### Abstract

Let K be a number field and fix a prime number p. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set B_{K,p} of primes of K satisfying that any elliptic curve over K with B_{K,p}-reduction has no p-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with B_{K,p} reduction and a p-torsion point. The action of the absolute Galois group on the p-torsion subgroup of E gives its associated Galois representation P_{E,p} modulo p. We also study the irreducibility and surjectivity of ρ_{E,p} for semistable elliptic curves with B_{K,p}-reduction.

Original language | English |
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Pages (from-to) | 83-96 |

Number of pages | 14 |

Journal | Bulletin of the Korean Mathematical Society |

Volume | 50 |

Issue number | 1 |

DOIs | |

Publication status | Published - Nov 11 2013 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Torsion points of elliptic curves with bad reduction at some primes II.** / Yasuda, Masaya.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Torsion points of elliptic curves with bad reduction at some primes II

AU - Yasuda, Masaya

PY - 2013/11/11

Y1 - 2013/11/11

N2 - Let K be a number field and fix a prime number p. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set BK,p of primes of K satisfying that any elliptic curve over K with BK,p-reduction has no p-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with BK,p reduction and a p-torsion point. The action of the absolute Galois group on the p-torsion subgroup of E gives its associated Galois representation PE,p modulo p. We also study the irreducibility and surjectivity of ρE,p for semistable elliptic curves with BK,p-reduction.

AB - Let K be a number field and fix a prime number p. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set BK,p of primes of K satisfying that any elliptic curve over K with BK,p-reduction has no p-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with BK,p reduction and a p-torsion point. The action of the absolute Galois group on the p-torsion subgroup of E gives its associated Galois representation PE,p modulo p. We also study the irreducibility and surjectivity of ρE,p for semistable elliptic curves with BK,p-reduction.

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UR - http://www.scopus.com/inward/citedby.url?scp=84887096343&partnerID=8YFLogxK

U2 - 10.4134/BKMS.2013.50.1.083

DO - 10.4134/BKMS.2013.50.1.083

M3 - Article

VL - 50

SP - 83

EP - 96

JO - Bulletin of the Korean Mathematical Society

JF - Bulletin of the Korean Mathematical Society

SN - 1015-8634

IS - 1

ER -