Open Access
September, 2011 Finiteness of endomorphism algebras of CM modular abelian varieties
Josep González
Rev. Mat. Iberoamericana 27(3): 733-750 (September, 2011).


Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2(\Gamma_1(N))^{\operatorname{new}}$. We prove that for any integer $n > 1$ the set of pairs of endomorphism algebras $\big( \operatorname{End}_{\overline{\mathbb{Q}}}(A_f) \otimes \mathbb{Q}, \operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q} \big)$ obtained from all normalized newforms $f$ with complex multiplication such that $\dim A_f=n$ is finite. We determine that this set has exactly 83 pairs for the particular case $n=2$ and show all of them. We also discuss a conjecture related to the finiteness of the set of number fields $\operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q}$ for the non-CM case.


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Josep González . "Finiteness of endomorphism algebras of CM modular abelian varieties." Rev. Mat. Iberoamericana 27 (3) 733 - 750, September, 2011.


Published: September, 2011
First available in Project Euclid: 9 August 2011

zbMATH: 1256.11037
MathSciNet: MR2895332

Primary: 11G18 , 14K22

Keywords: Complex Multiplication , modular abelian varieties

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.27 • No. 3 • September, 2011
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