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

TL;DR

This paper derives a formula for local root numbers of genus two curves over 5-adic fields with maximal ramification, extending known results from dimension one to a specific complex case.

Contribution

It provides the first explicit formula for local root numbers of genus two curves with maximal ramification at 5, using invariants from Weierstrass equations.

Findings

Derived a formula for local root numbers in the maximal ramification case

Identified criteria for curves with specific inertia action

Connected root numbers to invariants of Weierstrass equations

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.