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ZEROS OF CERTAIN COMBINATIONS OF EISENSTEIN SERIES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 August 2017

Sarah Reitzes ,
Polina Vulakh  and
Matthew P. Young
Sarah Reitzes
Affiliation:
Department of Mathematics, The University of Chicago, 5734 S University Ave, Chicago IL 60637, U.S.A. email [email protected]
Polina Vulakh
Affiliation:
Bard College, U.S.A. email [email protected]
Matthew P. Young
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A. email [email protected]
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Abstract

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_{k}(z)E_{\ell }(z)-E_{k+\ell }(z)$ in the standard fundamental domain lie on the boundary. We, moreover, find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $Re(z)=\pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 666 - 695
Copyright
Copyright © University College London 2017 

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