TY - JOUR

T1 - Cyclic reduction densities for elliptic curves

AU - Campagna, Francesco

AU - Stevenhagen, Peter

N1 - Funding Information: Francesco Campagna is supported by ANR-20-CE40-0003 Jinvariant. Peter Stevenhagen was funded by a research grant of the Max-Planck-Institut für Mathematik in Bonn. Both authors thank the institute in Bonn for its financial support and its inspiring atmosphere.

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

KW - Artin’s primitive root conjecture

KW - Cyclic reduction

KW - Elliptic curves

UR - http://www.scopus.com/inward/record.url?scp=85166742287&partnerID=8YFLogxK

U2 - 10.1007/s40993-023-00463-9

DO - 10.1007/s40993-023-00463-9

M3 - Article

AN - SCOPUS:85166742287

VL - 9

JO - Research in Number Theory

JF - Research in Number Theory

IS - 3

M1 - 61

ER -