ARTICLE IN PRESS

Journal of Number Theory 105 (2004) 333–343

http://www.elsevier.com/locate/jnt

An application of Mumford’s gap principle
Jung-Jo Lee and M. Ram Murty,1Department of Mathematics, Queen’s University, Kingston, Ont., Canada K7L 3N6
Received 3 May 2003
Communicated by A. Granville

Abstract
We study a Dirichlet series attached to a polynomial ﬁrst deﬁned by Rubin and Silverberg intheir study of ranks of quadratic twists of a ﬁxed elliptic curve. We apply Mumford’s gap
principle to show that the series converges if the associated polynomial has distinct roots anddegree at least 5.
r 2003 Elsevier Inc. All rights reserved.

1. Introduction
Let E be an elliptic curve deﬁned over Q: Suppose that E is deﬁned by theWeierstrass equationy2 ¼ x3 þ ax þ b;a; bAZ:

The quadratic twist, denoted ED ; of E is given by
Dy2 ¼ x3 þ ax þ b:
By a celebrated theorem of Mordell, the group of rational points, ED ðQÞ; of ED isa ﬁnitely generated abelian group. Its rank has been the study of numerous papers
such as [G,H,RS1,RS2].
Honda [H] has made the surprising conjecture that rankZ ED ðQÞ is bounded as Dvaries over all integers. At present, there is no evidence for or against this conjecture,
although in the case of function ﬁelds over ﬁnite ﬁelds, there are analogous works byCorresponding author.
E-mail address: murty@mast.queensu.ca (M.R. Murty).
1
Research partially supported by an NSERC grant.
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.jnt.2003.11.006

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Shafarevich and Tate [ST] and Ulmer [U]. In a related context, Goldfeld [G] hasconjectured that ‘‘on average’’ rankZ ED ðQÞ is 12: More precisely, Goldfeld
conjectured that
#fjDjpx : rankZ ED ðQÞ ¼ 0gB#fjDjpx : rankZ ED ðQÞ ¼ 1gB12 #fjDjpxg
and#fjDjpx : rankZ ED ðQÞX2g ¼ oðxÞas x-N:
Perhaps motivated by these conjectures, Rubin and Silverberg [RS1] derived thefollowing equivalent formulation of Honda’s conjecture.
For each natural number n; letYn¼
pnp ðnÞ
pbe its unique factorization as a product of prime powers. Deﬁne the square-free partof n byYsfðnÞ ¼p:
np ðnÞ odd

If n is a negative integer, we let sfðnÞ ¼
Now letsfð nÞ:gðxÞ ¼ x3 þ ax þ band set
Gðu; vÞ ¼ v4 gu
¼ vðu3 þ auv2 þ bv3 Þ:v

For ðu; vÞ ¼ 1; deﬁne the height hðu=vÞ ¼ maxf1; log juj; log jvjg: It is the ‘‘naive’’height with only a ﬁnite number of exceptions so that it takes positive values. Deﬁnethe series
X1Sg ðj; kÞ :¼:
kjsfðGðu;vÞÞjhðu=vÞjðu;vÞ¼1Gðu;vÞa0Theorem 1 (Rubin, Silverberg [RS1,RS2]). If j is a positive real number, then thefollowing conditions are equivalent:(a) rankZ ED ðQÞo2j for every DAZ\f0g;(b) Sg ðj; kÞ converges for some kX1;(c) Sg ðj; kÞ converges for every kX1:

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Their theorem suggests the study of the cognate series deﬁned as follows.
Let f ðxÞ be a monic polynomial of degree r with coefﬁcients in Z and with distinct
roots. Let8  u>< vr f
if r is even;v F ðu; vÞ ¼
u>: vrþ1 f
if r is odd:v
Consider the sum
Sf ðj; kÞ ¼

X

1kj
ðu;vÞ¼1 jsfðF ðu; vÞÞj hðu=vÞ

:F ðu;vÞa0We prove:
Theorem 2. If rX5; then Sf ðj; kÞ converges for every k41 and any positive real
number j:
We remark (as noted in [RS1]) that Theorem 2 is an immediate consequence of aconjecture of Caporaso et al. [CHM], namely that the number of rational points on acurve (deﬁned over Q say) of genus g is bounded by a constant depending only on g;
for gX2: They prove that their conjecture is a consequence of the Bombieri–Lang–
Vojta conjecture on distribution of rational points on varieties of general type.
Indeed, if we write m ¼ sfðF ðu; vÞÞ; then
01Sf ðj; kÞ ¼

NX
m2 ðmÞBB
mk @

m¼1

X

1CC;

jA

hðu=vÞðu;vÞ¼10aF ðu;vÞ¼7my2

where m is the Möbius function, and the inner sum is uniformly bounded because u y 2if r is even
¼ 7m r=2fvv
andfu y 2¼ 7m ðrþ1Þ=2vv

if r is odd

has a uniformly bounded number of solutions assuming the Caporaso, Harris andMazur conjecture. Thus the series converges for k41 and jX0:
Our goal in this paper is to prove Theorem 2 unconditionally. We will give two
proofs of Theorem 2. The ﬁrst one will have fewer prerequisites. The second will use
the deeper work of Vojta [V] on effective versions of Faltings theorem concerning
Mordell’s conjecture. Our main tool will be Mumford’s gap principle (see [La,p. 120], [Mu]) which we now describe.

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2. Mumford’s gap principle
Let K=Q be a number ﬁeld, C=K a hyperelliptic curve of genus gX2; J=K theJacobian variety of C: Lete : C+J
be an embedding of C into J of the form
P/½ðPÞ

ðP0 Þ%
for a ﬁxed base point P0 ACðKÞ:We assume that P0 is chosen so that Y ¼ eðC g 1 Þ is a symmetric divisor on J: Letĥ be the logarithmic canonical height on J with respect to Y: (See [La, p. 113] fordetails.)% corresponding to ĥ given by
We have an inner product on JðKÞ/P; QS ¼ 12ðĥðP þ QÞĥðPÞĥðQÞÞ:% is deﬁned accordingly as
The norm on the points of JðKÞpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
jPj ¼ /P; PS:
From the well-known property of the canonical height that ĥðkPÞ ¼ k2 ĥðPÞ forany natural number k; we havepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
jPj ¼ /P; PS ¼ 12ðĥð2PÞ 2ĥðPÞÞ ¼ ĥðPÞ:% with their images in JðKÞ:%Having ﬁxed e; we identify points on CðKÞProposition 3 (Mumford [Mu]). With notations as above, there is a constant g2 ¼
% we haveg2 ðC=KÞ such that for any two distinct points P; QACðKÞ;jPj2 þ jQj2
þ g2 :/P; QSp2g
As is well-known, Mumford’s gap principle implies that rational points are thinlydistributed. More precisely, Mumford [Mu] showed that if fxn g is a sequence of% lying in some ﬁnitely generated subgroup of JðKÞ% anddistinct points in CðKÞordered by increasing height then there is an integer N and a number a41 such thatjxnþN jXajxn j:Mumford then deduced that jxn j grows exponentially. In this sense, the points of% are sparsely distributed in JðKÞ:%CðKÞ

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For our purpose, we need to make the argument of Mumford more explicit. To
this end, the following lemma, due to Silverman [Si] will be useful.
Lemma 4. Let 0oy0 op2 be a fixed angle, and let z1 ; y; zN ARd be a collection ofpoints in Rd : IfN4

ðd1Þpd 1

2 sin;
ðy0 =2Þthen there are distinct indices iaj; such that/zi ; zj SXjzi jjzj j cos y0 :Proof. See [Si, p. 388]. A related packing argument can also be found in [K]. &If we deﬁne the cosine between two vectors v and w bycosðv; wÞ :¼/v; wS;
jvjjwjwe may write the Mumford’s inequality as
1 jPj jQjg2g cosðP; QÞpþ;þ2 jQj jPjjPjjQj
provided jPj; jQj are non-zero.
Now ﬁx y0 as in Lemma 4. We apply the lemma to the elements xi withR þ 1pipR þ N
where R is any non-negative integer and"#ðd 1Þp
N¼
þ 1:
2 sind 1 ðy0 =2Þ
By the lemma, there is a pair RoiojpR þ N such thatcos y0 pcosðxi ; xj Þ:Thus, by Mumford’s inequality

1 jxi j jxj jg2
g cos y0 pg cosðxi ; xj Þpþ:
þ
2 jxj j jxi jjxi jjxj j
The function f ðsÞ ¼ 12ðs þ s 1 Þ is easily analyzed and we see that it takes its
minimum (for s40) at s ¼ 1 with f ð1Þ ¼ 1:

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pﬃﬃﬃﬃﬃ
If we insist that jxi j42 g2 (say), then with y0 chosen close to 0; we obtain
1 jxi j jxj j1
þgð1 eÞ ¼ g cos y0 pþ :2 jxj j jxi j
4As gX2; this implies there is an a (depending only on e) so thatjxj jXajxi j:
Moreover, a41: [In fact, if jxi j is very large, e small, then a is approximately thenumber s such that s þ s 1 ¼ 2g: ] The important point is that a depends only on g2(with e chosen sufﬁciently small). We record our observation in the following.
Lemma 5. Let C be a curve of genus gX2 defined over a number field K: Lete : C+J
be the embedding of C into its Jacobian as before. Let ĥ be the canonical height on J
with respect to Y: There exists a constant g2 depending only on C and an absolute
constant a41 such that if L is any finite extension of K; and fxn g is the sequence ofpﬃﬃﬃﬃﬃ
points of CðLÞ ordered by increasing height, then for jxn j42 g2 ; we havejxnþN jXajxn jfor N explicitly given in terms of d ¼ rankJðLÞ: In fact, we have N^dcd1 for someconstant c1 which depends only on y0 of Lemma 4.
Proof. This follows immediately from the preceding discussion. It is an effective
version of the Mumford’s gap principle, analogous to Theorem 8.1 of [La, p. 135].
Remark. Several authors have discussed effective versions of Mumford’s theorem.
The ﬁrst such statement seems to be in [Si] and later in [Di, p. 92]. As these latterpresentations do not state the result in the form we need it, we have given thediscussion above, closely following [La,Si].
We now apply this to the case of a hyper-elliptic curve. Let f ðxÞ be a polynomialwith Z-coefﬁcients, of degree rX5; and with distinct roots. Then the curve
C : y2 ¼ f ðxÞ
has genus gX2 (see [Si2, p. 44]).
If we are interested in the rational points of the twists
Cm : my2 ¼ f ðxÞ;pﬃﬃﬃﬃ
then these can be viewed as points of CðLÞ with L ¼ Qð mÞ: The Mordell–Weiltheorem tells us that if Jm is the Jacobian of Cm ; then Jm ðLÞ is ﬁnitely generated.

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Moreover, if ya0; then the height of a rational point P ¼ ðx; yÞACm ðQÞ satisﬁes(see [Si3])ĥðPÞblog jmj:
We will need the following well-known result of Northcott in later discussions.
Proposition 6 (Northcott). Let K be a number field, V =K a projective variety and%h : V ðKÞ-Ran absolute logarithmic height relative to an ample divisor on V : Then for
all numbers d; and H; the set% : ½KðPÞ : K pdfPAV ðKÞand hðPÞpHgis finite.
Proof. See [La, p. 59].
With a view to proving Theorem 2, we record below the consequence of thepreceding discussion to our situation.
Proposition 7. Let f ðxÞAZ½x be a polynomial of degree rX5 and with distinct roots.
Let8  u>< vr fv F ðu; vÞ ¼u>: vrþ1 fv

if r is even;if r is odd:

There are constants g3 and c; depending only on f ; such that for jmj4g3 ;we haveX15c
hðu=vÞjðu;vÞ¼10aF ðu;vÞ¼my2rankZ Jm ðQÞfor any positive real number j:
Proof. First, note that ya0: We choose m sufﬁciently large to ensure all rationalpﬃﬃﬃﬃﬃ
points of Cm have height 42 g2 so that we can apply Lemma 5. The points ðu; vÞwith u; v coprime and satisfying
F ðu; vÞ ¼ my2

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correspond to the points
P¼

pﬃﬃﬃﬃ myu; 7 r=2v
v

(if r is even) and
pﬃﬃﬃﬃ myu; 7 ðrþ1Þ=2P¼vv
pﬃﬃﬃﬃ(if r is odd) in the set CðQð mÞÞ of the hyper-elliptic curve
C : y2 ¼ f ðxÞ:pﬃﬃﬃﬃ
We order the countable set of points in CðQð mÞÞ in the order of increasingheight to have a sequence fxn g: (Of course, by Faltings theorem, this is a ﬁnite
sequence but we are not assuming Faltings theorem.)
Let hx ðPÞ denote the naive height on the x-coordinate of P: Thenhx ðPÞ^ĥðPÞ:
By Northcott’s theorem (Proposition 6),[JðKÞtor ;K

where the union is over all quadratic extensions of Q; is a ﬁnite set. Thus, if a torsion
pﬃﬃﬃﬃ
point lies in inﬁnitely many CðQ mÞ; then y ¼ 0 necessarily. In other words, if m issufﬁciently large, then the only torsion points of F ðu; vÞ ¼ my2 come from points
with y ¼ 0; and these have been eliminated from the sum under consideration. Thus
we see that the convergence of the sum in question is determined by
X1:
pﬃﬃﬃ
pﬃﬃﬃĥðxn Þjxn ACðQð mÞÞ CðQð mÞÞtor

We partition the indices n in this sum into residue classes (mod N) with N given asin Lemma 5. For each such residue class tðmod NÞ; and n ¼ qN þ t; we have byLemma 5ĥðxn Þ ¼ jxn j2 4a2q jxt j2 ¼ a2 ðn tÞ=N jxt j2 :As a41; we ﬁnd that for each residue class tðmod NÞ; the contribution is5jxt j 2j :Summing this for tðmod NÞ gives an estimate of OðNÞ as the bound for the sum inquestion. ThenrankZ Jm ðQÞN^ðrankZ Jm ðQÞÞc1

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gives us an upper boundN5crankZ Jm ðQÞwith c ¼ 2c1 as our ﬁnal estimate for the sum.

&3. Proof of Theorem 2
We will show that Sf ðj; kÞ converges for k41 and any positive real number j:
We write the sum asNX
m2 ðmÞm¼1mkX

1

hðu=vÞ
ðu;vÞ¼10aF ðu;vÞ¼7my2

j:

If m is sufﬁciently large, say m4g3 ; then we can apply the estimate of Proposition
7 to deduce the convergence of the series. If ĥðPÞ is sufﬁciently large, say ĥðPÞ44g2 ;then we still can apply Lemma 5 so that the arguments of Proposition 7 remain valid
to deduce the convergence. Here g2 and g3 are as given in Lemma 5 and Proposition
7. The sum of the remaining terms is ﬁnite by the result of Northcott. (SeeProposition 6.) From ĥðPÞ^hðPÞ; we know that there exists a constant H such thatif ĥðPÞp4g2 then hðPÞpH to apply Proposition 6.
Thus, in analyzing Sf ðj; kÞ; we may apply Proposition 7. We deduceSf ðj; kÞ5NX
crankZ Jm ðQÞm¼1mk:

There are several ways to complete the proof. The standard descent argument(see [Si2]) gives
rankZ Jm ðQÞ5nðmÞ þ Oð1Þ
where nðmÞ denotes the number of prime divisors of m (see [H, p. 95] or [Si], [Si1]).
We have the estimate, for any e40cnðmÞ 5me
which gives the desired result. This completes the proof of Theorem 2.

4. An alternate proof
As a consequence of his work on effective versions of Faltings theorem on
Mordell’s conjecture, Vojta [V], [Bo] proved the following bound. (See [HS, ExerciseARTICLE IN PRESS
J.-J. Lee, M.R. Murty / Journal of Number Theory 105 (2004) 333–343

342E.2].) Let C=K be a curve of genus X2 deﬁned over a number ﬁeld K; let h :%CðKÞ-Rbe a height function on C corresponding to a projective embedding of C;
and let J=K be the Jacobian variety of C: There is a constant g (depending only onC=K) such that for all extensions L=K;
#fPACðLÞ : hðPÞXggp#JðLÞtor ðlog2 gÞ10rankJðLÞ :If we vary over extensions L of ﬁxed degree, then Northcott’s theorem(Proposition 6) implies that #JðLÞtor is uniformly bounded. Thus, the sumappearing in Theorem 2 is bounded by5NX
10rankZ Jm ðQÞm¼1mk;

which by the previous argument, converges for k41 if we use the boundlog mnðmÞ ¼ O:
log log m
However, Vojta’s theorem is rather deep. Our purpose here was to prove Theorem
2 using the ‘‘simpler’’ result embodied in Mumford’s gap principle.

Acknowledgments
We are thankful to Joseph Silverman, Ernst Kani, Michael Roth and the referee
for providing comments and suggestions.

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[G]
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