Generalized Legendre curves and quaternionic multiplication
type: Item
instance of: scholarly article
title: Generalized Legendre curves and quaternionic multiplication
zbMATH Open document ID: 1336.33009
author: Alyson Deines-Schartz
author: Jenny G. Fuselier
author: Ling Long
author: Holly Swisher
author: Fang-Ting Tu
published in: Journal of Number Theory
publication date: 2015-12-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1412.6906
review text: By using results by \textit{J. Wolfart} [Invent. Math. 92, No. 1, 187--216 (1988; Zbl 0649.10022)], \textit{N. Archinard} [Can. J. Math. 55, No. 5, 897--932 (2003; Zbl 1038.11036)] and \textit{G. Wüstholz} [in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 476--483 (1987; Zbl 0679.10023)] on periods and decompositions of abelian varieties, or results by \textit{K. Yamamoto} [J. Comb. Theory 1, 476--489 (1966; Zbl 0173.03901)] on Gauss sums, the authors consider abelian varieties from generalized Legendre curves, particularly their corresponding Galois representations, periods and endomorphism algebras. For certain one parameter families of two-dimensional abelian varieties, the question whether the endomorphism algebra of each fiber defined over the algebraic closure of \(\mathbb{Q}\) contains a quaternion algebra, is answered.
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 6522674
zbMATH Keywords: abelian varieties
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: Galois representations
zbMATH Keywords: triangle groups
zbMATH Keywords: Shimura curves
zbMATH Keywords: counting points over finite fields
reviewed by: József Sándor
describes a project that uses: LMFDB
MaRDI profile type: Publication
OpenAlex ID: W1597986446
arXiv ID: 1412.6906
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DOI: 10.1016/J.JNT.2015.04.019
description: scientific article