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Constants in Titchmarsh divisor problems for elliptic curves

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Abstract

Inspired by the analogy between the group of units \(\mathbb {F}_p^{\times }\) of the finite field with p elements and the group of points \(E(\mathbb {F}_p)\) of an elliptic curve \(E/\mathbb {F}_p\), E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum \(\sum \nolimits _{p \le x} \tau (p + a) \sim C x\). In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums in place of the constant C above. Specifically, by analyzing the division fields of an elliptic curve \(E/\mathbb {Q}\), we prove bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves \(E/\mathbb {Q}\). Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

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Author's contributions

Acknowlegements

This research started during the Arizona Winter School 2016: Analytic Methods in Arithmetic Geometry, organized at the University of Arizona, Tucson, USA, during March 12-16, 2016. We thank the conference organizers Alina Bucur, David Zureick-Brown, Bryden Cais, Mirela Ciperiani, and Romyar Sharifi for all their time and support, and we thank the National Science Foundation for sponsoring our participation in this conference. Moreover, we thank the referees for carefully reading the original manuscript and for all their comments and suggestions, which enabled us to improve the results of the paper.

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Author notes

  1. Renee Bell

    Present address: Université Paris-Sud, Bât. 307, 91405, Orsay Cedex, France

Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, PA, USA

    Renee Bell

  2. Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang, Republic of Korea

    Clifford Blakestad

  3. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, Sector 1, 010702, Bucharest, Romania

    Alina Carmen Cojocaru

  4. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL, 60607, USA

    Alina Carmen Cojocaru & Nathan Jones

  5. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Alexander Cowan & Geoffrey Smith

  6. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Vlad Matei

  7. Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA, 94305-2125, USA

    Isabel Vogt

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  1. Renee Bell
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Correspondence to Alina Carmen Cojocaru.

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ACC’s work on this material was partially supported by the Simons Collaboration Grant under Award No. 318454. IV’s work was partially supported by the NSF Graduate Research Fellowship Program and Grant DMS-1601946

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Bell, R., Blakestad, C., Cojocaru, A.C. et al. Constants in Titchmarsh divisor problems for elliptic curves. Res. number theory 6, 1 (2020). https://doi.org/10.1007/s40993-019-0175-9

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