# Commentarii Mathematici Helvetici

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**Volume 86, Issue 2, 2011, pp. 317–351**

**DOI: 10.4171/CMH/225**

Published online: 2011-02-27

Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves)

Josep González Rovira^{[1]}and Joan-Carles Lario

^{[2]}(1) Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain

(2) Universitat Politècnica de Catalunya, Barcelona, Spain

Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2(\Gamma_1(N))$ and assume that $A_f$ has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in $S_2(\Gamma_1(N))$ corresponding to the pullbacks of the regular differentials of all elliptic quotients of $A_f$. For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case $N=p^2$, for primes $p>3$ and $p\equiv 3$ mod $4$. For this case we prove that if $f$ has CM, then all optimal elliptic quotients of $A_f$ are also optimal in the sense that its endomorphism ring is the maximal order of $\mathbb{Q}(\sqrt{-p})$. Moreover, if $f$ has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve $A(p)$ and its Galois conjugates. Among all modular parametrizations $J_0(p^2)\to A(p)$, we describe a canonical one and discuss some of its properties.

*Keywords: *Modular abelian varieties, complex multiplication, Grössencharacter, optimal elliptic quotient

González Rovira Josep, Lario Joan-Carles: Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves). *Comment. Math. Helv.* 86 (2011), 317-351. doi: 10.4171/CMH/225