Bounds on special values of Lfunctions of elliptic curves in an Artin–Schreier family
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Abstract
We study a certain Artin–Schreier family of elliptic curves over the function field Open image in new window . We prove an asymptotic estimate on the special values of their Lfunction in terms of the degree of their conductor; we show that the special values are, in a sense, ‘asymptotically as large as possible’. We also provide an explicit expression for their Lfunction. The proof of the main result uses this expression and a detailed study of the distribution of character sums related to Kloosterman sums. Via the BSD conjecture, the main result translates into an analogue of the Brauer–Siegel theorem for these elliptic curves.
Keywords
Elliptic curves over function fields in characteristic p Explicit computation of Lfunctions Special values of Lfunctions BSD conjecture Kloosterman sums Sato–Tate distributionMathematics Subject Classification
11G05 11G40 11M38 11F67 11L05 11J201 Introduction
Let \({\mathbb {F}}_q\) be a finite field of odd characteristic p and Open image in new window . Consider a nonisotrivial elliptic curve E defined over K, and its associated Lfunction^{1}L(E, T). Via a cohomological interpretation, Grothendieck has proven that, even though L(E, T) is a priori defined as a formal power series in T, it is actually a polynomial with integral coefficients, whose degree we denote by b(E). Moreover, L(E, T) satisfies the expected functional equation relating L(E, T) to \(L(E, 1/q^2T)\).
Define \(\rho (E)\) to be the order of vanishing of L(E, T) at the central point \(T=q^{1}\) and the special value of L(E, T) by Open image in new window . These invariants both appear in the conjecture of Birch and SwinnertonDyer^{2} through which they are related to ‘arithmetic’ invariants of E.
These questions are still wide open and, as far as the author knows, they have only been settled for a very limited number of special families Open image in new window (see [10, Theorem 7.12], [7, Corollary 5.1], [8, Theorem 4.2] and [6, Theorem 9]). These examples are known as ‘Kummer families’ of elliptic curves: one obtains them by pullingback an elliptic curve \(E_1/K\) by the map \(t\mapsto t^d\) for larger and larger integers d which are coprime to q. In those cases, the ratio in (1) does have a limit, and this limit is 0.
In this article, we answer the two questions above for an ‘Artin–Schreier family’ of elliptic curves over K. More precisely, we prove
Theorem A
The name of ‘Artin–Schreier family’ stems from its construction: starting with the elliptic curve \(E_{\gamma }/K\) given by Open image in new window , one obtains \(E_{a, \gamma }\) by pulling back \(E_{\gamma }\) by the Artin–Schreier map Open image in new window for any \(a\geqslant 1\). This family of elliptic curves \(\{E_{a, \gamma }\}_{a\geqslant 1}\) was previously studied in [18] where, among other things, the authors prove that \(E_{a, \gamma }\) satisfies the BSD conjecture. Following [18, Section 7.3], we note the ressemblance between \(E_{a, \gamma }\) and a Legendre elliptic curve. We refer to Theorem 7.1 for a more quantitative version of (2).
Once again, prior to Theorem A, the only known examples of families of elliptic curves exhibiting a behaviour such as (2) were Kummer families. We also remark that the strategy of proof of (2) significantly differs from the one used in our previous works [6, 7, 8]; we comment further on this difference and on the reason for adopting a new approach in Remark 7.2.
From Theorem A and from the BSD conjecture, we will deduce (see Theorem 8.1):
Theorem B
Following [10], we view this result as an analogue of the Brauer–Siegel theorem for the sequence of elliptic curves \(\{E_{a, \gamma }\}_{a\geqslant 1}\). We further comment on this result in Sect. 8.
Let us give an outline of the proof of Theorem A as we describe the structure of the paper and state the other results contained in it. In Sect. 2, we start by reviewing the construction of the elliptic curves \(E_{a, \gamma }\) and by computing some of their invariants. We also recall the definition of their Lfunction and state the BSD conjecture (proven by Pries and Ulmer for \(E_{a, \gamma }\), cf. [18]).
In the following two sections, we compute the Lfunction of \(E_{a, \gamma }\); the relevant objects are introduced in Sect. 3. In particular, a central role is played by angles of some Kloosterman sums. For the purpose of this introduction let us only say that, to any place \(v\ne 0, \infty \) of K, we will attach a character sum Open image in new window . The sum Open image in new window is a real number satisfying Open image in new window where \(d_v\) is the degree of v. Hence there exists an angle Open image in new window such that Open image in new window . The reader is referred to Sects. 3.2 and 4.4 for precise definitions. Section 4 is devoted to the calculation of the Lfunction itself, which results in the following expression:
Theorem C
This result is proven by a ‘pointcounting’ argument, directly from the definition of \(L(E_{a, \gamma }, T)\), through manipulations of character sums over finite fields. Given the paucity of tables of Lfunctions of elliptic curves over K of large conductor, such an explicit expression of \(L(E_{a, \gamma }, T)\) may be of independent interest.
As a byproduct, Theorem C yields a closed formula for the analytic rank Open image in new window . Using that the BSD conjecture is proven for \(E_{a, \gamma }\), we recover a result of [18] stating that the ranks of \(E_{a, \gamma }(K)\) are unbounded as \(a\rightarrow \infty \); more precisely, we show in Corollary 4.8 that Open image in new window .
Theorem D
 (i– Theorem 6.6) For all continuously differentiable functions Open image in new window ,
 (ii
– Corollary 5.5) There is a constant \(c>0\) such that Open image in new window for all Open image in new window .
By the work of Katz, it is known that the angles of Kloosterman sums become equidistributed in Open image in new window with respect to the Sato–Tate measure (see [11]). It turns out that the same statement holds for the angles Open image in new window (see Theorem 6.5). The proof relies on an adaptation of Katz’s method in [11, Chapter 3] and results of Fu and Liu in [5]. This equidistribution result, however, is not sufficient for our purpose: we need a more effective version such as Theorem D (i). The effective version (Theorem 6.6) follows from Theorem 6.5 after a more detailed analysis using tools from equidistribution theory (see [17]).
The main goal of Sect. 5 is to prove Theorem D (ii); we actually prove a more general result there (see Theorem 5.1). The proof has a Diophantine approximation flavour and the main tool is a version of Liouville’s inequality, as found in [16].
Throughout this article, we fix a finite field\({\mathbb {F}}_q\)of characteristic\(p\geqslant 3\), and we denote by Open image in new window the function field of the projective line\({\mathbb {P}}^1_{/{\mathbb {F}}_q}\).
2 The Artin–Schreier family of elliptic curves \(E_{a, \gamma }\)
In this section, we explain in some detail how the curves \(E_{a, \gamma }\) are constructed and we collect elementary facts about them. We also setup some notations and conventions that will be in force for the rest of the paper. For a nice account of the theory of elliptic curves over K, the reader may consult [21].
2.1 Bad reduction and invariants
For any place v of K, we denote by \(d_v\) or \(\deg v\) the degree of v and by \({\mathbb {F}}_{v}\) the residue field at v. We identify finite places of K with monic irreducible polynomials in Open image in new window ; we also identify the residue field at \(v\ne \infty \) with Open image in new window if Open image in new window is the monic irreducible polynomial corresponding to v.
Proposition 2.1
Let \(Z_{a, \gamma }\) be the set of places of K that divide Open image in new window . Then \(E_{a, \gamma }\) has good reduction outside \(S= Z_{a, \gamma }\cup \{\infty \}\). The reduction of \(E_{a, \gamma }\) at places \(v\in S\) is as follows:
Place v of K  Reduction type of \(E_{a, \gamma }\) at v  \(\mathrm{ord}_{v}\Delta _{\min }(E_{a, \gamma })\)  

Multiplicative (of type \(\mathbf I _{4}\))  4  1  
Multiplicative (of type \(\mathbf I _2\))  2  1  
\(\infty \)  Split multiplicative (of type \(\mathbf I _{4q^a}\))  \(4q^a\)  1 
In this table, for all places v of bad reduction for \(E_{a, \gamma }\), we have denoted by \(\mathrm{ord}_{v}\Delta _{\min }(E_{a, \gamma })\) (resp. Open image in new window ) the valuation at v of the minimal discriminant of \(E_{a, \gamma }\) (resp. of the conductor of \(E_{a, \gamma }\)). See [19, Chapter IV, Section 9], [21, Lecture 1, Section 8] for the definitions of these local invariants.
Remark 2.2
As is clear from comparing (8) and the third column of the table in Proposition 2.1, the discriminant \(\Delta \) of the Weierstrass model (6) has the same valuation as \({\Delta _{\min }(E_{a, \gamma })}\) at all finite places of K. Therefore, the model (6) is a minimal integral model of \(E_{a, \gamma }\) at all places \(v\ne \infty \) of K.
2.2 Definitions of Lfunction, analytic rank and special value
Definition 2.3
Remark 2.4
The special value \(L^*(E_{a, \gamma }, 1)\) is ‘usually’ defined as the first nonzero coefficient in the Taylor expansion around \(s=1\) of the function \(s\mapsto L(E_{a, \gamma }, q^{s})\). Our definition (12) differs from that more ‘usual’ one by a factor \((\log q)^\rho \). We prefer to use the normalisation (12) because it ensures that \(L^*(E_{a,\gamma }, 1)\in {\mathbb {Q}}^*\). This choice is consistent with our normalisation of \(\mathrm{Reg}(E_{a, \gamma })\), see Sect. 2.3.
2.3 The BSD conjecture
It has been conjectured by Birch, SwinnertonDyer and Tate that the ‘analytically defined’ quantities \(\rho (E_{a, \gamma })\) and \(L^*(E_{a, \gamma }, 1)\) have an arithmetic interpretation (see [20, Conjecture B]). Even though this conjecture is still open in general, it has been proven by Pries and Ulmer for \(E_{a, \gamma }\) in [18]. Let us state their result as follows:
Theorem 2.5

The Tate–Shafarevich group Open image in new window is finite.

The rank of \(E_{a, \gamma }(K)\) is equal to \(\rho (E_{a, \gamma })=\mathrm{ord}_{T=q^{1}}L(E_{a, \gamma }, T)\).
 Moreover, one has where \(\tau (E_{a, \gamma })\) denotes the Tamagawa number of \(E_{a, \gamma }\).
Proof
We only sketch a proof and refer the interested reader to [18, Section 3] for more details. As we have seen at the beginning of this section, \(E_{a, \gamma }\) is 2isogenous to \(E^\circ _{a, \gamma }\). Since \(E_{a, \gamma }\) and \(E^\circ _{a, \gamma }\) are linked by an isogeny of degree prime to the characteristic of K, [15, Chapter I, Theorem 7.3] implies that the BSD conjecture holds for \(E_{a, \gamma }\) if and only if it does for \(E^\circ _{a, \gamma }\). Hence Theorem 2.5 will follow if we prove that \(E^\circ _{a, \gamma }\) satisfies the BSD conjecture.
We have also shown that \(E^\circ _{a, \gamma }\) is birational to the curve Open image in new window which, by construction, is given in affine coordinates by an equation of the form Open image in new window where \(f_\gamma \) is a certain rational function on \(\mathbb {P}^1_{/K}\) and Open image in new window is a separable additive polynomial. Under these conditions, [18, Corollary 3.1.4] proves that \(E^\circ _{a, \gamma }\) satisfies the BSD conjecture.
The crucial point of their proof is the following: given the specific shape of the equation of \(Z_{a, \gamma }\) to which \(E^\circ _{a, \gamma }\) is birational, the minimal regular model Open image in new window over \({\mathbb {F}}_q\) of the curve \(E^\circ _{a, \gamma }/K\) is dominated by a product of curves Open image in new window over \({\mathbb {F}}_q\) (where Open image in new window is actually the curve defined in Remark 3.3). The Tate conjecture (T) asserts that the order of the pole of the zeta function of a surface \(S/{\mathbb {F}}_q\) equals the rank of the Néron–Severi group of S (see [20, Conjecture C], or [21, Lecture 2, Sections 10–13]). Conjecture (T) is proven for surfaces that are dominated by products of curves. In particular, (T) holds for the surface Open image in new window . On the other hand, it is known that conjecture (T) for Open image in new window is equivalent to the BSD conjecture for the generic fiber of Open image in new window , i.e., for the elliptic curve \(E^\circ _{a, \gamma }/K\) ([21, Lecture 2, Theorem 8.1]). Hence the result. \(\square \)
In Sect. 7 we give bounds on the special value \(L^*(E_{a, \gamma }, 1)\) on the lefthand side of (13) and, in Sect. 8, we deduce from these an estimate on the asymptotically significant quantities on the righthand side of (13). For completeness, let us recall here the following bounds (which we will need to prove Theorem 8.1).
Proposition 2.6
 (i)
\(E(K)_{{\mathrm{tors}}} \ll _q 1\),
 (ii)
Open image in new window as \(H(E)\rightarrow \infty \).
Proof
The first bound is an analogue for elliptic curves over Open image in new window of Merel’s uniform bound on torsion for elliptic curves over \({\mathbb {Q}}\). There are several proofs of (i) and we refer the reader to [21, Lecture 1, Section 7] for a survey and a sketch of proof (by a modular method). The bound (ii) on the Tamagawa number is a consequence of [10, Theorem 1.22] in the case when E is semistable or \(p>3\). A selfcontained (and elementary) proof for all elliptic curves over K can also be found in[6, Théorème 1.5.4]. \(\square \)
3 The sums Open image in new window and the sets Open image in new window
The goal of this section is to introduce the objects which appear in the Lfunction of \(E_{a, \gamma }\). We fix a finite field \({\mathbb {F}}_q\) of odd characteristic and a nontrivial additive character \(\psi _q\) on \({\mathbb {F}}_q\), which we assume to take values in the cyclotomic field \({\mathbb {Q}}(\zeta _p)\). For instance, a standard choice of \(\psi _q\) is the map Open image in new window where \(\zeta _p\) is a primitive \(p\hbox {th}\) root of unity and Open image in new window is the trace map.
For any finite extension \({\mathbb {F}}\) of \({\mathbb {F}}_q\), we denote by Open image in new window the relative trace and we ‘lift’ \(\psi _q\) to a nontrivial additive character Open image in new window on \({\mathbb {F}}\) by putting Open image in new window .
3.1 Kloosterman sums
Proposition 3.1
 (i)Open image in new window is a totally real algebraic integer in \({\mathbb {Q}}(\zeta _p)\), i.e.,
 (ii)Open image in new window satisfies ‘Salié’s formula’: where \(\lambda :{\mathbb {F}}^\times \!\rightarrow \{\pm 1\}\) is the unique multiplicative character on \({\mathbb {F}}^\times \) of exact order 2 (extended by Open image in new window to the whole of \({\mathbb {F}}\)).
 (iii)
If \({\mathbb {F}}\) contains \({\mathbb {F}}_q\), one has Open image in new window .
 (iv)There exist two algebraic integers Open image in new window and Open image in new window such that Open image in new window and, for any finite extension Open image in new window , The pair Open image in new window is uniquely determined by \({\mathbb {F}}, \psi , \alpha \).
 (v)
Open image in new window and Open image in new window have magnitude \({\mathbb {F}}^{1/2}\) in any complex embedding. In particular, in any complex embedding of \({\mathbb {Q}}(\zeta _p)\), the sum Open image in new window satisfies Open image in new window (‘Weil bound’).
 (vi)In any complex embedding of \({\mathbb {Q}}(\zeta _p)\), one has
Proof
The reader can confer [14, Chapter 5, Section 5] and [22, Section 3] for proofs of these classical results about Kloosterman sums: (i) and (iii) are easily checked; items (ii), (iv) and (v) are [14, Theorems 5.47, 5.43 and 5.44], respectively; (vi) is proven in [22, Corollary 3.2]. \(\square \)
3.2 The sums Open image in new window
Assume a parameter \(\gamma \in {\mathbb {F}}_q^\times \) is given. A place \(v\ne 0, \infty \) of K with degree \(d_v\) corresponds to a monic irreducible polynomial Open image in new window of degree \(d_v\), with \(B_v\ne t\). Choose a root \(\beta _v\in \overline{{\mathbb {F}}_q}{}^{\,\times }\) of \(B_v\): we claim that the value of the Kloosterman sum Open image in new window does not depend on the choice of \(\beta _v\). Indeed, given one such \(\beta _v\) the \(d_v1\) other choices are of the form \({\beta _v}^{q^{j}}\) (with Open image in new window ) because the \(d_v\) different roots of \(B_v\) in \(\overline{{\mathbb {F}}_q}\) are all conjugate under the action of the Galois group \(\mathrm{Gal}({\mathbb {F}}_v/{\mathbb {F}}_q)\). A repeated application of Proposition 3.1 (iii) then proves the claim. Therefore the following definition makes sense:
Definition 3.2
Note that Open image in new window depends on \({\mathbb {F}}_q\) and \(\psi _q\), but we chose not to include these in the notation for brevity.
Remark 3.3
3.3 The sets Open image in new window
Definition 3.4
For any integer \(a\geqslant 1\), we denote by Open image in new window the set of places v of K, with Open image in new window , whose degree \(d_v\) divides a. In the identification between finite places of K and monic irreducible polynomials, Open image in new window corresponds to the set Open image in new window . Equivalently, Open image in new window is the set of closed points on the multiplicative group \({\mathbb {G}}_m={\mathbb {P}}^1\smallsetminus \{0, \infty \}\) over \({\mathbb {F}}_q\) whose degree divides a.
In what follows, we will frequently need the following estimates, which we record here for convenience.
Lemma 3.5
 (i)
Open image in new window for all \(a\geqslant 1\).
 (ii)
Open image in new window for all \(a\geqslant 1\).
Proof
For all \(n\geqslant 1\), we denote by \(\pi _q(n)\) the number of places \(v\ne 0, \infty \) of K of degree \(d_v=n\). In other words, \(\pi _q(n)\) is the number of closed points of degree n of \({\mathbb {G}}_m\) over \({\mathbb {F}}_q\).
The ‘Prime Number Theorem’ for Open image in new window states that Open image in new window for all \(n\geqslant 1\) where the hidden constant can be given explicitly (see [2, Proposition 6.3] for example). On the other hand, it is clear from the definition that Open image in new window . The estimate of \(\pi _q(n)\) and this relation directly imply (i). From this, one easily deduces (ii). \(\square \)
4 The Lfunction
With the notations introduced in the previous section, we can now state our first main result:
Theorem 4.1
The proof of this theorem occupies the rest of the present section. Our strategy is loosely based on the computation in [3, Section 3.2]: to give an expression of \(L(E_{a, \gamma }, T)\), we rely on an explicit ‘pointcounting’ argument with character sums. This requires showing an identity for counting solutions to ‘Artin–Schreier equations’ in terms of character sums, as well as a relation between the character sums that appear in the argument and the sums Open image in new window introduced above. We first give in the next subsection a proof of these two facts, and then prove Theorem 4.1 in Sect. 4.2.
Remark 4.2
Before moving on to the proof of this theorem, we note the following:
1. Even though the sums Open image in new window for Open image in new window individually depend on the choice of a nontrivial additive character \(\psi _q\) on \({\mathbb {F}}_q\), the Lfunction Open image in new window does not. Indeed, changing the choice of \(\psi _q\) amounts to permuting the factors in (17).
2. Note that Open image in new window . Thus, as a polynomial in T, the Lfunction \(L(E_{a, \gamma }, T)\) has degree Open image in new window . This is consistent with the expected degree, see (11).
4. Recall the elliptic curves \(E^\circ _{a, \gamma }\) introduced at the beginning of Sect. 2 and given by (7). Since isogenous elliptic curves share the same Lfunction (by [15, Chapter I, Lemma 7.1]), Theorem 4.1 also shows that the Lfunction of \(E^\circ _{a, \gamma }\) is given by (17).
5. It is perhaps illuminating to comment on the appearance of Kloosterman sums in \(L(E_{a, \gamma }, T)\). Choosing a prime \(\ell \ne p\), we denote by Open image in new window the ith \(\ell \)adic étale cohomology group of a smooth projective variety \(X/{\mathbb {F}}_q\). Let Open image in new window denote the minimal regular model of \(E^\circ _{a, \gamma }\). The cohomological interpretation of Lfunctions of Grothendieck implies that \(L(E^\circ _{a, \gamma }, T)\) is essentially the ‘interesting part’ of the zeta function of the surface Open image in new window , i.e., \(L(E^\circ _{a, \gamma }, T)\) is a factor of the characteristic polynomial of the Frobenius \({\text {Frob}}_q\) acting on Open image in new window .
By the construction of \(E^\circ _{a, \gamma }\) in Sect. 2, Open image in new window is a smooth model of a quotient of Open image in new window by the action of a certain finite group \(G_a\), where Open image in new window is the curve introduced in Remark 3.3 (see [18, Section 7.3] for a more detailed presentation). In particular, Open image in new window can be seen as a subspace of Open image in new window , itself a subspace of Open image in new window . Hence, by Künneth’s formula, \(L(E^\circ _{a, \gamma }, T)\) divides the characteristic polynomial of \({\text {Frob}}_q\) acting on Open image in new window .
As was noted in Remark 3.3, the numerator of the zeta function of Open image in new window , i.e., the characteristic polynomial of \({\text {Frob}}_q\) acting on Open image in new window , involves Kloosterman sums Open image in new window . Hence, the eigenvalues of \({\text {Frob}}_q\) acting on Open image in new window are products of the form Open image in new window . Being a factor of the characteristic polynomial of \({\text {Frob}}_q\) acting on Open image in new window , the Lfunction \(L(E^\circ _{a, \gamma }, T)\) has to involve some of the products Open image in new window .
Working out the details of this sketchy computation could lead to a different proof of Theorem 4.1.
4.1 Pointcounting and character sums
For any finite finite field \({\mathbb {F}}\) of odd characteristic, we fix a nontrivial additive character \(\psi \) on \({\mathbb {F}}\), and we denote by \(\lambda :{\mathbb {F}}^\times \rightarrow \{\pm 1\}\) the unique quadratic character on \({\mathbb {F}}^\times \), extended to the whole of \({\mathbb {F}}\) by Open image in new window .
Proposition 4.3
In order to prove this identity, we begin by recording the following ‘pointcounting’ lemma:
Lemma 4.4
Proof
Let us start by splitting the set Open image in new window into the two disjoint subsets Open image in new window and Open image in new window . Computing \(X_0\) is straightforward: if \(z=0\), any pair (u, 0) with \(u\in {\mathbb {F}}\) belongs to \(X_0\) and, if \(z\ne 0 \), no such pair belongs to Open image in new window . Therefore, we have Open image in new window .
Proof of Proposition 4.3
In the proof of Theorem 4.1, we will also need the following ‘counting lemma’:
Lemma 4.5
Proof
4.2 Proof of Theorem 4.1
Lemma 4.6
Proof
Starting from definition (10) of the Lfunction of \(E_{a, \gamma }\), expanding the logarithm of \(L(E_{a, \gamma }, T)\) as a power series in T and rearranging terms yields the desired expression for \(L(E_{a, \gamma }, T)\). See [1, Section 2.2] or [3, Section 3.2] for more details. \(\square \)
4.3 Rank and special value
From the factored expression of \(L(E_{a, \gamma }, T)\) obtained in Theorem 4.1, one can deduce explicit expressions of the analytic rank \(\rho (E_{a, \gamma })\) and of the special value \(L^*(E_{a, \gamma }, 1)\) (as defined in Sect. 2.2).
Proposition 4.7
Proof
Corollary 4.8
(Pries–Ulmer) Let \({\mathbb {F}}_q\) be a finite field of odd characteristic and Open image in new window . For all \({\gamma \in {\mathbb {F}}_q^\times }\), the rank of \(E_{a, \gamma }(K)\) is unbounded as \(a\geqslant 1\) tends to infinity.
4.4 Angles of the sums Open image in new window
5 Small angles of Kloosterman sums
In this section, we work in the following setting. Let \({\mathbb {F}}\) be a finite field of odd characteristic p, and \(\psi \) be a nontrivial additive character on \({\mathbb {F}}\). We assume that \(\psi \) takes values in \({\mathbb {Q}}(\zeta _p)\) and we pick a complex embedding \({\mathbb {Q}}(\zeta _p)\hookrightarrow {\mathbb {C}}\).
Theorem 5.1
Before we start the proof, we recall for convenience the following version of Liouville’s inequality:
Theorem 5.2
(Liouville’s inequality) Let Open image in new window be a polynomial of degree N. For any algebraic number \(z\in {\overline{\mathbb {Q}}}\), let \(D_z\) be its degree over \({\mathbb {Q}}\) and h(z) denote its logarithmic absolute Weil height.
See the introduction of [16] and the proof of Lemma 5 in loc.cit. for this version and its proof.
Proof of Theorem 5.1
Lemma 5.3
The ratio z has degree \(D_z\leqslant 2(p1)\) and height \(h(z)\leqslant \log \sqrt{{\mathbb {F}}}\). Moreover, \(z\ne \pm 1\).
Proof
Finally, Proposition 3.1 (vi) shows that Open image in new window , and the last assertion easily follows. \(\square \)
Let us deduce two corollaries from Theorem 5.1. The first one can be viewed as a slight improvement on the Weil bound on Kloosterman sums (i.e., an effective version of Proposition 3.1 (vi)):
Corollary 5.4
Proof
The second corollary is more central to our study of the size of \(L^*(E_{a, \gamma }, 1)\), cf. Sect. 7.1.
Corollary 5.5
Proof
For all Open image in new window , the residue field \({\mathbb {F}}_v\) is a subfield of \({\mathbb {F}}_{q^a}\) and we may choose \(\beta _v\) as in Sect. 3.2. Upon noting that \({\mathbb {F}}_v\leqslant q^a\), the corollary immediately follows from Theorem 5.1 applied to \({\mathbb {F}}={\mathbb {F}}_{v}\), Open image in new window and \(\alpha =\gamma \beta _v^2\). \(\square \)
6 Distribution of the sums Open image in new window
In this section, we fix again a finite field \({\mathbb {F}}_q\) of odd characteristic p, an element \(\gamma \in {\mathbb {F}}_q^\times \) and a nontrivial additive character \(\psi _q\) on \({\mathbb {F}}_q\) with values in \({\mathbb {Q}}(\zeta _p)\). For any finite extension \({\mathbb {F}}/{\mathbb {F}}_q\), we continue denoting by Open image in new window the composition Open image in new window of \(\psi _q\) with the trace Open image in new window .
Loosely speaking, we show that, as \(a\rightarrow \infty \), the numbers Open image in new window with \(v\in P_q(a)\) (see Sect. 3.2) are asymptotically distributed in \([2, 2]\) as ‘the traces of random matrices in \({\text {SU}}(2, {\mathbb {C}})\)’. In order to make this statement more precise and to prove it, we begin by introducing the necessary notations and notions.
Choose a prime number \(\ell \ne p\), an algebraic closure \(\overline{{\mathbb {Q}}_\ell }\) of \({\mathbb {Q}}_\ell \), an embedding \({\overline{\mathbb {Q}}}\hookrightarrow \overline{{\mathbb {Q}}_\ell }\), and a field isomorphism \(\overline{{\mathbb {Q}}_\ell }\simeq {\mathbb {C}}\). Through this isomorphism, we view \(\psi _q\) as a \(\overline{{\mathbb {Q}}_\ell }\)valued additive character on \({\mathbb {F}}_q\).
We fix a separable closure \(K^{{\text {sep}}}\) of K. The set of places \(v\ne 0, \infty \) of K can be identified with the set of closed points of the multiplicative group Open image in new window over \({\mathbb {F}}_q\). For a finite extension \({\mathbb {F}}/{\mathbb {F}}_q\) and a point \(\alpha \in {\mathbb {G}}_m({\mathbb {F}})\), we denote by Open image in new window the geometric Frobenius of \({\mathbb {G}}_m\) at \(\alpha \), which we view as a conjugacy class in the profinite group \(\mathrm{Gal}(K^{{\text {sep}}}/K)\). For any closed point v of \({\mathbb {G}}_m\), we choose \(\beta _v\in v\) and we let Open image in new window .
6.1 Angles of Kloosterman sums
Let us start by redefining the angles Open image in new window from a representationtheoretic point of view. The reader is referred to [11, Chapter 3] or [4] for more detailed presentations.
In [11, Chapter 4], Katz has constructed a lisse \(\overline{{\mathbb {Q}}_\ell }\)sheaf Open image in new window on \({\mathbb {G}}_m\) whose Frobenius traces are Kloosterman sums ( Open image in new window is the socalled Kloosterman sheaf). Taking a suitable Tate twist, one obtains a lisse \(\overline{{\mathbb {Q}}_\ell }\)sheaf Open image in new window of rank 2 on \({\mathbb {G}}_m\) which is pure of weight 0.
Katz has shown that the image of \(\mathrm{Gal}(K^{{\text {sep}}}/K)\) under \(\kappa \) is contained in \({\text {SL}}(2, \overline{{\mathbb {Q}}_\ell })\) (in other words, the representation \(\kappa \) has trivial determinant, see [11, Chapter 11]). Via the chosen isomorphism \(\overline{{\mathbb {Q}}_\ell }\simeq {\mathbb {C}}\), we may view \(\kappa (\mathrm{Gal}(K^{{\text {sep}}}/K))\) as a subgroup of \({\text {SL}}(2, {\mathbb {C}})\). The special unitary group Open image in new window is a maximal compact subgroup of \({\text {SL}}(2, {\mathbb {C}})\) and, since \({\text {SL}}(2, {\mathbb {C}})\) is semisimple, such an H is uniquely determined up to conjugation. For any place \(v\ne 0, \infty \), let \(\kappa ({\text {Fr}}_v)^{\text {s.s.}}\) be the semisimplification of \(\kappa ({\text {Fr}}_v)\): the closure of the subgroup of \({\text {SL}}(2, {\mathbb {C}})\) generated by all the \(\kappa ({\text {Fr}}_v)^{\text {s.s.}}\) is compact and thus, up to conjugation in \({\text {SL}}(2, {\mathbb {C}})\), lies in H.
We denote by \(H^\natural \) the set of conjugacy classes of H and we equip \(H^\natural \) with the measure \(\mu ^\natural \) obtained as the direct image of the Haar measure on H normalised to have total mass 1. The trace of \(M\in H\) (or of any element in its conjugacy class) is the sum of two conjugate complex number of magnitude 1, so it is a real number in \([2,2]\). More precisely, a matrix \(M\in H\) is conjugate (in H) to a diagonal matrix Open image in new window for some unique Open image in new window and \({{\mathrm{Trace}}}\, M = 2\cos \theta _M\). Hence, the set \(H^\natural \) endowed with \(\mu ^\natural \) can be identified with the interval Open image in new window endowed with the Sato–Tate measure Open image in new window (see [11, Chapter 13]). We identify any angle \(\theta \in [0, \pi ]\) with the conjugacy class of Open image in new window , which we also denote by the same symbol \(\theta \).
We are now ready to (re)define angles of Kloosterman sums. For any finite extension \({\mathbb {F}}/{\mathbb {F}}_q\) and any \(\alpha \in {\mathbb {G}}_m({\mathbb {F}})\), the semisimplification of Open image in new window is \({\text {SL}}(2,{\mathbb {C}})\)conjugate to an element of H, and we can define Open image in new window to be the conjugacy class in H of this element. In the identification between \(H^\natural \) and Open image in new window , this gives us a welldefined angle Open image in new window , see [11, Section 3.3].
Definition 6.1
Fix a finite field \({\mathbb {F}}_q\) equipped with a nontrivial additive character \(\psi _q\) and \(\gamma \in {\mathbb {F}}_q^\times \). For any place \(v\ne 0, \infty \) of K, let \(\varvec{\theta }_\gamma (v)\) be the angle associated to the Kloosterman sum Open image in new window by the construction above. In other words, we put Open image in new window .
Remark 6.2
6.2 Statement of results
Denote by \(\mu _{\mathrm{ST}}\) the Sato–Tate measure \(\frac{2}{\pi }\sin ^2\theta {\,\text {d}}\theta \) on Open image in new window . A sequence of Borel measures Open image in new window on Open image in new window is said to converge weak\(*\)to\(\mu _{\mathrm{ST}}\) if, for every continuous \({\mathbb {C}}\)valued function f on Open image in new window , the sequence of integrals \(\int _{[0, \pi ]} f{\,\text {d}}\mu _i\) converges to \(\int _{[0,\pi ]}f{\,\text {d}}\mu _{\mathrm{ST}}\) as \(i\rightarrow \infty \).
Our results concern two sequences of probability measures that we now introduce.
Definition 6.3
Remark 6.4
In terms of the measure \(\nu _a\), Corollary 5.5 can be reinterpreted as follows: given \({\mathbb {F}}_q, \psi _q\) and \(\gamma \) as above, for any \(a\geqslant 1\) the support of the probability measure \(\nu _a\) on Open image in new window is contained in Open image in new window .
We can now state the two main results of this section. First we show that the angles Open image in new window are asymptotically equidistributed with respect to the Sato–Tate measure as \(a\rightarrow \infty \). Namely,
Theorem 6.5
Assume we are given a datum \({\mathbb {F}}_q, \psi _q, \gamma \) as above. Then the sequences Open image in new window and Open image in new window of Borel probability measures both converge weak\(*\) to the Sato–Tate measure \(\mu _{\mathrm{ST}}\) when \(a\rightarrow \infty \).
In the course of proving Theorem 7.3, we will need a more effective version of (31): indeed, we require an estimate of the rate at which \(\int _{[0,\pi ]} f {\,\text {d}}\nu _a\) converges to \(\int _{[0, \pi ]} f{\,\text {d}}\mu _{\mathrm{ST}}\), at least for a smaller class of functions f. This is the object of the second result in this section:
Theorem 6.6
This will follow from the proof of Theorem 6.5 coupled with tools from distribution theory (see [17]).
The proofs will also show that the constants in Theorems 6.5 and 6.6 are effective and depend at most on q (and neither on the choice of \(\psi _q\) nor on the value of \(\gamma \in {\mathbb {F}}_q^\times \)).
6.3 Equidistribution of \(\varvec{\theta }_\gamma (v)\)’s
Proposition 6.7
Proof
Remark 6.8
To complete the proof of Theorem 6.5, it remains to show that Open image in new window also converges (weak\(*\)) to \(\mu _{\mathrm{ST}}\).
Proposition 6.9
Proof
6.4 Effectivity of the equidistribution
The nontrivial irreducible representations of \(H={\text {SU}}(2, {\mathbb {C}})\) are exactly the symmetric powers \({\text {Symm}}^n({{\text {std}}})\) of the standard representation \({{\text {std}}}:H\hookrightarrow {\text {GL}}(2, {\mathbb {C}})\). Moreover, if \(\Lambda _n = {\text {Symm}}^n({{\text {std}}})\) for some \({n\geqslant 1}\), then \(\Lambda _n\) has dimension \(n+1\) and the trace function \({\mathrm{Trace}}\,\Lambda _n:H^\natural \rightarrow {\mathbb {R}}\) corresponds to the map^{7} Open image in new window in the identification of \(H^\natural \) with Open image in new window .
Theorem 6.10
This statement is essentially^{8} in [17, Corollary 2], the proof of which is based on an adaptation to the Sato–Tate context of the proof of Koksma’s inequality for the uniform measure (see [12, p. 143]).
Note that, for a continuously differentiable function g, Open image in new window . Therefore, Theorem 6.6 follows directly from Theorem 6.10 and the following:
Proposition 6.11
Proof
7 Bounds on the special value
Theorem 7.1
Remark 7.2
In [6, 7, 8], the author has also proven, for other families of elliptic curves, lower bounds on special values of Lfunctions which are similar to (39). However, the approach used in those papers for proving such bounds significantly differs from the strategy of proof of Theorem 7.1: let us investigate what comes out of our previous method for the sequence \(\{E_{a, \gamma }\}_{a\geqslant 1}\) at hand.
To do so, we keep track of the contribution to the exponent \(e_{a, \gamma }\) of each factor in the product \(\Pi _2\). Fix a prime ideal \({\mathfrak {P}}\) of \({\overline{\mathbb {Q}}}\) above p and denote by \(\mathrm{ord}_{\mathfrak {P}}\) the \(\mathfrak {P}\)adic valuation on \({\overline{\mathbb {Q}}}\) so normalised that \(\mathrm{ord}_{\mathfrak {P}}(q)=1\). For any Open image in new window , one has \(\mathrm{ord}_{\mathfrak {P}}(q^{d_v})=d_v\) and it can be shown that Open image in new window . Indeed, we know from Proposition 3.1 that Open image in new window and Open image in new window are algebraic integers whose product is \({\mathbb {F}}_v=q^{d_v}\) and whose sum is Open image in new window . Besides it is known that Open image in new window (see [22, Proposition 3.1 (v)] for instance). Hence one of Open image in new window or Open image in new window is a \(\mathfrak {P}\)adic unit, so that the other has \(\mathfrak {P}\)adic valuation \(d_v\).
7.1 Evaluation of a Sato–Tate limit
In this subsection, we show the following result, which is the crucial input in our proof of the lower bound in Theorem 7.1. For any integer \(a\geqslant 1\), we again denote by \(\nu _a=\nu ({\mathbb {F}}_q, \psi _q, \gamma ;a)\) the probability measure on Open image in new window introduced in Sect. 6.2.
Theorem 7.3
Proof
Lemma 7.4
 (i)
 (ii)
Proof of Lemma 7.4
Since both w and \(w_\epsilon \) are symmetric around \(\pi /2\), it is sufficient to prove (i) and (ii) where the integrals are replaced by integrals over Open image in new window . We also note that, for all \(t\in (0, \pi /2]\), one has Open image in new window . This follows from the classical estimate: \(\sin t\geqslant \frac{2t}{\pi }\) for Open image in new window .
7.2 Proof of Theorem 7.1
8 Application to an analogue of the Brauer–Siegel theorem
In this section, we deduce from Theorem 7.1 and from the BSD conjecture that the following theorem holds (stated as Theorem B in the introduction).
Theorem 8.1
Proof
Note added in proof
Ulmer recently uploaded a preprint [23] in which he develops an algebraic approach to proving an analogue of the BrauerSiegel theorem for some elliptic curves over K (see Sect. 8). His strategy to do so is based on finding lower bounds on an invariant he introduces: the ‘dimension of Open image in new window ’.
We would like to point out that the sequence \(\{E_{a, \gamma }\}_{a\geqslant 1}\) provides an example where the approach of [23] fails to be conclusive. Indeed, Theorem 8.1 shows that the BrauerSiegel ratio of \(E_{a, \gamma }\) is large; even though Proposition 4.2 in [23] and our Remark 7.2 imply that Open image in new window .
Footnotes
 1.
Since the base field K is fixed and all the invariants of E we consider are relative to K, we drop the dependency on K from the notations.
 2.
Hereafter abbreviated as BSD.
 3.
When \(E_{a, \gamma }\) has bad reduction at v, note that \(a_v\) equals 0 (resp. \(+1\), \(1\)) if \(E_{a, \gamma }\) has additive (resp. split multiplicative, nonsplit multiplicative) reduction at v.
 4.
Here we view \(\kappa ({\text {Fr}}_v)\in \overline{{\mathbb {Q}}_\ell }\) as an element of \({\mathbb {C}}\) by means of the chosen isomorphism \(\overline{{\mathbb {Q}}_\ell }\simeq {\mathbb {C}}\).
 5.
The measure \(\xi _a\) is the measure denoted by \(X_a\) in [11, Section 3.5] applied to our situation.
 6.
This is an analogue in the Sato–Tate context of Weyl’s criterion for uniform distribution (see [12, Chapter 4, Section 1]).
 7.
so that \({{\mathrm{Trace}}}\,\Lambda _n(\theta )=U_n(\cos \theta )\), where \(U_n\) is the \(n\hbox {th}\) Chebyshev polynomial of the second kind.
 8.
Instead of considering the measure \(\mu _{\mathrm{ST}}\) on Open image in new window , Niederreiter works on the interval \([1, 1]\) endowed with the direct image of \(\mu _{\mathrm{ST}}\) under \(t\mapsto \cos t\) (the ‘semicircle measure’); the translation to our setting is straightforward.
 9.
 10.
Note though that Lang uses a different normalisation of the height: his (naive) height has an exponent 1 / 12 instead of our exponent 1.
Notes
Acknowledgements
It is a pleasure to thank Marc Hindry and Douglas Ulmer for their encouragements and for some useful comments on earlier versions of this work. The author would also like to thank Bruno Anglès, Peter Bruin and Peter Koymans for fruitful discussions about various parts of the paper. Thanks are also due to the referee for their careful reading and valuable suggestions to improve the exposition.
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