Kummer Generators and Torsion Points of Elliptic Curves With Bad Reduction at Some Primes

Abstract

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field K with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over K(ζp), and in this paper, we study the ramification of such Kummer extensions using the Kummer generators directly computed by Verdure in 2006. For quadratic fields K, we also give unramified Kummer extensions over K(ζp) generated from elliptic curves over K having a p-torsion point with bad reduction at certain primes. Many of these unramified Kummer extensions have not appeared in the previous work using fundamental units of quadratic fields.