Arithmetic of Division Fields

TL;DR

This paper investigates the properties of division fields associated with semistable abelian varieties over the rationals, focusing on the Galois group of 2-division fields and classifying certain mod 2 representations under GRH.

Contribution

It provides new insights into the Galois structure of division fields and classifies small conductor mod 2 representations for semistable abelian varieties.

Findings

Galois group analysis of 2-division fields with odd, squarefree conductor

Classification of small conductor irreducible semistable mod 2 representations under GRH

Applications to paramodular abelian varieties of odd conductor

Abstract

We study the arithmetic of division fields of semistable abelian varieties A over the rationals. The Galois group of the 2-division field of A is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in "Paramodular abelian varieties of odd conductor," arXiv:1004.4699.