A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms

Author:
Jayce Getz

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2221-2231

MSC (2000):
Primary 11F11

Published electronically:
March 4, 2004

Corrigendum:
Proc. Amer. Math. Soc. 138 (2010), 1159

MathSciNet review:
2052397

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Abstract | References | Similar Articles | Additional Information

Abstract: Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series in the standard fundamental domain for lie on . In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc . Using this result we prove a speculation of Ono, namely that the zeros of the unique ``gap function" in , the modular form with the maximal number of consecutive zero coefficients in its -expansion following the constant , has zeros only on . In addition, we show that the -invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight .

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Additional Information

**Jayce Getz**

Affiliation:
4404 South Avenue West, Missoula, Montana 59804

Email:
getz@fas.harvard.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07478-7

Keywords:
Modular forms

Received by editor(s):
March 21, 2003

Published electronically:
March 4, 2004

Additional Notes:
The author thanks the University of Wisconsin at Madison for its support.

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2004
American Mathematical Society