Determinants of subquotients of Galois representations associated with abelian varieties. (July 2014)

• Record Type: Journal Article
• Authors: Larson, Eric; Vaintrob, Dmitry
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 13:Number 3(2014:Jul.)
• Page Start: 517
• Page End: 559
• Publication Date: 2014-07
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748013000182
• Languages: English
• ISSNs: 1474-7480

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>Given an abelian variety <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg45674" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$A$]]></tex-math></alternatives></inline-formula> of dimension <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg4566k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$g$]]></tex-math></alternatives></inline-formula> over a number field <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg4561t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$K$]]></tex-math></alternatives></inline-formula>, and a prime <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455w3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\ell $]]></tex-math></alternatives></inline-formula>, the <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg45585" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\ell }^{n} $]]></tex-math></alternatives></inline-formula>-torsion points of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455xn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$A$]]></tex-math></alternatives></inline-formula> give rise to a representation <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455qb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$]]></tex-math></alternatives></inline-formula>. In particular, we get a mod-<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455t0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\ell $]]></tex-math></alternatives></inline-formula><italic>representation</italic> <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455dc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$]]></tex-math></alternatives></inline-formula> and an <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455h1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\ell $]]></tex-math></alternatives></inline-formula>-<italic>adic representation</italic> <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg455vj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$]]></tex-math></alternatives></inline-formula>. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.</p> <p>Applying our results in dimension <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh2cg456g0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$g= 1$]]></tex-math></alternatives></inline-formula>, 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.</p> </abstract>