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The Sato–Tate distribution in thin parametric families of elliptic curves

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Abstract

We obtain new results concerning the Sato–Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.

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Notes

  1. We take the opportunity to note that in the proof of [5, Theorem 5.1], there are some absolute value symbols that should be brackets; this is inconsequential for the argument.

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Acknowledgements

The authors are grateful to Ping Xi for interesting discussions. The authors would like to thank the referee for valuable comments. The research of the first author was supported by an IUF junior, the second and third authors were supported by the Australian Research Council Grant DP130100237, and the research of the fourth author was supported by the Simons Foundation Grant #234591.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot, Sorbonne Paris Cité, UMR 7586, Case 7012, 75013, Paris, France

    Régis de la Bretèche

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Min Sha & Igor E. Shparlinski

  3. School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand

    José Felipe Voloch

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  1. Régis de la Bretèche
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  2. Min Sha
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  3. Igor E. Shparlinski
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  4. José Felipe Voloch
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Correspondence to Min Sha.

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de la Bretèche, R., Sha, M., Shparlinski, I.E. et al. The Sato–Tate distribution in thin parametric families of elliptic curves. Math. Z. 290, 831–855 (2018). https://doi.org/10.1007/s00209-018-2042-0

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  • DOI: https://doi.org/10.1007/s00209-018-2042-0

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