Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family
Richard Griffon
math.NT 11G05, 11G40, 11M38, 11F67, 11L05

fancy

[
    Christian Zillinger
    December 30, 2023
=======================



7cm0.5pt.



  
Abstract – 
We study a certain Artin-Schreier family of elliptic curves over the function field K=_q(t). 

We prove an asymptotic estimate on the special values of their L-function in terms of the degree of their conductor; 
we show that the special values are, in a sense, “asymptotically as large as possible”. 
In the paper we also provide an explicit expression for their L-function.

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 L-functions,
Special values of L-functions,
BSD conjecture,
Kloosterman sums, 
Sato–Tate distribution. 
 
2010 Math. Subj. Classification:  
11G05, 

11G40, 

11M38, 


11F67, 
11L05, 
11J20. 
 
E-mail: mailto:r.m.m.griffon@math.leidenuniv.nlr.m.m.griffon@math.leidenuniv.nl


7cm0.5pt.





§ INTRODUCTION

[0]IntroductionIntroduction 
tocsectionIntroduction





  

Let _q be a finite field of odd characteristic p and K=_q(t).
Consider a nonisotrivial elliptic curve E defined over K, and  its associated L-function
[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.] L(E, T).
Via a cohomological interpretation, Grothendieck has proved 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 ρ(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 L^∗(E, 1):= lim_T→ q^-1 (1-qT)^-ρ(E)· L(E, T).
These invariants both appear in the conjecture of Birch and Swinnerton-Dyer[Hereafter abbreviated as BSD] through which they are related to “arithmetic” invariants of E. 







We will be interested in comparing the size of the special value 
L^∗(E, 1) to the degree b(E) of the L-function.
It is relatively straightforward to prove  that 

    -1 ≤log L^∗(E, 1)/log(q^b(E))≤ 0 + o(1)      (as  b(E)→∞),

and it seems natural to ask about the optimality of such bounds.
 In other words, given a family ℰ of nonisotrivial elliptic curves E over K 
 with b(E)→∞, we investigate the asymptotic behaviour of the ratio 
log(L^∗(E, 1))/log(q^b(E)) as E runs through ℰ. 
Does this ratio have a limit? If so, what is this limit?


These questions are still wide open and, as far as the author knows, they have only been settled for six special families ℰ (see <cit.>, <cit.>, <cit.> and <cit.>).  
These six examples are known as “Kummer families” of elliptic curves: one obtains them by pulling-back an elliptic curve E_1/K by the map t↦ t^d for larger and larger integers d which are coprime to q. 

In those cases, the ratio in (<ref>) 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 


Let _q be a finite field of odd characteristic p  and K=_q(t). 
Fix γ∈_q^× and, for all  integers a≥ 1, let E_a,γ be the elliptic curve over K given by the affine Weierstrass model

    E_a,γ:   y^2=x(x+16γ)(x+℘_a(t)^2), 
       with ℘_a(t) =t^q^a -t∈_q[t].

Then, as a→∞, the limit below exists and is 0: 

    lim_a→∞log L^∗(E_a, γ, 1)/log(q^b(E_a, γ)) = 0.



The name of “Artin–Schreier family” stems from its construction: starting with the elliptic curve E_γ/K given by y^2=x(x+16γ)(x+t^2), one obtains E_a, γ by pulling back E_γ by the Artin–Schreier map t↦℘_a(t) for any a≥1.
This family of elliptic curves {E_a, γ}_a≥ 1  was previously studied in 
<cit.> where, among others, the authors prove that E_a, γ  satisfies the BSD conjecture.
Following <cit.>, we note the ressemblance between E_a, γ and a Legendre elliptic curve. 
We refer to Theorem <ref> for a more quantitative version of (<ref>).

Once again, prior to Theorem <ref>, the only known examples of families of elliptic curves 
exhibiting a behaviour such as (<ref>)
were Kummer families.




  
 
From Theorem <ref> and from the BSD conjecture, we will deduce (see Theorem <ref>): 

Let _q be a finite field of odd characteristic and K=_q(t). 
For any γ∈_q^× and any integer a≥ 1, the Tate–Shafarevich group (E_a, γ) is finite.
Furthermore, one has

    lim_a→∞log(|(E_a, γ)|·(E_a, γ) )/log H(E_a, γ) = 1.



Following <cit.>, we view this result as an analogue of the Brauer–Siegel theorem for the elliptic curves {E_a, γ}_a≥ 1.
We further comment on this result in section <ref>. 



  

Let us give an outline of the proof of Theorem <ref> as we describe the structure of the paper and state the other results contained in it.
In section <ref>, we start by reviewing the construction of the elliptic curves E_a, γ and by computing some of their invariants. 
We also recall the definition of their L-function and state the BSD conjecture (proved by Pries and Ulmer for E_a, γ,  <cit.>).

In the two following sections, we compute the L-function of E_a, γ; 
the relevant objects are introduced in section <ref>. 
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≠ 0, ∞ of K, we will attach a character sum _γ(v).  
The sum _γ(v) is a real number satisfying |_γ(v)|< 2q^d_v/2 where d_v is the degree of v. 
Hence there exists an angle θ_v∈ (0, π) such that _γ(v) = 2q^d_v/2·cosθ_v. 
The reader is referred to <ref> and <ref> for precise definitions.
Section <ref> is devoted to the calculation 
of the L-function itself, which 
results in the following expression: 

 For all integers a≥ 1, we denote by P_q(a) the set of places v≠ 0, ∞ of K whose degree d_v divides a. 
Then, for all γ∈_q^×, the L-function of E_a, γ is given by

    L(E_a, γ, T)
    = ∏_v∈ P_q(a)(1-(qT)^d_v)
    (1-^2iθ_v(qT)^d_v)
    (1-^-2iθ_v(qT)^d_v),

where,  for all v∈ P_q(a),  θ_v∈(0,π) is as above (see <ref> and <ref> for a precise definition).

This result is proved by a “point-counting” argument, directly from the definition of L(E_a, γ, T), through manipulations of character sums over finite fields. 
Given the paucity of tables of L-functions of elliptic curves over K of large conductor, such an explicit expression of L(E_a, γ, T) can be of independent interest.

As a by-product, Theorem <ref> yields a closed formula for the analytic rank _T=q^-1L(E_a, γ, T). 
Using that the BSD conjecture is proven for E_a, γ, we recover a result of <cit.> stating that the ranks of E_a, γ(K) are unbounded as a→∞; 
more precisely, we show in Corollary <ref> that:
E_a, γ(K) ≫_q q^a/a. 

From Theorem <ref>, we also derive (Proposition <ref> and (<ref>)) an explicit expression for the special value L^∗(E_a, γ, 1). 
In the notations of (<ref>), we obtain an expression of the shape

    log L^∗(E_a, γ, 1)/b(E_a, γ) 
    = log(positive
    integer)/b(E_a, γ)   + 1/b(E_a, γ)·∑_v∈ P_q(a)logsin^2θ_v.
 
Estimating the first term is straightforward: it is o(1) as a→∞ and thus does not play any role in the asymptotics. 
In order to prove the limit in Theorem <ref>, we have  to investigate the behaviour of the second term: specifically, we need to show that it is o(1) too. 
The size of this term visibly depends on how the angles {θ_v}_v∈ P_q(a) distribute in the interval [0,π].
Since t↦logsin^2 t tends to -∞ 
at 0 and π, this size also depends on how close the angles 
can  be to the endpoints of (0,π).


Therefore we devote sections <ref> and <ref> to studying the distribution of the angles {θ_v}_v∈ P_q(a)  in more detail. 
The two main results in these sections can be stated as follows: 


 Notations being as above,


  * - Theorem <ref>)
For all continuously differentiable functions g:[0, π]→, 

    |1/|P_q(a)|∑_v∈ P_q(a) g(θ_v) 
    - 2/π∫_0^π g(t) ·sin^2 t t|
    ≪_q a^1/2/q^a/4·∫_0^π |g'(t)| t.


  * - Corollary <ref>)
There is a constant c>0 such that min{θ_v, π-θ_v}≥ (q^a)^-c for all v∈ P_q(a). 



By the work of Katz, it is known that the angles of Kloosterman sums become equidistributed  in [0, π] with respect to the Sato–Tate measure (see <cit.>). 
It turns out that the same statement holds for the angles {θ_v}_v∈ P_q(a) (see Theorem <ref>). 
The proof relies on an adaptation of Katz's method in <cit.> and results of Fu and Liu in <cit.>.
This equidistribution result, however, is not sufficient for our purpose: we need a more effective version such as Theorem <ref>(<ref>). 

The effective version (Theorem <ref>) will follow from Theorem <ref> after a more detailed analysis using tools from equidistribution theory (see <cit.>).












The main goal of section <ref> is to prove Theorem <ref>(<ref>); 




we actually prove a more general result there  
(see Theorem <ref>). 
The proof has a diophantine approximation flavour and the main tool 
is a version of Liouville's inequality (as in <cit.>). 







  

Combining the two results in Theorem <ref> and approximating t↦logsin^2 t by sufficiently regular functions,




we prove (Theorem <ref>) that 


    1/|P_q(a)|∑_v∈ P_q(a)logsin^2θ_v 2/π∫_0^πlogsin^2 t ·sin^2 t t  = log(/4).


Theorem <ref> then follows rather easily (see Theorem <ref>), 
since (<ref>) implies that 
the second term in (<ref>) is indeed o(1) as a→∞.  








Finally, section <ref> is devoted to the proof of Theorem <ref> (see Theorem <ref>).



equationsection    


§ THE ARTIN-SCHREIER FAMILY OF ELLIPTIC CURVES E_A, Γ
 




Throughout this article, we fix a finite field _q of characteristic p≥ 3, 

and we denote by K=_q(t) the function field of the projective line ^1/_q.



In this section, we explain in some detail how the curves E_a, γ 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 can confer <cit.>. 
 
For all integers a≥ 1, we let ℘_a(t) = t^q^a-t∈_q[t]. 
For any γ∈_q^× and any a≥ 1, we consider the elliptic curve E_a, γ defined over K by the affine Weierstrass model:

    E_a,γ:   y^2=x(x+16γ)(x+℘_a(t)^2).


The sequence {E_a, γ}_a≥ 1 is called an Artin-Schreier family of elliptic curves over K. 
This terminology comes from the following observations. 
Let E_γ/K be the elliptic curve given by y^2=x(x+16γ)(x+t^2). Then, for all a≥ 1, the curve E_a, γ is the pullback of E_γ under the Artin-Schreier map t↦℘_a(t) of ^1_/_q. 
Hence, studying E_a, γ over K=_q(t) is equivalent to studying E_γ over the Artin-Schreier extension _q(u_a) of _q(t), where ℘_a(u_a)=t.



We remark that E_a, γ is “almost” a Legendre curve. 
More precisely, in the setting of <cit.>, the curve E_a, γ can be obtained as follows. 
Starting from the Legendre elliptic curve E_0, γ/K given by

    E_0, γ :    y^2 = x(x-1)(1-(16γ)^-1· t^2),

one takes a quadratic twist by -(16γ)^-1, obtaining E'_0, γ defined by y^2 = x(x+1)(1+(16γ)^-1· t^2). 
Pulling back E'_0, γ along ℘_a:^1→^1, one recovers the curve E_a, γ defined by (<ref>).




  

These elliptic curves E_a, γ were studied in <cit.> where, among others, it was shown that they satisfy the BSD conjecture (see <ref>). 
In their paper, Pries and Ulmer describe an alternative construction of the curves E_a, γ, which we now recall for later use. 
For any γ∈_q^×, we let f_γ:^1_/K→^1_/K be the map [x_0:x_1]↦ x_0/x_1+γ x_1/x_0. 
Consider the curve Z_a, γ⊂^1_/K×^1_/K defined over K by 

    f_γ([x_0:x_1])-f_γ([y_0:y_1]) =℘_a(t),

in the  ([x_0: x_1], [y_0:y_1])-coordinates  on ^1_/K×^1_/K. 
This curve is smooth of  genus 1 and admits one K-rational point ([0:1], [0:1]). 
The curve Z_a, γ is given in the affine (x,y)-coordinates  on 𝔸^2_/K by 
 
    (x-y)(xy-γ)=℘_a(t)· xy.

 The change of coordinates
 (x, y)↦ (u,v)=( -γ·℘_a(t) -x+y/x-y, γ℘_a(t)·y(℘_a(t)+y-x) +2γ/x^2-y^2)
 then brings Z_a, γ into the affine Weierstrass form

    E^∘_a, γ:     v^2 -℘_a(t)· uv = u^3-2γ· u^2 +γ^2· u.

One finally passes from E^∘_a, γ to E_a, γ by means of the 2-isogeny  ϕ:E^∘_a, γ→ E_a, γ given by

    (u, v)↦ (x, y)=( 4v(v-℘_a(t)u)/u^2, 4(2v-℘_a(t)u)(u^2-γ^2)/u^2).

  
From the model (<ref>), it is straightforward to compute the j-invariant j(E_a, γ) of E_a, γ and obtain that

    j(E_a, γ) = (℘_a(t)^4 - 16γ·℘_a(t)^2 + 2^8 γ^2)^3/γ^2 ·℘_a(t)^4·(℘_a(t)^2 - 16γ)^2∈ K.

As a rational function of t, the j-invariant j(E_a, γ) is visibly nonconstant and separable. 
In particular, the curve E_a, γ is not isotrivial.




 §.§ Bad reduction and invariants



For any place v of K, we denote by d_v or v the degree of v and _v the residue field at v (an extension of _q of degree d_v). 
We identify finite places of K with monic irreducible polynomials in _q[t];
we also identify the residue field at v≠∞ with _q[t]/(B_v) if B_v∈_q[t] is the monic irreducible polynomial corresponding to v. 

Let us describe the reductions of E_a, γ at places of K and compute its relevant invariants. 
A straightforward computation of the discriminant of the model (<ref>) of E_a, γ gives that

    Δ = 2^12γ^2 ·℘_a(t)^4·(℘_a(t)^2 - 16γ)^2.

The finite places of bad reduction of E_a, γ are then the monic irreducible divisors of Δ in _q[t]. 
By a routine application of Tate's algorithm (see <cit.> for instance), one can give a more precise description:


Let Z_a, γ be the set of places of K that divide ℘_a(t)·(℘_a(t)^2-16γ). 
Then E_a, γ has good reduction outside S= Z_a, γ∪{∞}.
The reduction of E_a, γ at places v∈ S is as follows:


 
Place v of K      Fiber of E_a, γ at v      _vΔ_min(E_a, γ)     _v(E_a, γ)  

v |℘_a(t)      I_4       4     1  
 v |℘_a(t)^2-16γ      I_2          2    1 
  ∞        I_4q^a          4q^a     1  
 


  

In this table, for all places  v   of bad reduction for E_a, γ, we have denoted by _vΔ_min(E_a, γ) (resp. _v(E_a, γ)) the valuation at v of the minimal discriminant of E_a, γ (resp. of the conductor of E_a, γ). 
See <cit.>, <cit.> for the definitions of these local invariants. 




  

From this local information, one deduces the values of the following global invariants (we refer to <cit.> for their definition). 
The minimal discriminant divisor Δ_min(E_a, γ) 
has degree Δ_min(E_a, γ) = 12 q^a, 
and the conductor (E_a, γ)∈Div(^1) has degree (E_a, γ) = 3q^a+1.
Indeed, since both ℘_a(t) and ℘_a(t)^2-16γ are squarefree in _q[t], one has 

    ∑_v|℘_a(t) v = ℘_a(t) = q^a    and   ∑_v|℘_a(t)^2-16γ v =  (℘_a(t)^2-16γ) = 2q^a.

Hence the exponential differential height H(E_a, γ)  is q^q^a since, by definition (see 2 in <cit.>), it is given by H(E_a, γ) := q^(Δ_min(E_a, γ))/12.
Summarising these calculations, we have 

    Δ_min(E_a, γ) = 12 q^a,   (E_a, γ) = 3q^a+1,    and    
    H(E_a, γ) = q^q^a.



As is clear from (<ref>) and Proposition <ref>, the discriminant Δ 

of the Weierstrass model (<ref>) 
has the same valuation as Δ_min(E_a, γ) at all finite places of K. 
Therefore, the model (<ref>) is a minimal integral model of E_a, γ at all places v≠∞ of K. 





 §.§ Definitions of L-function, analytic rank and special value



For any place v of K, with degree d_v and residue field _v, we denote by (E_a, γ)_v the reduction modulo v of a minimal integral model of E_a, γ at v: (E_a, γ)_v is thus a plane cubic curve over _v.
By definition, the L-function of E_a, γ is the power series in T given by

    L(E_a, γ, T) = ∏_v good (1 - a_v · T^d_v + q^d_v·  T^2d_v)^-1·∏_v bad (1 - a_v ·  T^d_v)^-1∈[[T]],

where the products are over places of K where E_a, γ has good (resp. bad) reduction, and where 

    a_v := 
    q^d_v +1 - |(E_a, γ)_v(_v)|
        if E_a, γ has good reduction at v, 
    
    0     if E_a, γ has additive reduction at v, 
     
    +1    if E_a, γ has split multiplicative reduction at v, 
     
    -1    if E_a, γ has nonsplit multiplicative reduction at v.

We refer to <cit.> and <cit.> for more details. 

Since E_a, γ is not isotrivial, a deep theorem of Grothendieck shows that L(E_a, γ, T) is actually a polynomial in T with integral coefficients whose degree is denoted by b(E_a, γ). 
Further, by the Grothendieck-Ogg-Shafarevich formula and our computation of the degree of (E_a, γ) (see (<ref>)),    
we know that

    b(E_a, γ) =   L(E_a, γ, T) = (E_a, γ) -4 = 3(q^a-1).

In section <ref>, we will compute the polynomial L(E_a, γ, T) explicitly. For now, we only note that it makes sense to define the following two quantities: 


Let ρ(E_a, γ) be the analytic rank of E_a, γ , the multiplicity of T=q^-1 as a root of L(E_a, γ, T). 
Further, define the special value of L(E_a, γ, T) at T=q^-1 to be

    L^∗(E_a, γ,1) := .L(E_a, γ, T)/(1-qT)^ρ|_T=q^-1∈[q^-1]∖{0},    where ρ = ρ(E_a, γ).




The special value L^∗(E_a, γ, 1) is “usually” defined as the first nonzero coefficient in the Taylor expansion around s=1 of the function s↦ L(E_a, γ, q^-s). 
Our definition (<ref>) differs from that more “usual” one by a factor (log q)^ρ. 
We prefer to use the normalisation (<ref>) because it ensures that L^∗(E_a,γ, 1)∈^∗. 
This choice is consistent with our normalisation of (E_a, γ), see <ref> below.





 §.§ The BSD conjecture



The Mordell–Weil theorem implies that E_a, γ(K) is a finitely generated abelian group ( <cit.>). 
Since the canonical Néron–Tate height h:E_a, γ(K) → is quadratic, it induces a -bilinear pairing ⟨·,·⟩: E_a, γ(K)× E_a, γ(K)→, which is nondegenerate modulo E_a, γ(K) ( <cit.>).
We can then define the Néron-Tate regulator of E_a, γ by 

    (E_a, γ) := |( ⟨ P_i, P_j ⟩)_1≤ i,j ≤ r| ∈^∗,

for any choice of a -basis P_1, …, P_r ∈ E_a, γ(K) of E_a, γ(K)/E_a, γ(K). 
Note that we normalise ⟨·,·⟩ to have values in : we may do so since, in our context, this height pairing has an interpretation as an intersection pairing on the minimal regular model of E_a, γ (see <cit.>).    

Let us also recall that the Tate–Shafarevich group of E_a, γ/K is defined by

    (E_a, γ) :=
    ( ^̋1(K, E_a, γ) ⟶∏_v^̋1(K_v, (E_a, γ)_v)),


see <cit.> for more details. 
In Theorem <ref> right below, we will see that (E_a, γ) is finite. 
 
It has been conjectured by Birch, Swinnerton-Dyer and Tate that the “analytic” quantities ρ(E_a, γ) and L^∗(E_a, γ, 1) have an arithmetic interpretation (see <cit.>). 
Even though this conjecture is still open in general, it has been proved by Pries and Ulmer for E_a, γ in <cit.>. 
Let us state their result as follows:

[Pries - Ulmer]
For all γ∈_q^× and all integers a≥ 1, the elliptic curve E_a, γ/K satisfies the full Birch and Swinnerton-Dyer conjecture. That is to say,


  * The Tate–Shafarevich group (E_a, γ) is finite.

  * The rank of E_a, γ(K) is equal to ρ(E_a, γ)=_T=q^-1L(E_a, γ, T).

  * Moreover, one has

    L^∗(E_a, γ, 1) 
    = |(E_a, γ)| ·(E_a, γ)/H(E_a, γ)·(E_a, γ)· q/|E_a, γ(K)|^2,

where (E_a, γ) denotes the Tamagawa number of E_a, γ.



     
    We only sketch a proof and refer the interested reader to <cit.> for more details.
    As we have seen at the beginning of this section, E_a, γ is 2-isogenous to E^∘_a, γ. 
    Since E_a, γ and E^∘_a, γ are linked by an isogeny of degree prime to the characteristic of K, Theorem 7.3 in <cit.> implies that the BSD conjecture holds for E_a, γ if and only if it does for E^∘_a, γ. 
    
    
    Hence Theorem <ref> will follow 
    if we prove 
    that E^∘_a, γ satisfies the BSD conjecture.
    
    
    We have also shown that E^∘_a, γ is birational to the curve Z_a, γ⊂^1×^1 which,
    by construction,  is given in affine coordinates  by an equation of the form
    f_γ(x)-f_γ(y)=℘_a(t)
    where f_γ is a certain rational function on ^1 over K and ℘_a(t)∈_q[t] is a separable additive polynomial. 
    Under these conditions, Corollary 3.1.4 of <cit.> proves that E^∘_a, γ satisfies the BSD conjecture. 
    
    The crucial point of the proof is the following:  given the specific shape of the equation of Z_a, γ to which E^∘_a, γ is birational, the minimal regular model ^∘_a, γ→^1 over _q of the curve E^∘_a, γ/K is dominated by a product of curves _a, γ×_a, γ^∘_a, γ over _q (where _a, γ is actually the curve defined in Remark <ref> below). 
    The Tate conjecture (T) asserts that the order of the pole of the zeta function of a surface S/_q equals the rank of the Néron–Severi group of S (see <cit.>, or  10–13 in <cit.>).
    Conjecture (T) is proved for surfaces that are dominated  by products of curves. 
    In particular, (T) holds for the surface ^∘_a, γ/_q. 
    On the other hand, it is known that  conjecture (T) for  ^∘_a, γ/_q is equivalent to the BSD conjecture for the generic fiber of ^∘_a, γ→^1 , for the elliptic  curve E^∘_a, γ/K (Theorem 8.1 in <cit.>).
    Hence the result.
    

In section <ref> we give bounds on the special value L^∗(E_a, γ, 1) on the left-hand side of (<ref>) and, in section <ref>, 
we deduce from these an estimate on the asymptotically significant quantities on the right-hand side of (<ref>).
For completeness, let us recall the following bounds (which we will need to prove Theorem <ref>).
 
Let E be a nonisotrivial elliptic curve over K. Then
2


  * |E(K)| ≪_q 1,

  * log(E) = o( log H(E)) as H(E)→∞.




     
    The first bound is the analogue for elliptic curves over  K=_q(t) of Merel's uniform bound on torsion for elliptic curves over . 
    There are several proofs of (i) and we refer the reader to <cit.> for a survey and a sketch of proof (by a modular method). 
    The bound (ii) on the Tamagawa number is a consequence of 
    Theorem 1.22 of <cit.> 
    in the case when E is semistable or p>3.
    A self-contained (and elementary) proof for all elliptic curves over K can also be found in <cit.>.
    





§ CHARACTERS AND KLOOSTERMAN SUMS



The goal of this section is to introduce the objects which appear in the L-function of E_a, γ.

We fix a finite field field _q of odd characteristic and  a nontrivial additive character ψ_q on _q, which we assume to take values in the cyclotomic field (ζ_p).
For instance, a standard choice of ψ_q is the map 
ψ_q: x↦ζ_p^__q/_p(x) where ζ_p is  a primitive p-th root of unity and __q/_p:_q→_p is the trace map.
 
 
For any finite extension  of _q, we denote by _/_q : →_q the relative trace and we “lift” ψ_q to a nontrivial additive character ψ_:→(ζ_p)^× on  by putting ψ_ := ψ_q∘_/_q.




 §.§ Kloosterman sums
 


For a finite field  of odd characteristic p, a nontrivial additive character ψ on  with values in the cyclotomic field (ζ_p), and a parameter α∈^×, we define the Kloosterman sum _(ψ;α) by:

    _(ψ; α) := - ∑_x∈^×ψ(x+α/x).

As a sum of p-th roots of unity, _(ψ;α) is an algebraic integer in (ζ_p). 
For our purpose, it is convenient to normalise the sum by a -1 sign.
Let us gather in one proposition several classical facts about the Kloosterman sums that will be useful in this article.


 Let , ψ and α be as above. Then:

0.1em 

  *  _(ψ; α) is a totally real algebraic integer in (ζ_p) , _(ψ;α)∈[ζ_p+ζ_p^-1].


  *  _(ψ;α) satisfies “Salié's formula”: 

    _(ψ; α) = - ∑_y∈λ(y^2-4α) ·ψ(y),

where λ:^×→{±1} is the unique multiplicative character on ^× of exact order 2 (extended by λ(0):=0 to the whole of ).


  *  If  contains _q, one has _(ψ_q∘_/_q;α) = _(ψ_q∘_/_q; α^q).


  *  
There exist two algebraic integers _(ψ;α) and '_(ψ;α) such that for any finite extension '/, 

    _(ψ; α)·'_(ψ; α) = ||   and   _'(ψ∘_'/;  α) = _(ψ; α)^[':] + '_(ψ; α)^[':].

The pair {_(ψ;α), '_(ψ;α)} is uniquely determined by , ψ, α.

  *  
 _(ψ;α) and '_(ψ;α) have magnitude ||^1/2 in any complex embedding.  


  *  In any complex embedding of (ζ_p), one has

    0 < |_(ψ;α)| < 2||^1/2.






The reader can confer <cit.>  and <cit.> for proofs of these 
classical results about Kloosterman sums: (<ref>) and (<ref>) are easily checked; 
items (<ref>),  (<ref>) and (<ref>) 
are Theorems 5.47, 5.43 and 5.44 in  <cit.>, respectively; (<ref>) is proved in Corollary 3.2 of  <cit.>. 






 §.§ The sums _γ(v)
 


Assume a parameter γ∈_q^× is given. 
A place v≠ 0, ∞ of K with degree d_v 
corresponds to a monic irreducible polynomial B_v ∈_q[t] of degree d_v, with B_v≠ t. 
Choose a root β_v∈_̅q̅^× of B_v: we claim that the value of the Kloosterman sum __v(ψ__v; γβ_v^2) does not depend on the choice of β_v. 
Indeed, given one such β_v the d_v-1 other choices are of the form β_v^q^j (with j∈{1, 2, …, d_v-1}) because the d_v different roots of B_v in _̅q̅ are all conjugate under the action of the Galois group (_v/_q). 
A repeated application of  Proposition <ref>(<ref>) proves the claim.
Therefore, the following definition makes sense:  

 
Let _q be a finite field of characteristic p, ψ_q be a nontrivial additive character on _q and γ∈_q^×. For any place v≠ 0, ∞ of K=_q(t) corresponding to a monic irreducible B_v∈_q[t], we let 

    _γ(v) := __v(ψ__v;γβ_v^2) = -∑_x∈_v^×ψ_q∘__v/_q(x+γ·β_v^2/x),

for any choice of β_v∈_̅q̅^× such that B_v(β_v)=0. 

Note that _γ(v) depends on _q and ψ_q,  but we chose not to include these in the notation for brevity.



  

For any place v≠0, ∞, Proposition <ref>(<ref>)-(<ref>) shows that there exist a unique pair {_γ(v),'_γ(v)} of conjugate algebraic integers, which have magnitude |_v|^1/2 = q^d_v/2 in any complex embedding and such that

    _γ(v) 
    = _γ(v) +'_γ(v).

In other words, we denote by {_γ(v),'_γ(v)} the pair of algebraic integers {__v(ψ__v;γβ_v^2), '__v(ψ__v;γβ_v^2)}. 

 
These sums _γ(v) appear in the zeta function of a curve over _q. 
Namely, consider the hyperelliptic curve _a, γ over _q defined as a smooth projective model of the affine curve ℘_a(y)= x+γ/x. 
A computation, which probably goes back to Weil, shows that the zeta function of _a, γ/_q is given by 

    Z(_a, γ/_q; U) = ∏_v(1- _γ(v)· U^d_v)(1- '_γ(v)· U^d_v) /(1-U)(1-q· U),

where the product is over all places v≠ 0, ∞ of K whose degrees divide a (see <cit.>). 

 



 §.§ Number of solutions of “Artin-Schreier” equations


In the process of computing the L-function of E_a, γ, we will need the following “counting lemma”:

 
Let _q be a finite field and ψ_q be a nontrivial additive character on _q. 
For any finite extension   of _q and  any z∈, one has

    |{τ∈ : ℘_a(τ)=z}| 
    =∑_β∈_q^a∩ψ_(β· z).



    
     For the duration of the proof, we write =_q^n and we let b=(a,n); note that _q^b=_q^a∩_q^n. 
     The map ℘_a:_q^n→_q^n is an _q^b-linear endomorphism of _q^n whose kernel is _q^b. 
     In particular, the image of ℘_a must have dimension __q^b(_q^n)-1=n/b-1.
    The trace t := _q^n/q^b : _q^n→_q^b is a surjective _q^b-linear map, so that its kernel H is a sub-_q^b-vector space of _q^n of dimension n/b -1. 
    Recall that t∘℘_a = 0 on _q^n, thus we have Im ℘_a⊂ H. 
    Since these two subspaces have the same dimension, they coincide. 
    This shows that, for z∈_q^n, 
    
    |{τ∈_q^n : ℘_a(τ)=z}| 
        = 
        0    if  t(z)≠0, 
     
        |℘_a| = q^b      if  t(z)=0.

    On the other hand, for a given z∈_q^n, we notice that
    
    ∑_β∈_q^bψ__q^n (β· z)
           =∑_β∈_q^bψ__q^b∘_q^n/q^b (β· z)
        =∑_β∈_q^bψ__q^b∘( β·_q^n/ q^b(z)) 
        = 
        0    if  t(z)≠0, 
     
        q^b      if  t(z)=0,

    because t=_q^n/q^b is _q^b-linear. 
    Combining the two displayed equalities, we obtain the result.
    




 §.§ Preliminary estimates



For all n≥1, we denote by π_q(n) the number of places v≠ 0, ∞ of K of degree d_v=n. 
Equivalently, π_q(n) is the number of closed points of degree n of the multiplicative group _m over _q , the number of orbits of _m(_̅q̅) under the action of (_̅q̅/_q) which have cardinality n.
We also let P_q(n) be the set of closed points of _m whose degree divides n. 
Clearly, one has |P_q(n)| = ∑_d| nπ_q(d). 

In what follows, we will frequently need the following estimates, which we record here for convenience.

 Given a finite field _q, one has 
2


  *  
q^n ≪_q n·π_q(n) ≪_q q^n for all n≥ 1.

  *  
q^n ≪_q n· |P_q(n)| ≪_q q^n for all n≥ 1.


The involved constant are all effective and depend at most on q.


     
    The  “Prime Number Theorem” for _q[t] directly implies (<ref>). 
    Being a bit more careful, one can even give explicit constants (as in <cit.> for example). 
    From this estimate and the relation between π_q(n) and |P_q(d)|, one easily deduces (<ref>). 
    





§ THE L-FUNCTION

   

For any integer a≥ 1, we denote by P_q(a) the set of places v of K, with v∉{0, ∞}, whose degrees d_v   divide a. 
 In the identification between finite places of K and monic irreducible polynomials, P_q(a) corresponds to {B∈_q[t] : B  monic, irreducible s.t.  B| a  and B≠ t }.
Equivalently, P_q(a) is the set of closed points on _m whose degree divides a.

 

  
 We can now state the first main result of this article:
 
  Let _q be a finite field of odd characteristic p and K=_q(t). For any γ∈_q^× and any integer a≥ 1, consider the elliptic curve E_a, γ/K given by (<ref>). 

 The L-function of E_a, γ  is given by 

    L(E_a, γ, T) 
    = ∏_v∈ P_q(a)(1-q^d_v· T^d_v) (1-_γ(v)^2 · T^d_v)(1- '_γ(v)^2 · T^d_v),

 where 
 _γ(v), '_γ(v) are the algebraic integers associated to _γ(v) (see (<ref>) 
 and (<ref>)).
 
 The proof of this theorem occupies the rest of the present section. 
Our strategy is loosely based on the computation in <cit.>: to give an expression of L(E_a, γ, T), 
we rely on an explicit “point-counting” argument with character sums.
This requires proving an identity between the character sums that appear in the argument and the Kloosterman sums introduced above. 
We first give in the next subsection a proof of this identity, and then prove Theorem <ref> in <ref>.
 



 

 Before proving this theorem, we note the following:

0.1em

  * Even though the sums _γ(v) for v∈ P_q(a) depend on a choice of nontrivial additive character ψ_q on _q, the L-function L(E_a, γ, T)∈[T] does not. 
Indeed, changing the choice of ψ_q amounts to permuting the factors in (<ref>).


  * Note that ∑_v∈ P_q(a) d_v = |_m(_q^a)| = q^a-1. 
 Thus, as a polynomial in T, the L-function L(E_a, γ, T) has degree 3(q^a-1) = (E_a, γ)-4. This is consistent with the expected degree  (see <ref>).







  * For any integer a≥ 1, the L-function of the base change of  E_a, γ to K_a:=_q^a(t) admits a somewhat simpler expression.
Indeed, for all γ∈_q^× and a≥ 1, the L-function of E_a, γ/K_a is given by 

    L(E_a, γ/K_a, T) 
    = ∏_β∈_q^a^×(1-q^a· T) (1-(_β)^2 · T)(1- ('_β)^2 · T),

 where, for all β∈_q^a^×,   _β and  '_β are the two algebraic integers associated to the Kloosterman sum __q^a(ψ__q^a;γβ^2).  
 This follows directly from (<ref>).



  * Recall the elliptic curves E^∘_a, γ introduced at the beginning of <ref> and given by (<ref>). 
Since isogenous elliptic curves share the same L-function (by <cit.>), Theorem <ref> also shows that the L-function of E^∘_a, γ is given by (<ref>). 


  * It is perhaps illuminating to comment on the appearance of Kloosterman sums in L(E_a, γ, T). 

Choosing a prime ℓ≠ p, we denote by ^̋i(X)=^̋i_ét(X ×__q_̅q̅, _ℓ) the i-th ℓ-adic étale cohomology group of a smooth projective variety X/_q.
Let ^∘_a, γ→^1 denote the minimal regular model of E^∘_a, γ. 
By Grothendieck cohomological interpretation of L-functions, L(E^∘_a, γ, T) is  essentially the “interesting part” of the zeta function of the surface _a, γ/_q , L(E^∘_a, γ, T) is a factor of the characteristic  polynomial of the Frobenius _q acting on H^2(^∘_a, γ).


By the construction of E^∘_a, γ in <ref>, ^∘_a, γ is a smooth model of a quotient of _a, γ×_a, γ by the action of a certain finite group G_a, where _a, γ/_q is the curve introduced in  Remark <ref>   (see <cit.> for a more detailed presentation). 
In particular, ^̋2(^∘_a, γ) can be seen as a subspace of ^̋2((_a, γ×_a, γ)/G_a), itself a subspace of ^̋2(_a, γ×_a, γ). 
Hence, by Künneth's formula, L(E^∘_a, γ, T) divides the characteristic  polynomial of _q acting on ^̋1(_a, γ) ⊗^̋1(_a, γ).
 
As was noted in Remark <ref>, the numerator of the zeta function of _a, γ , the characteristic polynomial of _q acting on ^̋1(_a, γ), involves Kloosterman sums _γ(v).
Hence, the eigenvalues of _q acting on ^̋1(_a, γ) ⊗^̋1(_a, γ) are products of the form _γ(v_1)_γ(v_2). 
Being a factor of the characteristic polynomial of _q acting on ^̋1(_a, γ) ⊗^̋1(_a, γ), the L-function L(E^∘_a, γ, T) has to involve some of the products _γ(v_1)_γ(v_2).

Working out the details of this 
sketchy computation could lead to a different proof of Theorem <ref>.







 §.§ An identity between character sums



 For any finite finite field  of odd characteristic, we fix a nontrivial additive character ψ on , and we denote by λ:^×→{±1} the unique quadratic character on ^×, extended to the whole of  by λ(0):=0.  

 For any γ∈^× and β∈, we consider the following double character sum
 
    M_(β, γ) := ∑_x∈∑_z∈λ( x(x+16γ)(x+z^2))·ψ(β z),

  Up to a trivial term, we identify the character sum M_(β, γ) as the square of a Kloosterman sum:
 

 Notations being as above, we have
 
    M_(β, γ) =
    
    1    if β=0 
    _(ψ ; γ·β^2)^2-||
       if β≠0.



In order to prove this identity, we begin by recording the following “point-counting” lemma:


 
 For any z∈ and γ∈^×, consider 

    X_, γ(z) :={ (u,v)∈^2 : v^2 - (u^2-uz-4γ) v + γ z^2 = 0}⊂^2.


We have



    | X_, γ(z) | = ||· (1+δ_z,0)- 1  + ∑_x∈λ(x(x+16γ)(x+z^2)),

where δ_z,0=1 if z=0 and δ_z,0=0 otherwise.

  
     We split the set X_, γ(z) into two disjoint parts X_0:={(u,v)∈ X_, γ(z)  : v=0} and X_1 := X_, γ(z) ∖ X_0.
    Computing |X_0| is straightforward: if z=0, any pair (u,0) with u∈  belongs to X_0 and, if z≠0, no such pair belongs to X_, γ(z). 
    Therefore, we have |X_0| = ||·δ_z,0.
    
    
    
    
    
    To count the number of elements in X_1, let us introduce an auxiliary set
    
    Y_, γ(z) := {(x,y)∈^2 : y^2=x(x+16γ)(x+z^2)}⊂^2,

    which we also split into two parts: 
    Y_0:={ (x,y)∈ Y_, γ(z): x=0} and Y_1 := Y_, γ(z) ∖ Y_0.
    The two maps
    
    [                               X_1                ⟶              Y_1          and   .                               Y_1                ⟶              X_1;                             (u,v)                ⟼   (4v, 4v(2u-z))                                              (x,y)                ⟼ (y+z x/2x, x/4). ]

    are easily checked to be well-defined and inverse to each other. Thus, we have |X_1|=|Y_1| and the Lemma will be proved once we have related |Y_1| to the character sum in (<ref>). Grouping points (x,y)∈ Y_1 according to their x-coordinates, we obtain that:
    
    |Y_1|
        = ∑_x∈^×( 1+ λ(x(x+16γ)(x+z^2)) )
    
        = || -1 + ∑_x∈λ(x(x+16γ)(x+z^2)).

    
    
    
    
    
    
    Since |X_, γ(z)|  =   |X_0| + |Y_1|, we conclude that relation (<ref>) holds.
    

     
    We treat the case where β=0 first: we have to prove that
    
    ∑_x∈∑_z∈λ( x(x+16γ)(x+z^2)) = 1.

    This identity follows easily upon using the following equality (see <cit.>):
    
    ∀ b,c∈,   ∑_y∈λ(y^2 + b y +c) = 
        ||-1    if  b^2=4c, 
     
         -1    otherwise.

    It remains to prove the Proposition in the case where β≠ 0.
    If β≠ 0, the character x↦ψ(β x) is nontrivial, and we can
    expand _(ψ, γβ^2)^2 as a double character sum  using Salié's formula (<ref>): we obtain that
    
    _(ψ, γβ^2)^2
           = (∑_u∈λ(u^2-4γ)·ψ(β u))^2
        = ∑_u_1∈∑_u_2∈λ((u_1^2-4γ)(u_2^2-4γ))·ψ(β(u_1+u_2)) 
       = ∑_z∈(∑_u∈λ((u^2-4γ)((u-z)^2-4γ)))·ψ(β z).

    The last line follows from the first by letting (u, z) =(u_1, u_1+u_2).  
    For a given z∈, notice that
    
    ∀ u ∈,      (u^2-4γ)((u-z)^2-4γ) = (u^2-zu-4γ)^2 - 4γ z^2,

    which is the discriminant of the quadratic equation
    V^2 -(u^2-zu-4γ)· V + γ z^2=0
    with unknown V. Hence, the number of solutions v∈ to this equation is given by
    
    |{ v∈  : v^2 -(u^2-zu-4γ)· v +γ z^2=0 }| 
        = 1 +λ( (u^2-zu-4γ)^2 - 4γ z^2).

    Therefore we can  rewrite the inner sum (over u∈) in (<ref>) as
     
    ∑_u∈λ((u^2-4γ)((u-z)^2-4γ))
           = -||+∑_u∈(1+ λ( (u^2-zu-4γ)^2 - 4γ z^2) ) 
       = - || +∑_u ∈| { v∈  : v^2 -(u^2-zu-4γ)· v +γ z^2=0 }| 
       = -||+| X_, γ(z)|,

    where X_, γ(z) is the set introduced in Lemma <ref>. In that same Lemma <ref>, we have proved that 
    
    ∀ z∈,     | X_, γ(z)| - ||  = ||·δ_z,0- 1  + ∑_x∈λ(x(x+16γ)(x+z^2)).

    On multiplying this identity by ψ(β z) and summing over all z∈, we deduce from (<ref>) that
    
    _(ψ; γβ^2)^2 
        = ∑_z∈ (||·δ_z,0- 1)ψ(β z) 
        + M_(β, γ) 
        = || + M_(β, γ).

    Indeed, the sum ∑_z∈ψ(β z) vanishes because x↦ψ(β x) is nontrivial. This conclude the proof. 
    
    



 §.§ Proof of Theorem <ref>


 
We start by giving a “concrete” expression of L(E_a, γ, T) in terms of the number of rational points on the reductions of E_a, γ at places of K. To that end, we introduce the following notations: for any n≥ 1 and any τ∈^1(_q^n), denote by v_τ the place of K corresponding to τ and by (E_a, γ)_τ the reduction of a integral minimal model of E_a, γ at v_τ (a cubic plane curve over _v_τ, not necessarily smooth). 

We then let

    A_a, γ(τ, q^n) := q^n+1 - |(E_a, γ)_τ(_q^n)|.

With these notations, we have:

The L-function of E_a, γ is given by 

    log L(E_a, γ, T) = ∑_n=1^∞(∑_τ∈^1(_q^n)A_a, γ(τ, q^n)) ·T^n/n.



     Starting from the definition (<ref>) of the L-function of E_a, γ,  expanding log L(E_a, γ/K, T) as a power series in T and rearranging terms yields the desired  expression of L(E_a, γ, T). 
    
    See <cit.> or <cit.> for more details. 
    


The next step is to find a more tractable expression of the inner sums in (<ref>).
For any finite extension _q^n of _q, we again let λ :_q^n^×→{±1} 

be the unique nontrivial character of _q^n^× of order 2. 

For any τ∈^1(_q^n), we choose an affine model y^2=f_τ(x) of (E_a, γ)_τ 

(with f_τ(x)∈_v_τ[x] of degree 3): 
a standard computation then yields that 

    A_a, γ(τ, q^n)  
    = q^n+1 - |(E_a, γ)_τ(_q^n)| 
    = q^n - ∑_x∈_q^n(1 + λ( f_τ(x)) ) 
    =- ∑_x∈_q^nλ( f_τ(x)).

Since E_a, γ has split multiplicative reduction at ∞ (see Proposition <ref>), we have A_a, γ(∞, q^n) = 1.
Moreover, by Remark <ref>,  we may choose f_τ(x) = x(x+16γ)(x+℘_a(τ)^2) for any τ∈_q^n.
Summing identity 
(<ref>) over all τ∈_q^n for this choice of f_τ(x), we obtain that

    - ∑_τ∈_q^nA_a, γ(τ, q^n)
    =  ∑_x∈_q^n∑_τ∈_q^nλ( x(x+16γ)(x+℘_a(τ)^2)).

We know from Lemma <ref> 
that, for any z∈_q^n, 

    |{τ∈_q^n : ℘_a(τ)=z}| 
    =∑_β∈_q^a∩_q^nψ_q^n 
    (β· z).

Therefore, we deduce that

    - ∑_τ∈_q^nA_a, γ(τ, q^n)
       =  ∑_x∈_q^n∑_z∈_q^n|{τ∈_q^n : ℘_a(τ)=z}| ·λ( x(x+16γ)(x+z^2)) 
       = ∑_β∈_q^a∩_q^n( ∑_x∈_q^n∑_z∈_q^nλ( x(x+16γ)(x+z^2))·ψ_q^n(β· z) )
    = 
     ∑_β∈_q^a∩_q^n M_q^n(β, γ),

where M_q^n(β, γ) is the double character sum that we studied in subsection <ref> (with =_q^n).

Proposition <ref>, together with our computation of A_a, γ(∞, q^n), then leads to
 
    ∑_τ∈^1(_q^n)A_a, γ(τ, q^n)
    = - ∑_β∈(_q^a∩_q^n)∖{0}(_q^n(ψ_q^n, γβ^2)^2 - q^n).



For each β∈_q^n^×, Proposition <ref>(<ref>) proves the existence of two algebraic integers __q^n(ψ_q^n; γβ^2) and '__q^n(ψ_q^n; γβ^2) whose product is |_q^n|=q^n and whose sum is _q^n(ψ_q^n; γβ^2). We thus have 

    _q^n(ψ_q^n, γβ^2)^2 - q^n 
    = (_q^n(ψ_q^n; γβ^2) + '_q^n(ψ_q^n; γβ^2))^2 -q^n
    = _q^n(ψ_q^n; γβ^2)^2 + '_q^n(ψ_q^n; γβ^2)^2 + q^n.

Hence, for all n≥ 1, (<ref>) can be rewritten as  
    ∑_τ∈^1(_q^n)A_a, γ(τ, q^n)
    = - ∑_β∈(_q^a∩_q^n)∖{0}(_q^n(ψ_q^n; γβ^2)^2 + '_q^n(ψ_q^n; γβ^2)^2 + q^n)



For any β∈_q^a^×, we let d_β := [_q(β):_q] be the degree of β over _q. Such a β also belongs to _q^n if and only if d_β divides n , if and only if _q^n is an extension of _q^d_β. Therefore, by Proposition <ref>(<ref>) one has

    _q^n(ψ_q^n, γβ^2)
    = _q^n(ψ_q^d_β∘_q^n/q^d_β;γβ^2)
    = _q^d_β(ψ_q^d_β; γβ^2)^n/d_β + '_q^d_β(ψ_q^d_β; γβ^2)^n/d_β.

For brevity, we temporarily write 
_1(β) := _q^d_β(ψ_q^d_β; γβ^2)^2, 
_2(β) := '_q^d_β(ψ_q^d_β; γβ^2)^2 and
_3(β) := q^d_β.  



We deduce from (<ref>) that

    ∑_τ∈^1(_q^n)A_a, γ(τ, q^n)
    = - ∑_β∈(_q^a∩_q^n)∖{0}(_1(β)^n/d_β + _2(β)^n/d_β +_3(β)^n/d_β).

Upon multiplying both sides of this identity by T^n/n and summing over all integers n≥1, we arrive at


    - log L(E_a, γ, T)
    
    =   ∑_n≥ 1(∑_β∈(_q^a∩_q^n)∖{0}(_1(β)^n/d_β + _2(β)^n/d_β +_3(β)^n/d_β))·T^n/n,

and exchanging the order of summation leads to

    - log L(E_a, γ, T)
       = ∑_β∈_q^a^×(
    ∑_m≥ 1(_1(β)^m + _2(β)^m +_3(β)^m)·T^m d_β/m d_β)
       = ∑_β∈_q^a^× - 1/d_β·log(∏_i=1^3 (1-_i(β)· T^d_β) )


Moreover, as we have explained in <ref>, for each β∈_q^a^× with degree d_β,  the triple {_1(β), _2(β), _3(β)}

 is constant along the Galois orbit {β, β^q, …, β^q^d_β-1} , the closed point of _m corresponding to β. Consequently, 
for a closed point v of _m whose degree divides a, we may define _i(v) to be _i(β) for any choice of β∈ v.
Each term log(1-_i(v)T^d_v) (for i=1, 2, 3) appears d_v=d_β times in (<ref>) and we may thus rewrite the sum over β∈_q^a^× there as a sum over closed points of _m whose degrees divide a:

    log L(E_a, γ, T)
    =  ∑_ v∈ P_q(a)log(∏_i=1^3 (1-_i(v)· T^d_v) )






Exponentiating this identity and replacing the _i(v)'s by their value concludes the calculation of the L-function of E_a, γ over K. 


 §.§ Rank and special value

 
 From the factored expression of L(E_a, γ, T) obtained in Theorem <ref>, one can deduce explicit expressions of the analytic rank ρ(E_a, γ) and of the special value L^∗(E_a, γ, 1) (as defined in <ref>). 

  
 For any γ∈_q^× and any integer a≥ 1, we have
 
    ρ(E_a, γ)    
     = |P_q(a)| 
    and     L^∗(E_a, γ, 1) 
       = ∏_v∈ P_q(a)4 v ·∏_v∈ P_q(a)(1 -  _γ(v)^2/4q^ v).

 
  
     
    For each v∈ P_q(a), let 
    
    g_v(T) := (1-q^d_v· T^d_v) (1-_γ(v)^2 · T^d_v)(1- '_γ(v)^2 · T^d_v)
 
    be the corresponding factor of L(E_a, γ, T) (as in Theorem <ref>).
    The factor 1-q^d_v· T^d_v has a simple zero at T=q^-1  and neither of the other two factors of g_v(T) vanishes at T=q^-1. Indeed, neither of _γ(v)^2 and '_γ(v)^2 can equal q^d_v 
    since |_γ(v)| = |_γ(v)+'_γ(v)| is strictly smaller than 2 q^d_v/2 by Proposition <ref>(<ref>).
    Hence, g_v(T) has a simple zero at T=q^-1 and we have 
    
    ρ(E_a, γ) = _T=q^-1L(E_a, γ, T) 
        = ∑_v∈ P_q(a)_T=q^-1g_v(T) 
        = |P_q(a)|,

    which proves (<ref>). 
    Next, by definition ( (<ref>)) of the special value L^∗(E_a, γ, 1), we have
    
    L^∗(E_a, γ, 1)
        =  .L(E_a, γ, T)/(1-qT)^ρ|_T=q^-1
        = ∏_v∈ P_q(a)(.g_v(T)/1-qT|_T=q^-1).

    Furthermore, by a straightforward computation, we obtain that
    
    .g_v(T)/1-qT|_T=q^-1
        = d_v ·( 1 - _γ(v)^2/q^d_v)
        ( 1 - '_γ(v)^2/q^d_v)
        = d_v ·4 q^d_v - _γ(v)^2/q^d_v.

    Combining the last two displayed identities directly leads to the desired expression of L^∗(E_a, γ, 1). 
    

Since the curve E_a, γ satisfies the BSD conjecture (see Theorem <ref>), Proposition <ref> implies that E_a, γ(K)= |P_q(a)|. 
It is worthwhile to note that the Mordell-Weil rank of E_a, γ(K) is actually independent of γ∈_q^× (this is evident from (<ref>)).
 By Lemma <ref>(<ref>), we therefore have

    q^a/a≪_q  E_a, γ(K) ≪_q
    q^a/a.

 In particular, we retrieve the following “unbounded rank” result (see <cit.>). 

Let _q be a finite field of odd characteristic p and K=_q(t).
For all γ∈_q^×,  
the rank of E_a, γ(K) is unbounded as a≥ 1 tends to infinity.

One might further compare (<ref>) to the upper bound of 
<cit.>, which states that 

    E_a, γ(K) ≪_q (E_a, γ)/log(E_a, γ).

From <ref>, we know that q^a≪(E_a, γ)≪ q^a; therefore the lower bound in (<ref>) shows that E_a, γ(K) attains Brumer's upper bound 
(up to constants depending on q).





 §.§ Angles of the sums _γ(v)
 


For any v∈ P_q(a), 
we know that _γ(v) is a 
real algebraic integer with |_γ(v)|≤ 2q^d_v/2 in any complex embedding (see items (<ref>), (<ref>), (<ref>) in Proposition <ref>). 
Thus, there exists a unique angle _γ(v)∈ [0, π] such that

    _γ(v) = 2q^d_v/2·cos_γ(v).

Note that the individual angles _γ(v) depend on a choice of complex embedding (ζ_p), 
but that the set {_γ(v)}_v∈ P_q(a) does not. 

We can then rewrite the expression of L^∗(E_a, γ, 1) obtained in Proposition <ref> in terms of these angles:

    L^∗(E_a, γ, 1) 
    = ∏_v∈ P_q(a)4 v ·∏_v∈ P_q(a)sin^2(_γ(v)).

In section <ref>, we will prove upper and lower bounds on the size of L^∗(E_a, γ, 1) in terms of the degree b(E_a, γ) of the L-function. 
From the above expression, it is obvious that this size crucially depends on how the angles {_γ(v)}_v∈ P_q(a) are distributed in [0, π] when a→∞. 
Therefore, we spend the next two sections describing this distribution in some detail.






§ SMALL ANGLES OF KLOOSTERMAN SUMS


 
 In this section, we work in the following setting.
 Let  be a finite field of odd characteristic p, and  ψ be a nontrivial additive character on . 
 We assume that ψ takes values in (ζ_p) and we pick a complex embedding (ζ_p). 
 
For any α∈^× the Kloosterman sum _(ψ;α) is a real algebraic integer with |_(ψ;α)|≤ 2||^1/2 (see Proposition <ref>(<ref>), (<ref>) and (<ref>)). 
Thus there is a well-defined angle θ_(ψ,α)∈[0,π] associated to the Kloosterman sum by 

    _(ψ;α) := 2||^1/2·cosθ_(ψ;α).

Further, as noted in Proposition <ref>(<ref>), the angle θ_(ψ;α)  cannot be 0 or π since _(ψ;α)  “never attains the Weil bound”. 
In this section, we investigate how close  θ_(ψ;α) can be to 0 and π; we prove the following result, which may be of independent interest.

 
There exists an effectively computable constant c_p>0 (depending at most on p) such that the following holds:
for any finite field  of characteristic p,  any nontrivial additive character ψ on  and any α∈^×, one has

    |θ_(ψ;α)| > ||^-c_p     and     
    |π-θ_(ψ;α)| > ||^-c_p.

Moreover c_p=2(p-1) is a suitable value of the constant.


Before we start the proof, we recall for convenience the following version of  Liouville's inequality:
[Liouville's inequality]
Let P∈[X] be a polynomial of degree N.
For any algebraic number z∈, let D_z be its degree over  and h(z) denote its logarithmic absolute Weil height.

Either P(z)= 0 or

    |P(z)| ≥P_1^-D_z+1·exp(-N· D_z · h(z)),

in any complex embedding of (z), where P_1 is the sum of the absolute values of the coefficients of P.


See the introduction of <cit.> and the proof of Lemma 5 in loc.cit. for this version and its proof.

    
    We let , ψ and α be as in the statement of Theorem <ref>; we choose an embedding  which is compatible with our choice of (ζ_p).  By Proposition <ref>(<ref>), we can write
    
    _(ψ;α) = _(ψ; α) + '_(ψ;α),

    for two algebraic integers _(ψ; α), '_(ψ;α) of magnitude ||^1/2 whose product is ||. 
    In the given complex embedding, one of _(ψ; α) and '_(ψ;α) equals ||^1/2·^i_(ψ;α) by Proposition <ref>(<ref>). Without loss of generality, we assume that _(ψ;α)=||^1/2·^i_(ψ;α).
    
    We let z∈ be 
    the ratio z:=_(ψ;α)/||^1/2 =^iθ_(ψ;α)
    and L:=(z)⊂. 
    We have 
    
     
    z has degree D_z≤ 2(p-1) and height h(z)≤log√(||). Moreover, z≠±1.
    
         
        By Proposition <ref>(<ref>), we have _(ψ; α)∈(ζ_p+ζ_p^-1). 
        Further,  Proposition <ref>(<ref>) implies that 
        _(ψ; α) and '_(ψ;α) are the roots of X^2 - _(ψ;α) · X +||; it is then clear that the degree of _(ψ; α) over  is ≤ 2 [(ζ_p+ζ_p^-1):] = p-1.
        
        Thus we have
        
    D_z = [L:] ≤ [(||^1/2, _(ψ; α)):] ≤ 2[(ζ_p+ζ_p^-1):]≤ 2(p-1),

        as was to be shown. 
        Since |_(ψ; α)|=||^1/2, 
        we infer that |z|=1 in any complex embedding of L,  therefore the archimedean places of L do not contribute to h(z).
        We deduce further that z is a unit at all finite places of L which are not above p; hence these places do not contribute to h(z) either. 
        It remains to consider finite places 𝔓 of L lying above p:
        since both _(ψ;α) and '_(ψ;α) = ||/_(ψ;α) are algebraic integers, the contributions of 𝔓 to h(z) is 
        ≤1/2·_𝔓(||)/[L:]. 
        By summing these, we conclude that h(z)≤1/2log ||. 
        
        Finally, Proposition <ref>(<ref>) shows that θ_(ψ;α)∉{ 0, π}, and 
        the last assertion easily follows.
        
        
        
        
        
        
        
    
    Upon applying Liouville's inequality (<ref>) to z with P=X±1 and using Lemma <ref>, we obtain that
    
    log|z ± 1|   ≥ (-D_z+1)·log 2  - D_z · h(z)
        ≥ -D_z·(log 2  +log√(||)) 
       > - D_z ·log ||≥ -2(p-1)·log ||.

    Noting that the the graph of t↦ |sin t| lies below its tangents at t=π/2, one sees that |^i t -1| ≤ |t| for all t∈[0,π].
    
    
    
    From the lower bound (<ref>) on |z-1|, we deduce from this inequality that 
    
    
    
    |θ_(ψ;α)| ≥ |^iθ_(ψ;α)-1| =|z-1|> ||^-c_p,

    where c_p = 2(p-1). To get the lower bound on |θ_(ψ;α) -π|, we use the same inequality with t replaced by t'=π - t and the lower bound on |z+1| in (<ref>).  This concludes the proof of Theorem <ref>.
    
    
    
    








Let us deduce two corollaries from Theorem <ref>. The first one can be viewed as a slight improvement on the Weil bound on Kloosterman sums (, an effective version of Proposition <ref>(<ref>)):
 For any finite field  of characteristic p≥ 3, any nontrivial additive character ψ on  and any α∈^×, we have

    |_(ψ;α)| 
    
    ≤ 2||^1/2·(1-2/π^2· ||^-2c_p),

where c_p is the constant in Theorem <ref>. 



     
    By construction, we have |_(ψ;α)| = 2||^1/2·|cosθ_(ψ;α)|. Theorem <ref>  implies that θ_(ψ; α) lies in [Q, π-Q] with Q=(q^a)^-c_p.
    The Corollary follows from the elementary observation that 
    
    ∀θ∈[Q, π-Q],    
        |cosθ| = √(1-sin^2θ)≤ 1-sin^2 θ/2≤ 1 - 2min{θ, π-θ}^2/π^2≤ 1- 2Q^2/π^2. □

    
    
    
    

The second corollary is more central to our study of the size of L^∗(E_a, γ, 1),  <ref>. 
  

Let _q be a finite field of characteristic p≥ 3 equipped with a nontrivial additive character ψ_q.  
For any γ∈_q^× and any integer a≥ 1, the angles _γ(v) for v∈ P_q(a) defined in <ref> satisfy

    (q^a)^-c_p≤_γ(v)≤π - (q^a)^-c_p,

where c_p is the constant 
in Theorem <ref> above.

 
    
    For all v∈ P_q(a), the residue field _v is a subfield of _q^a and we may choose β_v as in <ref>. 
    Upon noting that |_v|≤ q^a,    the Corollary follows immediately from Theorem <ref> applied to =_v, ψ = ψ_q∘__v/_q and α=γβ_v^2. 
    
 




§ DISTRIBUTION OF THE SUMS _Γ(V)


    
In this section, we fix again a finite field _q of odd characteristic p, an element γ∈_q^× and a nontrivial additive character ψ_q on _q with values in (ζ_p). 

For any finite extension /_q, we continue denoting by ψ_ the composition ψ_q∘_/_q of  ψ_q with the trace _/_q : →_q.

Loosely speaking, we show that, asymptotically as a→∞, the numbers q^-d_v/2·_γ(v) with v∈ P_q(a) (see <ref>) are distributed in [-2, 2] as “the traces of random matrices in (2, )”. 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 ℓ≠ p, an algebraic closure  of _ℓ, an embedding , and 

a field isomorphism ≃. 
 Through this isomorphism, we view ψ_q as a -valued additive character on _q.





We fix a separable closure K of K. 
The set of places 
v≠ 0, ∞ of K can be identified with the set of closed points of the multiplicative group _m=^1∖{0, ∞} over _q. 
For a finite extension  /_q and a point α∈_m(), 
we denote by _, α the geometric Frobenius of _m at α, which we view as a conjugacy class in the profinite group (K/K). 
For any closed point v of _m, we choose β_v∈ v and we let _v:=__v, β_v. 






  
  



 §.§ Angles of Kloosterman sums



Let us start by redefining the angles _(ψ_;α) from a representation-theoretic point of view.
The reader is referred to <cit.> or <cit.> for more detailed presentations.


In Chapter 4 of <cit.>, Katz has constructed 

a lisse -sheaf ^∘ on _m whose Frobenius traces are Kloosterman sums (^∘ is the so-called Kloosterman sheaf). Taking a suitable Tate twist, one obtains 
a lisse -sheaf =^∘(1/2)  of rank 2 on _m 
which is pure of weight 0. 


By definition,  “is” a continuous 2-dimensional -representation κ: (K/K)→(2, ) 
  which is unramified outside {0, ∞} and which satisfies the following.  
   For all places v≠ 0, ∞ of K,   the eigenvalues of κ(_v) have magnitude[Here we view κ(_v)∈ as an element of  by means of the chosen isomorphism ≃] 1 (“pure of weight 0”) and the trace  of κ(_v) is:
   

 


    (κ(_v) )
    
    = |_v|^-1/2·__v(ψ__v;β_v),

where β_v∈_m(_̅q̅) is a choice of element in the closed point of _m corresponding to v (as in <ref>). 
Note that, even though _v is only defined up to conjugation in (K/K), its (κ(_v) ) is well-defined.
 
Katz has shown that the image of (K/K) under κ is contained in (2, ) (in other words, the representation κ has trivial determinant, see <cit.>).

Via the chosen isomorphism ≃, 
we may view 
κ((K/K)) as a subgroup of (2, ).
The special unitary group K:=(2, ) is a maximal compact subgroup of (2, ) and, since (2, ) is semisimple, such a K is uniquely determined up to conjugation.

For any place v≠ 0, ∞, let 
κ(_v)ß be the semisimplification of κ(_v): the closure of the subgroup of  (2, ) generated by all the κ(_v)ß is compact and thus, up to conjugation in (2, ), 
lies in K.





We denote by K^♮ the set of conjugacy classes of K and we equip K^♮ with 
the measure μ^♮ obtained as the direct image 

of the Haar measure on K normalised to have total mass 1. 
The trace of M∈ K (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∈ K is conjugate (in K) to a diagonal matrix  Diag(^iθ_M, ^-iθ_M) for some unique θ_M∈[0,π] and M = 2cosθ_M.

Hence, the set K^♮ endowed with μ^♮ can be identified with the interval [0, π] endowed with the Sato–Tate measure := 2/πsin^2θθ (see <cit.>). 
We identify any angle θ∈[0, π] with the conjugacy class of Diag(^iθ, ^-iθ) ∈ K^♮, which we also denote by the same symbol θ.
 
 We are now ready to (re)define angles of Kloosterman sums.
 For any finite extension /_q and any α∈_m(), the semisimplification of κ(_, α) ∈(2,) is (2,)-conjugate 
 to an element of K, and we can define θ_(ψ_; α)∈ K^♮ to be the conjugacy class (in K) of this element. 
  In the identification between K^♮ and [0, π], this gives us a well-defined angle θ_(ψ_; α)∈[0,π], see <cit.>. 
  
For any finite extension /_q and any α∈_m(), we thus have
  
    2·cosθ_(ψ_; α)=  (κ(_, α)ß)= (κ(_, α) )
    = ||^-1/2·_(ψ_;α),

so that the new definition of cosθ_(ψ_; α) coincides with the one given at the beginning of section <ref>.

     
    Fix a finite field _q equipped with a nontrivial additive character ψ_q and γ∈_q^×. 
    For any place v≠ 0, ∞ of K, let _γ(v) be the angle associated to the Kloosterman sum _γ(v)=__v(ψ__v; γβ_v^2) by the construction above. 
    In other words, we put _γ(v) := θ__v(ψ__v;γβ_v^2)∈ K^♮.
    


Let  be the finite extension of _q with [:_q]=a. 
For an element α∈_m(), let w be the closed point of _m corresponding to α (, w is the (_̅q̅/_q)-orbit of α).
The residue field _w is then a subfield of  and _, α = (_w)^[:_w] as conjugacy classes in (K/K). 
Therefore κ(_, α) = κ(_w)^[:_w] and

    a w·θ__w(ψ__w;α)  = [:_w] ·θ__w(ψ__w;α) ≡θ_(ψ_;α) π.

In particular, when v is a closed point of _m whose degree divides a and when β_v∈ v, by  our definition  _γ(v) = __v(ψ__v; γβ_v^2), the relation above reads

    a v·_γ(v) 
    
    ≡_(ψ_; γβ_v^2) π.






 §.§ Statement of results


  
Denote by  the Sato–Tate measure 2/πsin^2θθ on [0,π]. 
A sequence of Borel measures {μ_i}_i≥ 1 on [0,π] is said to converge weak-∗ to  if, for every continuous -valued function f on [0, π], the sequence of ∫_[0, π] fμ_i converges to ∫_[0,π]f as i→∞.
 


  

Our results concern two sequences of probability measures that we now introduce. 


We fix a finite field _q of characteristic p≥ 3, a nontrivial additive character ψ_q on _q and γ∈_q^×.
For an integer a≥ 1, we again denote by P_q(a) the set of closed points of _m whose degrees divide a. For all integers a≥ 1, we define

    ν(_q, ψ_q, γ; a)
    := 1/|P_q(a)|·∑_v∈ P_q(a)δ{_γ(v)},

where δ{x} denotes the Dirac delta measure at x∈ [0,π].
For all finite extensions /_q, we also define

    ξ(_q, ψ_q, γ; )
    
    := 1/|_m()|·∑_β∈_m()δ{θ_(ψ_; γβ^2)}.

In what follows, we abbreviate ν(_q, ψ_q, γ; a) by ν_a and   ξ(_q, ψ_q, γ; _q^a) by ξ_a.


Clearly, both ν_a and ξ_a are Borel measures on [0,π] with total mass 1. 
It follows from the discussion in <ref> that we may view  ν_a and ξ_a as measures on K^♮; we use both points of view interchangeably.
Moreover, we note that ξ_a is also given by[The measure ξ_a is the measure denoted by X_a in <cit.> applied to our situation.]  



    ξ_a = ξ(_q, ψ_q, γ; _q^a)
    
    = 1/|_m(_q^a)|·∑_v∈ P_q(a) v·δ{a v·_γ(v)},

where a v·_γ(v) is to be understood modulo π (see Remark <ref>). 


In terms of the measure ν_a, Corollary <ref> can be reinterpreted as follows: given  _q, ψ_q and γ as above, for any a≥ 1 the support of the probability measure ν_a on [0, π]  is contained in [(q^a)^-c_p, π  -  (q^a)^-c_p]. 




  

We can now state the two main results of this section. 
First we show that the angles {_γ(v)}_v∈ P_q(a) are asymptotically equidistributed with respect to the Sato–Tate measure as a→∞. 
Namely, 

 
Assume we are given a datum _q, ψ_q, γ as above. 
Then the sequences {ξ_a}_a≥ 1 and {ν_a}_a≥ 1 of Borel probability measures both converge weak-∗ to the Sato–Tate measure  when a→∞.


This statement concretely means that, for all continuous functions f on [0,π], we have

    1/|P_q(a)|·∑_v∈ P_q(a) f(_γ(v))
    =∫_[0,π] f ν_a ∫_[0, π] f  = 2/π∫_0^π f(t)sin^2(t) t.

It will be proven in Propositions <ref> and <ref> by a suitable adaptation of the arguments in <cit.> and <cit.>. 

In the course of proving Theorem <ref>, we will need a more effective version of (<ref>): indeed, we require an estimate of the rate at which  ∫_[0,π] f ν_a converges to ∫_[0, π] f, at least for a smaller class of functions f. 
This is the object of the second result in this section: 

 
Assume we are given a datum _q, ψ_q, γ as above. For any continuously differentiable function g on [0,π], we have

    |∫_[0,π] g ν_a - ∫_[0, π] g| ≪_q a^1/2/q^a/4·∫_0^π |g'(t)| t      (as  a→∞).



This will follow from the proof of Theorem <ref> coupled with tools from distribution theory (see <cit.>).

We remark that the constants in Theorems <ref> and  <ref> depend at most on q (and neither on the choice of ψ_q  nor on the value of γ∈_q^×).




 §.§ Equidistribution of _γ(v)'s


  
In this subsection, we prove Theorem <ref> in two steps (Propositions <ref> and <ref>). 
Let us first make a reduction (see <cit.> for more details). 
We need to show that, for all complex-valued continuous functions f on [0,π], we have

    ∫_[0,π] f ξ_a,   ∫_[0,π] f ν_a ∫_[0, π] f.

Since K is compact and since [0,π] can be identified with K^♮, there is a natural correspondence between the space of continuous functions on [0,π] and the space 𝒞^0_cent(K) of continuous central functions on K=(2, ).
When 𝒞^0_cent(K) is endowed with the topology of the supremum norm, the Peter-Weyl theorem asserts that the vector subspace generated by characters of irreducible finite-dimensional representations of K is dense in 𝒞^0_cent(K). 
By density, Theorem <ref> will follow if we can show that, for all irreducible finite-dimensional representations Λ of K,

    ∫_K^♮Λξ_a,   ∫_K^♮Λν_a ∫_K^♮Λμ^♮.

If Λ_0 is the trivial representation of K, Λ_0 is the trivial character  on K and the above limits trivially hold because the measures ν_a, ξ_a and μ^♮ on K^♮ all have total mass 1. 
By orthogonality of characters, the integral ∫_K^♮Λμ^♮ on the right-hand side vanishes when Λ is a nontrivial irreducible finite-dimensional representation of K.

Hence,  the proof of Theorem <ref> reduces to that of the following statement, which is an analogue to Weyl's criterion for uniform distribution (see <cit.>) in the Sato–Tate context: for any nontrivial irreducible finite-dimensional representation Λ of (2, ), one has

    ∫_K^♮Λξ_a,   ∫_K^♮Λν_a  0.
  
We actually prove slightly more precise estimates.
 

Fix _q, ψ_q and γ∈_q^× as in <ref>.
Let Λ be a nontrivial irreducible representation of K=(2, ). For all a≥ 1, one has

    | ∫_K^♮Λξ_a|
    ≪_q   Λ/q^a/2.



	  
	For any finite extension /_q, notice that 
	
    ∑_β∈^×Λ(θ_(ψ_; γβ^2))
    	   = ∑_α'∈^×(1+λ_(α'))·Λ(θ_(ψ_; γα')) 
       = ∑_α∈^×Λ(θ_(ψ_;α)) 
    	+ λ_(γ^-1) ·∑_α∈^×λ_(α)·Λ(θ_(ψ_;α)),

	where λ_ denotes the unique nontrivial character of order 2 on ^×. 
	Therefore, one has
	
    |∑_β∈^×Λ(θ_(ψ_; γβ^2))|
    	≤|∑_α∈^×Λ(θ_(ψ_;α)) | + |∑_α∈^×λ_(α)·Λ(θ_(ψ_;α))|.

	For any multiplicative character χ on _q^× and any finite extension /_q, we denote by χ_:=χ∘_/_q the character on ^× “lifted” by the norm _/_q:→_q.  
	The crucial input is a result of Fu and Liu (see Lemma 4 in <cit.>) who have proved that, for every multiplicative character χ on _q, one has
	
    |∑_α∈_m()χ_(α)·Λ(θ_(ψ_;α))| ≤Λ/2· ||^1/2.

	Applying this inequality successively to both multiplicative characters on _q whose order divides 2, we deduce from (<ref>) that 
	
    |∑_β∈^×Λ(θ_(ψ_; γβ^2))| ≤ (Λ)· ||^1/2.

	Therefore, for any finite extension /_q, we have proved that
	
    | ∫_K^♮Λξ(_q, ψ_q, γ; )| 
    	= | 1/|_m()|∑_β∈^×Λ(θ_(ψ_; γβ^2))|
    	≪_q Λ· ||^-1/2.

	Specialising to =_q^a yields the desired estimate since ξ_a=ξ(_q, ψ_q, γ; _q^a). 
	

 
Let us suggest an alternative way of proving  Proposition <ref>.


Denote again by  the Kloosterman sheaf (suitably twisted to be pure of weight 0) whose existence was proved by Katz. 

For a given γ∈_q^×, consider the morphism f:_m→_m given by β↦γβ^2, and put _γ:=f^∗. 

Then _γ is again a lisse -sheaf of rank 2 on _m which is pure of weight 0. 
In the case where γ=1, Remark 1 in <cit.> sketches a proof that _γ satisfies the assumptions of <cit.>. 
One could therefore prove Proposition <ref> 
by making use of <cit.>. 

This argument should carry over to the case of an arbitrary γ≠ 0.













To complete the proof of Theorem <ref>, it remains to show that {ν_a}_a≥1 also converges (weak-∗) to .


Fix a datum _q, ψ_q and γ∈_q^× as in <ref>.
Let Λ be a nontrivial irreducible representation of (2, ). For all a≥ 1, one has

    | ∫_K^♮Λν_a|
    ≪_q Λ·a/q^a/2.




	
	Consider the measure W_a := |_m(_q^a)|·ξ_a - a|P_q(a)|·ν_a on K^♮ (or [0,π]); in other words, by (<ref>),
	
    W_a     = ∑_ v∈ P_q(a) 
     v <a(  v ·δ{a v_γ(v)} 
    	- a ·δ{_γ(v)})
    	= ∑_  b | a 
     b <a∑_ v  s.t.
     v= b( b ·δ{ab_γ(v)} 
    	- a ·δ{_γ(v)}).

	As is clear from the second expression, W_a is a sum of ∑_  b | a 
 b <aπ_q(b)· (b+a)
	
	Dirac δ measures supported at points of K^♮. 
	For any z∈ K, the eigenvalues of Λ(z) all have magnitude 1, therefore we have |Λ(z)|≤Λ.
	Hence we find that
	
    |∫_K^♮Λ W_a|
    	≤( ∑_  b | a 
     b <aπ_q(b)· (b+a) ) ·Λ≪_q Λ· a · q^a/2

	Indeed, straightforward estimates using Lemma <ref>(<ref>) show that
	
    ∑_  b | a 
     b <aπ_q(b)· (b+a)
    	≤ 2 a·∑_  b | a 
     b <aπ_q(b)
    	
    	≪_q a  ∑_1≤ b≤ a/2 q^b
    	≪_q a · q^a/2.


	Notice that 
	
    ν_a = |_m(_q^a)|/a|P_q(a)|·ξ_a 
    	+ 1/ a|P_q(a)|· W_a,

	where a|P_q(a)|≫_q q^a and 
	|_m(_q^a)|≪_q a|P_q(a)|, again by Lemma <ref>. 
	
	Therefore, combining  (<ref>)  in the previous Proposition 
	and inequality (<ref>), we deduce that
	
    |∫_K^♮Λν_a|
    	   ≤|_m(_q^a)|/a|P_q(a)|·|∫_K^♮Λξ_a| 
    	+1/ a|P_q(a)||∫_K^♮Λ W_a| 
       ≪_q Λ·( q^-a/2 + a · q^a/2/q^a) 
    	≪_q Λ·a/q^a/2.

	This concludes the proof of the Proposition  and, by the discussion at the beginning of this subsection, that of Theorem <ref>.
	




 §.§ Effectivity of the equidistribution



The nontrivial irreducible representations 
of K=(2, ) are exactly the symmetric powers ^n(std) 
of the standard representation std:(2, )(2, ). 
Moreover, if Λ_n = ^n(std) for some n≥ 1, then Λ_n has dimension n+1 and the trace function Λ_n:K^♮→ corresponds to the map
[so that Λ_n(θ)=U_n(cosθ), where U_n is the n-th Chebyshev polynomial of the second kind.] 
θ↦sin((n+1)θ)/sinθ in the identification of K^♮ with [0, π].




It is convenient to denote by _n(a), the “n-th moment” of {_γ(v)}_v∈ P_q(a) ,

    _n(a):= ∫_K^♮Λ_n  ν_a 
    = 1/|P_q(a)|·∑_v∈ P_q(a)sin((n+1)_γ(v))/sin_γ(v).

With this notation, the result of Proposition <ref> can be rewritten as follows. 
Given a datum _q, ψ_q,γ∈_q^× as in <ref> and an integer n≥ 1, one has 

    ∀ a≥ 1,     | _n(a)|
    ≪_q (n+1)·aq^-a/2.
 



  

To measure “how far” from being perfectly equidistributed with respect to the Sato–Tate distribution are  the angles _γ(v),

 it is customary to introduce the star discrepancy _q, γ(a):

    _q, γ(a) :=  
    sup_x| ∫_[0, x)ν_a - ∫_[0, x)|
    = sup_x| | { v∈ P_q(a) : _γ(v)∈[0,x) }|/|P_q(a)| - ∫_[0, x)|,

where the supremums are taken over x∈[0,π]. 
This definition is the direct analogue of the  star discrepancy for the uniform measure (see <cit.>) in the context of .
The interest of finding good upper bounds on _q, γ(a) is exemplified by the following result, which is similar to Koksma's inequality (see Theorem 5.1 in <cit.>).

[Niederreiter]
For any function  g:[0,π]→ of total bounded variation g, one has

    | ∫_[0,π] gν_a - ∫_[0,π] g| ≤_q, γ(a) ·g.



This statement is essentially Corollary 2 in <cit.>, 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 <cit.>).

Note that, for a continuously differentiable function g, one has g = ∫_0^π |g'(t)| t. 
Therefore, Theorem <ref> follows directly from Theorem <ref> and the following: 

 
The star discrepancy of {_γ(v)}_v∈ P_q(a) is bounded by

    _q, γ(a)
    ≪_q a^1/2/q^a/4.



	 
	Niederreiter (see <cit.>) has proved a variant of the “Erdös–Turán inequality” in the Sato–Tate context.
	Just as the Erdös–Turán theorem  (see <cit.>), his result gives an upper bound on _q, γ(a) in terms of “exponential sums”, here the moments _n(a) defined in  (<ref>). 
	Let us state Lemma 3 in <cit.> as follows[Niederreiter works on the interval [-1, 1] endowed with the direct image of μ_ST under t↦cos t (the “semi-circle measure”); the translation to our setting is straightforward. Note that <cit.>   actually gives explicit constants in (<ref>).]: 
	for any odd positive integer N, we have
	
    _q, γ(a) ≪1/N + ∑_n=1^2N-1n+1/n(n+2)· |_n(a)|,

	As was noted in (<ref>), Proposition <ref> reads: | _n(a)|≪_q (n+1)· a q^-a/2.
	Also remark that
	
    ∑_n=1^2N-1(n+1)^2/n(n+2)≤ 2N - 1 +∑_n=1^∞1/n^2+2n  = 2N-1+3/4 ≪ N.

	Hence, for all odd N≥ 1, (<ref>) leads to _q, γ(a) ≪_q  N^-1 + a q^-a/2· N.
	Choosing N to be the largest odd integer smaller than (a^-1q^a/2)^1/2, we have N^-1≪ a^1/2q^-a/4 and we obtain the desired bound. 
	





§ BOUNDS ON THE SPECIAL VALUE

 

  By definition (<ref>), the special value L^∗(E_a, γ, 1) is the value at T=q^-1 of a polynomial with integral coefficients of degree ≤ b(E_a, γ). Therefore, L^∗(E_a, γ, 1)  is of the form 
  n/q^b(E_a, γ) for some integer n≥ 1, and we deduce the following “trivial” lower bound on L^∗(E_a, γ, 1):
  
    log L^∗(E_a, γ, 1)/log( q^b(E_a, γ))= log n/log( q^b(E_a, γ)) - 1 ≥ - 1.



On the other hand, using techniques from classical complex analysis, Hindry and Pacheco 
show 
the following upper bound on L^∗(E_a, γ, 1)  (see Theorem 7.5 in <cit.>): 

    log L^∗(E_a, γ, 1)/log( q^b(E_a, γ))≪_q log b(E_a, γ)/b(E_a, γ).

 In this section, we prove the main theorem of this article (Theorem <ref> in the introduction), which provides a significant improvement on (<ref>):
 

 Let _q be a finite field of odd characteristic p and K=_q(t).
There exist positive constants C_1, C_2, which depend at most on q and p such that the following holds. 
For all γ∈_q^× and all integers a≥ 1, the special value L^∗(E_a, γ, 1) satisfies: 
 
    - C_1·1/log b(E_a, γ)≤log L^∗(E_a, γ, 1)/log(q^b(E_a, γ))≤ C_2·log b(E_a, γ)/ b(E_a, γ)   (as a→∞).

  



We will prove this Theorem in <ref>. 
 
The upper bound in (<ref>) does not radically improve on the upper bound of Hindry and Pacheco (loc. cit.) 
but, since our proof is rather short and elementary, 
we decided to include it here for completeness.
Our proof of the lower bound in (<ref>), on the other hand, is much more involved: the crucial step is the computation of a  “Sato–Tate limit”, using the results of sections <ref> and <ref> (see the next subsection). 
For later use (in section <ref>), we note that Theorem <ref> implies that

    |log L^∗(E_a, γ, 1)| = o(b(E_a, γ))  
         (as  a→∞).





 §.§ 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 <ref>.
For any integer a≥ 1, we again denote by ν_a=ν(_q, ψ_q, γ;a) the probability measure on [0,π] introduced in <ref>.

 
Let _q be a finite field equipped with a nontrivial additive character ψ_q, and γ∈_q^×.
Then 

    ∫_[0,π]logsin^2 ν_a  ∫_[0,π]logsin^2.


More concretely, this statement means that

    1/|P_q(a)|∑_v∈ P_q(a)log(sin^2 _γ(v) )
    2/π∫_0^πlog(sin^2t )·sin^2 t   t = loge/4,

the evaluation of the integral on the right-hand side being a routine exercise in calculus.

	 
	For conciseness, we denote by w:[0,π]→ the function given by w(t) := -log(sin^2 t) if t≠ 0, π and w(0)=w(π):=0.
	Choose a  nondecreasing continuously differentiable function  ϕ_0:[0,1]→ such that ϕ_0 ≡ 0 on [0,1/3] and ϕ_0≡ 1 on [2/3, 1].
	
	
	
	
	
	For all ϵ∈(0, 1), we define a function ϕ_ϵ:[0,π]→ by
	
    ϕ_ϵ(t) =
    	ϕ_0(t/ϵ)     if  t∈[0,ϵ], 
    
    	1    if  t∈[ϵ, π-ϵ], 
    ϕ_0((π-t)/ϵ)     if  t∈[π-ϵ, π],

	and we let w_ϵ := w·ϕ_ϵ.  By construction, w_ϵ is a continuously differentiable function on [0,π] such that w_ϵ≤ w on [0,π], w≡ w_ϵ on [ϵ, π-ϵ], and w_ϵ≡ 0 on [0, ϵ/3]∪ [π-ϵ/3, π]. 
	
	
	Furthermore, we have the following analytic estimates:

	
	 Notations being as above, for all ϵ∈ (0, 1), we have
	2
	
	
  *  ∫_0^π |w'_ϵ(t)| t ≪ |logϵ|,
	
  *  ∫_0^π (w(t)-w_ϵ(t))·sin^2(t) · t ≪ϵ |logϵ|.
	
	
	The constants depend only on the choice of ϕ_0. 
	

	We postpone the proof of this Lemma until the end of the subsection, and we now prove that 
	∫_[0, π] w ν_a converges to  ∫_[0, π] w when a→∞. 
	For any ϵ∈ (0,1), note that 
	
    |∫  w ν_a - ∫  w |
    	   ≤∫ |w-w_ϵ| ν_a_:=T_1
    	+  |∫  w_ϵν_a  - ∫ w_ϵ|_:=T_2
    	+  ∫ | w_ϵ - w| _:=T_3.

	Let us bound each of these three terms using the results in sections <ref> and <ref>. 

	For ϵ>0 sufficiently small, the first term T_1 vanishes. 
	Indeed, w≡ w_ϵ on [ϵ, π-ϵ] and, as we have proved,  the support of ν_a is contained in [(q^a)^-c_p, π - (q^a)^-c_p] (see Remark <ref>). 
	Therefore, for any ϵ < (q^a)^-c_p, we have T_1=0.

	The function w_ϵ being continuously differentiable on [0,π], we can use our effective equidistribution result (Theorem <ref>) to control the second term T_2. 
	Precisely, Theorem <ref> yields
	
    T_2 = |∫  w_ϵν_a  - ∫ w_ϵ|
    	≪_q a^1/2/q^a/4·∫_0^π |w'_ϵ(t)| t 
    	≪_q a^1/2/q^a/4· |logϵ|,

	where the rightmost upper bound is Lemma <ref>(<ref>).
	Finally, Lemma <ref>(<ref>) proves that T_3 ≪ϵ |logϵ|. 

	In summary, 
	for all ϵ>0 such that ϵ < (q^a)^-c_p, we have
	
    |∫_[0,π]  w ν_a - ∫_[0,π]  w | 
    	≪_q a^1/2/q^a/4· |logϵ| + ϵ|logϵ|.

	Upon choosing ϵ of the form ϵ= (q^a)^ -γ for some γ>max{c_p, 4^-1} (whose choice can be made to depend at most on p), 
	we conclude that
	
    |∫_[0,π]  w ν_a - ∫_[0,π]  w | 
    	≪_q γ a  ·   a^1/2/q^a/4 + γ/q^a·γ≪_q, pa^3/2/q^a/4,

	which proves Theorem <ref>. 
	We even obtain a more quantitative version of (<ref>): 
	
    1/|P_q(a)|∑_v∈ P_q(a)log(sin^2 _γ(v))
    	= log(/4) + O(a^3/2/q^a/4)    (as  a→∞),

	where the implicit constant depends at most on q, p and the choice of the auxiliary function ϕ_0.
	

 
	
	Since both w and w_ϵ are symmetric around π/2, it is sufficient to prove (<ref>) and (<ref>) where the integrals are replaced by integrals over [0, π/2]. 
	We also note that, for all t∈(0, π/2], one has 0≤ w(t)≤ 2logπ/2t. This follows from the classical estimate:  sin t≥2t/π for t∈ [0, π/2].
	 
	To prove (<ref>), we study separately the integrals over (0,ϵ) and over [ϵ, π/2].
	
	Since w and w_ϵ coincide on [ϵ, π/2], we have 
	
    ∀ t∈[ϵ, π/2],   
    	|w'_ϵ(t)| = |w'(t)|  =2|cos t/sin t| ≤2/|sin t|≤π/t,

	by the classical estimate mentioned above. 
	Hence we have
	∫_ϵ^π/2 |w'_ϵ(t)| t 
	≤π∫_ϵ^π/2t^-1 t
	≪ |logϵ|.
	On the interval (0,ϵ), we use the fact that w_ϵ = wϕ_ϵ to deduce that
	
    ∀ t∈(0,ϵ),    |w'_ϵ(t)| 
    	≤ |w'(t)|·ϕ_ϵ(t) + |w(t)| · |ϕ'_ϵ(t)|
    	≤π/t·ϕ_ϵ(t) + |w(t)| ·ϕ'_0_∞/ϵ,

	where ϕ_0'_∞ denotes the supnorm of ϕ'_0 on [0,π]. 
	By the upper bound on w(t) at the beginning of the proof and the fact that ϕ_ϵ≡ 0 on [0,ϵ/3), we deduce that 
	
	
    ∀ t∈(0,ϵ),    |w'_ϵ(t)| 
    	≤3π/ϵ + 2ϕ'_0_∞/ϵ·logπ/2t,

	and from there, that 
	
    ∫_0^ϵ |w'_ϵ(t)| t 
    	≤   3π  + 2ϕ'_0_∞/ϵ·∫_0^ϵlogπ/2t t
    	 ≪ 1 + ϕ'_0_∞/ϵ·ϵ|logϵ| 
    	 ≪ |logϵ|.

	Summing the contributions of ∫_0^ϵ and ∫_ϵ^π/2, we conclude that (<ref>) holds, with a constant depending only on ϕ_0.  We now show that (<ref>) holds: by the symmetry of w-w_ϵ  and 
	since w≡ w_ϵ on [ϵ, π/2], it suffices to prove that ∫_0^ϵ |w(t)-w_ϵ(t)|sin^2t t ≪ϵ|logϵ|. By construction of ϕ_ϵ, we notice that 
	
    ∫_0^ϵ |w(t)-w_ϵ(t)|·sin^2t  t 
    	≤∫_0^ϵ  w(t) ·sin^2t  t  
    	≤∫_0^ϵ  w(t)    t 
    	(*)≤ 2 ∫_0^ϵlogπ/2t t
    	≪ϵ |logϵ|,

	where inequality (*) follows from the upper bound on w(t) at the beginning of the proof. 
	




 §.§ Proof of Theorem <ref>



 In (<ref>) 
 and (<ref>), we have proved that 
 
    L^∗(E_a, γ, 1) = ∏_v∈ P_q(a)4 v ·∏_v∈ P_q(a)(1 -  _γ(v)^2/4q^ v) =   ∏_v∈ P_q(a)4 v ·∏_v∈ P_q(a)sin^2(_γ(v)).


Therefore, we have


    log L^∗(E_a, γ, 1) /log(q^b(E_a, γ))
    = 1/log(q^b(E_a, γ))·∑_v∈ P_q(a)log(4 v) + 1/log(q^b(E_a, γ))∑_v∈ P_q(a)log(sin^2 _γ(v)),

and we estimate the two terms separately. First of all, let us prove that 
 
    0≤1/log(q^b(E_a, γ))·∑_v∈ P_q(a)log(4 v) 
     ≪_q log a/ a,       (as a→∞).

 The lower bound is clear, and the upper bound follows from Lemma <ref>(<ref>): indeed, we have b(E_a, γ) ≫ q^a (see (<ref>)) and thus
  
    ∑_v∈ P_q(a)log(4 v) 
     ≤log (4a) · |P_q(a)|
     ≪_q  log a · q^a/a,


The upper bound in (<ref>) implies the upper bound in Theorem <ref> since the second term in (<ref>)  is negative. To conclude the proof, it thus remains to prove that 

    |1/log(q^b(E_a, γ))∑_v∈ P_q(a)log(sin^2 _γ(v))| ≪_q, p1/log b(E_a, γ).

Let us write that

    |1/log(q^b(E_a, γ))∑_v∈ P_q(a)log(sin^2 _γ(v) )| 
    ≤|P_q(a)|/log(q^b(E_a, γ))·|1/|P_q(a)|∑_v∈ P_q(a)log(sin^2 _γ(v) )|.

By (<ref>) again, we have b(E_a, γ)≫ q^a, 
and Lemma <ref>(<ref>) yields that |P_q(a)|≪_q q^a/a. Moreover, by Theorem <ref>, we have

    1/|P_q(a)|∑_v∈ P_q(a)log(sin^2 _γ(v) ) = ∫_[0,π]logsin^2ν_a = log(/4) + o(1)     (as  a→∞).

Putting these together, we obtain that 

    |1/log(q^b(E_a, γ))∑_v∈ P_q(a)log(sin^2 _γ(v) )| 
    ≪_q, p1/a≪_q, p1/log b(E_a, γ).

This concludes the proof of Theorem <ref>.





§ APPLICATION TO AN ANALOGUE OF THE BRAUER–SIEGEL THEOREM


 
 In this section, we deduce from Theorem <ref> and from the BSD conjecture that the following theorem holds (stated as Theorem <ref> in the introduction).
 

Let _q be a finite field of odd characteristic and K:=_q(t).
For all γ∈_q^× and all integers a≥ 1, consider the elliptic curve E_a, γ/K as above. 
Then the Tate–Shafarevich group (E_a, γ) is finite and, as a→∞,

    log(|(E_a, γ)|·(E_a, γ)) ∼log H(E_a, γ).



Alternatively, (<ref>) can be rewritten under the form

    ∀ϵ>0,   
     H(E_a, γ)^1-ϵ≪_q, ϵ
     |(E_a, γ)|·(E_a, γ) 
     ≪_q, ϵ  H(E_a, γ)^1+ϵ    (as a→∞).

The upper bound in (<ref>) was essentially conjectured by Lang for elliptic curves over  with finite Tate–Shafarevich groups[Note though that Lang uses a different normalisation of the height: his (naïve) height has an exponent 1/12 instead of our exponent 1.] (see <cit.>). 
Theorem <ref> thus provides 
an unconditional example where this conjecture holds for elliptic curves over K=_q(t). 
The lower bound in (<ref>) further proves that the exponent 1 of the height is optimal, in the sense that 1 cannot be replaced by any smaller number. 
 







 
 

  

One may also view Theorem <ref> as an analogue of the Brauer–Siegel theorem for the elliptic curves E_a, γ.   The Brauer–Siegel theorem states that, as F runs through a sequence of number fields of given degree n over  and whose discriminants Δ_F tend, in absolute value, to +∞, 
one has

    ∀ϵ>0,   Δ_F^1/2-ϵ≪_n, ϵ
     |Cl(F)|· R(F)
     ≪_n, ϵΔ_F^1/2+ϵ    (as  |Δ_F|→∞),

where Cl(F) denotes the class group of F and R(F) its regulator of units. 
At least formally, (<ref>) is very similar to (<ref>). A more detailed analogy is explained in <cit.> and <cit.>.

 
	
	We know from Theorem <ref> that the BSD conjecture holds for E_a, γ. 
	In particular, the Tate–Shafarevich group (E_a, γ) is indeed finite and the special value L^∗(E_a, γ, 1) satisfies  (<ref>). 
	The BSD formula (<ref>) and Proposition <ref> then imply the estimate: 
	
    log(|(E_a, γ)|·(E_a, γ))/log H(E_a, γ) = 1 + log L^∗(E_a, γ, 1)/log H(E_a, γ) + o(1) 
    	     (as a→∞).
 
	Therefore, to conclude the proof of Theorem <ref>, it remains to prove that |log L^∗(E_a, γ, 1) | = o( log H(E_a, γ)) or, alternatively,  that 
	
    |log L^∗(E_a, γ, 1) | = o( b(E_a, γ)) 
    	     (as a→∞)

	because log H(E_a, γ) and b(E_a, γ) have the same order of magnitude as a→∞ (see <ref>). 
	But we have already showed in Theorem <ref> that this asymptotic estimate holds.
	




7cm0.5pt.



  
Acknowledgements 
 It is a pleasure to thank Marc Hindry and Douglas Ulmer for their encouragements and for their 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. 





This work has been carried out at Universiteit Leiden whose financial support and perfect working conditions are gratefully acknowledged.
The author is also partially supported by the ANR Grant ANR-17-CE40-0012 (FLAIR).





alpha
 



7cm0.5pt

Mathematisch Instituut, Universiteit Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands.
 
E-mail: mailto:r.m.m.griffon@math.leidenuniv.nlr.m.m.griffon@math.leidenuniv.nl

