The existence of an abelian variety over \(\overline{\mathbb{Q}}\) isogenous to no Jacobian
type: Item
instance of: scholarly article
title: The existence of an abelian variety over \(\overline{\mathbb{Q}}\) isogenous to no Jacobian
zbMATH Open document ID: 1250.14032
author: Jacob Tsimerman
published in: Annals of Mathematics. Second Series
publication date: 2012-08-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.2181
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 6074023
zbMATH Keywords: abelian variety
zbMATH Keywords: André-Oort conjecture
zbMATH Keywords: complex multiplication
zbMATH Keywords: Jacobian
zbMATH Keywords: Shimura subvariety
reviewed by: Dan Erik Petersen
MaRDI profile type: Publication
OpenAlex ID: W2142201959
arXiv ID: 1010.2181
cites work: The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy
cites work: Abelian varieties isogenous to a Jacobian
cites work: Equidistribution of special subvarieties
cites work: Travaux de Shimura
cites work: Q3877792
cites work: Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces
cites work: The number of extensions of a number field with fixed degree and bounded discriminant
cites work: Q4108427
cites work: Big symplectic or orthogonal monodromy modulo \(\ell\)
cites work: Q4830109
cites work: Q4135671
cites work: Q4371936
cites work: Q4836066
cites work: A conjecture of Yves André's
DOI: 10.4007/ANNALS.2012.176.1.12
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review text: In [ibid. 176, No. 1, 589--635 (2012; Zbl 1263.14032)], \textit{Ch.-L. Chai} and \textit{F. Oort} proved the following theorem: let \(A_g\) be the moduli space of principally polarized abelian varieties of dimension \(g\) over \(\mathbb{\overline Q}\), and \(X \subset A_g\) a proper closed subvariety. If the André-Oort conjecture holds, then there is a point \([A] \in A_g(\mathbb{\overline Q})\) such that \(A\) is not isogenous to \(B\), for any \([B] \in X\) [\textit{C. Chai} and \textit{F. Oort}, Ann. Math. 176, 589--635 (2012; Zbl 1263.14032)]. (The title takes its name from the special case when \(g \geq 4\) and \(X\) is the Torelli locus.)NEWLINENEWLINEIn this article, the same statement is proven, without use of the André-Oort conjecture. The strategy is to modify Klingler-Yafaev's conditional proof of André-Oort assuming the GRH [\textit{B. Klingler} and \textit{A. Yafaev}, ``The André-Oort conjecture'', preprint (2008)]. They need the GRH to produce `many' `small' split primes for certain CM fields. What Tsimerman does is prove in this particular case the existence of sufficiently many CM fields with enough small split primes `by hand', using powerful equidistribution results, after which he can carry out Klingler-Yafaev's proof unconditionally.
description: scientific article; zbMATH DE number 6074023