Jacobians of etale covers of the projective line minus three points

Persistent Link:
http://hdl.handle.net/10150/284336
Title:
Jacobians of etale covers of the projective line minus three points
Author:
Rasmussen, Christopher Jorgen
Issue Date:
2004
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:
We consider the outer pro-2 Galois representation on the algebraic fundamental group of the projective line minus three points. This representation has a kernel, whose fixed field Ω₂, is a pro-2 extension of Q(μ₂∞), unramified away from 2. The fields of 2-power torsion of the Jacobians of curves defined over Q, possessing good reduction away from 2, are also pro-2 extensions of Q(μ₂∞), unramified away from 2. In this dissertation, we show that these fields are contained in O2 for certain choices of curves. In particular, the result is shown for all elliptic curves over Q with good reduction away from 2. In proving this theorem, we will demonstrate that these curves appear in the tower of finite etale 2-covers of the projective line minus three points. In the final chapter, we briefly consider three natural generalizations of the result and give partial results in these cases. Specifically, we consider the case of elliptic curves defined over certain extensions of Q, the case of the prime ℓ = 3, and the case of higher genus curves occurring as 2-covers.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Kim, Minhyong

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleJacobians of etale covers of the projective line minus three pointsen_US
dc.creatorRasmussen, Christopher Jorgenen_US
dc.contributor.authorRasmussen, Christopher Jorgenen_US
dc.date.issued2004en_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.abstractWe consider the outer pro-2 Galois representation on the algebraic fundamental group of the projective line minus three points. This representation has a kernel, whose fixed field Ω₂, is a pro-2 extension of Q(μ₂∞), unramified away from 2. The fields of 2-power torsion of the Jacobians of curves defined over Q, possessing good reduction away from 2, are also pro-2 extensions of Q(μ₂∞), unramified away from 2. In this dissertation, we show that these fields are contained in O2 for certain choices of curves. In particular, the result is shown for all elliptic curves over Q with good reduction away from 2. In proving this theorem, we will demonstrate that these curves appear in the tower of finite etale 2-covers of the projective line minus three points. In the final chapter, we briefly consider three natural generalizations of the result and give partial results in these cases. Specifically, we consider the case of elliptic curves defined over certain extensions of Q, the case of the prime ℓ = 3, and the case of higher genus curves occurring as 2-covers.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorKim, Minhyongen_US
dc.identifier.proquest3131634en_US
dc.identifier.bibrecord.b46709496en_US
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