Moments of zeta functions associated to hyperelliptic curves over finite fields

Abstract

Let $q$ be an odd prime power, and $H_{d,q}$ denote the set of square-free monic polynomials $D(x)\in F_q[x]$ of degree $d$. Katz and Sarnak showed that the moments, over $H_{d,q}$, of the zeta functions associated to the curves $y^2=D(x)$, evaluated at the central point, tend, as $q\to \infty$, to the moments of characteristic polynomials, evaluated at the central point, of matrices in $USp(2\lfloor (d-1)/2\rfloor )$. Using techniques that were originally developed for studying moments of $L$-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for $q$ fixed and $d\to \infty$. We provide theoretical and numerical evidence in favour of their conjecture. In some cases we are able to work out exact formulas for the moments and use these to precisely determine the size of the remainder term in the predicted moments.

32 pages

References in corpus (3)

• The Mean Value of $L(\frac{1}{2},\chi )$ in the Hyperelliptic Ensemble
• Conjectures for the integral moments and ratios of L-functions over function fields
• Weyl group multiple Dirichlet series constructed from quadratic characters

Cited by in corpus (1)

• Moments of quadratic Dirichlet $L$-functions over Function Fields