Cyclic reduction of Elliptic Curves
Publikation: Working paper › Forskning
Dokumenter
- cycred1
Indsendt manuskript, 358 KB, PDF-dokument
For an elliptic curve $E$ defined over a number field $K$, we study
the 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 of
a universal constant $A_\infty\approx .8137519$
and a rational correction factor $c_{E/K}$.
Unlike $\delta_{E/K}$ itself, $c_{E/K}$ is a
finite sum of rational numbers that
can 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.
the 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 of
a universal constant $A_\infty\approx .8137519$
and a rational correction factor $c_{E/K}$.
Unlike $\delta_{E/K}$ itself, $c_{E/K}$ is a
finite sum of rational numbers that
can 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.
Originalsprog | Engelsk |
---|---|
Udgiver | arXiv preprint |
Status | Udgivet - 2019 |
Antal downloads er baseret på statistik fra Google Scholar og www.ku.dk
Ingen data tilgængelig
ID: 244330248