Math @ Duke

Publications [#292895] of Jayce R. Getz
Papers Published
 Getz, J, A generalization of a theorem of Rankin and SwinnertonDyer on zeros of modular forms,
Proceedings of the American Mathematical Society, vol. 132 no. 8
(2004),
pp. 22212231, ISSN 00029939 [doi]
(last updated on 2017/12/16)
Abstract: Rankin and SwinnertonDyer (1970) prove that all zeros of the Eisenstein series E k 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 gexpansion following the constant 1, has zeros only on A. In addition, we show that the jinvariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight k.


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