The Change in Lambda Invariants for Cyclic p-Extensions of Z(p)-Fields

Persistent Link:
http://hdl.handle.net/10150/217113
Title:
The Change in Lambda Invariants for Cyclic p-Extensions of Z(p)-Fields
Author:
Schettler, Jordan Christian
Issue Date:
2012
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:
The well-known Riemann-Hurwitz formula for Riemann surfaces (or the corresponding formulas of the same name for curves/function fields) is used in genus computations. In 1979, Yûji Kida proved a strikingly analogous formula in [Kid80] for p-extensions of CM-fields (p an odd prime) which is similarly used to compute Iwasawa λ -invariants. However, the relationship between Kida’s formula and the statement for surfaces is not entirely clear since the proofs are of a very different flavor. Also, there were a few hypotheses for Kida’s result which were not fully satisfying; for example, Kida’s formula requires CM-fields rather than more general number fields and excludes the prime p = 2. Around a year after Kida’s result was published, Kenkichi Iwasawa used Galois cohomology in [Iwa81] to establish a more general formula (about representations) that did not exclude the prime p = 2 nor need the CM-field assumption. Moreover, Kida’s formula follows as a corollary from Iwasawa’s formula. We’ll prove a slight generalization of Iwasawa’s formula and use this to give a new proof of a result of Kida in [Kid79] and Ferrero in [Fer80] which computes λ-invariants in imaginary quadratic extensions for the prime p = 2. We go on to produce special generalizations of Iwasawa’s formula in the case of cyclic p-extensions; these formulas can be realized as statements about Q(p)-representations, and, in the cases of degree p or p², about p-adic integral representations. One upshot of these formulas is a vanishing criterion for λ-invariants which generalizes a result of Takashi Fukuda et al. in [FKOT97]. Other applications include new congruences and inequalities for λ-invariants that cannot be gleaned from Iwasawa’s formula. Lastly, we give a scheme theoretic approach to produce a general formula for finite, separable morphisms of Dedekind schemes which simultaneously encompasses the classical Riemann-Hurwitz formula and Iwasawa’s formula.
Type:
text; Electronic Dissertation
Keywords:
lambda; Mathematics; invariants; Iwasawa
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
McCallum, William G.

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleThe Change in Lambda Invariants for Cyclic p-Extensions of Z(p)-Fieldsen_US
dc.creatorSchettler, Jordan Christianen_US
dc.contributor.authorSchettler, Jordan Christianen_US
dc.date.issued2012-
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.abstractThe well-known Riemann-Hurwitz formula for Riemann surfaces (or the corresponding formulas of the same name for curves/function fields) is used in genus computations. In 1979, Yûji Kida proved a strikingly analogous formula in [Kid80] for p-extensions of CM-fields (p an odd prime) which is similarly used to compute Iwasawa λ -invariants. However, the relationship between Kida’s formula and the statement for surfaces is not entirely clear since the proofs are of a very different flavor. Also, there were a few hypotheses for Kida’s result which were not fully satisfying; for example, Kida’s formula requires CM-fields rather than more general number fields and excludes the prime p = 2. Around a year after Kida’s result was published, Kenkichi Iwasawa used Galois cohomology in [Iwa81] to establish a more general formula (about representations) that did not exclude the prime p = 2 nor need the CM-field assumption. Moreover, Kida’s formula follows as a corollary from Iwasawa’s formula. We’ll prove a slight generalization of Iwasawa’s formula and use this to give a new proof of a result of Kida in [Kid79] and Ferrero in [Fer80] which computes λ-invariants in imaginary quadratic extensions for the prime p = 2. We go on to produce special generalizations of Iwasawa’s formula in the case of cyclic p-extensions; these formulas can be realized as statements about Q(p)-representations, and, in the cases of degree p or p², about p-adic integral representations. One upshot of these formulas is a vanishing criterion for λ-invariants which generalizes a result of Takashi Fukuda et al. in [FKOT97]. Other applications include new congruences and inequalities for λ-invariants that cannot be gleaned from Iwasawa’s formula. Lastly, we give a scheme theoretic approach to produce a general formula for finite, separable morphisms of Dedekind schemes which simultaneously encompasses the classical Riemann-Hurwitz formula and Iwasawa’s formula.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectlambdaen_US
dc.subjectMathematicsen_US
dc.subjectinvariantsen_US
dc.subjectIwasawaen_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.advisorMcCallum, William G.en_US
dc.contributor.committeememberSharifi, Romyaren_US
dc.contributor.committeememberThakur, Dinesh S.en_US
dc.contributor.committeememberJoshi, Kirti N.en_US
dc.contributor.committeememberMcCallum, William G.en_US
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