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Determinants of subquotients of Galois representations associated with abelian varieties

  • Eric Larson and Dmitry Vaintrob

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.

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Determinants of subquotients of Galois representations associated with abelian varieties

  • Eric Larson and Dmitry Vaintrob

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