Publications [#292895] of Jayce R. Getz

Papers Published

1. Getz, J, A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms, Proceedings of the American Mathematical Society, vol. 132 no. 8 (January, 2004), pp. 2221-2231, American Mathematical Society (AMS), ISSN 0002-9939 [doi]
   (last updated on 2024/04/24)

   Abstract:
   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.