Vanishing of Dirichlet L-functions at the central point over function fields

2020/12/30 by Ravi Donepudi, Wanlin Li, Donepudi, Ravi +1 · 1 citation
Computer Science · Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.15319

openalex publication_date 2020/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a geometric criterion for Dirichlet L-functions associated to cyclic characters over the rational function field \mathbbF_q(t) to vanish at the central point s=1/2. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over \mathbbF_q. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over \mathbbF_q(t) whose L-functions vanish at the central point where q=p⁴ⁿ for any rational prime p ≡ 2 \bmod 3. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the L-functions of Dirichlet characters of other orders.

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• Murmurations of quadratic and cubic characters over function fields

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