Revista Matemática Iberoamericana

Finiteness of endomorphism algebras of CM modular abelian varieties

Josep González

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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.

Article information

Rev. Mat. Iberoamericana, Volume 27, Number 3 (2011), 733-750.

First available in Project Euclid: 9 August 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14K22: Complex multiplication [See also 11G15]

modular abelian varieties complex multiplication


González, Josep. Finiteness of endomorphism algebras of CM modular abelian varieties. Rev. Mat. Iberoamericana 27 (2011), no. 3, 733--750.

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