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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Periods of cusp forms and elliptic curves over imaginary quadratic fields

Authors: J. E. Cremona and E. Whitley
Journal: Math. Comp. 62 (1994), 407-429
MSC: Primary 11F67; Secondary 11F66, 11G05, 11G40
MathSciNet review: 1185241
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Abstract: In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups $ {\Gamma _0}(\mathfrak{n})$, where n is an ideal in the ring of integers R of K. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series $ L(F,s)$ at $ s = 1$ and compare with the value of $ L(E,1)$ which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that $ L(F,1) = 0$ whenever $ E(K)$ has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.

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Article copyright: © Copyright 1994 American Mathematical Society

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