# Moments of zeta functions associated to hyperelliptic curves

# Moments of zeta functions associated to hyperelliptic curves

I will discuss conjectures, theorems, and experiments concerning the moments, at the central point, of zeta functions associated to hyperelliptic curves over finite fields of odd characteristic. 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 $d$. Let $2g=d-1$ if $d$ is odd, and $2g=d-2$ if $d$ is even. 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 of matrices in $USp(2g)$, evaluated at the central point. Using techniques that were originally developed for studying moments of L-functions over number fields, Andrade and Keating have 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. We will also discuss uniform estimates, in both parameters $q,d$, for the moments. This is joint work with Kevin Wu.