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arXiv · 2102.07745

Root numbers of 5-adic curves of genus two having maximal ramification

Lukas Melninkas
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Abstract

The formulas for local root numbers of abelian varieties of dimension one are known. In this paper we treat the simplest unknown case in dimension two by considering a curve of genus 2 defined over a $5$-adic field such that the inertia acts on the first $\ell$-adic cohomology group through the largest possible finite quotient, isomorphic to $C_5\rtimes C_8$. We give a few criteria to identify such curves and prove a formula for their local root numbers in terms of invariants associated to a Weierstrass equation.

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Lukas Melninkas. 2021-02-15. Root numbers of 5-adic curves of genus two having maximal ramification. https://arxiv.org/abs/2102.07745

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