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October 2021 Vanishing of Dirichlet L-functions at the central point over function fields
Ravi Donepudi, Wanlin Li
Rocky Mountain J. Math. 51(5): 1615-1628 (October 2021). DOI: 10.1216/rmj.2021.51.1615

Abstract

We give a geometric criterion for Dirichlet L-functions associated to cyclic characters over the rational function field 𝔽q(t) to vanish at the central point s=12. 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 𝔽q. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over 𝔽q(t) whose L-functions vanish at the central point where q=p4n for any rational prime p2 mod3. 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.

Citation

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Ravi Donepudi. Wanlin Li. "Vanishing of Dirichlet L-functions at the central point over function fields." Rocky Mountain J. Math. 51 (5) 1615 - 1628, October 2021. https://doi.org/10.1216/rmj.2021.51.1615

Information

Received: 1 January 2021; Revised: 12 February 2021; Accepted: 12 February 2021; Published: October 2021
First available in Project Euclid: 17 February 2022

Digital Object Identifier: 10.1216/rmj.2021.51.1615

Subjects:
Primary: 11M38 , 11S40 , 11T22
Secondary: 14G10

Keywords: abelian varieties over finite fields , Carlitz extensions , Chowla’s conjecture , Cyclotomic function fields , L-functions , zeta functions of curves

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 5 • October 2021
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