Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields

Abstract

This paper concerns certain two-dimensional abelian varieties A which are Q-simple factors of J0(N) and have extra twist by the character associated to a quadratic number field k. Results of Ribet [22] and Momose [21] are used to give a simple necessary and sufficient condition for A to split over k. The L-series of A over k is the square of the Mellin transform of a cusp form of weight 2 over k with rational integer coefficients which, if A does not split over k, is thus not the L-series of any elliptic curve defined over it. This answers negatively a question raised by the author [5] and others [8,15] in relation to a Weil-Taniyama conjecture for imaginary quadratic number fields. All examples coming from J0(N) with N ⩽ 300 are given explicitly. The complex multiplication case is also considered in more detail: if A has CM by an imaginary quadratic order θ, it is shown that θ must have class number 1 or 2. An explicit construction is given for these (finitely many) cases.