Finiteness of endomorphism algebras of CM modular abelian varieties

Abstract

Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2({\Gamma }_1(N){)}^{\mathrm{new}}$. We prove that for any integer $n>1$ the set of pairs of endomorphism algebras $({\mathrm{End}}_{\overline{\mathbb{Q}}}(A_f)\otimes \mathbb{Q},{\mathrm{End}}_{\mathbb{Q}}(A_f)\otimes \mathbb{Q})$ 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 ${\mathrm{End}}_{\mathbb{Q}}(A_f)\otimes \mathbb{Q}$ for the non-CM case.