Joint IAS/Princeton University Number Theory Seminar

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.