Persistent Link:
http://hdl.handle.net/10150/193812
Title:
Tate-Shafarevich Groups of Jacobians of Fermat Curves
Author:
Levitt, Benjamin L.
Issue Date:
2006
Publisher:
The University of Arizona.
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:
Number Theory; Arithmetic Geometry; Tate-Shafarevich
Degree Name:
PhD
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
McCallum, William G.
Committee Chair:
McCallum, William G.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleTate-Shafarevich Groups of Jacobians of Fermat Curvesen_US
dc.creatorLevitt, Benjamin L.en_US
dc.contributor.authorLevitt, Benjamin L.en_US
dc.date.issued2006en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © 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.abstractFor 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.abstractQ(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.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectNumber Theoryen_US
dc.subjectArithmetic Geometryen_US
dc.subjectTate-Shafarevichen_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorMcCallum, William G.en_US
dc.contributor.chairMcCallum, William G.en_US
dc.contributor.committeememberThakurt, Dineshen_US
dc.contributor.committeememberJoshi, Kirtien_US
dc.contributor.committeememberLux, Klausen_US
dc.identifier.proquest1709en_US
dc.identifier.oclc659746293en_US
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