Determinants of Subquotients of Galois Representations Associated to Abelian Varieties

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