Root numbers of 5-adic curves of genus two having maximal ramification
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 -adic field such that the inertia acts on the first -adic cohomology group through the largest possible finite quotient, isomorphic to . 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.
Keywords
Cite
@article{arxiv.2102.07745,
title = {Root numbers of 5-adic curves of genus two having maximal ramification},
author = {Lukas Melninkas},
journal= {arXiv preprint arXiv:2102.07745},
year = {2021}
}
Comments
22 pages; major revision: changed the title, deleted the part on elliptic curves (to appear in another submission), changed the formulation of the main result, shortened Section 1, corrected 1.8, merged and shortened Sections 2 and 3, the discussion on discriminants and conductors was rewritten and is now contained in 3.12, the four examples in section 7 were replaced by a new one in 5.9