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Conjecture: 100% of Elliptic Surfaces Over \(\mathbb {Q}\) have Rank Zero

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Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))

Abstract

Based on an equation for the rank of an elliptic surface over \(\mathbb {Q}\) which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank 0 when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain L-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.

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Acknowledgements

We thank Noam Elkies, Bjorn Poonen, and Michael Snarski for helpful discussions. This work was supported by grant 550031 from the Simons Foundation.

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Authors and Affiliations

  1. Math Department, Harvard University, Cambridge, MA, USA

    Alex Cowan

Authors
  1. Alex Cowan
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Correspondence to Alex Cowan .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Noam Elkies

  3. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI, USA

    Brendan Hassett

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Bjorn Poonen

  5. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew V. Sutherland

  6. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    John Voight

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Cowan, A. (2021). Conjecture: 100% of Elliptic Surfaces Over \(\mathbb {Q}\) have Rank Zero. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_10

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