Iwasawa theory for elliptic curves

Karumbidza, Archiebold Chemwaida (2006-04)

Thesis (MSc) -- University of Stellenbosch, 2006.

Thesis

ENGLISH ABSTRACT: In this thesis we consider modules over the Iwasawa algebra, these naturally arise in the classical theory of class groups over cyclotomic fields as exposed by Kenkichi Iwasawa. We classify these modules up to - psuedo-isomorphism. We apply this classification to estimate the growth of the p-part of the ideal class groups along a Zp·extension. Class groups can be interpreted as "generalized Selmer groups", these include 7- the traditional Selmer groups attached to elliptic curves. We are thus natural!)> led to consider the growth of Selmer groups of elliptic curves along a Zp-extension. Under the hypothesis of good ordinary reduction these Selmer groups have a particulary simple and elegant description, we use this description to prove the so called "Control Theorem" of B. Mazur. As a consequence we are able to inyestigate the growth behavior of Selmer and Tate-Shafarevich groups along a Zp-extension. The original motivations for considering the Selmer groups of elliptic curves along a Zp·extension are however quite different from what has been suggested above. The Mordell-Weil theorem says the abelian group of rational points on an elliptic curve over Q is finitely generated. Tt is natural to wonder if the Mordell-Weil theorem still holds over infinite extensions of Q which are at least Galois. There is a simple criterion due to B. Mazur for the Mordell-Weil theorem to still hold, the hypotheses of this criterion, however, are hard to verify in any particular case. Consequences of the "Control Theorem" allow us to verify this hypothesis over a Zp-extension, thus the Mordell-Weil Theorem holds for a Zp·extension

AFRIKAANSE OPSOMMING: In hierdie tesis oorweeg ons modules oar die lwasawa algebra, hierdie kom natuurlik voor in die klassieke teorie van klas groepe oor sikliese liggame soos blootgestel deur Kenkichi Iwasawa. Ons beskryf hierdie modules tot pseudo-isomorfisme. Ons pas hierdie beskrywing toe om die groei van die p-deel van die ideaal klas groep in 'n Zv-uitbreiding te benader. Klas groepe kan as veralgemeende Selmer groepe beskou word, hierdie sluit in die tradisionele Selmer groepe verbind aan elliptiese kurwes. Dit is dus vanselfsprekend om die groei van Selmer groepe can elliptiese kurwes in 'n Zp-uitbreiding te oorweeg. Onder die hipotese van goeie gewone reduksie het hierdie Selmer groepe 'n besondere eenvoudige en elegante beskrywing, ons gebruik hierdie beskrywing om die sogenoemede "Beheer Stelling" van B. Mazur te bewys. As 'n afteiding is dit moontlik om die groei handing can Selmer en Tate-Shafarevich groepe in 'n Zv-uitbreiding te ondersoek. Die oorspronklike motivering vir die oorweging van Selmer groepe van elliptiese kurwes in 'n Zp-uitbreiding is egter heelwat anders as wat hierbo aangedui is. Die Mordell-Weil stelling se dat die Abelian groep van rasionale punte op 'n elliptiese kurwe oar Q eindig voortgebring is. Dit is algemeen om te wonder of die Mordell-Weil stelllig steeds hon oar oneindige uitbreidings van Q wat ten minste Galois is. Daar is 'n eenvoudige voorwaarde deur B. Mazur waaronder die Mardell-Weil stelling steeds staan, die hipotese van hierdie voorwaarde is egter moeilik om te kontroleer in enige spesiale voorbeeld. Gevolge van die "Beheer Stelling" laat ons toe om die hipotese te kontroleer oor 'n Zp-uitbreiding en dus staan die Mardell-Weil stelling vir 'n Zv-uitbreiding.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/50616
This item appears in the following collections: