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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Heegner zeros of theta functions


Authors: Jorge Jimenez-Urroz and Tonghai Yang
Journal: Trans. Amer. Math. Soc. 355 (2003), 4137-4149
MSC (2000): Primary 11G05, 11M20, 14H52
Published electronically: June 18, 2003
MathSciNet review: 1990579
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Abstract: Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant $-7$. This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.


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Additional Information

Jorge Jimenez-Urroz
Affiliation: Departamento de Matemática Aplicada IV, ETSETB, Universidad Politecnica de Catalunya, 08034 Barcelona, España
Email: jjimenez@mat.upc.es

Tonghai Yang
Affiliation: Department of Mathematics, University of Wisconsin Madison, Madison, Wisconsin 53717
Email: thyang@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03277-X
PII: S 0002-9947(03)03277-X
Keywords: Theta functions, elliptic curves, Heegner points
Received by editor(s): February 25, 2002
Received by editor(s) in revised form: December 20, 2002
Published electronically: June 18, 2003
Additional Notes: The first author was partially supported by PB90-0179 and Ramon y Cajal program of MCYT. The second author was partially supported by NSF grant DMS-0070476
Article copyright: © Copyright 2003 American Mathematical Society