Details
| Original language | English |
|---|---|
| Article number | 61 |
| Journal | Research in Number Theory |
| Volume | 9 |
| Issue number | 3 |
| Early online date | 31 Jul 2023 |
| Publication status | Published - 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
- Mathematics(all)
- Algebra and Number Theory
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In: Research in Number Theory, Vol. 9, No. 3, 61, 09.2023.
Research output: Contribution to journal › Article › Research › peer review
}
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 -