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On the representation of the number of integral points of an elliptic curve modulo a prime number

The Ramanujan Journal volume 36, pages 483–499 (2015)Cite this article

Abstract

In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third degree, in explicit non-exponential terms. In the process, we prove explicit formulas for the calculation of certain series involving the Riemann zeta function.

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Notes

  1. By f′(x) we denote the derivative df(x)/dx.

  2. By r′ we denote the derivative dr(x,p)/dx.

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Acknowledgements

I would like to thank my Ph.D. advisor Professor E. Kowalski who suggested to me this area of research and for his very useful advice.

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Authors and Affiliations

  1. Department of Mathematics, ETH-Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Michael T. Rassias

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  1. Michael T. Rassias
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Correspondence to Michael T. Rassias.

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Rassias, M.T. On the representation of the number of integral points of an elliptic curve modulo a prime number. Ramanujan J 36, 483–499 (2015). https://doi.org/10.1007/s11139-013-9524-9

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  • DOI: https://doi.org/10.1007/s11139-013-9524-9

Keywords

  • Elliptic curves
  • Integral points
  • Exponential sums
  • Riemann zeta function

Mathematics Subject Classification

  • 11L03
  • 11L07
  • 14H52
  • 11M06
  • 11Yxx
  • 11T71