TY - UNPB

T1 - Cyclic reduction of Elliptic Curves

AU - Campagna, Francesco

AU - Stevenhagen, Peter

PY - 2019

Y1 - 2019

N2 - For an elliptic curve $E$ defined over a number field $K$, we studythe density of the set of primes of $K$ for which $E$ has cyclic reduction. For $K=\mathbb{Q}$, Serre proved that, under GRH, the density equals an inclusion-exclusion sum $\delta_{E/\mathbb{Q}}$involving the field degrees of an infinite family of division fields of $E$.We extend this result to arbitrary number fields $K$, and prove that,for $E$ without complex multiplication,$\delta_{E/K}$ equals the product ofa universal constant $A_\infty\approx .8137519$and a rational correction factor $c_{E/K}$.Unlike $\delta_{E/K}$ itself, $c_{E/K}$ is afinite sum of rational numbers thatcan be used to study the vanishing of $\delta_E$, which is a non-trivial phenomenon over number fields $K\ne\mathbb{Q}$.We include several numerical illustrations.

AB - For an elliptic curve $E$ defined over a number field $K$, we studythe density of the set of primes of $K$ for which $E$ has cyclic reduction. For $K=\mathbb{Q}$, Serre proved that, under GRH, the density equals an inclusion-exclusion sum $\delta_{E/\mathbb{Q}}$involving the field degrees of an infinite family of division fields of $E$.We extend this result to arbitrary number fields $K$, and prove that,for $E$ without complex multiplication,$\delta_{E/K}$ equals the product ofa universal constant $A_\infty\approx .8137519$and a rational correction factor $c_{E/K}$.Unlike $\delta_{E/K}$ itself, $c_{E/K}$ is afinite sum of rational numbers thatcan be used to study the vanishing of $\delta_E$, which is a non-trivial phenomenon over number fields $K\ne\mathbb{Q}$.We include several numerical illustrations.

M3 - Working paper

BT - Cyclic reduction of Elliptic Curves

PB - arXiv preprint

ER -