Low-lying zeros in families of elliptic curve L-functions over function fields

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over ${\mathbb{F}}_q(T)$. In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.     

On the mean values of Dirichlet L-functions

We study the 2kth power moment of Dirichlet L-functions L(s,) at the centre of the critical strip , where the average is over all primitive characters (mod q). We extend to this case the hybrid Euler–Hadamardproduct results of Gonek, Hughes and Keating for the Riemannzeta function. This allows us to recover conjectures for themoments based on random matrix models, incorporating the arithmeticalterms in a natural way.     

On discrete mean values of Dirichlet L-functions

Czechoslovak Mathematical Journal - Let χ be a nonprincipal Dirichlet character modulo a prime number p ? 3 and let $${\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left(...     

On the special values of automorphic L-functions

Let π be a cuspidal representation of $\text{GL}{}_{\mathrm{n}}\text{(}\text{A}{}_{\mathrm{<mtext>Q</mtext>}}\text{)}$ with non-vanishing cohomology and denote by L(π,s) its L-function. Under a certain local non-vanishing assumption, we prove the rationality of the values of L(π?χ,0) for characters χ, which are critical for π. Note that conjecturally any motivic L-function should coincide with an automorphic L-function on GL_n; hence, our result corresponds to a conjecture of Deligne for motivic L-functions. To cite this article: J. Mahnkopf, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

有限域上多项式的指数和及其L-函数

L-函数蕴藏着深刻的算术信息,是数论中重要的研究对象.有限域上多项式的指数和及其L-函数在一般情形下难以计算.通过利用高斯和及多项式的次数矩阵的Smith标准形,得到了在特定情形下有限域上一类多项式的指数和及其L-函数的具体公式.     

On Jacquet-Langlands isogeny over function fields

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of D-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.     

On metaplectic forms over function fields

We propose a concept of half integral weight in the global function field context, and construct natural families of functions with given weight. An analogue of Shimura correspondence (between weight 2 functions and weight ${\frac{3}{2}}$ functions) via theta series from “definite” quaternion algebras over function fields is then established. From the study of Fourier coefficients of these theta series, we arrive at a Waldspurger-type formula. This formula is then applied to L-series coming from elliptic curves over function fields.     

Prime number theorems for Rankin-Selberg L-functions over number fields

In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal represen-tations of GLm(AE) × GLm (AF ) over cyclic algebraic number fields E and F which are invariant under the Galois action,by exploiting a result proved by Arthur and Clozel,and prove a prime number theorem for this L-function.     

On linear combinations of special values of L-functions

Using a specific evaluation method for L-functions, it is shown that character values and weighted averages of Gauss sums can be well approximated by linear combinations of the algebraic parts of special values of L-functions under correct parity conditions. This is a surprising universal property of these special values since the coefficients of the resulting combinations turn out to be independent of the character but its parity.     

On hyperelliptic modular curves over function fields

We prove that there are only finitely many modular curves of -elliptic sheaves over which are hyperelliptic. In odd characteristic we give a complete classification of such curves. The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

On central critical values of the degree four L-functions for {\text {GSp}}\left( 4\right) : a simple trace formula

We establish a simple relative trace formula for ${\text {GSp}}(4)$ and inner forms with respect to Bessel subgroups to obtain a certain Bessel identity. From such an identity, one can hope to prove a formula relating central values of degree four spinor $L$ -functions to squares of Bessel periods as conjectured by Böcherer. Under some local assumptions, we obtain nonvanishing results, i.e., a global Gross–Prasad conjecture for $(\text {SO}(5),\text {SO}(2))$ .     

Faddeev invariants for central simple algebras over rational function fields

Let p be a prime, k a field, containing a primitive pth root of unity, char k ≠ p. We give an upper bound for the Faddeev index of a central simple algebra of exponent p over the rational function field k(t) in the case where the ramification set of the algebra consists of rational points. This bound depends only on the number of ramification points and in certain cases turns out to be strict. In the case where p =  2 and the ramification set in \({\mathbb{A}_k^1}\) consists of three rational points we compute the Faddeev index, using the language of quadratic forms. Let X be a smooth geometrically irreducible complete curve over k. We show that there exist algebras of exponent p over k(X) with the prescribed Faddeev index, provided there are algebras of exponent p and arbitrarily large index over k. In the last section of the paper we consider another invariant of a central simple algebra of prime exponent p over k(t), the so called Faddeev cyclic length. In certain cases we compute this invariant, using triviality of the divided power operations on central simple cyclic algebras of exponent p.     

On the central value of symmetric square L-functions

Let S _k (N, χ) be the space of cusp forms of weight k, level N and character χ. For let L(s, sym² f) be the symmetric square L-function and be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym² f) and over a basis of S _k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym² f) does not vanish. The author was supported by NSERC grant 311664-05.