A simple method to extract zeros of certain Eisenstein series of small level

Abstract

This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level N where \(N=2,3,5\) and 7. The method is based on the observation that these Eisenstein series are integral over the graded algebra of modular forms on SL(2, Z) and their zeros are ‘controlled’ by those of \(E_4\) and \(E_6\) in the fundamental domain of \(\Gamma _0(N)\).

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Acknowledgements

The author would like to thank Council of Scientific and Industrial Research (CSIR) for the funding. She thanks the anonymous referee for suggestions and comments. This work would not have been possible without the help and guidance of Prof. Justin R David, Department of Centre for High Energy Physics, Indian Institute of Science, Bangalore and Prof. Soumya Das, Department of Mathematics, Indian Institute of Science, Bangalore. The author also thanks Ritwik Pal and Pramath Anamby, Department of Mathematics, Indian Institute of Science, Bangalore for useful discussions.

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Correspondence to Aradhita Chattopadhyaya.

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Communicating Editor: A Raghuram

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Chattopadhyaya, A. A simple method to extract zeros of certain Eisenstein series of small level. Proc Math Sci 130, 6 (2020). https://doi.org/10.1007/s12044-019-0530-8

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Keywords

  • Zeros of Eisenstein series
  • zeros of modular forms of \(\Gamma _0(N)\)

2000 Mathematics Subject Classification

  • 11F11
  • 11F03