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Determinants of Subquotients of Galois Representations Associated to Abelian Varieties (1110.0255v4)

Published 3 Oct 2011 in math.NT and math.AG

Abstract: Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell$, the $\elln$-torsion points of $A$ give rise to a representation $\rho_{A, \elln} : \gal(\bar{K} / K) \to \gl_{2g}(\zz/\elln\zz)$. 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.

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