On the root numbers of abelian varieties with real multiplication

TL;DR

This paper investigates the local root numbers of abelian varieties with real multiplication over p-adic fields, providing explicit formulas under certain reduction types and inertia action conditions.

Contribution

It establishes the reduction types of such abelian varieties and derives explicit formulas for their local root numbers in different cases.

Findings

Abelian varieties with real multiplication have either potentially good or totally toric reduction.

Explicit formulas for local root numbers are provided under specific inertia action assumptions.

The results apply to p-adic fields with p > 2, expanding understanding of local root numbers in this context.

Abstract

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.