Conjectures for the integral moments and ratios of L-functions over function fields

Abstract

We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of $L$-functions defined over number fields. Specifically, we give a heuristic for the moments and ratios of a family of $L$-functions associated with hyperelliptic curves of genus $g$ over a fixed finite field ${\mathbb{F}}_q$ in the limit as $g\to \infty$. Like in the number field case, there is a striking resemblance to the corresponding formulae for the characteristic polynomials of random matrices. As an application, we calculate the one-level density for the zeros of these $L$-functions.

40 pages

References in corpus (1)

• The Mean Value of $L(\frac{1}{2},\chi )$ in the Hyperelliptic Ensemble

Cited by in corpus (3)

• Sharp upper bounds for moments of quadratic Dirichlet $L$-functions
• The Ratios Conjecture and upper bounds for negative moments of $L$-functions over function fields
• Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic