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

Journal Article

Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A:= {eiθ:π/2 ≤ θ ≤ 2π/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 g-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.

Full Text

Duke Authors

Cited Authors

  • Getz, J

Published Date

  • January 1, 2004

Published In

Volume / Issue

  • 132 / 8

Start / End Page

  • 2221 - 2231

International Standard Serial Number (ISSN)

  • 0002-9939

Digital Object Identifier (DOI)

  • 10.1090/S0002-9939-04-07478-7

Citation Source

  • Scopus