Cyclic reduction of Elliptic Curves
Publikation: Working paper › Forskning
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Cyclic reduction of Elliptic Curves. / Campagna, Francesco; Stevenhagen, Peter.
arXiv preprint, 2019.Publikation: Working paper › Forskning
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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 -
ID: 244330248