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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: Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series $E_{k}$ in the standard fundamental domain for $\Gamma $ lie on $A:= \{ e^{i \theta } : \frac{\pi }{2} \leq \theta \leq \frac{2\pi }{3} \}$. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc $A$. Using this result we prove a speculation of Ono, namely that the zeros of the unique ``gap function" in $M_{k}$, the modular form with the maximal number of consecutive zero coefficients in its $q$-expansion following the constant $1$, has zeros only on $A$. In addition, we show that the $j$-invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight $k$.

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Additional Information

Jayce Getz
Affiliation: 4404 South Avenue West, Missoula, Montana 59804

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