Nonvanishing of hyperelliptic zeta functions over finite fields

Ellenberg, Jordan S.; Li, Wanlin; Shusterman, Mark

doi:10.48550/arxiv.1901.08202

Cited by 1 publication

(1 citation statement)

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“…Even though these results show that there are infinitely many hyperelliptic L-functions that vanish at s = 1 2 , it is believed that 100% of hyperelliptic L-functions do not vanish at s = 1 2 in a sense that will be made precise in Section 2. The work of Ellenberg-Li-Shusterman [6] provides evidence supporting this belief. Extending this work beyond hyperelliptic L-functions is the recent work of David-Florea-Lalín [5], where the authors show a positive proportion of L-functions associated to cubic characters do not vanish at the critical point s = 1/2 based on their previous work [4] in which they study moments of the central value of cubic L-functions.…”

Section: Introductionmentioning

confidence: 88%

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

Donepudi¹,

Li²

2020

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We give a geometric criterion for Dirichlet L-functions associated to cyclic characters over the rational function field F 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 F q . Using this geometric criterion, we obtain a lower bound on the number of cubic characters over F q (t) whose L-functions vanish at the central point where q = p 4n for any rational prime p ≡ 2 mod 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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Section: Introductionmentioning

confidence: 88%