Determinants of Subquotients of Galois Representations Associated to Abelian Varieties

TL;DR

This paper investigates the determinants of subquotients of Galois representations arising from the torsion points of abelian varieties over number fields, revealing surprising similarities between mod-$ ell$ and $ ell$-adic characters.

Contribution

It characterizes the possible determinants of subquotients of Galois representations associated to abelian varieties, highlighting the parallels between mod-$ ell$ and $ ell$-adic cases.

Findings

Possible mod-$ ell$ characters are similar to $ ell$-adic characters.

Not all mod-$ ell$ subquotients lift to $ ell$-adic representations.

Provides a classification of determinants for subrepresentations across all abelian varieties.

Abstract

Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell$, the $\ell^n$-torsion points of $A$ give rise to a representation $\rho_{A, \ell^n} : \gal(\bar{K} / K) \to \gl_{2g}(\zz/\ell^n\zz)$. In particular, we get a mod-$\ell$ representation $\rho_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\ff_\ell)$and an $\ell$-adic representation $\rho_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\zz_\ell)$. In this paper, we describe the possible determinants of subrepresentations (or more generally, subquotients) of these two representation for $\ell$ a prime number, as $A$ varies over all $g$-dimensional abelian varieties. Note that it is certainly not the case that any mod-$\ell$ subquotient lifts to an $\ell$-adic one. Nevertheless, the list of possible mod-$\ell$ characters turns out to be remarkably similar to the list of possible $\ell$-adic characters.