replace (u1, u2, u3) by
(−u1,−u2, u3). Tracing through our argument, one sees that we have made
the transformation 
t1 = −s1u21u2u3v2y21,
t2 = −s2u1u22u3v2y22,
t3 = u
3y1y2y3,
t4 = s1s2s3.
In particular, it is clear that the height condition |x| 6 B is equivalent to
Ψ(v, s,u,y) 6 B, in the notation of (4.5). We now observe that under
this transformation the equation (4.1) becomes (4.4), and the coprimality
relations (4.6) and (4.7) can be rewritten
gcd(s2s3, u1u2u3vy1) = gcd(s3, u1u2u3vy2) = gcd(u1y1, u2y2) = 1,
|µ(u1u2u3)| = 1, gcd(s1, u1u2u3vy2 gcd(y3, s2)) = 1.
We can combine these relations with (4.4) to simplify them still further. In
fact, once combined with (4.4), we claim that they are equivalent to the
conditions
|µ(u1u2u3)| = 1, gcd(s1s2s3, u1u2u3v) = gcd(yi, yj) = gcd(yi, sj , sk) = 1,
appearing in the statement of the lemma. To establish the forward impli-
cation, it suffices to show that gcd(y1, y3) = gcd(y2, y3) = 1, the remaining
conditions being immediate. But these two conditions follow on combining
(4.4) with the fact that gcd(y1, s2u2y2) = 1. To see the reverse implication,
the conditions are all immediate apart from
gcd(y1, s2s3) = gcd(y2, s1s3) = gcd(u1, y2) = gcd(u2, y1) = 1.
But each of these is an easy consequence of the assumed coprimality rela-
tions, and (4.4). Finally, we leave it as an exercise to the reader to check
that each (v, s,u,y) counted in the right hand side of Lemma 4.3 produces
a primitive solution of (4.1) with t3, t4 > 1. This completes the proof of
Lemma 4.3. �
In what follows let us write i for a generic element of the set {1, 2, 3}. Fix
a choice of v ∈ N and Si, Ui, Yi > 0, and write
N = Nv(S;U;Y) (4.8)
for the total contribution to NU (B) in Lemma 4.3 from s,u,y contained in
the intervals
Si/2 < si 6 Si, Ui/2 < |ui| 6 Ui, Yi/2 < yi 6 Yi. (4.9)
Write
S = S1S2S3, U = U1U2U3, Y = Y1Y2Y3.
48 T.D. BROWNING
If N = 0 there is nothing to prove, and so we assume henceforth that the
dyadic ranges in (4.9) produce a non-zero value of N . In particular we must
S ≪ B U2Y ≪ B/v3, SiUUiY 2i ≪ B/v2. (4.10)
In this set of lecture notes we will provide two upper bounds for NU (B).
The object of our first bound is to merely establish linear growth, without
worrying about the factor involving logB that we expect to see. By ignoring
some of the technical machinery needed to get better bounds it is hoped that
the overall methodology will be brought into focus. Later we will indicate
how the expected upper bound can be retrieved with a little more work.
4.2. A crude upper bound. We begin by establishing linear growth for
NU (B). Note that (4.10) forces the inequalities Si, Ui, Yi ≪ B. We proceed
to establish the following upper bound.
Lemma 4.4. We have
NU (B) ≪ (logB)9
v6B1/3
Si,Ui,Yi>0
Nv(S;U;Y),
where the maximum is over Si, Ui, Yi > 0 such that (4.10) holds.
Proof. Our starting point is Lemma 4.3. It follows from (4.5) that v 6 B1/3
for any v, s,u,y that contributes to the right hand side. Let us fix a choice
of v ∈ N such that v 6 B1/3, and cover the ranges for s,u,y with dyadic
intervals. Thus for fixed integers σi, νi, ηi > 0, we write
Si = 2
σi , Ui = 2
νi , Yi = 2
and consider the contribution from s,u,y in the range (4.9). But this is
just N = Nv(S;U;Y). Now we have already seen that N = 0 unless (4.10)
holds. Finally, since each Si, Ui, Yi is O(B), it follows that the number of
dyadic intervals needed is O((logB)9). This completes the proof of the
lemma. �
We may now restrict our attention to bounding Nv(S;U;Y) for fixed
values of Si, Ui, Yi > 0 such that (4.10) holds, and fixed v 6 B
1/3. In the
arguments that follow it will be necessary to focus attention on primitive
vectors s ∈ N3. To enable this we draw out possible common factors between
s1, s2, s3, obtaining
Nv(S;U;Y) =
N ∗v (k−1S;U;Y). (4.11)
Here N ∗v (S;U;Y) is defined as for Nv(S;U;Y) but with the extra condition
that gcd(s1, s2, s3) = 1. Let us write S
i = Si/k and S
′ = k−1S.
Recall the equation (4.4) that we must count solutions to, which it will be
convenient to denote by T , and which we will think of as defining a variety
in P2 ×P2 ×P2, with homogeneous coordinates s,u,y. The key idea will be
to count points on the fibres of projections π : T → P2 × P2. This amounts
to fixing six of the variables and estimating the number of points on the
resulting plane curve. Since this family of curves will vary with B, so it is
vital to obtain bounds that are completely uniform in the coefficients of the
defining equation.
THE MANIN CONJECTURE IN DIMENSION 2 49
Let us begin by fixing the variables u,y, and estimating the corresponding
number of vectors s. Now it follows from the coprimality conditions in
Lemma 4.3 that
gcd(u1y
1 , u2y
2, u3y
3) = 1.
For fixed u,y, (4.4) defines a line in P2. We clearly have
N ∗v (S′;U;Y) 6
M1(a;S
in the notation of (4.3), with ai = uiy
i . Since a is primitive, it therefore
follows from Lemma 4.1 that
N ∗v (S′;U;Y) ≪
k2 maxSiUiY
≪ UY + k−2S2/3U2/3Y 1/3.
Here we have used the trivial lower bound max{a, b, c} > (abc)1/3, valid for
any a, b, c > 0. Using (4.10) we conclude that
N ∗v (S′;U;Y) ≪ UY +
. (4.12)
The second term here will be satisfactory from our point of view, but the
first is disastrous, since we will run into trouble when it comes to summing
over k in (4.11).
It turns out that an altogether different bound is required to handle the
contribution from really small values of S′. For this we will fix values of s,u
in (4.4), and count points on the resulting family of conics. First we need
to record the coprimality relation
gcd(siui, sjuj) = 1,
which we claim holds for any of the vectors s,u,y in which we are interested.
But this follows on noting that gcd(si, sj) = 1 for any primitive vector s ∈ Z3
such that (4.4) holds and gcd(si, u1u2u3) = gcd(yi, sj, sk) = 1. We now have
N ∗v (S′;U;Y) 6
M2(a;Y),
in the notation of (4.3), with ai = siui. In particular ai is non-zero and
gcd(ai, aj) = 1 in the statement of Lemma 4.2, whence
N ∗v (S′;U;Y) ≪
k3/2Y 1/2
S1/2U1/2
2ω(s1s2s3u1u2u3).
In view of the bounds S,U ≪ B, we clearly have
2ω(s1s2s3u1u2u3) ≪ε (s1s2s3u1u2u3)ε ≪ε (SU)ε ≪ε B2ε.
Once inserted into our bound for N ∗v (S′;U;Y), and combined with (4.10),
we deduce that
N ∗v (S′;U;Y) ≪ε B2ε
S1/2U1/2Y 1/2
SUB2ε
B1+2ε
k3/2v3/2
(4.13)
Here the second term will provide a satisfactory contribution, and we will
balance the first term with our earlier estimate (4.12).
50 T.D. BROWNING
Note that
k3/2v3/2
by (4.10). It therefore follows from (4.13) and (4.12) that
N ∗v (S′;U;Y) ≪ε
B1+2ε
k3/2v
Once inserted into (4.11), and then into the statement of Lemma 4.4, we
may conclude that
NU (B) ≪ε (logB)9
v6B1/3
B1+2ε
k3/2v
≪ε B1+2ε(logB)10
≪ε B1+3ε.
Recall that NU2(B) 6 NU (B), where U ⊂ S is the open subset associated to
the surface (4.1), and NU2(B) is the counting function associated to (1.12).
On redefining the choice of parameter ε > 0, we have therefore established
the following result.
Theorem 4.1. We have NU2(B) ≪ε B1+ε, for any ε > 0.
The reader will note that there many places in our argument where we
have been wasteful. The most damaging has been in our use of the trivial
bound 2ω(n) = Oε(n
ε), in the deduction of (4.13). Using the fact that 2ω(n)
has average order ζ(2)−1 log n, it is not particularly difficult to replace the
Bε in Theorem 4.1 with a large power of logB.
Exercise 19. By analysing the proof of Theorem 4.1, find an explicit value
of A > 6 such that NU2(B) ≪ B(logB)A.
In the next section we will be able to show that the value A = 6 is an
admissible exponent, as claimed in Theorem 1.2.
4.3. A better upper bound. Crucial to the proof of Theorem 4.1 was an
investigation of the density of integer solutions to the equation (4.4). It is
in our treatment of this equation that we will hope to gain some saving.
Lets put the problem on a more general footing. For any A,B,C ∈ R3>1,
let M(A,B,C) denote the number of a,b, c ∈ Z3∗ such that
a1b1c
1 + a2b2c
2 + a3b3c
3 = 0 (4.14)
|ai| 6 Ai, |bi| 6 Bi, |ci| 6 Ci,
gcd(ai, cj) = gcd(ci, cj) = 1 (4.15)
|µ(a1a2a3)| = 1, gcd(ai, bj , bk) = 1. (4.16)
Here, we recall that Z3∗ denotes the set of primitive vectors in Z
3 with all
components non-zero. It will be convenient to set
A = A1A2A3, B = B1B2B3, C = C1C2C3.
THE MANIN CONJECTURE IN DIMENSION 2 51
Arguing exactly as in the previous section, it is not difficult to deduce from
Lemmas 4.1 and 4.2 that
M(A,B,C) ≪ε Amin{C,AεB1+ε}+A2/3B2/3C1/3 +A1/2+εB1/2+εC1/2
for any ε > 0, whence
M(A,B,C) ≪ε A2/3B2/3C1/3 +A1+εB1/2+εC1/2. (4.17)
By working a little harder, we would like to replace the terms Aε, Bε with
something rather smaller.
The main problem to be faced emerges in the application of Lemma 4.2,
which gives
M(A,B,C) ≪
|a1a2a3b1b2b3|1/2
2ω(a1a2a3b1b2b3).
Rather than using the trivial bound 2ω(n) = Oε(n
ε), as above, we can try
to make use of the fact that 2ω(n) has average order ζ(2)−1 log n in order to
get some saving. Following this line of thought it is fairly straightforward
to show that AεBε can be replaced by (logA)3/2(logB)3 in (4.17). However
this would still not be enough to deduce the best possible upper bound for
NU (B) that we would like. Let us simplify matters by considering only the
contribution
S(A,B) =
2ω(a1a2a3b1b2b3),
to the above estimate for M(A,B,C). Then S(A,B) has exact order of
magnitude
(logAi)(logBi),
so how can we hope to do better than this? The crucial observation comes
in noting that we are only interested in summing over values of a,b for
which the corresponding conic (4.14) has a non-zero solution c ∈ Z3, with
gcd(ci, cj) = 1. If we denote this finer quantity by S∗(A,B), then it is
actually possible to show that
S∗(A,B) ≪ AB. (4.18)
This is established in [13, Lemma 1], and is simply a facet of the well-known
fact that a random plane conic doesn’t have a rational point. This should
be compared with the work of Serre [58]. Using the large sieve inequality,
Serre has shown
#{y ∈ Z3 : |y| 6 Y, (−y1y3,−y2y3)Q = 1} ≪
(log Y )3/2
where
(a, b)Q =
1, if ax2 + by2 = z2 has a solution (x, y, z) 6= 0 in Q3,
−1, otherwise,
denotes the Hilbert symbol. Guo [32] has established an asymptotic for-
mula for the corresponding quantity in which one counts only odd values of
y1, y2, y3 such that the product y1y2y3 is square-free.
52 T.D. BROWNING
Thus in addition to considering the density of integer solutions to diagonal
quadratic equations, as in the previous section, we also need to consider how
often such an equation has at least one non-trivial integer solution in order
to derive sufficiently sharp bounds. The outcome of this investigation is the
following result, which is established in [13, Lemma 2].
Lemma 4.5. For any ε > 0, we have
M(A,B,C) ≪ε A2/3B2/3C1/3 + στAB1/2C1/2,
where
σ = 1 +
min{A,B}ε
min{BiBj}1/16
, τ = 1 +
min{BiBj}1/16
It is clear that this constitutes a substantial sharpening over our earlier
estimate (4.17) for M(A,B,C). Nonetheless this is still not enough on
its own, and we will need an alternative estimate when B1, B2, B3 have
particularly awkward sizes. The following result is rather easy to establish.
Lemma 4.6. We have
M(A,B,C) ≪ ABiBj(Ck + CiCjA−1k )(logAC)
for any permutation {i, j, k} of the set {1, 2, 3}.
Proof. For fixed integers a, b, q, let ρ(q; a, b) denote the number of solutions
to the congruence at2 + b ≡ 0 mod q. We then have
ρ(q; a, b) 6
|µ(d)|
. (4.19)
It will clearly suffice to establish Lemma 4.6 in the case (i, j, k) = (1, 2, 3),
say. Now it follows from (4.14) that for given ai, b1, b2, c3, and each corre-
sponding solution t of the congruence
a1b1t
2 + a2b2 ≡ 0 (mod a3c23),
we must have c1 ≡ tc2 mod a3c23 in any solution to be counted. This gives
rise to an equation of the form h.w = 0, with h = (1,−t, a3c23) and w =
(c1, c2, k). Upon recalling that gcd(c1, c2) = 1 from (4.15), an application of
Lemma 4.1 therefore yields the bound
≪ ρ(a3c23; a1b2, a2b2)
|a3c23|
for the number of possible b3, c1, c2 given fixed choices of ai, b1, b2 and c3. It
now follows from (4.19) that
M(A,B,C) ≪
ai,b1,b2,c3
ρ(a3c
3; a1b2, a2b2)
|a3c23|
ai,b1,b2,c3
d|a3c3
|µ(d)|
(−a1a2b1b2
|a3c23|
ai,b1,b2,c3
τ(a3)τ(c3) + C1C2
ai,b1,b2,c3
τ(a3)τ(c3)
|a3c23|
THE MANIN CONJECTURE IN DIMENSION 2 53
A simple application of partial summation now reveals that
M(A,B,C) ≪
AB1B2C3 +A1A2B1B2C1C2
(logAC)2,
as required to complete the proof of Lemma 4.6. �
We are now ready to combine Lemmas 4.5 and 4.6 to get a sharper upper
bound for NU (B). Taking Lemma 4.3 as our starting point we need to
bound the quantity N = Nv(S;U;Y) defined in (4.8), for fixed choices of
v ∈ N and Si, Ui, Yi > 0. As previously we will need to extract common
factors from s1, s2, s3, leading to the equality (4.11). Writing S
i = Si/k and
S′ = k−1S, as before, it is a simple matter to check that we have
N ∗v (S′;U;Y) 6 M(U,S′,Y),
with (a,b, c) = (u, s,y). Indeed we plainly have
gcd(ui, yj) = gcd(yi, yj) = 1, |µ(u1u2u3)| = gcd(ui, sj , sk) = 1,
and u, s,y ∈ Z4∗ , for any vectors counted by N ∗v (S′;U;Y), as required for
M(U,S′,Y). It now follows from (4.11) and Lemma 4.5 that
Nv(S;U;Y) ≪ε
(U2/3S2/3Y 1/3
+ k2/16στ
US1/2Y 1/2
≪ε U2/3S2/3Y 1/3 + στUS1/2Y 1/2,
for any ε > 0, where
σ = 1 +
min{S,U}ε
min{SiSj}1/16
, τ = 1 +
min{SiSj}1/16
In order to obtain our final estimate for NU (B) we need to sum this bound
over all positive integers v 6 B1/3, as in Lemma 4.4, and over all possible
dyadic intervals for Si, Ui, Yi, subject to (4.10).
Suppose for the moment that we want to sum over all possible dyadic
intervals X 6 |x| < 2X, for which |x| 6 X . Then in deducing Lemma 4.4
we employed the basic bound O(logX ) for the number of possible choices for
X. In the present investigation we will be more efficient and take advantage
of the easily established estimates
Xδ ≪δ
1, if δ < 0,
X δ, if δ > 0,
where the sum is over dyadic intervals for X 6 X . We will make frequent
use of these bounds without further mention.
Returning to our estimate for Nv(S;U;Y), we may now conclude from
the bound Yi 6 B
1/2/(v2SiUUi)
1/2 in (4.10) that
NU (B) ≪ε
v6B1/3
Si,Ui,Yi
U2/3S2/3Y 1/3 + στUS1/2Y 1/2
≪ε B1/2
v6B1/3
Si,Ui
v6B1/3
Si,Ui,Yi
στUS1/2Y 1/2
≪ε B(logB)6 +
v6B1/3
Si,Ui,Yi
στUS1/2Y 1/2.
54 T.D. BROWNING
The first term on the right-hand side is clearly satisfactory, and it remains
to deal with the second term, which we denote by R for convenience. We
would like to show that R ≪ B(logB)6.
Suppose without loss of generality that S1 6 S2 6 S3, so that in particular
min{SiSj} = S1S2 in σ and τ. If there is a constant A > 0 such that
S3 6 (S1S2)
A, then it follows that
σ 6 (S1S2)
ε−1/16Sε3 6 (S1S2)
(1+A)ε−1/16 ≪ 1,
provided that ε is sufficiently small. Taking τ ≪ logB, we may then argue
as above to conclude that there is a contribution of O(B(logB)6) to R from
this case. Suppose now that there exists A′ > 0 such that U 6 (S1S2)
Then we have σ ≪ 1 and τ ≪ logB, so that there is a contribution of
O(B(logB)6) to R in this case too.
Finally it remains to consider the contribution to NU (B) from Si, Ui, Yi
such that
S1S2 6 min{S3, U}δ, (4.20)
for some small value of δ > 0, with S1 6 S2 6 S3. Let us denote this
contribution N0, say. To estimate N0 we will return to the task of estimat-
ing Nv(S;U;Y) for fixed v, Si, Ui, Yi, but this time apply Lemma 4.6 with
(i, j, k) = (1, 2, 3). This gives
Nv(S;U;Y) ≪ (logB)2
US1S2Y3 + U1U2S1S2Y1Y2
We must now sum over dyadic intervals for Si, Ui, Yi. Thus it follows from
the bound Yi 6 B
1/2/(v2SiUUi)
1/2 in (4.10) that
N0 ≪ (logB)2
v6B1/3
Si,Ui,Yi
US1S2Y3 + U1U2S1S2Y1Y2
≪ (logB)2
v6B1/3
Si,Ui
Y1,Y2
B1/2U1/2S1S2
Si,Ui,Y3
B(S1S2U1U2)
Since U2 ≪ B/(v3Y1Y2) in (4.10), and S1S2 ≪ Sδ3 by (4.20), we therefore
deduce that the overall contribution from the first inner sum is
≪ (logB)2
v6B1/3
Si,Y1,Y2
U2,U3
B3/4S1S2
v7/4S
S2,S3,Y1,Y2
U2,U3
B3/4(logB)2
1/2−δ
Turning to the contribution from the second inner sum, we deduce from a
second application of (4.20) that
N0 ≪ B + (logB)2
Si,Ui,Y3
B(S1S2U1U2)
≪ B + (logB)2
S2,S3,Ui,Y3
U (1−δ)/2
≪ B(logB)5.
THE MANIN CONJECTURE IN DIMENSION 2 55
Once combined with our earlier work, this therefore concludes the proof of
Theorem 1.2.
Exercise 20. By mimicking the argument in [13, §5], establish the lower
bound NU2(B) ≫ B(logB)6.
Acknowledgements. The author is grateful to Roger Heath-Brown for
a number of useful conversations relating to the contents of §2.3, and to
Michael Harvey for spotting several typographical errors in an earlier draft.
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School of Mathematics, University of Bristol, Bristol BS8 1TW
E-mail address: t.d.browning@bristol.ac.uk
	1. Introduction
	1.1. Notation
	1.2. The Manin conjectures
	1.3. Degree 3 surfaces
	1.4. Degree 4 surfaces
	1.5. Degree 5 surfaces
	1.6. Universal torsors
	2. Further heuristics
	2.1. The lines on a cubic surface
	2.2. Cubic characters and Jacobi sums
	2.3. The Hardy–Littlewood circle method
	3. The A1 del Pezzo surface of degree 6
	3.1. Elementary considerations
	3.2. The asymptotic formula
	4. The D4 del Pezzo surface of degree 3
	4.1. Elementary considerations
	4.2. A crude upper bound
	4.3. A better upper bound
	References
