Finiteness of endomorphism algebras of CM modular abelian varieties

Abstract

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.