Kummer generators and torsion points of elliptic curves with bad reduction at some primes

Research output: Contribution to journalArticle

2 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1743-1752
Number of pages10
JournalInternational Journal of Number Theory
Volume9
Issue number7
DOIs
Publication statusPublished - Nov 1 2013
Externally publishedYes

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Torsion Points
Elliptic Curves
Generator
Quadratic field
Fundamental Units
Primitive Roots
Ramification
Roots of Unity
Number field
Torsion
Subgroup
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All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Kummer generators and torsion points of elliptic curves with bad reduction at some primes. / Yasuda, Masaya.

In: International Journal of Number Theory, Vol. 9, No. 7, 01.11.2013, p. 1743-1752.

Research output: Contribution to journalArticle

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