Endomorphism algebras of motives attached to elliptic modular forms.
type: Item
instance of: scholarly article
title: Endomorphism algebras of motives attached to elliptic modular forms.
zbMATH Open document ID: 1050.11062
author: Alexander F. Brown
author: Eknath Ghate
publication date: 2004-01-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116082
review text: The authors study the endomorphism algebras of motives attached to cusp forms. Let \(f=\sum a_n q^n\) be a normalized cuspidal eigen newform of weight \(k\geq 2\), level \(N\geq 1\), and nebentypus \(\varepsilon\). When \(k>2\), let \(M_f\) denote the motif attached to \(f\) constructed by \textit{A. J. Scholl} [Invent. Math., 100, 419-430 (1990; Zbl 0760.14002)]. When \(k=2\), let \(M_f\) be the Abelian variety \(A_f\) defined by Shimura and Eichler. Let \(X_f\) denote the endomorphism algebra of \(M_f\). Suppose that \(f\) does not have CM, that is, it is not attached to a Hecke character of an imaginary quadratic field, and \(f\) has an extra twist in the sense that \(f\) is a twist of its conjugate. Following \textit{K. A. Ribet} [Math. Ann. 253, 43--62 (1980; Zbl 0421.14008)], the authors consider a simple algebra \(X\) associated to 2-cocycles in terms of Jacobi sums. As one of the main theorems, the authors show that \(X\) is isomorphic to a subalgebra of \(X_f\). This is an analogon of a \textit{M. Momose} [J. Fac. Sci. Univ. Tokyo 28, 89-109 (1981; Zbl 0482.10023)] where \(X_f\) is replaced by the endomorphism algebras of the Betti realization of \(M_f\). The algebra \(X\) is expected to be \(X_f\), which is a consequence of the Tate conjecture. The equality is proved by Ribet when \(k=2\), (loc. cit.) and it also follows from Faltings' theorem. The remaining part of this paper is to determine the simple algebra \(X\). It is shown that \(X\) is either a matrix algebra over a totally real field \(F\) or a matrix algebra of a central quaternion division algebra over \(F\). The authors relate the local invariants of \(X\) and the valuation of the Fourier coefficients \(a_p\) and the character values \(\varepsilon(p)\). Among other results, \(X\) is completely determined in the case where \(\varepsilon\) is real and \(p\) does not divide \(2N\) (Theorem 4.1.11). The approach is to reduce the problems to information of the filtered \((\phi,N)\)-modules associate…
reviewed by: Chia-Fu Yu
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F32
zbMATH DE Number: 2021759
zbMATH Keywords: endomorphism algebras
zbMATH Keywords: modular motives
zbMATH Keywords: Tate conjecture
zbMATH Keywords: filtered \((\phi
zbMATH Keywords: N)\)-modules
zbMATH Keywords: Newton polygons
zbMATH Keywords: symbols
MaRDI profile type: Publication
OpenAlex ID: W2472082289
cites work: Twists of newforms and pseudo-eigenvalues of \(W\)-operators
cites work: Motives for Hilbert modular forms
cites work: Q5607047
cites work: Lectures on \(p\)-divisible groups
cites work: Les Schémas de Modules de Courbes Elliptiques
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Q4858020
cites work: Q4490714
cites work: Motives, numerical equivalence, and semi-simplicity
cites work: Q3939877
cites work: La classe de Brauer de l'algèbre d'endomorphismes d'une variété abélienne modulaire
cites work: Twists of modular forms and endomorphisms of abelian varieties
cites work: Q3919818
cites work: Motives for modular forms
cites work: Quelques applications du théorème de densité de Chebotarev
cites work: On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields
cites work: On the factors of the Jacobian variety of a modular function field
cites work: \(\ell\)-adic representations associated to elliptic curves on \(\mathbb{Q}_p\)
DOI: 10.5802/AIF.1989
Recommended article: Adjoint lifts and modular endomorphism algebras
Recommended article: Q5461861
Recommended article: Finiteness of endomorphism algebras of CM modular Abelian varieties
Recommended article: Crossed product algebras attached to weight one forms
Recommended article: Twists of central simple algebras and endomorphism algebras of some abelian varieties
Recommended article: Irreducibility of Hecke polynomials
Recommended article: Q3724459
Recommended article: Some results on Galois representations attached to motives of \(\mathrm{GL}_2\)-type
Recommended article: Siegel modular forms of degree three and the cohomology of local systems
Recommended article: Periods of Hecke characters
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2021759