Nonvanishing of L--functions associated to fixed order characters over function fields

2025/06/09 by Chantal David, Alexandra Florea, David, Chantal +3 · 2 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2506.07815

Abstract

We show that a positive proportion of the values L(1/2,χ_c) are non-zero, where χ_c is the ℓ^th residue symbol for ℓ ≥ 3 over \mathbbF_q[t], when averaging over square-free polynomials c in \mathbbF_q[t], as q ≡ 1(\textrmmod 2ℓ) is fixed and the degree of c goes to infinity. In the case of ℓ=3, we show that at least 1/6 of L(1/2,χ_c)≠ 0, while for ℓ>3, the proportion depends on the order of the character. This improves a previous result of Ellenberg, Li, and Shusterman showing that there are infinitely many χ of (prime) order ℓ such that L(1/2, χ) ≠ 0 (with completely different techniques). Our result is achieved by computing the one-level density of zeros in the family of L--functions and surpassing the (-1,1) barrier for the support of the Fourier transform of the test function, necessary to obtain a positive proportion of non-vanishing result. Using similar techniques, we also prove a result towards the equidistribution of the angles of the order ℓ shifted Gauss sums when summing over prime arguments, a result which may be of independent interest.

Cited by

• Twisted second moment of primitive cubic L-functions
• Murmurations of quadratic and cubic characters over function fields

Related

• Unconditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros
• One-level density and non-vanishing for cubic L-functions over the Eisenstein field
• Non-vanishing for cubic L--functions
• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189
• A one-formula proof of the nonvanishing of L-functions of real\n characters at 1