Determining monodromy groups of abelian varieties
type: Item
instance of: scholarly article
title: Determining monodromy groups of abelian varieties
zbMATH Open document ID: 1498.14116
author: David Zywina
published in: Research in Number Theory
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2009.07441
review text: Given an abelian variety over a number field one can associate to it its motivic Galois group, its Mumford-Tate group (defined by means of Hodge decomposition), its \(\ell\)-adic monodromy groups (where \(\ell\) is a rational prime), and its Sato-Tate group. The author assumes the Mumford-Tate conjecture and shows that from two well chosen Frobenius polynomials of the abelian variety (corresponding to two ``random'' primes), one can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. He also proposes a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular one obtains a probabilistic algorithm to compute the dimension of the Hodge classes of the abelian variety for any fixed degree.
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 7608402
zbMATH Keywords: abelian varieties
zbMATH Keywords: Galois representations
zbMATH Keywords: Mumford-Tate groups
zbMATH Keywords: monodromy groups
reviewed by: Vladimir P. Kostov
describes a project that uses: Magma
describes a project that uses: GitHub
MaRDI profile type: Publication
OpenAlex ID: W3087407610
arXiv ID: 2009.07441
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DOI: 10.1007/S40993-022-00391-0
description: scientific article