arXiv subjects

Lukas Melninkas

Publications and source records attributed to Lukas Melninkas.

2 recordsLinked to original sources

arXiv2021

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

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.

math.NT↗

arXiv2021

On the root numbers of abelian varieties with real multiplication

Let $A/K$ be an abelian variety with real multiplication defined over a $p$-adic field $K$ with $p>2$. We show that $A/K$ must have either potentially good or potentially totally toric reduction. In the former case we give formulas of the local root number of $A/K$ under the condition that inertia acts via an abelian quotient on the associated Tate module; in the latter we produce formulas without additional hypotheses.

math.NT↗