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

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 $k$.