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

2014/04/08 by J.C. Andrade, Julio Andrade, J. C. Andrade +3 · 47 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #Function (biology) #Mathematical analysis #Mathematics #Pure mathematics #math.NT #msc:11G20 #msc:11M50 #msc:14G10

paper · pdf · doi:10.1016/j.jnt.2014.02.019

published in Journal of Number Theory 142, 102-148 (Elsevier BV) · 40 pages

openalex publication_date 2014/04/08 · arxiv created 2014/07/11 · arxiv updated 2014/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 \mathbbF_q in the limit as g→∞. 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.

Citations

• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189

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• The fourth moment of quadratic Dirichlet L-functions
• The Ratios Conjecture and upper bounds for negative moments of L-functions over function fields
• Hybrid Euler-Hadamard Product for Dirichlet L-functions with Prime conductors over Function Fields
• Rudnick and Soundarajan's Theorem over Prime Polynomials for the Rational Function Field
• Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic
• Hybrid Euler-Hadamard product for quadratic Dirichlet L-functions in function fields
• Number fields and function fields: coalescences, contrasts and emerging applications. [europepmc]
• Moments of zeta functions associated to hyperelliptic curves over finite fields. [europepmc]
• Moments of derivatives of quadratic Dirichlet L -functions with prime conductor. [europepmc]

Related

• Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic
• The Ratios Conjecture and upper bounds for negative moments of L-functions over function fields
• The Integral Moments and Ratios of Quadratic Dirichlet L-Functions over Monic Irreducible Polynomials in \mathbbF_q[T]
• Applications of theL-functions ratios conjectures
• Secondary terms in the asymptotics of moments of L-functions