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Journal of the London Mathematical Society 1992 s2-45(3):404-416; doi:10.1112/jlms/s2-45.3.404
© 1992 by London Mathematical Society
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© Oxford University Press

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

J. E. Cremona

Department of Mathematics, University of Exeter North Park Road, Exeter EX4 4QE

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 {theta}, it is shown that {theta} must have class number 1 or 2. An explicit construction is given for these (finitely many) cases.


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