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Determinants of subquotients of Galois representations associated with abelian varieties

Part of: Arithmetic algebraic geometry Abelian varieties and schemes

Published online by Cambridge University Press:  18 July 2013

Eric Larson  and
Dmitry Vaintrob
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Abstract

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 ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. 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 $g= 1$, 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.

Keywords

abelian varietiesgalois representations

MSC classification

Secondary: 11G05: Elliptic curves over global fields 14K02: Isogeny 11G15: Complex multiplication and moduli of abelian varieties 14K15: Arithmetic ground fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 3 , July 2014 , pp. 517 - 559
Copyright
©Cambridge University Press 2013 

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