Elliptic curves with large Tate–Shafarevich groups over Fq (t)Richard Griffon and Guus de Wit
Abstract. Let Fq be a finite field of odd characteristic p. We exhibit elliptic curves over the rational function field K = Fq (t) whose Tate-Shafarevichgroups are large. Precisely, we consider certain infinite sequences of explicit
elliptic curves E the Tate–Shafarevich group X(E) of which, we prove, is finite and satisfies |X(E)| = H(E)1+o(1) as H(E) → ∞, where H(E) denotes
the exponential differential height of E. We further show that, despite being
large, these Tate–Shafarevich groups have trivial p-primary part.
The proof involves explicitly computing the L-functions of these elliptic
curves, proving the BSD conjecture for them, and obtaining estimates on thesize of the central value of their L-function.

Introduction
0.1. The size of Tate–Shafarevich groups of elliptic curves. Fix a finite field Fq of odd characteristic p and let K := Fq (t) denote the rational function
field over Fq . Let E be an elliptic curve over K which, we assume, is non-isotrivial(i.e., its j-invariant j(E) ∈ K does not lie in Fq ). One attaches to E its Tate–
Shafarevich group, denoted by X(E). Among other reasons, the arithmetic significance of X(E) stems from its measuring, in a certain sense, how badly the
local–global principle fails for E. The Tate–Shafarevich group remains a mysterious object; for instance, the finiteness of X(E) is still conjectural in general, eventhough it has been proved in a number of cases. Let us assume for now that X(E)is indeed finite: what can then be said about its size? There are several natural
choices of numerical invariants of E to compare |X(E)| to. Here, we choose theexponential differential height H(E) and the conductor N (E): these are defined by
1

H(E) := q 12 deg ∆min (E)andN (E) := q deg N (E) ,where ∆min (E) and N (E) denote the minimal discriminant and the conductor
divisors of E, respectively. Goldfeld and Szpiro [1] have proven upper bounds on
the order of X(E) in terms of these two invariants.
Let us quote a special case of their result:
Theorem 1 (Goldfeld–Szpiro [1]). Let E be a non-isotrivial elliptic curve
over K. Assume that E has finite Tate–Shafarevich group.
2010 Mathematics Subject Classification. Primary 11G05, 11G40; Secondary 14G10, 11L05.
Key words and phrases. Elliptic curves over function fields, Tate-Shafarevich groups, Explicit
computation of L-functions, BSD conjecture, Gauss and Kloosterman sums.
Correspondence to be sent to richard. griffon@ unibas. ch .
1

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RICHARD GRIFFON AND GUUS DE WIT(i) Then, for all ε > 0, one has |X(E)| q,ε H(E)1+ε .
(ii) If, moreover, j(E) ∈ K is not a p-th power in K, then,for all ε > 0, one has |X(E)| q,ε N (E)1/2+ε .
In this statement1, item (ii) follows from item (i) by applying Szpiro’s inequality
for elliptic curves with separable j-invariants. It is then natural to wonder about
the optimality of (i) and (ii): apart from the ε’s, are the exponents of the height (1)and of the conductor (1/2) in these upper bounds best possible? In other words,
as E ranges over all elliptic curves over K, what is the largest power of H(E)– or N (E) – that does appear in |X(E)|, up to a ±ε for all ε > 0?
In the analogous setting of elliptic curves over Q, the analogue of Theorem 1
is known to be a consequence of the ABC conjecture (see §1–3 in [1]). Moreover,de Weger [21] conjectures that the exponents 1 and 1/2 should indeed be optimal inthat context (see Conjectures 2 and 4 in [21]). Following the analogy between thearithmetics of elliptic curves over Q and over K, one can translate the statementof de Weger’s conjecture (the quantities H(E) and N (E) here correspond to the
1/12-th power of the minimal discriminant and to the conductor of an elliptic curve
over Q, respectively). This translation results in the following:
Conjecture 2 (de Weger [21]). Assuming finiteness of the relevant Tate–
Shafarevich groups,(i) For any ε > 0, there are infinitely many elliptic curves E/K such that
|X(E)| q,ε H(E)1−ε .
(ii) For any ε > 0, there are infinitely many elliptic curves E/K such that
|X(E)| q,ε N (E)1/2−ε .
In the same paper, de Weger proves, conditionally to the Birch and SwinnertonDyer (BSD) conjecture, that (the analogue of) Conjecture 2(i) holds for elliptic
curves over Q. Prior to [21], Mai and Murty [10, Theorem 2] had shown, again
conditionally to the BSD conjecture, a weaker version of Conjecture 2(ii) for elliptic
curves over Q, with the exponent 1/2 replaced by 1/4. Both of these results rely
on considering sequences of well-chosen quadratic twists of a given elliptic curve.
In the context of elliptic curves over K, the above-stated Conjecture 2(i) is notdifficult to prove. One can indeed take advantage of the existence of inseparable
isogenies of large degree to construct sequences of elliptic curves over K with large
Tate–Shafarevich groups (see §1.5, where we build one such example). In suchsequences, the inseparability degree of the j-invariant grows to infinity. Moreover,by construction of these sequences, the elliptic curves therein are K-isogenous: one
cannot, therefore, hope to deduce from these any result towards Conjecture 2(ii).
One also notices that the p-primary parts of the Tate–Shafarevich groups of theelliptic curves belonging to these sequences are “large”; and that, actually, the orderof the p-primary part already accounts for the observed “large X” phenomenon.
We are therefore led to ask the following questions:
Question 1 For a given ε > 0, are there infinitely many pairwise non K-isogenous
elliptic curves E/K such that |X(E)| q,ε H(E)1−ε ?
1For two functions f (x), g(x) defined on [0, ∞), we use Vinogradov’s notation “f (x)  g(x)”ato mean that there exists a constant C > 0, depending at most on the parameters a, such that|f (x)| ≤ Cg(x) for x → ∞. This is synonymous with Landau’s notation “f (x) = Oa (g(x))”.

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)3Question 2 If so, is the fact that their Tate–Shafarevich groups are “large” always
explained by a “large p-primary part of X” phenomenon?
Question 3 In the first question, can one also require that the j-invariants of theinvolved elliptic curves have bounded inseparability degree?
In this paper, we give positive answers to the first and third questions:
Theorem A. For all ε > 0, there are infinitely many pairwise non K-isogenous
elliptic curves E/K with separable j-invariant and finite Tate–Shafarevich group,
such that|X(E)| ≥ H(E)1−ε .
We also prove a result in direction of Conjecture 2(ii), with the same exponent 1/4 as in [10]:Theorem B. For all ε > 0, there are infinitely many pairwise non K-isomorphic
elliptic curves E/K with separable j-invariant and finite Tate–Shafarevich group,
such that|X(E)| ≥ N (E)1/4−ε .
In contrast to the aforementioned results concerning elliptic curves over Q,
Theorems A and B are unconditional, and the involved elliptic curves are notquadratic twists of each other. Our proof is constructive and effective in that
we exhibit sequences of elliptic curves over K satisfying these properties and we
provide explicit bounds on the order of their Tate–Sharafarevich groups.
Finally, for all the elliptic curves in these sequences, we prove that the p-primaryparts of their Tate–Shafarevich groups are trivial, thus answering in the negative
the second question raised above.
0.2. Elliptic curves with large Tate–Shafarevich groups. Theorems A
and B both follow from our main theorem, which we now state. Let Fq be a finite
field of odd characteristic p. For any parameter γ ∈ F×
q and any integer a ≥ 1, weawrite ℘a (t) := tq − t ∈ Fq [t] and consider the elliptic curve Eγ,a defined over K by
the Weierstrass model:(1)Eγ,a :
y 2 = x x2 + ℘a (t)x + γ .
The main result of this paper is the following:
Theorem C. In the above setting,(1) As γ ∈ F×
q and a ≥ 1 vary, the curves Eγ,a are pairwise neither K-isomorphic
nor K-isogenous. Moreover, the j-invariant j(Eγ,a ) is separable.
(2) For any γ ∈ F×
q and a ≥ 1, the Tate–Shafarevich group X(Eγ,a ) is finite.
(3) Given γ ∈ F×,q as a → ∞, we have|X(Eγ,a )| = H(Eγ,a )1+o(1) .
(4) For any γ ∈ F×
q and a ≥ 1, the p-primary part of X(Eγ,a ) is trivial.
One can reformulate (3) as follows: given ε > 0, for any γ ∈ F×
q and any large
enough integer a ≥ 1 (depending on ε), we haveH(Eγ,a )1−ε ≤ |X(Eγ,a )| ≤ H(Eγ,a )1+ε .

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RICHARD GRIFFON AND GUUS DE WIT

It is then clear that, for a given ε > 0, all but finitely many of the Ea,γ ’s satisfy|X(Eγ,a )| ≥ H(Eγ,a )1−ε . Further, we will see that H(Eγ,a ) = N (Eγ,a )1/4 , so that
|X(Eγ,a )| ≥ N (Eγ,a )1/4−ε holds for all but finitely many of the Eγ,a ’s.
In particular, Theorem C does imply Theorems A and B.
0.3. Organisation of the paper. We now explain the overall strategy ofthe proof of Theorem C as we describe the plan of the paper. The first section
briefly recalls the definitions and main properties of the objects that will be usedthroughout the paper. In the last subsection §1.5, we show how the Frobenius isogenies allow one to construct sequences of elliptic curves with large Tate–Shafarevichgroups (thus proving Conjecture 2(i) above). We found it worthwhile to include
this construction since the argument in §1.5 illustrates on a simple example thegeneral structure of the proof of Theorem C.
The elliptic curves Eγ,a are introduced in section 2, where we also calculate
some of their more easily accessible invariants: their height and conductor, their
Tamagawa number, as well as the torsion subgroup of Eγ,a (K). The curves Eγ,awere first studied by Pries and Ulmer in [13]: we recall in §2.2 how they wereconstructed there, and some of the results of [13]. Item (1) of Theorem C is provedin Proposition 2.4.
A large part of the proof of Theorem C is analytic, in that we rely on a detailed
study of the relevant L-functions. Our first main goal is therefore to obtain an
explicit expression for the L-function of Eγ,a . To that end, we introduce somenotation in section 3. We define a certain finite set Pq (a) of places of K and, toeach place v ∈ Pq (a) we attach in §3.3 two character sums over the residue fieldof K at v: a Gauss sum g(v) and a Kloosterman sum Klγ (v). We then show (seeTheorem 4.1) that, for any integer a ≥ 1 and any γ ∈ F×
q , the L-function L(Eγ,a , T )of Eγ,a admits the following expression:Y(2)
L(Eγ,a , T ) =1 − g(v)Klγ (v)T deg v + g(v)2 q deg v T 2 deg v .
v∈Pq (a)This identity is proved in section 4 by an elementary method based on the definition of L(Eγ,a , T ) and manipulation of character sums. This result, which is
instrumental in the proof of our main theorem, may be of independent interest.
Using (2) and arithmetic properties of Gauss and Kloosterman sums, we elucidate in Theorem 5.2 the p-adic valuations of the zeros of L(Eγ,a , T ). As a consequence, we deduce that L(Eγ,a , T ) does not vanish at the central point T = q −1 forits functional equation (see Theorem 5.3). This non-vanishing result is enough to ensure that the BSD conjecture holds for Eγ,a (Corollary 5.4) which, in turn, has several important corollaries for our study. First, the Tate–Shafarevich group X(Eγ,a )is indeed (unconditionally) finite, as claimed in Theorem C(2). Secondly, we derivefrom the BSD formula and from our computations in section 2 that
(3)

|X(Eγ,a )| = q −1 H(Eγ,a ) · L(Eγ,a , q −1 ).

Lastly – even though this is less central to our point – the Mordell–Weil groupEγ,a (K) is finite and, taking into account our description of its torsion subgroup,
we conclude that Eγ,a (K) = O, (0, 0) .
In light of the link (3) between the order of the Tate–Shafarevich group and thecentral value of L(Eγ,a , T ), we estimate the size of |X(Eγ,a )| in terms of H(Eγ,a )ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)5by proving adequate upper and lower bounds on L(Eγ,a , q −1 ). Specifically, in orderto show that Theorem C(3) holds, we need to prove that(4)−o(1) ≤

log L(Eγ,a , q −1 )≤ o(1)log H(Eγ,a )(as a → ∞).

The proof of these inequalities is carried out in section 7. Proving the lower boundin (4) is the crucial step. After evaluating expression (2) for L(Eγ,a , T ) at T = q −1 ,straightforward analytical considerations yield that L(Eγ,a , q −1 ) can be boundedfrom below byX
Klγ (v)−1, where w(x) := log |x2 (1 − x2 )|.
log L(Eγ,a , q ) ≥w
2q deg v/2v∈Pq (a)For any place v ∈ Pq (a), the algebraic number Klγ (v)/2q deg v/2 is known to be
totally real; moreover, its image in any complex embedding of Q lies in the interval [−1, 1] by the Weil bound for Kloosterman sums. It is apparent that the size ofthe right-hand side of the previous display depends on how the set
Θγ,a := {Klγ (v)/2q deg v/2 }v∈Pq (a)is distributed in [−1, 1]. We study the relevant aspects of the distribution of Θγ,ain section 6. Namely, we first recall from [3] an effective version of an asymptotic
equidistribution statement, the qualitative form of which says that, asymptoticallyas a → ∞,√ the set Θγ,a becomes equidistributed in [−1, 1] with respect to themeasure π2 1 − x2 dx. Next we prove a quantitative version of the fact that Θγ,a“avoids” the points −1, 0 and 1, which are the poles of x 7→ w(x) on [−1, 1].
We then conclude the proof of the lower bound in (4) by coupling these twofacts about Θγ,a with a suitable approximation of w on [−1, 1]. The resulting
estimates (4) constitute Theorem 7.1. We combine these to (3) in §8.2 to complete
the proof of Theorem C(3).
Finally, in order to prove Theorem C(4), we show that |X(Eγ,a )| is relatively
prime to p (see Theorem 8.1). To do so, we start from the BSD formula (3): insteadof estimating the “archimedean size” of the central value L(Eγ,a , q −1 ) as we did
above, we now estimate its “p-adic size”. The p-adic valuation of L(Eγ,a , q −1 )visibly depends on the p-adic valuations of the zeros of L(Eγ,a , T ). The abovementioned Theorem 5.2 allows us to obtain the desired result, in §8.1.
1. Invariants of elliptic curves over function fields
Let Fq be a finite field of characteristic p and K := Fq (t) denote the field ofrational functions on the projective line P1/Fq . For simplicity, and since we will notneed the general case, we assume throughout that p 6= 2. Let MK denote the setof all places v of K; the set MK may be identified with the set of closed points of◦P1/Fq . Among those is the degree-1 closed point ∞ = [0 : 1]; we write MK
for the set
1
1
MK \ {∞} of closed points of A/Fq = PFq \ {∞}. There is a 1-to-1 correspondence◦
between MK
and the set of monic irreducible elements of Fq [t].
In this section, we introduce the relevant invariants of elliptic curves over K.
For a more detailed introduction to the arithmetic of elliptic curves over function
fields and their invariants, the reader is referred to [19]. Some of the results quoted
in this section are valid more generally for elliptic curves (or even abelian varieties)
over a function field of positive characteristic, but we only state the special cases

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RICHARD GRIFFON AND GUUS DE WIT

that are required for our purpose. We do not mention K in the notation used forthe invariants of elliptic curves over K which we consider: unless explicitly noted
otherwise, these invariants are relative to K.
1.1. Conductor and height. Let E be a non-isotrivial elliptic curve over K(i.e., its j-invariant j(E) is not a constant rational function). We denote by ∆min (E)its minimal discriminant divisor, which is a divisor on P1/Fq . The exponential differential height H(E) is defined by
1

H(E) := q 12 deg ∆min (E) .
The conductor divisor N (E) of E is also a divisor on P1/Fq (see [16, Chap. IV, §10]
for the precise definition). We then let
N (E) := q deg N (E) .
We will call N (E) the “(numerical) conductor ” of E. An easy inequality between
the conductor and the minimal discriminant divisors yields that N (E)1/12 ≤ H(E).
In the other direction, Szpiro’s inequality states that
H(E) ≤ N (E)1/2 ,provided that j(E) ∈ K is not a p-th power in K (see [1, Thm. 3] and the referencesin that article). Note that two K-isogenous elliptic curves have the same numericalconductor (see §1.3).
1.2. The Tate–Shafarevich group. Fix a separable closure K sep of K andconsider the Galois cohomology group H1 (K, E) := H1 (Gal(K sep /K), E(K sep )).
For any place v ∈ MK , there is a similarly defined group H1 (Kv , E), where Kv
denotes the completion of K at v. Recall that the Tate–Shafarevich group of E isdefined by:!
Y
1
1
X(E) := ker H (K, E) −→H (Kv , E) ,v∈MK

where the product runs over all places v of K and the arrow is the product of thecanonical restriction maps H1 (K, E) → H1 (Kv , E).
Perhaps a more illuminating way of thinking about X(E) is to observe that it
is in bijection with the set of K-isomorphism classes of pairs (C, φ), where C/K isa curve of genus 1 which has at least one Kv -rational point for all places v ∈ MK
and φ is a K-isomorphism between E and Jac(C) (see [17, Chap. X, §3-4] for moredetails). One can show that the class of such a pair (C, φ) is trivial in X(E) if andonly if C has a K-rational point. This point of view makes it clearer that X(E)measures a local-global obstruction.
By construction, the Tate–Shafarevich group X(E) is a torsion abelian group.
It is conjectured, but not known in general, that X(E) is finite (in the cases westudy, we will show that it is indeed finite).
1.3. The L-function. For any place v ∈ MK , we denote by Fv the residue
field of K at v, which is a finite extension of Fq , and by deg v = [Fv : Fq ] the degree
e v denote the reduction of E at v. This plane cubic curveof v. For such a v, let (E)over Fv is not necessarily smooth: we say that E has good reduction at v if it is,and that E has bad reduction at v otherwise. In both cases, it makes sense to countELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)7e v . For any v ∈ MK , put av (E) := |Fv | + 1 − |(E)e v (Fv )|Fv -rational points on (E)and define the L-function of E by the formal Euler productY
Lv (E, T )−1 ,(1.1)
L(E, T ) :=v∈MK

where the product runs over all places of K and where(1 − av (E)T deg v + q dv T 2 deg v if E has good reduction at v,(1.2) Lv (E, T ) =1 − av (E)T deg v
if E has bad reduction at v.
By the Hasse bound, the Euler product in (1.1) a priori converges on the complex disk T ∈ C : |T | < q −3/2 . However, deep results of Grothendieck andDeligne (among others) actually prove that L(E, T ) is a polynomial with integral coefficients in T , with constant coefficient 1. The degree b(E) of that polynomial is given by the Grothendieck–Ogg–Shafarevich formula, which states thatdeg L(E, T ) = deg N (E) − 4. Moreover, the L-function satisfies the expected functional equation: L(E, T ) = ±1 · (qT )b(E) · L(E, 1/q 2 T ). Lastly, L(E, T ) satisfies the
Riemann Hypothesis i.e., the complex zeros of z 7→ L(E, z) have magnitude q −1 .
In particular, the value of L(E, T ) at the central point for its functional equation(viz. T = q −1 ) is a rational number, and the Riemann Hypothesis allows one to seethat L(E, q −1 ) is non-negative.
Recall that two K-isogenous elliptic curves E and E 0 have the same L-function:see [5, App. C] for a proof. In particular, their L-functions have the same degree and
the Grothendieck–Ogg–Shafarevich formula implies that deg N (E) = deg N (E 0 ), sothat the numerical conductors N (E) and N (E 0 ) coincide.
By the results recalled in the previous paragraph, the L-function of E can be
written as a product of the formb(E)L(E, T ) =Y(1 − zk T ),k=1

for some algebraic integers zk . These zk ’s are the inverse zeros of the polynomial
×L(E, T ). We choose a prime ideal P of Q above p and denote by ordP : Q → Q
the corresponding discrete valuation, so normalised that ordP (q) = 1. For allk ∈ {1, . . . , b(E)}, we let λk = ordP (zk ) ∈ Q. The functional equation of L(E, T )implies that both zk and q 2 /zk are algebraic integers, so that 0 ≤ λk ≤ 2 for all k.
Renumbering the zk ’s if necessary, we can assume that λ1 ≤ λ2 ≤ · · · ≤ λb(E) . Thus
ordered, the sequence {λ1 , λ2 , . . . , λb(E) } will be called the p-adic slope sequence ofL(E, T ). It then follows from the functional equation satisfied by L(E, T ) that one
has λb(E)−k = 2 − λk for any k ∈ {1, . . . , b(E)}.
1.4. The BSD conjecture. Birch and Swinnerton-Dyer (and Tate, in thiscontext) conjectured that the analytic behaviour of the L-function z 7→ L(E, z)around z = q −1 encodes arithmetic information about E. The reader is referred to[18], [1, §4] or [19] for more detailed accounts of the BSD conjecture.
In the case that L(E, T ) does not vanish at T = q −1 , which is the most relevant
for our purpose, their conjecture is entirely proved. We state the end result as
follows:

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RICHARD GRIFFON AND GUUS DE WIT

Theorem 1.1 (BSD conjecture for elliptic curves of analytic rank 0). Let E
be a non-isotrivial elliptic curve over K. Assume that L(E, q −1 ) 6= 0. Then thefollowing statements hold:(1) The Mordell–Weil group E(K) is finite (i.e., E(K) is torsion).
(2) The Tate–Shafarevich group X(E) is finite.
(3) Moreover, one has(1.3)

L(E, q −1 ) =|X(E)| q · τ (E),·
H(E) |E(K)|2where τ (E) denotes the Tamagawa number of E (i.e., the product over allplaces v of K of the number of connected components of the fiber over v of theNéron model of E).
We sketch a proof of this theorem. Write ρE = ordT =q−1 L(E, T ) for the
analytic rank of E i.e., the multiplicity of q −1 as a root of L(E, T ). Tate [18]has proved that 0 ≤ rankZ E(K) ≤ ρE . By assumption we have ρE = 0 here, sothat rankZ E(K) vanishes too: hence the finiteness of E(K). Moreover, we haveρE = rankZ E(K) (this equality is the so-called “weak BSD conjecture”).
On the other hand, deep results by Tate [18] and Milne [12] show that the“weak BSD conjecture” implies the whole BSD conjecture. More precisely, they
prove the equivalence:ρE = rankZ E(K) ⇐⇒ ∃ ` prime : X(E)[`∞ ] is finite⇐⇒ the BSD conjecture holds for E/K.
Milne [12] assumes that p 6= 2, but note that the equivalence is now known to holdeven in even characteristic, by subsequent work of Kato and Trihan [8].
Therefore, the equality ρE = rankZ E(K) implies (2) and (3).
1.5. Large X via Frobenius isogenies. This section is inspired by [20, §5]:
our main purpose here is to illustrate, on a very simple example, the steps that weshall follow to prove that some elliptic curves have a large Tate–Shafarevich group.
Let E be a non-isotrivial elliptic curve over K = Fq (t) and, for any integern ≥ 1, let En denote the base change of E/K under the pn -th power Frobenius
morphism Frpn : K → K. In other words, we put En = E ×K K where theunderlying map K → K is Frpn . The induced map Fpn : E → En is a purely
inseparable isogeny of degree pn (see [19, p. 225]). We obtain in this manner ansequence (En )n≥1 of elliptic curves over K. One obviously has j(En ) = j(E)p ,so that the inseparability degree of j(En ) is unbounded as n varies. Moreover, wehave N (En ) = N (E) since the curves are K-isogenous (see §1.1). For the same
reason, we have L(En , T ) = L(E, T ) for all n ≥ 1. On the other hand, the proof of
Theorem 5.1 in [20] shows that the height H(En ) tends to infinity as n → ∞.
Now let us assume that E is chosen so that L(E, q −1 ) is non-zero. Clearly, thecentral value L(En , q −1 ) is then also non-zero for any n ≥ 1. This implies thatthe full BSD conjecture holds for all the elliptic curves En in the sequence (seeTheorem 1.1). In particular, the Tate–Shafarevich groups X(En ) are all finite.
Furthermore, the BSD formula (1.3) states that
L(En , q −1 ) =|X(En )|
q · τ (En )·.
H(En ) |En (K)tors |2

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)9The growth of τ (En ) can be estimated in termsof H(En ): [6, Thm. 1.22] or [2,Thm. 1.5.4] yield that log τ (En ) = o log H(En ) as n → ∞. Further, there is a
uniform bound on torsion subgroups of elliptic curves over K (see Proposition 7.1
in [19, Lect. 1] for instance): here, this shows that |En (K)tors | remains bounded as
n → ∞. From the last displayed identity and these two bounds, we deduce that:log |X(En )|log L(En , q −1 )=1+
+ o(1)log H(En )log H(En )(as n → ∞).

Since L(En , q −1 ) = L(E, q −1 ) = O(1), we therefore obtain thatlog |X(En )| ∼ log H(En )(as n → ∞).

In other words, we have |X(En )| = H(En )1+o(1) as n → ∞, so that the elliptic
curves En do have “large Tate–shafarevich groups”.
Let us denote by vp : Q× → Q the p-adic valuation on Q, normalised so thatvp (q) = 1. We now evaluate the p-adic valuations of both sides of the BSD formula(instead of their logarithm, as we just did). The above
mentioned upper boundon τ (En ) also yields that vp (τ (En )) = O vp (H(En )) as n → ∞. Hence, we havevp (L(En , q −1 ))vp (|X(En )|)≥1++ o(1) = 1 + o(1).
vp (H(En ))vp (H(En ))Besides, the order of X(En )[p∞ ] equals q vp (|X(En )|) . Since vp (H(En )) is none otherthan log H(En )/ log q, we deduce from the above that
|X(En )[p∞ ]| ≥ H(En )1+o(1)(as n → ∞).

Therefore, the “large X” phenomenon for the elliptic curves in the sequence (En )n≥1
is “explained” by a large p-primary part of X.
The set of elliptic curves E to which this construction applies is non-empty. For
instance, consider the Legendre elliptic curve E/K given by y 2 = x(x − 1)(x − t).
This curve is non-isotrivial, and a straighforward application of Tate’s algorithm
shows that the conductor divisor N (E) has degree 4. The Grothendieck–Ogg–
Shafarevich formula then predicts that deg L(E, T ) = 0. This forces the L-function
to be trivial (i.e., L(E, T ) = 1); in particular, L(E, q −1 ) is non-zero.
The previous paragraph shows that we have:
Proposition 1.2. For any integer n ≥ 1, consider the elliptic curve En definedover K bynEn :y 2 = x x − 1 x − tp .
For all n ≥ 1, the Tate–Shafarevich group X(En ) is finite and, as n → ∞, one has|X(En )| = H(En )1+o(1) .
This proposition proves that Conjecture 2(i) in the introduction is true. To prove
Theorem C, we rely on a somewhat similar strategy: the various steps we follow
in the rest of the paper are but more elaborate versions of the ones in the proof of
Proposition 1.2.
Remark 1.3. In case the characteristic p of K is congruent to 1 modulo 6,
the authors of [4] have very recently produced a sequence of elliptic curves over K
with large Tate–Shafarevich groups, the p-primary parts of which are trivial. In
contrast to the elliptic curves studied in the present paper, though, the elliptic
curves forming the sequence of [4], being sextic twists of y 2 = x3 + 1 over K, are10

RICHARD GRIFFON AND GUUS DE WIT

all isotrivial and are all K-isomorphic (they have j-invariant 0). Theorem C is also
independent of the congruence of p modulo 6.
2. The elliptic curves Eγ,aLet Fq be a finite field of characteristic p ≥ 3, K = Fq (t) and MK be the set◦of places of K. Recall that we identify, as we may, the set MK= MK \ {∞} bothwith the set of monic irreducible polynomials in Fq [t] and with the set of closed
points of A1/Fq = P1/Fq \ {∞}.
a
For any integer a ≥ 1, we let ℘a (t) := tq − t ∈ Fq [t].
2.1. Definition. Let γ be a nonzero element of Fq and a ≥ 1 be an integer.
We let Eγ,a be the elliptic curve over K given by the affine Weierstrass model(2.1)

Eγ,a :

y 2 = x3 + ℘a (t)x2 + γx.

Its j-invariant is easily computed to be(2.2)

j(Eγ,a ) = 28 ·(℘a (t)2 − 3γ)3,γ 2 (℘a (t)2 − 4γ)

and the discriminant of the model (2.1) is ∆γ,a = 24 γ 2 (℘a (t)2 − 4γ). Note that the
j-invariant is not constant, hence the curve Eγ,a is non-isotrivial. We also remarkthat j(Eγ,a ) is not a p-th power in K i.e., j(Eγ,a ) is separable.
Even though the model (2.1) still makes sense when p = 2 or γ = 0, theexpression of its discriminant ∆γ,a shows that the curve Eγ,a defined by (2.1) is
not smooth in these cases. Hence our assuming that p 6= 2 and γ 6= 0 to ensurethat Eγ,a is a bona fide elliptic curve.
2.2. Construction of Eγ,a . The sequences {Eγ,a }a≥1 are essentially some of
the titular Artin–Schreier families of elliptic curves studied in [13] (see §6.2 there).
We briefly give more details about how the sequences {Eγ,a }a≥1 were constructedin loc. cit. and “compare” them to other similarly constructed sequences.
The basic input of the construction in [13] is a pair (f, g) of rational functions
P1/Fq → P1/Fq whose divisors satisfy mild conditions. The output is a sequence,indexed by powers Q of q, of smooth projective surfaces XQ over Fq , each equippedwith a natural fibration πQ : XQ → P1/Fq and admitting a dominant rational map
Cf,Q × Cg,Q 99K XQ from the product of two curves Cf,Q , Cg,Q defined over Fq interms of f and g. One of the main results of [13] is that the Jacobian variety of
the generic fiber of πQ satisfies the BSD conjecture.
Pries and Ulmer further classify (see Proposition 3.1.5 and §4.2 in [13]) thevarious polar behaviours of f and g for which the resulting sequence (XQ )Q is asequence of elliptic surfaces (i.e., for which the generic fiber of πQ is an elliptic
curve over K). This classification results in seven “types” of pairs (f, g).
The elliptic curve Eγ,a defined by (2.1) corresponds to the case (2, 1+1) of theirconstruction. Specifically, one starts with f (u) = u2 (one double pole at ∞ ∈ P1/Fq )and g(v) = v + γ/v (one simple pole at 0 and one simple pole at ∞). One obtains,for any power Q of q, a smooth projective surface XQ over Fq which is birational
to the affine surface YQ defined in affine coordinates (u, v, t) byYQ :f (u) − g(v) = tQ − t.

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)11The surface XQ is equipped with a surjective morphism πQ : XQ → P1/Fq , which
extends the natural projection YQ → A1/Fq given by (u, v, t) 7→ t. Writing Q = q a
for some integer a ≥ 1, the elliptic curve Eγ,a over K then arises as the genericfiber of πQ . The model (2.1) for Eγ,a /K indeed appears naturally by putting theequation for the generic fiber of πQ into Weierstrass form.
We wish to remark that one other case of the Pries–Ulmer construction is,for a good choice of parameters, “isogenous” to the case (2, 1 + 1). Given an1
1element γ ∈ F×
q , consider the two rational maps f, g : P/Fq → P/Fq given by
f (u) = u2 and g(v) = v 2 + γ/v 2 . In the terminology of [13], this input data has
type (2, 2 + 2) since f has a single double pole at ∞, and g has double poles at 0
and at ∞. For any power Q of q, let XQ be the smooth projective surface over Fq ,fibered over P1/Fq via πQ : XQ → P1/Fq , which is birational to the affine surface YQ
over Fq given by
YQ :f (u) − g(v) = tQ − tand such that the morphism πQ : XQ → P1/Fq extending (u, v, t) ∈ YQ 7→ t ∈ A1/Fq is0minimal. Writing Q = q a for some integer a ≥ 1, we denote by Eγ,a/K the generic0fiber of πQ . By construction, after clearing denominators, the curve Eγ,ais given2
in affine coordinates (u, v) ∈ A/K by
0
Eγ,a
:

u2 v 2 = v 4 + ℘a (t)v 2 + γ.

The change of coordinates (u, v) 7→ (x, y) = 2v(u + v), 4v uv + v 2 + ℘a (t) then0
provides the following affine Weierstrass model for Eγ,aover K:


0
Eγ,a
:
y 2 = x + ℘a (t) x2 − 4γ .
Our main observation is:0
Proposition 2.1. For any γ ∈ F×
q and any a ≥ 1, the elliptic curve Eγ,adefined over K by the affine Weierstrass model


0
Eγ,a
:
y 2 = x + ℘a (t) x2 − 4γ

is 2-isogenous to Eγ,a over K.
Proof. Using the formulae in [17, Chap. III, Ex. 4.5], we find that the map0
φ : (x, y) ∈ Eγ,a 7→ y 2 /x2 − ℘a (t), y 1 − γ/x2 ∈ Eγ,a,which is clearly defined over K, provides the desired 2-isogeny.



0Since they are K-isogenous, the elliptic curves Eγ,a and Eγ,ahave the same
conductor, they share the same L-function, and their Mordell-Weil ranks are equal.
Therefore, our main results about the elliptic curves Eγ,a (such as the explicit
expression for their L-function, and the fact that they have large Tate-Shafarevich
0groups) are also valid for the curves Eγ,a.
Furthermore, we know by [13, Coro. 3.1.4] that, for all γ ∈ F×
q and a ≥ 1,0
the elliptic curves Eγ,a and Eγ,asatisfy the BSD conjecture. The “geometric”
calculations of [13, §6.2] also show that the Mordell-Weil group Eγ,a (K) has rank 0.
We will recover these two facts via a different method, as corollaries of our explicit
expression for the L-function of Eγ,a (see Theorem 5.3 and Corollary 5.4).

12

RICHARD GRIFFON AND GUUS DE WIT◦
2.3. Bad reduction and invariants. We let Bγ,a ⊂ MK
denote the subset
corresponding to monic irreducible polynomials in Fq [t] which divide ℘a (t)2 − 4γ.
◦
Equivalently, Bγ,a is the set of places v ∈ MK
dividing the discriminant of theWeierstrass model (2.1) for Eγ,a .

Proposition 2.2. For any γ ∈ F×
q and any integer a ≥ 1, the elliptic curve Eγ,ahas good reduction outside Bγ,a ∪ {∞}. Moreover, its places of bad reduction are
as follows:
Place vReduction at v

v ∈ Bγ,aMultiplicative (fiber of type I1 )∞δv (Eγ,a ) νv (Eγ,a )Additive (fiber of type I∗4qa )1

1

4q a + 62

where, for any place v ∈ MK , we have denoted by δv (Eγ,a ) (resp. by νv (Eγ,a ))the order at v of the minimal discriminant divisor of Eγ,a (resp. of its conductor
divisor).
Proof. This follows from a routine application of Tate’s algorithm to computethe type of the fibers of bad reduction (as explained in [16, §IV.9]).

From the above proposition, we deduce explicit expressions for the exponential
differential height H(Eγ,a ) and the “numerical” conductor N (Eγ,a ) of Eγ,a (asdefined in §1.1):a1(2.3) H(Eγ,a ) = q 12 deg ∆min (Eγ,a ) = q (q +1)/2 ,anda

N (Eγ,a ) = q deg N (Eγ,a ) = q 2(q +1) .

Indeed, since p 6= 2 and γ 6= 0, the polynomial ℘a (t)2 − 4γ ∈ Fq [t] is squarefree,
and therefore we haveXXdeg v =
deg v = deg ℘a (t)2 − 4γ = 2q a .
v∈Bγ,av|℘a (t)2 −4γ

It is clear from (2.3) that we have H(Eγ,a ) = N (Eγ,a )1/4 .
The Tamagawa number τ (Eγ,a ) of Eγ,a is the product over all places v ∈ MKof the number of components in the special fiber of the Néron model of Eγ,a at v.
It is also readily calculated from the above proposition with the help of the tableon p. 365 of [16]. We find that the only contribution to τ (Eγ,a ) comes from thefiber above ∞: since the special fiber of Eγ,a at ∞ has Kodaira type I∗4qa , its groupof components is isomorphic to (Z/2Z)2 . We obtain that
(2.4)

τ (Eγ,a ) = 4.

Remark 2.3. For the computation of the L-function in section 4, it will be◦
useful to have minimal integral models of Eγ,a at places in MK
at our disposal. For◦a place v ∈ MK , we find by comparing the valuation at v of the discriminant ∆γ,aof the model (2.1) with the valuation at v of the minimal discriminant in the abovetable, that the Weierstrass model (2.1) is minimal and integral at v.
This computation of the invariants of Eγ,a yields the following:
Proposition 2.4. As γ varies in F×
q and as a ≥ 1 varies among integers, theelliptic curves Eγ,a are pairwise non K-isomorphic and pairwise non K-isogenous.

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)13Proof. Two elliptic curves over K are K-isomorphic if and only if they havethe same j-invariant. It is obvious from (2.2) that deg j(Eγ,a ) = 4a is strictlyincreasing when a ≥ 1 grows. Also apparent on (2.2) is the fact that, for a fixed a,the position of the poles of j(Eγ,a ) varies with γ ∈ F×
q . Hence the first assertion.
In a similar vein, it follows from (2.3) that deg N (Eγ,a ) increases with a ≥ 1.
Since two K-isogenous elliptic curves have equal conductor divisors, we conclude
that Eγ,a and Eγ,a0 cannot be K-isogenous for any a 6= a0 . For a given a ≥ 1 anddistinct γ, γ 0 ∈ F×
q , the curves Eγ,a and Eγ 0 ,a are not K-isogenous either, since thesets Bγ,a ∪ {∞} and Bγ 0 ,a ∪ {∞} of places where they have bad reduction differ(i.e. their conductor divisors have different supports).

2.4. Torsion subgroup. We conclude this section by elucidating the structure of the torsion subgroup of the Mordell-Weil group Eγ,a (K):
Theorem 2.5. For any γ ∈ F×
q and any integer a ≥ 1, let P0 ∈ Eγ,a (K)be the point withcoordinates
(0,
0)
in
the Weierstrass model (2.1). Then we haveEγ,a (K)tors = O, P0 .
We will show later on (Corollary 5.4) that the whole group Eγ,a (K) is torsion, sothat the above result provides a complete list of the K-rational points on Eγ,a .
Proof. We see on (2.2) that the j-invariant of Eγ,a is not a p-th power in K,
so that Proposition 7.1 in [19, Lect. I] ensures that Eγ,a (K)tors contains no pointwith p-th power order.
As was shown in the previous subsection, Eγ,a has additive reduction at ∞.
We may thus make use of Lemma 7.8 of [15], which asserts that the prime-to-p
part of Eγ,a (K)tors embeds into the group G∞ of components of the special fiber
at ∞ of the Néron model of Eγ,a . Since the reduction at ∞ is of type I∗4qa , thetable in §7.2 of [15] tells us that G∞ ' (Z/2Z)2 .
Combining the last two paragraphs, we deduce that Eγ,a (K)tors is isomorphic
to a subgroup of (Z/2Z)2 . In particular, the torsion subgroup of Eγ,a (K) consists
only of 2-torsion points and we infer that Eγ,a (K)tors = Eγ,a (K)[2].
The 2-torsion subgroup of Eγ,a (K) is readily computed: it consists of 4 points,
given in the homogenised version of (2.1) by
O = [0 : 1 : 0], P0 = [0 : 0 : 1],ihpP+ = ℘a (t) + ℘a (t)2 − 4γ : 0 : −2 ,hipP− = ℘a (t) − ℘a (t)2 − 4γ : 0 : −2 .

Since p 6= 2 and γ 6= 0, the polynomial ℘a (t)2 −4γ ∈ Fq [t] is squarefree, so that only
the first two p
points are K-rational (the latter two are rational over the quadratic
extension K( ℘a (t)2 − 4γ) of K). We deduce that Eγ,a (K)[2] = {O, P0 }.

3. Preliminaries on character sums
In the next section (see Theorem 4.1), we will compute an explicit expression for
the L-function of the curve Eγ,a introduced above. To carry out this computation,
we first need to set up some notation and conventions. These will be in force forthe rest of the paper.

14

RICHARD GRIFFON AND GUUS DE WIT

3.1. Gauss sums and Kloosterman sums. Let F be a finite field of odd
characteristic p. Any additive character ψ on F may and will be assumed to takevalues in the p-th cyclotomic field Q(ζp ). For any finite extension F0 /F, we denotethe trace map by TrF0 /F : F0 → F. If ψ is an additive character on F, then thecomposition ψ ◦ TrF0 /F is an additive character on F0 .
×

We denote by λ : F× → Q (or λF if confusion is likely to arise) the uniquenontrivial multiplicative character of order 2 on F× . We extend λ to the whole of F
by setting λ(0) := 0.
Definition 3.1. For any additive character ψ on F, define the quadratic Gauss
sum GF (ψ, λ) byX
GF (ψ, λ) := −λ(x)ψ(x).
x∈F

Note our choice of normalising GF (ψ, λ) by multiplying the sum by −1. By
construction, the sum GF (ψ, λ) is an algebraic integer in the cyclotomic field Q(ζp ).
Recall the following facts about Gauss sums:(Ga 1) For any nontrivial additive character ψ on F, any α ∈ F× and any finite(α)(α)
extension F0 /F, define ψF0 by x 7→ ψ ◦ TrF0 /F (αx). (This map ψF0 is a
nontrivial additive character on F0 .) Letting α0 := α|F| , one has
(α0 )

(α)

GF0 (ψF0 , λF0 ) = GF0 (ψF0 , λF0 ).
(Ga 2) For any nontrivial additive character ψ on F and any finite extension F0 /F,
one has0
GF0 (ψ ◦ TrF0 /F , λF0 ) = GF (ψ, λF )[F :F] .
(Ga 3) For any nontrivial additive character ψ on F, one has |GF (ψ, λ)| = |F|1/2 inany complex embedding of Q(ζp ).
(Ga 4) For any nontrivial additive character ψ on F, the quotient GF (ψ, λ)/|F|1/2
is a 4th root of unity (which may be explicitly determined).
These results are quite classical, and the reader is referred to [9, Chap. V, §2] for
their proofs.
Definition 3.2. For α ∈ F and an additive character ψ on F, define theKloosterman sum KlF (ψ; α) by
X 
α
.
(3.1)
KlF (ψ; α) := −
ψ x+
x
×
x∈F

Again, we point out our choice of normalising the sum by multiplying it by −1.
One can show that the Kloosterman sum KlF (ψ; α) is a totally real algebraic integerin Q(ζp ) i.e., KlF (ψ; α) ∈ Z[ζp + ζp−1 ]. We remind the reader of the following facts
about Kloosterman sums:(Kl 1) For any nontrivial additive character ψ on F and any α ∈ F× , one has theidentity (sometimes called Salié’s formula):X
KlF (ψ; α) = −λ(y 2 − 4α)ψ(y).
y∈F(Kl 2) For any nontrivial additive character ψ on F, any α ∈ F and any finite
extension F0 /F, letting α0 = α|F| , one has
KlF0 (ψ ◦ TrF0 /F ; α0 ) = KlF0 (ψ ◦ TrF0 /F ; α).

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)15(Kl 3) For any nontrivial additive character ψ on F and any α ∈ F× , there is aunique pair {klF (ψ; α), kl0F (ψ; α)} of algebraic integers, whose product is |F|and such that, for any finite extension F0 /F, one has0

0

KlF0 (ψ ◦ TrF0 /F ; α) = klF (ψ; α)[F :F] + kl0F (ψ; α)[F :F] .
(Kl 4) For any nontrivial additive character ψ on F and any α ∈ F× , one has|klF (ψ; α)| = |kl0F (ψ; α)| = |F|1/2 in any complex embedding of Q. In particular, one has |KlF (ψ; α)| ≤ 2|F|1/2 in any complex embedding of Q(ζp ).
These results are classical: see [9, Chap. V, §5] for proofs thereof. For convenience,
we also state here a fact that will only be proved later on (see Lemma 5.1 andRemark 6.2):(Kl 5) For any nontrivial additive character ψ on F and any α ∈ F× , KlF (ψ; α) is ap-adic unit in Q(ζp ). In particular, one has 0 < |KlF (ψ; α)| < 2|F|1/2 in any
complex embedding of Q(ζp ).
3.2. Places of degree dividing a. Let Fq be a finite field of odd characteristic, and let K := Fq (t) denote the rational function field over Fq .
Definition 3.3. For any integer a ≥ 1, let Pq (a) denote the set of placesv ∈ MK such that v 6= 0, ∞ and deg v | a. That is to say, Pq (a) is the set of closed
points of the multiplicative group Gm/Fq = P1Fq \ {0, ∞} whose degree divides a.
◦and monic irreducible elements
In the usual identification between places in MK
of Fq [t], the set Pq (a) corresponds to the set of monic irreducible polynomials
B ∈ Fq [t] such that B 6= t and deg B | a.
The latter interpretation allows for an easy estimation of the cardinality |Pq (a)|.
Indeed the Prime Number Theorem for Fq [t] states that, for any integer n ≥ 1,
the number πq (n) of monic irreducible polynomials in Fq [t] of degree n satisfies:
q n /n−q n/2 ≤ πq (n) ≤Pq n /n (see, for instance, [14, Thm. 2.2] and its proof). Notingthat |Pq (a)| = −1 + n|a πq (n), we deduce the existence of constants cq , c0q > 0,
depending at most on q, such thatqaqa
≤ |Pq (a)| ≤ cq · .
(3.2)∀a ≥ 1,c0q ·aa3.3. The sums g(v) and Klγ (v). Let Fq be a finite field of odd characteristic p, and endow Fq with a nontrivial additive character ψq taking values in thep-th cyclotomic field Q(ζp ). For any finite extension F/Fq we “lift” ψq to a nontrivial character ψF on F by composing ψq with the relative trace map; i.e., welet ψF := ψq ◦ TrF/Fq .
Definition 3.4. Let γ ∈ F×
q and a ≥ 1 be an integer. For any place v ∈ Pq (a),we denote by Fv the residue field of K at v, and by dv := [Fv : Fq ] the degree of v.
Viewing v as the Gal(Fq /Fq )-orbit of an Fq -rational point of Gm/Fq , we may pick×

an element βv ∈ F×v ⊂ Fq representing that orbit v. (The choices of representatives
2
dv −1
×
(β )
of v in Fq are then βv , βvq , βvq , . . . , βvq.) The map ψFvv : x 7→ ψq ◦TrFv /Fq (βv x)
defines a nontrivial additive character on Fv .
(β )• We denote the Gauss sum GFv (ψFvv , λFv ) by g(v). A repeated application of (Ga 1) shows that the definition of g(v) makes sense, in that the(β )value of the sum GFv (ψFvv , λFv ) does not depend on a particular choiceof representative βv for the orbit v.

16

RICHARD GRIFFON AND GUUS DE WIT(β )• We denote the Kloosterman sum KlFv (ψFvv ; γ) by Klγ (v), and we let{klγ (v), kl0γ (v)} be the pair of algebraic integers attached to the Kloosterman sum Klγ (v) as in (Kl 3). Again, these definitions make sense: repeated(β )applications of (Kl 2) imply that the value of KlFv (ψFvv ; γ), and hence thepair of algebraic integers attached to it by (Kl 3), do not depend on the
choice of βv in v.
4. The L-function of Eγ,aThe main objective of this section is to provide an explicit expression for the
L-function of the elliptic curve Eγ,a . The resulting formula will prove instrumental
for the proof of our main theorem. In the notation set up in the previous section,
the result is as follows:
Theorem 4.1. Let Fq be a finite field of odd characteristic. For any γ ∈ F×q
and any a ≥ 1, consider the elliptic curve Eγ,a over K = Fq (t) defined by (2.1).
The L-function L(Eγ,a , T ) ∈ Z[T ] of Eγ,a is given by:Y
1 − g(v)klγ (v)T deg v 1 − g(v)kl0γ (v)T deg v ,(4.1)
L(Eγ,a , T ) =v∈Pq (a)where Pq (a) denotes the set of places defined in §3.2, and g(v), klγ (v), kl0γ (v) denotethe algebraic integers attached to any v ∈ Pq (a) in §3.3.
The proof of this theorem occupies the rest of the section: the next subsection proves a useful identity between character sums, and the following subsection
contains the computation leading to Theorem 4.1.
Remark 4.2. (1) Given the definition of {klγ (v), kl0γ (v)} and formula (Kl 3),
an equivalent way of formulating (4.1) is:
Y

(4.2)
L(Eγ,a , T ) =1 − g(v)Klγ (v)T deg v + g(v)2 q deg v T 2 deg v .
v∈Pq (a)(2) To define the sums g(v) and Klγ (v) for a place v ∈ Pq (a), we started by fixing
a nontrivial additive character ψq on Fq . We remark that the expression for
the L-function of Eγ,a obtained in Theorem 4.1 is independent of this choice.
Indeed, a different choice of ψq has the sole effect of permuting the factorsin (4.1), for the L-function
P L(Eγ,a , T ) has integral coefficients.
aa(3) By definition, we havev∈Pq (a) deg v = |Gm (Fq )| = q − 1. Hence, as apolynomial in T , the L-function of Eγ,a has degreeb(Eγ,a ) = deg L(Eγ,a , T ) = 2(q a − 1).
This is compatible with the Grothendieck–Ogg–Shafarevich formula and thevalue of deg N (Eγ,a ) found in (2.3). Note that b(Eγ,a ) is even.
4.1. An identity between character sums. For this subsection, we let F
be a finite field of odd characteristic, which we equip with an additive character ψ.
Denote by λF = λ : F× → {±1} the unique quadratic character on F× , extendedby λ(0) := 0. For any γ ∈ F× , define the double character sum
XX
S(F, ψ, γ) :=λ(x3 + zx2 + γx)ψ(z).
z∈F x∈F

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)17Note that the terms with x = 0 do not contribute to the sum since λ(0) = 0. We
may therefore disregard these terms and use the multiplicativity of λ to get:
)
(

X
X 
x2 + γ2
S(F, ψ, γ) =
λ(x )
λ z+
ψ(z) .
xz∈F

x6=0

For a given x ∈ F× , we put u = z + (x2 + γ)/x in the sum displayed between
brackets: using the additivity of ψ and the definition of GF (ψ, λ), we obtain that



X 
γX
γ
x2 + γ· GF (ψ, λ).
ψ(z) = ψ −x −λ(u)ψ(u) = −ψ −x −
λ z+
x
x
x
u∈F

z∈F

Multiplying this identity by λ(x ), and summing over all x ∈ F× then yields that
Xγλ(x)2 ψ −x −S(F, ψ, γ) = −GF (ψ, λ) ·
.
x
×
2

x∈F

2

Noting that λ(x) = 1 for all x 6= 0 (because λ has order 2) and reindexing the sum
by setting y = −x entails thatX γS(F, ψ, γ) = −GF (ψ, λ) ·ψ y+
= GF (ψ, λ) · KlF (ψ; γ).
y×y∈FWe have thus proved:
Lemma 4.3. In the above setting, one has S(F, ψ, γ) = GF (ψ, λ) · KlF (ψ; γ),
where the Gauss and Kloosterman sums are as defined in §3.1.
Additionally, it is immediate to check that all three sums in the above identity
vanish when ψ is trivial.
4.2. Proof of Theorem 4.1. Fix a parameter γ ∈ F×
q and an integer a ≥ 1
as in the statement of the theorem. Our starting point for the computation isexpression (4.3) below for the L-function of Eγ,a .
◦For any τ ∈ Fq , we denote by vτ ∈ MK
the corresponding place of K; wepick a minimal integral Weierstrass (affine) model of Eγ,a /K at vτ of the form
y 2 = fτ (x, t) where fτ (x, t) is a monic cubic polynomial in x, with coefficientsin Fq [t]. Recall that λFqn : Fqn → {±1} denotes the unique character of exact
order 2 on F×
q n , extended by λFqn (0) := 0 to the whole of Fq n . With this notation,
one has the equality of formal power series:


∞
X
X X
 Tn

(4.3)
log L(Eγ,a , T ) = −λFqn fτ (x, τ ) 
.
n
n=1
n
n
τ ∈Fq

x∈Fq

This expression can be derived from the definition (1.1)-(1.2) of the L-function, justas in [3] (see Lemma 4.6 and the following paragraph there) or [4, §4]. Here, wehave implicitly used the fact that Eγ,a has additive reduction at the place ∞ (seeProposition 2.2) to ignore the local term corresponding to this place. At a place ofadditive reduction, the local Euler factor of L(Eγ,a , T ) in (1.2) is indeed trivial.
We next aim at giving a more explicit expression of the inner double sumsin (4.3). As was pointed out in Remark 2.3, one can choose, for any τ ∈ Fq ,fτ (x, t) = x3 + ℘a (t)x2 + γx.

18

RICHARD GRIFFON AND GUUS DE WIT

For any integer n ≥ 1, the inner double sum in (4.3) can thus be rewritten asXXλFqn fτ (x, τ )τ ∈Fqn x∈Fqn

=

X

X 

τ ∈ Fqn : ℘a (τ ) = z· λFqn x3 + zx2 + γx .

z∈Fqn x∈Fqn

Moreover, Lemma 4.5 in [3] asserts that, for any z ∈ Fqn ,Xψq ◦ TrFqn /Fq (βx).
τ ∈ Fqn : ℘a (τ ) = z =β∈Fqn ∩Fqa

For all β ∈ Fqn , the map z 7→ ψq ◦ TrFqn /Fq (βz) is an additive character on Fqn ,(β)

which we denote by ψqn . Hence, for any integer n ≥ 1, we haveXXλFqn fτ (x) =τ ∈Fqn x∈Fqn

X(β)

S(Fqn , ψqn , γ),β∈Fqn ∩Fqa
(β)

X

where S(Fqn , ψqn , γ) =X

(β)λFqn (x3 + zx2 + γx)ψqn (z).

z∈Fqn x∈Fqn
(β)

(β)

(β)

Lemma 4.3 above now yields that S(Fqn , ψqn , γ) = GFqn (ψqn , λ) · KlFqn (ψqn ; γ)
for all β ∈ Fqn , where the Gauss and Kloosterman sums are as in §3.1. Combining
the above equalities, we obtain that


∞
X
X
Tn
(β)
(β)

GFqn (ψqn , λ) · KlFqn (ψqn ; γ)
(4.4)− log L(Eγ,a , T ) =
.
n
n=1
n
a
β∈Fq ∩Fq(β)(β)

When β = 0, the additive character ψqn is trivial, so both GFqn (ψqn , λ) and(β)

KlFqn (ψqn ; γ) vanish. For any β ∈ (Fqa ∩ Fqn ) \ {0}, we denote the place of K◦
(equivalently, vβ is the Gal(Fq /Fq )-orbit of β). The
containing β by vβ ∈ MK◦
place vβ is not 0 ∈ MK and has degree deg vβ = [Fq (β) : Fq ], which divides both aand n. In particular, vβ belongs to Pq (a).
(β)

Lemma 4.4. In the notation of §3.3, one has GFqn (ψqn , λ) = g(vβ )n/ deg vβ and(β)

KlFqn (ψqn ; γ) = klγ (vβ )n/ deg vβ + kl0γ (vβ )n/ deg vβ .
Proof. For brevity, we write d = deg vβ (recall that d divides n). By multiplicativity of the trace in towers of extensions and because β ∈ Fqd , we have(β)(β)ψFqn = ψF d ◦ TrFqn /Fqd . Moreover, λ = λFqn coincides with λFqd ◦ NFqn /Fqd .
q
Hence, the Hasse–Davenport relation for Gauss sums (Ga 2) implies that
(β)

(β)

(β)

GFqn (ψqn , λ) = GFqn (ψqd ◦ TrFqn /Fqd , λFqd ◦ NFqn /Fqd ) = GFqd (ψqd , λFqd )[Fqn :Fqd ] .
(β)

Since, by definition, we have g(vβ ) = GFqd (ψqd , λFqd ), the first identity is proved.
The second identity is proved in a similar fashion, using the Hasse–Davenport
relation for Kloosterman sums (Kl 3).
ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)19Plugging the identities of Lemma 4.4 into (4.4) and exchanging the order ofsummation yields that− log L(Eγ,a , T )X
X n/ deg vβ
n/ deg vβ  T ng(vβ )klγ (vβ )+ g(vβ )kl0γ (vβ )=.
nβ∈Fqa \{0}n≥1deg vβ |nUpon reindexing the second sum (by setting m = n/ deg vβ ), we obtain that( ∞X
X (g(vβ )klγ (vβ )T deg vβ )m1− log L(Eγ,a , T ) =deg vβ m=1
m
×
β∈Fqa

+

∞
X
(g(vβ )kl0γ (vβ )T deg vβ )m
m=1

)

m

Whence we have, by the formal power series identity − log(1 − u) =.

um
m=1 m ,

P∞

log L(Eγ,a , T )X=1
log (1 − g(vβ )klγ (vβ )T deg vβ )(1 − g(vβ )kl0γ (vβ )T deg vβ ) .
deg vβ×

β∈Fqa

Finally we group the indices β ∈ F×
q a corresponding to the same v ∈ Pq (a), and get:log L(Eγ,a , T )
X X
=


1
log (1 − g(v)klγ (v)T deg v )(1 − g(v)kl0γ (v)T deg v )deg vv∈Pq (a) β∈v
X=log (1 − g(v)klγ (v)T deg v )(1 − g(v)kl0γ (v)T deg v ) .
v∈Pq (a)To conclude the proof of Theorem 4.1, there only remains to exponentiate thisidentity.

5. Non-vanishing of L(Eγ,a , T ) at the central point and consequences
As before, let Fq be a finite field of odd characteristic p and K := Fq (t). For
any parameter γ ∈ F×
q and any integer a ≥ 1, we consider the elliptic curve Eγ,adefined over K by (2.1). In this section, we describe the behaviour of the L-function
z 7→ L(Eγ,a , z) around z = 1 (see Theorem 5.3). We first gather some information
about its p-adic slope sequence from Theorem 4.1 and results about Gauss andKloosterman sums. We then derive an important corollary of this analytic study,
namely the BSD conjecture for the elliptic curves Eγ,a (see Corollary 5.4).
5.1. p-adic slopes of L(Eγ,a , T ). We choose, once and for all, a prime ideal P
×of Q above p. We denote by ordP : Q → Q the corresponding discrete valuation,
so normalised that ordP (q) = 1. The goal of this subsection is to compute thep-adic slope sequence of L(Eγ,a , T ) explicitly (see §1.3).
We first prove a probably well-known lemma for which we could not find a
proof in the literature:

20

RICHARD GRIFFON AND GUUS DE WIT

Lemma 5.1. Let F be a finite extension of Fq , and ψ be a nontrivial additive
character on F. For any α ∈ F× , consider the Kloosterman sum KlF (ψ; α) and thepair {klF (ψ; α), kl0F (ψ; α)} of algebraic integers associated to it by (Kl 3). Then wehave{ordP klF (ψ; α), ordP kl0F (ψ; α)} = 0, [F : Fq ] .
Equivalently, one has ordP KlF (ψ; α) = 0.
The last statement is equivalent to the first assertion of (Kl 5). In the same setting,the P-adic valuation of the Gauss sum GF (ψ, λ) is easily determined: (Ga 4) showsthat the sums GF (ψ, λ) has the same P-adic valuation as |F|1/2 , so that
(5.1)

ordP GF (ψ, λ) =[F : Fq ].
2

Proof. By construction, KlF (ψ; α) is an element of Q(ζp ). The unique prime
ideal of Q(ζp ) above p is the principal ideal I = (ζp − 1), so that p · Z[ζp ] = I p−1 .
Since ψ(y) is a power of ζp for all y ∈ F, we have ψ(y) ≡ 1 mod I and we find thatX
1 ≡ |F| − 1 ≡ −1 mod I,−KlF (ψ; α) ≡x∈F×because |F| is a power of p. In particular, KlF (ψ; α) 6≡ 0 mod p in Q, and we indeed
have ordP KlF (ψ; α) = 0.
On the other hand, (Kl 3) implies that the algebraic integers klF (ψ; α) and
kl0F (ψ; α) satisfy:klF (ψ; α) · kl0F (ψ; α) = |F|andklF (ψ; α) + kl0F (ψ; α) = KlF (ψ; α).

The pair {v, v 0 } formed by their P-adic valuations thus satisfies the following constraints: v, v 0 ≥ 0, v + v 0 = ordP |F| = [F : Fq ], and min{v, v 0 } ≤ ordP KlF (ψ; α) = 0.
The only pair {v, v 0 } fulfilling these requirements is {0, [F : Fq ]}.
Hence the result.

We can now state and prove the following:
Theorem 5.2. Let γ ∈ F×
q and a ≥ 1. Consider the elliptic curve Eγ,a /K
defined by (2.1), and write b = deg L(Eγ,a , T ). Recall from Remark 4.2(3) thatb = 2(q a − 1) is even. The L-function L(Eγ,a , T ) has p-adic slope sequence
λ1 =

1
1
1
3
3
3
, λ2 = , . . . , λb/2 = , λb/2+1 = , λb/2+2 = , . . . , λb = .
2
2
2
2
2
2

Proof. By definition (see §1.3), the p-adic slope sequence of L(Eγ,a , T ) isthe (suitably indexed) multiset of P-adic valuations of the inverses of zeros of the
L-function z 7→ L(Eγ,a , z).
Upon staring at the expression for L(Eγ,a , T ) obtained in Theorem 4.1, one immediately sees that an algebraic number z ∈ Q is the inverse of a zero of L(Eγ,a , T )if and only if there exists a place v ∈ Pq (a) with z deg v ∈ {g(v)klγ (v), g(v)kl0γ (v)}.
In particular, if z is the inverse of a zero of L(Eγ,a , T ), there exists v ∈ Pq (a) suchthat ordP (z deg v ) equals one of ordP (g(v)klγ (v)) or ordP (g(v)kl0γ (v)).
As can be seen from applying formula (5.1) in the case where GF (ψ, λ) = g(v),we have ordP g(v) = deg v/2. The previous lemma applied to KlF (ψ; α) = Klγ (v)ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)21further yields that ordP klγ (v), ordP kl0γ (v) = {0, deg v}. We therefore haveordP z =ordP (z deg v )
deg v
(
) 

0
ordP g(v) + ordP klγ (v) ordP g(v) + ordP klγ (v)
1 3
∈
=
,
,
.
deg v
deg v
2 2

As was recalled in §1.3, the p-adic slope sequence {λk }bk=1 of L(Eγ,a , T ) admits
the following symmetry: for a given s ∈ [0, 2], there are as many indices k suchthat λk = s as there are indices k 0 such that λk0 = 2 − s. From this observationand from the above display, we immediately conclude that, among the b slopes of
the L-function L(Eγ,a , T ), half of them equal 1/2 and the other half equal 3/2.
The result follows by choosing a suitable numbering of these slopes.

5.2. Non-vanishing at the central point. The following is a direct consequence of Theorem 5.2:
Theorem 5.3. For any γ ∈ F×
q and any a ≥ 1, the L-function L(Eγ,a , T ) does
not vanish at T = q −1 . In other words, one has ordT =q−1 L(Eγ,a , T ) = 0.
Proof. Given the expression for L(Eγ,a , T ) of Theorem 4.1, it suffices to show
that, for any v ∈ Pq (a), the two factors
1 − g(v)klγ (v)q − deg v

and1 − g(v)kl0γ (v)q − deg vof L(Eγ,a , q −1 ) are nonzero. As we have seen in the previous subsection, for allplaces v ∈ Pq (a), we have
deg v 3 deg v
0,.
ordP (g(v)klγ (v)), ordP (g(v)klγ (v)) =22
Since ordP (q deg v ) = deg v, neither of g(v)klγ (v) or g(v)kl0γ (v) can equal q deg v , andwe are done.
Remark 6.2 outlines an alternative proof of Theorem 5.3: there, we obtain the
non-vanishing by relying on the “angular distribution” of the Gauss and Kloosterman sums.
5.3. The BSD conjecture for Eγ,a . Given the non-vanishing of L(Eγ,a , T )at T = q −1 , we deduce from the BSD result (see Theorem 1.1) that the BSD
conjecture holds for all the elliptic curves Eγ,a .
Corollary 5.4. For all γ ∈ F×
q and all integers a ≥ 1, one has:
(1) The Mordell–Weil group Eγ,a (K) consists of the two points O and P0 = (0, 0).
(2) The Tate–Shafarevich group X(Eγ,a ) is finite.
(3) The following identity holds:(5.2)

L(Eγ,a , q −1 ) =|X(Eγ,a )|.
−1
q · H(Eγ,a )That the elliptic curves Eγ,a satisfy the BSD conjecture is not new: as was
already mentioned in §2.2, it was first proved by Pries and Ulmer in [13], where
the proof relies on the geometry of the minimal regular model of Eγ,a . Our proof
appears to be largely independent of theirs. Pries and Ulmer have further shown(also by a geometric argument) that Eγ,a (K) has rank 0 (see §6.2 in [13]).

22

RICHARD GRIFFON AND GUUS DE WIT

Proof. By Theorem 1.1(1), the Mordell–Weil group Eγ,a (K) is finite, hencecoincides with its torsion subgroup. Theorem 2.5 then entails the first assertion.
The second assertion is copied verbatim from the second item of Theorem 1.1. As
for identity (5.2), it suffices to plug the values |Eγ,a (K)| = 2 and τ (Eγ,a ) = 4 (bothcalculated in §2.3) into the BSD formula (1.3).

6. Distribution of the Kloosterman sums Klγ (v)In this section, we work in the following setting: we fix a finite field Fq of odd
characteristic p, endowed with a nontrivial additive character ψq ; we also choose,
once and for all, an embedding ι : Q ,→ C. For any γ ∈ F×
q and any a ≥ 1, we
consider the elliptic curves Eγ,a defined by (2.1).
In the next section, we will describe the asymptotic behaviour of L(Eγ,a , q −1 )as a → ∞. Doing so will require some knowledge about the “angular distribution”
in the complex plane of (the images under ι of) the algebraic integers klγ (v), kl0γ (v),for v ∈ Pq (a). We therefore lay out, in this section, the relevant facts about that
distribution.
6.1. Angles of Gauss and Kloosterman sums. For a place v ∈ Pq (a), theWeil bounds (Ga 3) and (Kl 4) for Gauss and Kloosterman sums reveal that theimages under ι of the algebraic integers g(v), klγ (v) and kl0γ (v) lie on the complex
circle {z ∈ C : |z| = q deg v/2 }. We may therefore introduce the following angles:
Definition 6.1. For a place v ∈ Pq (a),• We know from (Ga 4) that the algebraic number g(v)/q deg v/2 is a 4th rootof unity. Thus, there is a unique ε(v) ∈ {0, π/2, π, 3π/2} such that
ι(g(v)) = q deg v/2 eiε(v) .
• By (Kl 4), the complex number ι(Klγ (v)) is real with |ι(Klγ (v))| ≤ 2q deg v/2 .
Hence there exists a unique angle θγ (v) ∈ [0, π] such that
ι(Klγ (v)) = 2q deg v/2 cos θγ (v).
For all v ∈ Pq (a), we remark that one hasnoι(klγ (v)), ι(kl0γ (v)) = q deg v/2 eiθγ (v) , q deg v/2 e−iθγ (v) .
With this new notation, the expression for L(Eγ,a , T ) obtained in Theorem 4.1
becomes:
(6.1)


Y 
L(Eγ,a , T ) =1 − ei(ε(v)+θγ (v)) (qT )deg v 1 − ei(ε(v)−θγ (v)) (qT )deg v .
v∈Pq (a)Note that, whilst the angles θγ (v) individually depend on the choice of ι, the set{θγ (v)}v∈Pq (a) does not: a different choice of ι only permutes the various θγ (v),since L(Eγ,a , T ) has integral coefficients.
Evaluating both sides of the above equality at T = q −1 yields that

Y 
(6.2)
L(Eγ,a , q −1 ) =1 + e2iε(v) − 2eiε(v) cos θγ (v) .
v∈Pq (a)ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)23Remark 6.2. (1) For any v ∈ Pq (a), Lemma 5.1 implies that the angle θγ (v)does not lie in {0, π/2, π}. Indeed, were θγ (v) to hit one of 0, π/2 or π, thenthe sum Klγ (v) would equal +2q deg v/2 , 0 or −2q deg v/2 , respectively. This isincompatible with Lemma 5.1 since Klγ (v) would then not be a p-adic unit.
More generally, with the same construction as in Definition 6.1, one canassociate an angle θF,ψ,α ∈ [0, π] to any of the Kloosterman sums KlF (ψ; α)introduced in §3.1. A straightforward adaptation of the above argument showsthat θF,ψ,α ∈/ {0, π/2, π}, which directly implies the second assertion in (Kl 5).
(2) The previous item actually provides a second proof of the non-vanishing of
the L-function L(Eγ,a , T ) at T = q −1 , as follows. It is clear that, for a placev ∈ Pq (a), the factor indexed by v in the product (6.1) vanishes at T = q −1
if and only if ε(v) ≡ ±θγ (v) mod 2π. By Definition 6.1 above, we know thatε(v) ∈ {0, π/2, 3π/2, 2π}. What we have shown in item (1) proves that the v-th
factor in (6.1) does not vanish at T = q −1 . Hence L(Eγ,a , q −1 ) is nonzero.
6.2. Angular distribution of the sums Klγ (v). Recall that the Sato–Tatemeasure µ∞ on [0, π] is defined by dµ∞ := π2 sin2 θ dθ. We denote by µa thediscrete probability measure on [0, π] supported on the set {θγ (v)}v∈Pq (a) of angles
introduced in the previous subsection. In other words, we put
X
1
µa :=
δ{θγ (v)},|Pq (a)|v∈Pq (a)where δ{x} denotes the Dirac delta measure at x ∈ [0, π]. Note that µa depends
neither on the choice of a nontrivial additive character ψq on Fq nor on that of anembedding ι : Q ,→ C. The measure µa does depend on Fq and γ though, but wechose not to reflect this in the notation for brevity.
In a previous work, the first-named author has proved that, as a → ∞, thesequence of measures (µa )a≥1 converges weak-∗ to µ∞ in a quantitative way. Morespecifically, Theorem 6.6 in [3] states that:
Theorem 6.3. In the above setting, given any continuously differentiable function g : [0, π] → C and any γ ∈ F×
q , as a ≥ 1 tends to +∞, one has
Z
Z
Z π
a1/2
(6.3)
g dµa −
g dµ∞ q a/4 ·|g 0 (t)| dt,
q
[0,π]
[0,π]
0
where the implicit constant is effective and depends at most on q.
The reader is referred to section 6 of [3] for a detailed proof of that statement.
Note that µa is the same measure as the one denoted by νa there.
6.3. ‘Small’ angles of Kloosterman sums. As we have seen in Remark 6.2,
the fact that L(Eγ,a , T ) does not vanish at T = q −1 is equivalent to the statementthat none of the angles θγ (v) lies in {0, π/2, π}. In this subsection, we obtain aquantitative version of the fact that the set {θγ (v)}v∈Pq (a) “avoids” {0, π/2, π}.
Theorem 6.4. Let p be an odd prime number. There exists a constant σp > 0,
depending at most on p, such that the following holds. Let Fq be a finite fieldof characteristic p endowed with a nontrivial additive character ψq , and fix anembedding ι : Q ,→ C. For any parameter γ ∈ F×
q and any integer a ≥ 1, one has{θγ (v)}v∈Pq (a) ⊂ εa , π2 − εa ∪ π2 + εa , π − εa where εa = (q a )−σp .

24

RICHARD GRIFFON AND GUUS DE WIT In other words, the measure µa is supported on εa , π2 − εa ∪ π2 + εa , π − εa .
Corollary 5.5 in [3] already shows that θγ (v) ∈ εa , π − εa ] for all v ∈ Pq (a); so we
actually only need to prove that |θγ (v) − π/2| ≥ εa . However, for convenience, wegive a complete proof of Theorem 6.4.
The main tool to prove this result is the following version of Liouville’s inequality, which is taken from the introduction of [11]. Let P ∈ Z[X] be a polynomial of
degree N . For any algebraic number κ ∈ Q, let ht(κ) denote its absolute logarithmic
Weil height. If P (κ) 6= 0, thenlog |P (κ)|≥ − log kP k1 − N · ht(κ),[Q(κ) : Q]

(6.4)

in any complex embedding of Q(κ). Here we have denoted by kP k1 the sum of
the absolute values of the coefficients of P . The reader is referred to [11] for thisstatement and its proof.
Proof of Theorem 6.4. In the setting of the theorem, we claim that there
exists a constant σp > 0 such that, for all v ∈ Pq (a), one hasmin{θγ (v), |θγ (v) − π/2|, π − θγ (v)} ≥ (q deg v )−σp .
Since deg v divides a for all v ∈ Pq (a), the theorem clearly follows from this claim.
In order to prove the claim, consider the algebraic number κ := klγ (v)q − deg v/2 .
Up to replacing klγ (v) by kl0γ (v), we can assume that ι(κ) = eiθγ (v) . It is relatively
easy to show that [Q(κ) : Q] ≤ 2(p − 1) and that ht(κ) ≤ log(q deg v/2 ), see [3, Lem.
5.3] for details. Let us now apply Liouville’s inequality to κ with the following threepolynomials: P1 = X − 1, P2 = X + 1 and P3 = X 4 − 1.
By (Kl 5) or Remark 6.2, we know that κ ∈/ {1, i, −1}, so that Pj (κ) 6= 0 forj ∈ {1, 2, 3}. Combined with our estimates of the degree and of the height of κ,
Liouville’s inequality (6.4) yields that, for j ∈ {1, 2, 3},
log |ι(Pj (κ))| ≥ −[Q(κ) : Q](log 2 + 4ht(κ)) ≥ −2(p − 1)(log 2 + 2 log q deg v )≥ −6(p − 1) log q deg v .

(6.5)

Besides, a quick analysis shows that, for any θ ∈ [0, π], one has|P1 (eiθ )| = |eiθ − 1| ≤ |θ| = θ,
|P2 (eiθ )| = |eiθ + 1| = |ei(θ−π) − 1| ≤ |θ − π| = π − θ,and|P3 (eiθ )| = |e4iθ − 1| = |eiθ − i| · |eiθ + i|·

Applying these inequalities to θ = θγ (v) and using (6.5), we directly obtain thatθγ (v) ≥ |P1 (κ)| ≥ (q deg v )−6(p−1) ,and that

π − θγ (v) ≥ |P2 (κ)| ≥ (q deg v )−6(p−1) .

By further noting that q 2 deg v ≥ 4, we also get that
|π/2 − θγ (v)| ≥ |P3 (κ)|/4 ≥ (q deg v )−6(p−1) /4 ≥ (q deg v )−(6p−4) .
The claim now follows immediately, with σp = 6p − 4 > 0 being a suitable choiceof constant.
ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)25

7. Estimates of the central value L(Eγ,a , q −1 )The goal of this section is to prove a precise asymptotic estimate on the central
value L(Eγ,a , q −1 ) as a → ∞. This will provide the crucial input for our bounds
on the size of X(Eγ,a ) in the next section.
The result is as follows:
Theorem 7.1. Let Fq be a finite field of odd characteristic, and K = Fq (t).
There are positive constants c1 , c2 (depending at most on q) such that: for any−1γ ∈ F×) of the L-function of theq and any a ≥ 1, the central value L(Eγ,a , q
elliptic curve Eγ,a satisfies:
−

(7.1)

log |L(Eγ,a , q −1 )|
c2
c1
≤
≤ .
a
log H(Eγ,a )aThe proof of this theorem will be given in §7.2 after proving an intermediate
result in the following subsection. The upper bound in (7.1) provides a slight
improvement on the generic upper bound on the central value: for any non-isotrivial
elliptic curve E over K with L(E, q −1 ) 6= 0, it is known that
log log log H(E)log L(E, q −1 )≤ c3 ·log H(E)
log log H(E)(as H(E) → ∞),for some constant c3 > 0. This can be proved by using the fact that the zerosof L(E, T ) become uniformly equidistributed on the circle {z ∈ C : |z| = q −1 } as
H(E) → ∞ (see [6, Thm. 7.5]). For easier comparison, note that log log H(Eγ,a )has the same order of magnitude as a, as was shown in (2.3).
The lower bound in (7.1) is, on the other hand, much stronger than thegeneric lower bound on the central value. For a non-isotrivial elliptic curve Ewith L(E, q −1 ) 6= 0, the latter only yields that−1 +c4
log L(E, q −1 )≤log H(E)log H(E)(as H(E) → ∞),for some c4 < 0. Via the BSD formula, this translates into a lower bound on theorder of X(E) which is weaker than the one we require to prove Theorem C(3).
7.1. Convergence of a certain sequence. Let a ∈ Z≥1 and γ ∈ F×
q . Wekeep the notation introduced in section 6. Consider the non-negative function
W : [0, π] → R defined by(− log sin2 θ · cos2 θ
if θ ∈ [0, π] \ {0, π/2, π},
W (θ) :=
0if θ ∈ {0, π/2, π}.
The function θ 7→ W (θ) is continuously differentiable on (0, π/2) ∪ (π/2, π) and, forall θ ∈ [0, π], it satisfies W (π − θ) = W (θ). Moreover, a routine check shows that
W is integrable on [0, π] for the Lebesgue measure, as well as for the Sato–Tate measure µ∞ . By Remark 6.2, the function W is also integrable for themeasure µa intro
R
duced in §6.2: it thus makes sense to consider the sequence IW = [0,π] W dµa.
a≥1

Even though we know, by Theorem 6.3, that (µa )a≥1 converges weak-∗ to µ∞ , wecannot directly conclude that the sequence IW converges because W is not continuous on [0, π].
Nonetheless, the goal of this subsection is to show the following:

26

RICHARD GRIFFON AND GUUS DE WIT

R
Proposition 7.2. In the above setting, the sequence
W dµa
con[0,π]a≥1Rverges to [0,π] W dµ∞ . More precisely, one hasZZa3/2
W dµa −
W dµ∞ q a/4(as a → ∞),
q
[0,π]
[0,π]
where the implicit constant is effective and depends at most on q.
Even though the exact
R value of the limit is of little importance to us, one can actually compute that [0,π] W dµ∞ = log(16). Using the definition of the measure µa ,the above result can thus be rewritten in a more explicit form:X
1
log sin2 θγ (v) · cos2 θγ (v)|Pq (a)|v∈Pq (a) 3/2 a
(as a → ∞).
= − log(16) + Oq
q a/4
Proof. We pick, once and for all, a nondecreasing continuously differentiable
function β0 : [0, 1] → [0, 1] such that β0 (x) = 0 for all x ∈ [0, 1/3], and β0 (x) = 1
for all x ∈ [2/3, 1]. For a small enough ε > 0, we define a “smoothing” function
βε : [0, π] → [0, 1] as follows: define βε on [0, π/2] byif θ ∈ [0, ε],β0 (θ/ε)βε (θ) := 1
if θ ∈ [ε, π/2 − ε],
β0 ((π/2 − θ)/ε) if θ ∈ [π/2 − ε, π/2],
and extend it to [0, π] by requiring that βε (π − θ) = βε (θ) for all θ ∈ [0, π].
For any ε > 0, we put Wε := W · βε . The function Wε : [0, π] → R is
continuously differentiable on the whole interval [0, π], it coincides with W on[ε, π/2 − ε] ∪ [π/2 + ε, π − ε], and it satisfies Wε (π − θ) = Wε (θ) for all θ ∈ [0, π].
Let us also record the following estimates:
Lemma 7.3. For all ε ∈ (0, 1/4), one has
Z
Z π

(b)
(a)
|Wε0 (t)| dt = O | log ε| ,|W − Wε | dµ∞ = O ε| log ε| .

[0,π]

0

We postpone proving these two bounds until the end of the section, and carry on
with the proof of Proposition 7.2. In the notation introduced above, the triangle
inequality yields that
Z
Z
Z
(7.2)
W dµa −
W dµ∞ ≤|W − Wε | dµa
[0,π]

[0,π]

[0,π]

Z

Z
Wε dµa −

+
[0,π]

Z
|W − Wε | dµ∞ .

Wε dµ∞ +
[0,π]

[0,π]

We treat each of the three terms appearing in the right-hand side of this inequality
separately.
R
The third term [0,π] |W − Wε | dµ∞ can be directly bounded with the help of
Lemma 7.3(b).

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)27

To show that the middle term is small when a → ∞, we apply Theorem 6.3
to Wε , which is permissible since Wε is continuously differentiable on [0, π]. We
obtain that
Z
Z
Z π
a1/2
a1/2
Wε dµa −Wε dµ∞ q a/4 ·|Wε0 (t)| dt q a/4 · | log ε|,
q
q
[0,π]
[0,π]
0
where the last inequality follows from Lemma
R 7.3(a).
Finally, we claim that the first term [0,π] |W − Wε | dµa on the right-handside of (7.2) vanishes provided that ε > 0 is small enough. Indeed, we know by
Theorem 6.4 that the support of µa is contained in [εa , π/2 − εa ] ∪ [π/2 + εa , π − εa ]where εa = (q a )−σp for some constant σp > 0. By the proof of that theorem, one
can take σp = 6p − 4. On the other hand, as was noted in a previous paragraph,
W and
R Wε coincide on [ε, π/2 − ε] ∪ [π/2 + ε, π − ε]. Hence, choosing ε < εa , wehave [0,π] |W − Wε | dµa = 0.
Summing up these three contributions, inequality (7.2) yields thatZZ
a1/2
W dµa −
W dµ∞ q 0 + a/4 · | log ε| + ε| log ε|,
q
[0,π]
[0,π]
for all ε < εa . Note that (q a )−6q < εa because 6q > 6p − 4 = σp . We may therefore
take ε = (q a )−6q , and this choice provides the desired bound.

Proof of Lemma 7.3. Both W and Wε are periodic of period π/2 and aresymmetric around π/4 (i.e., W (π/2 − θ) = W (θ) for all θ ∈ [0, π/2], and similarly
for Wε ). The same holds for |Wε0 | and |W − Wε |. We are thus reduced to proving
the following two bounds:
Z
Z π/4
0
(b’)
|W − Wε | dµ∞  ε| log ε|.
(a’)|W (t)| dt  | log ε|,ε[0,π/2]0

To that end, we first gather a few useful remarks:(i) Since Wε = W · βε and since 0 ≤ βε (t) ≤ 1, we have∀t ∈ [0, π],|Wε0 (t)| ≤ |W 0 (t)| + |βε0 (t)|W (t).

(ii) By construction of βε , we have |βε0 (t)| = 0 for all t ∈ [ε, π/4] and |βε0 (t)| ≤ M/εfor t ∈ [0, ε], where M := supx∈[0,1] |β00 (x)|.
(iii) The function Wε is constant (equal to 0) on [0, ε/3], so that Wε0 (t) = 0 for allt ∈ (0, ε/3).
(iv) The functions W and Wε are symmetric around π/4, and they coincide on[ε, π/2 − ε]. Moreover, we have∀t ∈ [0, π/2],|W (t) − Wε (t)| = (1 − βε (t))W (t) ≤ W (t).

(v) For t ∈ (0, π/4], classical convexity inequalities imply that (sin t)−1 ≤ π/(2t)and (cos t)−1 ≤ π/(π − 2t). We deduce that



π2
π
−1
W (t) = 2 log (sin t · cos t)≤ 2 log≤ 2 log ,2t(π − 2t)t
for all t in this interval. In particular, this yields that
Z ε
Z ε
t
W (t) dt ≤ −2
log dt  ε| log ε|.
π
0
0

28

RICHARD GRIFFON AND GUUS DE WIT(vi) For all t ∈ (0, π/4], a simple calculation shows that


sin t
cos t
2 cos(2t)W 0 (t) = 2
−
=−
.
cos t
sin t
sin t · cos t
Thus, using the same classical inequalities as in the previous item, we obtainthat, for all t ∈ (0, π/4],|W 0 (t)| ≤

2π 2
2π
2
≤
≤
.
sin t · cos t2t(π − 2t)t

We now combine the above items to derive the desired inequalities. First, we have
Z ε
Z ε
Z ε
Z ε
by (i)by (iii)|Wε0 (t)| dt =|Wε0 (t)| dt ≤|W 0 (t)| dt +W (t)|βε0 (t)| dt
0

ε/3
by (vi), (ii)

≤

ε/3

Z ε

2π
dt +
ε/3 t

ε/3

Z ε

by (v)M
W (t) dt  | log ε|.
εε/3Furthermore, we have
Z π/4
Z π/4
Z π/4
by (i)|Wε0 (t)| dt ≤|W 0 (t)| dt +W (t)|βε0 (t)| dtεεε

by (vi), (ii)

Z π/4

≤
ε

2π
dt  | log ε|.
tSumming the last two displays proves (a’). Finally, we notice thatZ
|W − Wε | dµ∞ =[0,π/2]

2
≤
π

2
π

Z π/2

|W (t) − Wε (t)| sin2 (t) dt

0

Z π/2

by (iv)|W (t) − Wε (t)| dt
0

≤

4
π

Z ε

by (v)

W (t) dt  ε| log ε|,0which proves (b’) and concludes the proof of the lemma.



7.2. Proof of Theorem 7.1. With Proposition 7.2 at hand, we may now
proceed to prove Theorem 7.1. In the notation introduced in the previous sections,
we begin by taking the logarithm of identity (6.2):(7.3)

log L(Eγ,a , q −1 ) = log |L(Eγ,a , q −1 )|X=
log 1 + e2iε(v) − 2eiε(v) cos θγ (v)v∈Pq (a)=X

log |Fv (θγ (v))|,v∈Pq (a)where the functions Fv : [0, π] → R are defined as follows:2 − 2 cos θ if ε(v) = 0,−2i cos θif ε(v) = π/2,∀v ∈ Pq (a),Fv : θ 7→

2
+
2
cos
θ
if
ε(v) = π,
2i cos θif ε(v) = 3π/2.

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)29Note that |Fv (θγ (v))| > 0 for all v ∈ Pq (a) because θγ (v) ∈/ {0, π/2, π} by (Kl 5)(see Remark 6.2). Straightforward analytic estimates show that, for any placev ∈ Pq (a), one hassin2 θ · cos2 θ ≤ |Fv (θ)| ≤ 4.
In particular, writing W (θ) = − log sin2 θ · cos2 θ as in §7.1, we obtain that∀θ ∈ [0, π],∀v ∈ Pq (a),−W (θγ (v)) ≤ log |Fv (θγ (v))| ≤ log 4.

Summing this chain of inequalities over all v ∈ Pq (a) and using equality (7.3) yields:
X
1
log L(Eγ,a , q −1 )|Pq (a)|·−W (θγ (v)) ≤≤ log 4 ·.
log H(Eγ,a )log H(Eγ,a )log H(Eγ,a )v∈Pq (a)It is clear from these inequalities that Theorem 7.1 will be proved once we show
the following two assertions:(C1 ) There exists a constant c1 > 0 such that1log H(Eγ,a )X

W (θγ (v)) ≤v∈Pq (a)(C2 ) There exists a constant c2 > 0 such thatc1,a|Pq (a)|c2≤ .
log H(Eγ,a )aWe thus now prove these two claims. The second one (C2 ) is a direct consequenceof the following two estimates: we know from (2.3) that log H(Eγ,a ) q q a , andfrom (3.2) that |Pq (a)| q q a /a.
Next we turn to the proof of (C1 ): by Proposition 7.2, we have
 3/2 ZX
1a
W (θγ (v)) =W dµ∞ + Oq,
|Pq (a)|
q a/4
[0,π]
v∈Pq (a)as a → ∞, where the implicit constant depends at most on q. Therefore, making
use of the estimate in (C2 ) which we have just proved, we obtain that0≤1
log H(Eγ,a )Xv∈Pq (a)W (θγ (v)) =|Pq (a)|1·log H(Eγ,a ) |Pq (a)|

c2
≤
a

X

W (θγ (v))

v∈Pq (a) 3/2 !
ac1
W dµ∞ + Oq
≤ ,
a/4
a
q
[0,π]

Z

for some constant c1 > 0, depending at most on q. This proves the first claim (C1 )and concludes the proof of Theorem 7.1.

8. Proof of Theorem C
In this section, we gather our results so far to prove our main result (Theorem C in the introduction). We have already proved assertions (1) and (2) of thattheorem: see Proposition 2.4 and Corollary 5.4(2), respectively. In the following
two subsections, we prove assertions (4) and (3) of Theorem C, in this order.

30

RICHARD GRIFFON AND GUUS DE WIT

8.1. The p-primary part of X(Eγ,a ). Let us first prove assertion (4) of
Theorem C, concerning the p-primary part of X(Eγ,a ):Theorem 8.1. Let Fq be a finite field of odd characteristic p, and K = Fq (t).
For any γ ∈ F×
q and any integer a ≥ 1, consider the elliptic curve Eγ,a over K
defined by (2.1). Then the p-primary part of X(Eγ,a ) is trivial. In other words,
the integer |X(Eγ,a )| is relatively prime to p.
×

Proof. Recall from §5.1 that ordP : Q → Q denotes an extension of thep-adic value to Q, normalised so that ordP (q) = 1. Taking P-adic valuation ofboth sides of the BSD formula (5.2), we obtain that
(8.1)

ordP |X(Eγ,a )| = ordP L(Eγ,a , q −1 ) + ordP H(Eγ,a ) − 1.

Our formula (2.3) for the height of Eγ,a implies that ordP H(Eγ,a )−1 = (q a − 1)/2.
To conclude the proof, it thus suffices to see that ordP L(Eγ,a , q −1 ) = −(q a − 1)/2.
Indeed, one would then deduce from (8.1) that ordP |X(Eγ,a )| = 0, and this would
directly show that X(Eγ,a ) has trivial p-primary part (by the structure theorem
for finite abelian groups).
We now proceed to compute ordP L(Eγ,a , q −1 ). Evaluating at T = q −1 theexpression for L(Eγ,a , T ) obtained in Theorem 4.1, and taking P-adic valuations
on both sides of the resulting identity yields that!
X
g(v)kl0γ (v)g(v)klγ (v)−1.
+ ordP 1 −ordP L(Eγ,a , q ) =ordP 1 −q deg v
q deg vv∈Pq (a)The results proved in §5.1 imply that, for all places v ∈ Pq (a), we have





deg v deg v
0− deg v− deg v
ordP g(v)klγ (v)q
, ordP g(v)klγ (v)q
= −
,
.
2
2
Using the cases of equality in the non-archimedean triangle inequality, we thenobtain that




X
deg v
deg v
1 X
ordP L(Eγ,a , q −1 ) =min 0, −+ min 0,=−deg v.
22
2v∈Pq (a)v∈Pq (a)As has already been observed, we have v∈Pq (a) deg v = |Gm (Fqa )| = q a − 1. We
therefore conclude that ordP L(Eγ,a , q −1 ) = −(q a − 1)/2, as was to be shown. PRemark 8.2. (1) In a very recent paper [20], Ulmer introduces the notion ofdimension of X for abelian varieties over Fq (t) whose Tate–Shafarevich groupis finite. In the special case of Eγ,a , this invariant is defined as follows: for all
integers n ≥ 1, let Kn := Fqn (t) and consider 
log X Eγ,a ×K Kn Kn [p∞ ].
dim X(Eγ,a ) := limn→∞log q n
Proposition 4.1 of [20] shows that the limit exists and is a nonnegative integer,
called the dimension of X of Eγ,a . Further, [20, Prop. 4.2] provides an expression for dim X(Eγ,a ) in terms of the p-adic slope sequence of the L-functionof Eγ,a . Using that expression and our Theorem 5.2, an easy calculation yields
that dim X(Eγ,a ) = 0.

ELLIPTIC CURVES WITH LARGE TATE–SHAFAREVICH GROUPS OVER Fq (t)31(2) The previous item shows that the order of the p-primary part of the Tate–
Shafarevich group of the base-changed elliptic curve Eγ,a ×K Kn over Kn grows
slowly with n (its log is o(n)). By replacing q by q n in the proof of Theorem 8.1,
one can actually prove the stronger assertion that, for all n ≥ 1,
|X(Eγ,a ×K Kn /Kn )[p∞ ]| = 1.
8.2. The size of X(Eγ,a ). Finally, we prove assertion (3) of Theorem C
about the size of X(Eγ,a ). We actually show a slightly more precise result:
Theorem 8.3. Let Fq be a finite field of odd characteristic and K = Fq (t). For
any γ ∈ F×
q and any integer a ≥ 1, consider the elliptic curve Eγ,a over K definedby (2.1). Then, as a → ∞, we have|X(Eγ,a )| = H(Eγ,a )1+Oq (1/a) ,where the implicit constant is effective and depends at most on q.
Recall from (2.3) that H(Eγ,a ) = N (Eγ,a )1/4 . Hence, the above implies:
Corollary 8.4. In the same setting, as a → ∞, we have|X(Eγ,a )| = N (Eγ,a )1/4+Oq (1/a) .
Proof (of Theorem 8.3). We note that, by (2.3), log log H(Eγ,a ) and ahave the same order of magnitude when a → ∞, the involved constants dependingat most on q. The elliptic curve Eγ,a satisfies the BSD conjecture (see Corollary 5.4):
taking the logarithm of both sides of the BSD formula (5.2) and reordering terms
yields thatlog |X(Eγ,a )|log qlog L(Eγ,a , q −1 )=1−+.
log H(Eγ,a )log H(Eγ,a )log H(Eγ,a )
The term log q/log H(Eγ,a ) is clearly o 1/ log log H(Eγ,a ) = o(a−1 ) as a → ∞. To
control the right-most term, we use our bound on the central value L(Eγ,a , q −1 ):we deduce from Theorem 7.1 that
log L(Eγ,a , q −1 )= O 1/ log log H(Eγ,a ) = O(1/a),log H(Eγ,a )(8.2)

as a → ∞, where the implicit constant depends at most on q. Plugging these two
estimates into (8.2) completes the proof.
Remark 8.5. Theorem 8.3 can be interpreted as an analogue of the Brauer–
Siegel theorem for the sequences {Eγ,a }a≥1 . We refer the reader to [6, 7] for a
detailed description of the analogue we have in mind: let us simply recall that
Hindry and Pacheco have introduced the Brauer–Siegel ratio of an abelian variety
with finite Tate–Shafarevich group and that, in the case at hand, the Brauer–Siegel
ratio is given by Bs(Eγ,a ) = log |X(Eγ,a )| log H(Eγ,a ). (For an elliptic curve withpositive Mordell–Weil rank, the Brauer–Siegel ratio also includes the Néron–Tate
regulator.) With this notation, Theorem 8.3 can be rewritten in a compact form:(8.3)
Bs(Eγ,a ) = 1 + Oq 1/a(as a → ∞).
There is only a handful of sequences of elliptic curves over K for which one canprove that the Brauer–Siegel ratio has a limit and that this limit is 1 (see [6, 2, 3, 7]
and references therein). The families studied in the present paper therefore provide
further examples of that behaviour. However, it seems interesting to remark that

32

RICHARD GRIFFON AND GUUS DE WIT

the sequences {Eγ,a }a≥1 consist of rank 0 elliptic curves, whereas previous articles
considered sequences with unbounded rank.

Acknowledgements
This article is partly based on GdW’s Leiden Master’s thesis [22], written
under the supervision of RG. The authors would like to thank Bas Edixhoven,
Marc Hindry and Douglas Ulmer for fruitful conversations about this work anduseful comments on earlier versions thereof.
GdW received funding from the ALGANT Master Programme during his studies at Università degli studi di Milano and Universiteit Leiden. Work on this project
was started at Universiteit Leiden. RG now works at Universität Basel, with funding from the Swiss National Science Foundation (SNSF Professorship #170565
awarded to Pierre Le Boudec). These institutions are gratefully thanked for creating great working conditions, and for their financial support. RG also acknowledges funding from Agence Nationale de la Recherche (Grant ANR-17-CE40-0012
FLAIR).
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elliptiques. PhD thesis, Université Paris Diderot, July 2016.
[3] Richard Griffon. Bounds on special values of L-functions of elliptic curves in an ArtinSchreier family. European Journal of Mathematics, 5(2):476–517, 2018.
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algant/algant_theses.php).
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH4051 Basel (Switzerland)E-mail address: richard.griffon@unibas.ch
Mathematisch Instituut, Universiteit Leiden, PO Box 9512, 2300RA Leiden (theNetherlands)
E-mail address: dewit.guus@gmail.com