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Constants in Titchmarsh divisor problems for elliptic curves


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


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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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).

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  • Titchmarsh divisor
  • Divisor sum
  • Serre curve
  • Elliptic curve
  • Galois representation

Mathematics Subject Classification

  • 11A25: arithmetic functions, related numbers, inversion formulas
  • 11G05: elliptic curves over global fields
  • 11G20: curves over finite and local fields
  • 11N37: asymptotic results on arithmetic functions
  • 11Y60: evaluation of constants