Cyclic reduction of Elliptic Curves

Publikation: Working paperForskning

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  • cycred1

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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.
OriginalsprogEngelsk
UdgiverarXiv preprint
StatusIkke-udgivet - 2021

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