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

This paper concerns certain two‐dimensional abelian varieties A which are Q‐simple factors of J 0 ( 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 J 0 ( 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.