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.

arXiv comment40 pages

References in corpus (6)

• Autocorrelation of Random Matrix Polynomials
• The Mean Value of $L(\frac{1}{2},\chi )$ in the Hyperelliptic Ensemble
• Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field
• Statistics for traces of cyclic trigonal curves over finite fields
• A note on the fourth moment of Dirichlet L-functions
• The sixth power moment of Dirichlet L-functions

Cited by in corpus (17)

• Moments of quadratic twists of elliptic curve L-functions over function fields
• Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-Functions
• Moments of zeta functions associated to hyperelliptic curves over finite fields
• Hybrid Euler-Hadamard product for quadratic Dirichlet $L$-functions in function fields
• The fourth moment of quadratic Dirichlet $L$-functions
• Sharp upper bounds for moments of quadratic Dirichlet $L$-functions
• A representation theory approach to integral moments of L-functions over function fields
• Moments of quadratic Dirichlet $L$-functions over Function Fields
• Rudnick and Soundararajan's Theorem for Function Fields
• Hybrid Euler-Hadamard Product for Dirichlet $L$-functions with Prime conductors over Function Fields
• A Refined Conjecture for the Variance of Gaussian Primes Across Sectors
• Optimal sums of three cubes in ${\mathbb{F}}_q[t]$
• 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
• The First Moment of $L(\frac{1}{2},\chi )$ for Real Quadratic Function Fields
• Quadratic Weyl group multiple Dirichlet series of Type $D_4^{(1)}$
• Rudnick and Soundarajan's Theorem over Prime Polynomials for the Rational Function Field