Cyclic reduction densities for elliptic curves

Abstract

AbstractFor 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 $$\delta _{E/K}$$ δ E / K involving the degrees of the m-division fields $$K_m$$ 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 $$\delta _{E/K}$$ 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 $$\mathcal {O}$$ O , we show that $$\delta _{E/K}$$ δ E / K admits a similar ‘factorization’ in which the Artin type product also depends on $$\mathcal {O}$$ O . For E admitting CM over $$\overline{K}$$ K ¯ by an order $$\mathcal {O}\not \subset K$$ O ⊄ K , which occurs for $$K=\textbf{Q}$$ K = Q , the entanglement of division fields over K is non-finite. In this case we write $$\delta _{E/K}$$ δ E / K as the sum of two contributions coming from the primes of K that are split and inert in $$\mathcal {O}$$ 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.