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

DOI:
https://doi.org/10.1090/S0002-9947-03-03277-X

Published electronically:
June 18, 2003

MathSciNet review:
1990579

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Abstract | References | Similar Articles | Additional Information

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 . 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:
https://doi.org/10.1090/S0002-9947-03-03277-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