JournalsrmiVol. 27, No. 3pp. 733–750

Finiteness of endomorphism algebras of CM modular abelian varieties

  • Josep González Rovira

    Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain
Finiteness of endomorphism algebras of CM modular abelian varieties cover
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Abstract

Let AfA_f be the abelian variety attached by Shimura to a normalized newform fS2(Γ1(N))newf\in S_2(\Gamma_1(N))^{\operatorname{new}}. We prove that for any integer n>1n > 1 the set of pairs of endomorphism algebras (EndQ(Af)Q,EndQ(Af)Q)\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 ff with complex multiplication such that dimAf=n\dim A_f=n is finite. We determine that this set has exactly 83 pairs for the particular case n=2n=2 and show all of them. We also discuss a conjecture related to the finiteness of the set of number fields EndQ(Af)Q\operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{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