Computing local constants for CM elliptic curves

TL;DR

This paper derives a formula for local constants of CM elliptic curves at primes of good reduction and uses it to identify field extensions where the O-rank of the curve increases.

Contribution

It introduces a new explicit formula for the arithmetic local constant of CM elliptic curves at primes of good reduction, enabling analysis of rank growth over extensions.

Findings

Derived a general formula for local constants of CM elliptic curves.

Identified field extensions over which the O-rank of the curve increases.

Applied the formula specifically to CM curves over Q.

Abstract

For E/k an elliptic curve with CM by O, we determine a formula for (a generalization of) the arithmetic local constant of [4] at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.