Determinants of subquotients of Galois representations associated with abelian varieties. (July 2014) - Larson, Eric; Vaintrob, Dmitry
Abstract Given an abelian variety <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg45674" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$A$]]> </alternatives> of dimension <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg4566k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$g$]]> </alternatives> over a number field <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg4561t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$K$]]> </alternatives>, and a prime <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455w3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\ell $]]> </alternatives>, the <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg45585" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\ell }^{n} $]]> </alternatives>-torsion points of <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455xn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$A$]]> </alternatives> give rise to a representation <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455qb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$]]> </alternatives>. In particular, we get a mod-<alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455t0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\ell $]]> </alternatives>representation <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455dc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$]]> </alternatives> and an <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455h1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\ell $]]> </alternatives>-adic representation <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455vj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$]]> </alternatives>. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar. Applying our results in dimension <alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg456g0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$g= 1$]]> </alternatives>, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur's bound on rational isogenies of prime degree to number fields.