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

Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A:= {e^iθ: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.

Duke Scholars

Author Jayce Robert Getz Mathematics