Cyclic reduction densities for elliptic curves

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Authors

  • Francesco Campagna
  • Peter Stevenhagen

External Research Organisations

  • Leiden University
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Details

Original languageEnglish
Article number61
JournalResearch in Number Theory
Volume9
Issue number3
Early online date31 Jul 2023
Publication statusPublished - Sept 2023

Abstract

For an elliptic curve E defined over a number field K, the heuristic density of the set of primes of K for which E has cyclic reduction is given by an inclusion–exclusion sum δE/K involving the degrees of the m-division fields Km of E over K. This density can be proved to be correct under assumption of GRH. For E without complex multiplication (CM), we show that δE/K is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of E and a universal infinite Artin-type product. For E admitting CM over K by a quadratic order O , we show that δE/K admits a similar ‘factorization’ in which the Artin type product also depends on O . For E admitting CM over K¯ by an order O⊄ K , which occurs for K= Q , the entanglement of division fields over K is non-finite. In this case we write δE/K as the sum of two contributions coming from the primes of K that are split and inert in O . The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities.

Keywords

    Artin’s primitive root conjecture, Cyclic reduction, Elliptic curves

ASJC Scopus subject areas

Cite this

Cyclic reduction densities for elliptic curves. / Campagna, Francesco; Stevenhagen, Peter.
In: Research in Number Theory, Vol. 9, No. 3, 61, 09.2023.

Research output: Contribution to journalArticleResearchpeer review

Campagna, F & Stevenhagen, P 2023, 'Cyclic reduction densities for elliptic curves', Research in Number Theory, vol. 9, no. 3, 61. https://doi.org/10.1007/s40993-023-00463-9
Campagna, F., & Stevenhagen, P. (2023). Cyclic reduction densities for elliptic curves. Research in Number Theory, 9(3), Article 61. https://doi.org/10.1007/s40993-023-00463-9
Campagna F, Stevenhagen P. Cyclic reduction densities for elliptic curves. Research in Number Theory. 2023 Sept;9(3):61. Epub 2023 Jul 31. doi: 10.1007/s40993-023-00463-9
Campagna, Francesco ; Stevenhagen, Peter. / Cyclic reduction densities for elliptic curves. In: Research in Number Theory. 2023 ; Vol. 9, No. 3.
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