Cyclic reduction densities for elliptic curves

TL;DR

This paper investigates the heuristic density of primes where an elliptic curve has cyclic reduction, providing explicit formulas and factorizations for different types of elliptic curves, and analyzing conditions for the density to vanish.

Contribution

It offers a detailed factorization of the cyclic reduction density for elliptic curves over number fields, including cases with and without complex multiplication, and addresses non-finite entanglement scenarios.

Findings

Explicit formulas for density involving division fields

Factorization of density into rational and Artin-type components

Conditions under which the density vanishes

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 $\delta_{E/K}$ 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 $\delta_{E/K}$ 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 ${\mathcal{O}}$, we show that $\delta_{E/K}$ admits a similar `factorization' in which the Artin type product also depends on ${\mathcal{O}}$. For $E$ admitting CM over $\bar K$ by an order ${\mathcal{O}}\not\subset K$, which occurs for $K={\bf Q}$, the entanglement of division fields over $K$ is non-finite. In this case we write $\delta_{E/K}$ as the sum of two contributions coming from the primes of $K$ that are split and inert in ${\mathcal{O}}$. 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.