# Finiteness of endomorphism algebras of CM modular abelian varieties

### Josep González Rovira

Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain

## Abstract

Let $A_{f}$ be the abelian variety attached by Shimura to a normalized newform $f∈S_{2}(Γ_{1}(N))_{new}$. We prove that for any integer $n>1$ the set of pairs of endomorphism algebras $(End_{Q}(A_{f})⊗Q,End_{Q}(A_{f})⊗Q)$ obtained from all normalized newforms $f$ with complex multiplication such that $dimA_{f}=n$ 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 $End_{Q}(A_{f})⊗Q$ for the non-CM case.

## Cite this article

Josep González Rovira, Finiteness of endomorphism algebras of CM modular abelian varieties. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 733–750

DOI 10.4171/RMI/651