### Abstract

Let E be the elliptic curve y^{2} = x(x + 1)(x + t) over the field Fp(t), where p is an odd prime. We study the arithmetic of E over extensions Fq(t1/d), where q is a power of p and d is an integer prime to p. The rank of E is given in terms of an elementary property of the subgroup of (ℤ/dℤ)× generated by p. We show that for many values of d the rank is large. For example, if d divides 2(p^{f} - 1) and 2(p ^{f} - 1)/d is odd, then the rank is at least d/2. When d = 2(pf - 1), we exhibit explicit points generating a subgroup of E(Fq(t^{1/d})) of finite index in the "2-new" part, and we bound the index as well as the order of the "2-new" part of the Tate-Shafarevich group.

Original language | English (US) |
---|---|

Pages (from-to) | 261-280 |

Number of pages | 20 |

Journal | Mathematical Research Letters |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*21*(2), 261-280. https://doi.org/10.4310/MRL.2014.v21.n2.a5