Cyclic reduction densities for elliptic curves

verfasst von

Francesco Campagna, Peter Stevenhagen

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 δ_E/K involving the degrees of the m-division fields K_m 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 δ_E/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 δ_E/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 δ_E/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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

Leiden University

Typ

Artikel

Journal

Research in Number Theory

Band

9

Publikationsdatum

09.2023

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.1007/s40993-023-00463-9 (Zugang: Offen)