Chebotarev-Sato-Tate distribution for abelian surfaces potentially of \(\mathrm{GL}_2\)-type
type: Item
instance of: scholarly article
title: Chebotarev-Sato-Tate distribution for abelian surfaces potentially of \(\mathrm{GL}_2\)-type
arXiv ID: 2203.11498
author: Mohammed Amin Amri
published in: European Journal of Mathematics
publication date: 2023-10-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/2203.11498
review text: The Sato-Tate conjecture in the case of abelian varieties can be extended to a conjecture predicting the equidistribution with respect to a certain probability measure of Frobenius conjugacy classes in the Chebotarev-Sato-Tate group associated to an abelian variety and a finite Galois extension of the underlying algebraic number field. The main result of this paper is the proof of the conjecture under certain technical assumptions in certain cases of abelian surfaces potentially of \(\mathrm{GL}_2\)-type over a totally real number field. In order to explain the result more precisely, some notation is required. Let \(A\) be a polarized abelian variety of dimension \(g\) over an algebraic number field \(F\). Given a rational prime \(\ell\), the \(\ell\)-adic rational Tate module \(V_\ell(A)\) is defined as the vector space \[ V_\ell(A)=T_\ell(A)\otimes_{\mathbb{Z}}\mathbb{Q} \] of dimension \(2g\) over \(\mathbb{Q}_\ell\), where \(T_\ell(A)\) is the usual \(\ell\)-adic Tate module. The symplectic basis of this vector space with respect to the Weil pairing arises from a fixed symplectic basis on the singular homology \(H_1(A(\mathbb{C}),\mathbb{Q})\). Then, the action of the absolute Galois group \(G_F\) of the underlying number field \(F\), gives the representation \[ \rho_{A,\ell}:G_F\to \mathrm{GSp}_{2g}(\mathbb{Q}_\ell). \] Let \(G_\ell\) denote its image, \(G_\ell^{\mathrm{Zar}}\) its Zariski closure, and \(G_\ell^{1,\mathrm{Zar}}\) the kernel of the similitude character of \(G_\ell^{\mathrm{Zar}}\). For simplicity, one may fix an isomorphism of the algebraic closure of \(\mathbb{Q}_\ell\) and \(\mathbb{C}\), and view these groups as subgroups of the complex group \(\mathrm{GSp}_{2g}(\mathbb{C})\). The Sato-Tate group \(ST_A\) of the abelian variety \(A\) is defined as the maximal compact subgroup of the complex Lie group \(G_\ell^{1,\mathrm{Zar}}(\mathbb{C})\), which lies inside \(\mathrm{USp}(2g)\), i.e., the intersection of the unitary group and the symplectic …
reviewed by: Neven Grbac
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 7753451
zbMATH Keywords: Sato-Tate distibution
zbMATH Keywords: Galois representations
zbMATH Keywords: abelian varieties
MaRDI profile type: Publication
OpenAlex ID: W4386883740
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DOI: 10.1007/S40879-023-00682-5
description: scientific article; zbMATH DE number 7753451