Vanishing of Dirichlet L-functions at the central point over function fields

Non-vanishing of Dirichlet L-functions at the Central Point

Zeros of Dirichlet L-functions over Function Fields

Random matrix theory has successfully modeled many systems in physics and mathematics, and often analysis in one area guides development in the other. Hughes and Rudnick computed 1-level density statistics for low-lying zeros of the family of primitive Dirichlet L-functions of fixed prime conductor Q, as Q→∞, and verified the unitary symmetry predicted by random matrix theory. We compute 1and 2...

Vanishing of L-functions of Elliptic Curves over Number Fields

Let E be an elliptic curve over Q, with L-function LE(s). For any primitive Dirichlet character χ, let LE(s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions LE(s, χ) at the central value s = 1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that LE(1, χ)...

Multiple Dirichlet Series over Rational Function Fields

We explicitly compute some double Dirichlet series constructed from n order Gauss sums over rational function fields. These turn out to be rational functions in q−s1 and q−s2 , where q is the size of the constant field. Key use is made of the group of 6 functional equations satisfied by these series.

Double Dirichlet Series over Function Fields

We construct a nite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a nite eld Fq , q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s;w. W...