### Abstract

For a prime p, let ζ_{p} 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(ζ_{p}), 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(ζ_{p}) 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 language | English |
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Pages (from-to) | 1743-1752 |

Number of pages | 10 |

Journal | International Journal of Number Theory |

Volume | 9 |

Issue number | 7 |

DOIs | |

Publication status | Published - Nov 1 2013 |

Externally published | Yes |

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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.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Kummer generators and torsion points of elliptic curves with bad reduction at some primes

AU - Yasuda, Masaya

PY - 2013/11/1

Y1 - 2013/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84887056090&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887056090&partnerID=8YFLogxK

U2 - 10.1142/S1793042113500541

DO - 10.1142/S1793042113500541

M3 - Article

AN - SCOPUS:84887056090

VL - 9

SP - 1743

EP - 1752

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 7

ER -