Finiteness of endomorphism algebras of CM modular Abelian varieties
type: Item
instance of: scholarly article
title: Finiteness of endomorphism algebras of CM modular Abelian varieties
zbMATH Open document ID: 1256.11037
DOI: 10.4171/RMI/651
author: Josep González
published in: Revista Matemática Iberoamericana
publication date: 2011-10-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1312906776
review text: This intriguing article presents some evidence for a conjecture attributed to Coleman on the endomorphism algebras associated to general abelian varieties, particularly in the setting of certain modular abelian varieties. To state the general conjecture, let \(A\) be an abelian variety defined over a number field \(L\). We write \(\text{End}_L(A)\) to denote the ring of endomorphisms of \(A\) defined over \(L\), and \(\text{End}_L^0(A) = \text{End}_L(A) \otimes {\mathbb{Q}}\) its associated endomorphism algebra. Given integers \(n, m \geq 1\), it is conjectured that there exist ony finitely many \({\mathbb{Q}}\)-algebras \(M\) such that \(M \approx \text{End}_L^0(A)\) for some abelian variety \(A\) of dimension \(n\) defined over a number field \(L\) of degree \(m\). Inspired by this conjecture, the author proves the following result. Let \(f\) be a normalized cuspidal newform of weight \(2\), level \(\Gamma_1(N)\), and trivial Nebentype. Let \[ f(z)= \sum_{n \geq 1} a_n(f) q^n, \qquad q = \exp(2 \pi i z) \] denote its Fourier series expansion, and \(L_f = {\mathbb{Q}}(a_v(f))\) the number field obtained by adjoining to \({\mathbb{Q}}\) the coefficients \(a_v(f)\) for any positive density set of primes \(v\). A classical construction associates to \(f\) an abelian variety \(A_f\) defined over \({\mathbb{Q}}\) of dimension \([L_f: {\mathbb{Q}}]\). This abelian variety occurs as a simple quotient over \({\mathbb{Q}}\) of the Jacobian of the modular curve \(X_1(N)\), and its endomorphism algebra \(\text{End}_L^0(A_f)\) is isomorphic to \(L_f\). Anyhow, for each integer \(n \geq 1\), consider the set \(\mathcal{S}_n\) of pairs of endomorphism algebras \[ \left( \text{End}^0_{\overline{\mathbb{Q}}}(A_f), \text{End}^0_{{\mathbb{Q}}}(A_f) \right), \] where \(f\) ranges over the set of normalized newforms of weight \(2\), level \(N\), and trivial Nebentype such that \(\dim(A_f)=n\). It is easy to see from the theory of complex multiplication for elliptic curves that \(\math…
reviewed by: Jeanine Van Order
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 5965412
zbMATH Keywords: modular abelian varieties
zbMATH Keywords: complex multiplication
MaRDI profile type: Publication
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OpenAlex ID: W1572759640
description: scientific article; zbMATH DE number 5965412