A Generalization of a Theorem of Rankin and Swinnerton-dyer on Zeros of Modular Forms

Rankin and Swinnerton-Dyer [R, S-D] 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 Mk, 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 the weight k.