Principally polarized abelian surfaces with surjective Galois representations on \(l\)-torsion
type: Item
instance of: scholarly article
zbMATH Open document ID: 1320.11052
author: Erik Wallace
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.3635
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 6355580
zbMATH Keywords: Galois representations
zbMATH Keywords: torsion on abelian varieties
zbMATH Keywords: polarized abelian surfaces
MaRDI profile type: Publication
OpenAlex ID: W2140795343
arXiv ID: 1212.3635
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DOI: 10.1112/JLMS/JDU033
title: Principally polarized abelian surfaces with surjective Galois representations on \(l\)-torsion
published in: Journal of the London Mathematical Society. Second Series
review text: Let \(E\) be an elliptic curve over \(\mathbb Q\), without complex multiplication, and \(l\) be a prime number. Then the absolute Galois group acts on the \(l\)-torsion sub-group \(E[l]\) as a subgroup of \(\mathrm{Aut}(E[l])\simeq \mathrm{GL}_2(\mathbb{F}_l)\). It is expected that this subgroup is in fact the full group \(\mathrm{GL}_2(\mathbb{F}_l)\) and this is the case for all but finitely many primes \(l\). Also, a theorem of \textit{W. Duke} [C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 8, 813--818 (1997; Zbl 1002.11049)] states that the density of elliptic curves defined over \(\mathbb Q\) admitting at least one `exceptional' prime \(l\), i.e. a prime \(l\) for which the mentioned representation is not surjective, is zero. A generalization over number fields was provided by \textit{D. Zywina} [Bull. Lond. Math. Soc. 42, No. 5, 811--826 (2010; Zbl 1221.11136)].NEWLINENEWLINEIn the paper under review, the author considers the natural analogue for abelian surfaces: namely, given a family of abelian surfaces over an algebraic variety, it is proved that for almost all of them the Galois action on their points of \(l\)-torsion, for every prime \(l\), is given by the full group \(\mathrm{GSP}_{2g}({\mathbb F}_l)\) of symplectic similitudes.NEWLINENEWLINEMore precisely: let \(K\) be a number field, \(V\) a smooth geometrically irreducible affine variety of dimension \(3\), \(\mathcal{A}\to V\) a principally polarized abelian scheme over \(V\) of relative dimension \(2\), inducing a dominant map to the moduli space of principally polarized abelian surfaces. Let us fix a height function on \(V(\bar{K})\) and consider, for each \(x\geq 1\), the set \(E_K(x)\) of points \(t\in V(K)\) of height \(\leq x\) such that the abelian surface \(A_t\) has the following property: for at least one prime \(l\) the Galois action on \({\mathcal A}_t[l]\) is not surjective on the group \(\mathrm{GSP}_{2g}({\mathbb F}_l)\). Finally, denote by \(B_K(x)\) the set of all the \(K\)-ration…
reviewed by: Pietro Corvaja
description: scientific article; zbMATH DE number 6355580