Cyclic reduction densities for elliptic curves

2019/12/31 by Campagna, Francesco, Stevenhagen, Peter
#11G05 #11R45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2001.00028

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\not⊂ K, which occurs for K=\bf 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.

Related

• Character sums for elliptic curve densities
• The distribution and density of cyclic groups of the reductions of an elliptic curve over a function field
• Divisibility of orders of reductions of elliptic curves
• Entanglement of elliptic curves upon base extension
• Positive density of primes of ordinary reduction for abelian varieties of simple signature