Swan Modules and Elliptic Functions

Srivastav, Anupam

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Description

Title

Swan Modules and Elliptic Functions

Author(s)

Srivastav, Anupam

Issue Date

1987

Doctoral Committee Chair(s)

Ullom, Stephen V.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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 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 ${\cal U}$.

Graduation Semester

1987

Type of Resource

text

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