Mathematical Proceedings of the Cambridge Philosophical Society



On the Zeros of Certain Cusp Forms


SANOLI GUN a1
a1 Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India. e-mail: jhulan@mri.ernet.in

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Abstract

F.K.C. Rankin and H.P.F. Swinnerton–Dyer proved that all the zeros of the Eisenstein Series $E_k$ contained in the standard fundamental domain $\mathcal{F}$ lie on the arc $A\,{=}\, \{ e^{i\theta}| {\pi}/{3} \,{\le}\, \theta \,{\le}\, {\pi}/{2}\}$. Recently, J. Getz has generalized the method of Rankin and Swinnerton–Dyer to show that modular forms under certain conditions have similar properties. In this paper we prove similar results for certain types of cusp forms, motivated by the work of R.A. Rankin. Further, we give a closed formula for the zeros of a class of cusp forms in terms of the Fourier coefficients following the method of Kohnen.

(Published Online September 28 2006)
(Received May 18 2005)
(Revised July 18 2005)