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Determinants of subquotients of Galois representations associated with abelian varieties
Published online by Cambridge University Press: 18 July 2013
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.
- 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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