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Ullom, Stephen V. Srivastav, Anupam 2014-12-16T06:18:21Z 2014-12-16T06:18:21Z 10000-01-01 1987 1987 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 Lubin-Tate 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 ${\cal U}$. Made available in DSpace on 2014-12-16T06:18:21Z (GMT). No. of bitstreams: 1 8803209.pdf: 2710520 bytes, checksum: 9089fdd2e447e69b821678d285c855f9 (MD5) Previous issue date: 1987 Embargo set by: Seth Robbins for item 71428 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 112 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. http://hdl.handle.net/2142/71262 (UMI)AAI8803209 Mathematics Swan Modules and Elliptic Functions text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.