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# Tate-Shafarevich Groups of Jacobians of Fermat Curves

- Persistent Link:
- http://hdl.handle.net/10150/193812
- Title:
- Tate-Shafarevich Groups of Jacobians of Fermat Curves
- Author:
- Issue Date:
- 2006
- Publisher:
- Rights:
- Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
- Abstract:
- For a fixed rational prime p and primitive p-th root of unity ζ, we consider the Jacobian, J, of the complete non-singular curve give by equation yᵖ = xᵃ(1 − x)ᵇ. These curves are quotients of the p-th Fermat curve, given by equation xᵖ+yᵖ = 1, by a cyclic group of automorphisms. Let k = Q(ζ) and k(S) be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k(S) over k studied by W. McCallum and R. Sharifi in [MS02] to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into X(k, J).; Q(zeta) and k_S be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k_S over k studied by W. McCallum and R. Sharifi to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into Shah(k,J).
- Type:
- text; Electronic Dissertation
- Keywords:
- Degree Name:
- PhD
- Degree Level:
- doctoral
- Degree Program:
- Degree Grantor:
- University of Arizona
- Advisor:
- Committee Chair:
- McCallum, William G.

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.language.iso | EN | en_US |

dc.title | Tate-Shafarevich Groups of Jacobians of Fermat Curves | en_US |

dc.creator | Levitt, Benjamin L. | en_US |

dc.contributor.author | Levitt, Benjamin L. | en_US |

dc.date.issued | 2006 | en_US |

dc.publisher | The University of Arizona. | en_US |

dc.rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. | en_US |

dc.description.abstract | For a fixed rational prime p and primitive p-th root of unity ζ, we consider the Jacobian, J, of the complete non-singular curve give by equation yᵖ = xᵃ(1 − x)ᵇ. These curves are quotients of the p-th Fermat curve, given by equation xᵖ+yᵖ = 1, by a cyclic group of automorphisms. Let k = Q(ζ) and k(S) be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k(S) over k studied by W. McCallum and R. Sharifi in [MS02] to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into X(k, J). | en_US |

dc.description.abstract | Q(zeta) and k_S be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k_S over k studied by W. McCallum and R. Sharifi to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into Shah(k,J). | en_US |

dc.type | text | en_US |

dc.type | Electronic Dissertation | en_US |

dc.subject | Number Theory | en_US |

dc.subject | Arithmetic Geometry | en_US |

dc.subject | Tate-Shafarevich | en_US |

thesis.degree.name | PhD | en_US |

thesis.degree.level | doctoral | en_US |

thesis.degree.discipline | Mathematics | en_US |

thesis.degree.discipline | Graduate College | en_US |

thesis.degree.grantor | University of Arizona | en_US |

dc.contributor.advisor | McCallum, William G. | en_US |

dc.contributor.chair | McCallum, William G. | en_US |

dc.contributor.committeemember | Thakurt, Dinesh | en_US |

dc.contributor.committeemember | Joshi, Kirti | en_US |

dc.contributor.committeemember | Lux, Klaus | en_US |

dc.identifier.proquest | 1709 | en_US |

dc.identifier.oclc | 659746293 | en_US |

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