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Determination of modular elliptic curves by Heegner points

Luo, Wenzhi and Ramakrishnan, Dinakar (1997) Determination of modular elliptic curves by Heegner points. Pacific Journal of Mathematics, 181 (3). pp. 251-258. ISSN 0030-8730. http://resolver.caltech.edu/CaltechAUTHORS:LUOpjm97

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Abstract

For every integer N ≥ 1, consider the set K(N) of imaginary quadratic fields such that, for each K in K(N), its discriminant D is an odd, square-free integer congruent to 1 modulo 4, which is prime to N and a square modulo 4N. For each K, let c = ([x]−[∞]) be the divisor class of a Heegner point x of discriminant D on the modular curve X = X(0)(N) as in [GZ]. (Concretely, such an x is the image of a point z in the upper half plane H such that both z and Nz are roots of integral, definite, binary quadratic forms of the same discriminant D ([B]).) Then c defines a point rational over the Hilbert class field H of K on the Jacobian J = J(0)(N) of X. Denote by cK the trace of c to K.


Item Type:Article
Additional Information:© Copyright 1997, Pacific Journal of Mathematics. To the memory of Olga Taussky-Todd. This Note is dedicated to the memory of Olga Taussky-Todd. Perhaps it is fitting that it concerns heights and special values, as it was while attending the lectures of B. Gross on this topic in Quebec in June 1985 that the second author first met Olga. We would like to thank B. Gross and W. Duke for comments on an earlier version of the article. Thanks are also due to different people, Henri Darmon in particular, for suggesting that a result such as Theorem A above might hold by a variant of [LR]. Both authors would also like to acknowledge the support of the NSF, which made this work possible.
Record Number:CaltechAUTHORS:LUOpjm97
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:LUOpjm97
Alternative URL:http://pjm.math.berkeley.edu/1997/181-3/p13.html
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ID Code:833
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Oct 2005
Last Modified:26 Dec 2012 08:41

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