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Title:  Swan Modules and Elliptic Functions 
Author(s):  Srivastav, Anupam 
Doctoral Committee Chair(s):  Ullom, Stephen V. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  M. J. Taylor has described the additive Galois module structure of rings of integers of certain Kummer extensions with respect to an elliptic group law. He has obtained elliptic analogues of cyclotomic results. The arithmetic nature of elliptic resolvents, the elliptic analogue of the Gauss sum conductor formula and the strength of the elliptic group law being a LubinTate formal group law enabled Taylor to show that the ring of algebraic integers is free over the associated order, if and only if, a certain elliptic analogue of a Swan module is a principal ideal of the associated order. In this thesis we find an explicit generator for the square of this elliptic Swan module in quite a general case. This generator is a product of elliptic resolvent elements. For a number field F, Let O$\sb{\rm F}$ denote its ring of integers. Let p be an odd rational prime. Let K be a quadratic imaginary number field with discriminant less than $4$. Moreover, assume the prime 2 splits in K/${\rm I\!Q}$ and p is inert in K/${\rm I\!Q}$. Set ${\rm l\!p}$ = pO$\sb{\rm K}$. We fix positive integers r $>$ m and let N (respectively, L) denote K ray classfield mod $4{\rm {l\!p}\sp{m+r}}$ (respectively, $4{\rm {l\!p}\sp{r}}$). We let $\Gamma$ = Gal(N/L) and ${\cal U} = \{$x $\in$ L$\Gamma$: O$\sb{\rm N}$ $\cdot$ x $\subseteq$ O$\sb{\rm N}\}$, the associated order of N/L. Let $\Sigma$ = $\sum\sb{\gamma\in\Gamma}\gamma$ and set I$\sb2$ = (2,p$\sp{\rm m}\Sigma){\cal U}$ = $2{\cal U}$ + p$\sp{\rm m}\Sigma{\cal U}$, a locally free ideal of ${\cal U}$. Taylor has shown that O$\sb{\rm N}$ is ${\cal U}$free, if and only if, the elliptic Swan module I$\sb2$ is a principal ${\cal U}$ideal. We show that I$\sb2$ = $(2,\Sigma){\cal U}$ and so it is obtained from the usual Swan module by an extension of rings. The main result of this thesis is: Theorem. If p $\equiv$ $\pm$1 mod 8, then I$\sb2$ is a principal ideal of the associated order ${\cal U}$. 
Issue Date:  1987 
Type:  Text 
Description:  112 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1987. 
URI:  http://hdl.handle.net/2142/71262 
Other Identifier(s):  (UMI)AAI8803209 
Date Available in IDEALS:  20141216 
Date Deposited:  1987 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois