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 E-k in the standard fundamental domain for Gamma lie on A := {e(itheta):pi/ less than or equal to theta less than or equal to 2pi/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.