Determinants of Subquotients of Galois Representations Associated to Abelian Varieties

Details Determinants of Subquotients of Galois Representations Associated to Abelian Varieties Determinants of Subquotients of Galois Representations Associated to Abelian Varieties

2011-10-03

arxiv.org/abs/1110.0255v4

Mathematics

Number Theory

Scientific paper

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.

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