Date of Award

2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Sands, Jonathan

Abstract

Stark's Conjectures were formulated in the late 1970s and early 1980s. The most general version predicts that the leading coe cient of the Maclaurin series of an Artin L-function should be the product of an algebraic number and a regulator made up of character values and logarithms of absolute values of units. When known, Stark's conjecture provides a factorization of the analytic class number formula of Dirichlet. Stark succeeded in formulating a \re ned abelian" version of his conjecture when the L-function in question has a rst order zero and is associated with an abelian extension of number elds. In the spirit of Stark, Rubin and Popescu formulated analogous \re ned abelian" conjectures for Artin L-Functions which vanish to arbitrary order r at s = 0. These conjectures are identical to Stark's own re ned abelian conjecture when restricted to order of vanishing r = 1. We introduce Popescu's Conjecture C(L=F; S; r): We prove Popescu's Conjecture for multiquadratic extensions when the set of primes S of the base eld is minimal given minor restrictions on the S-class group of the base eld. This extends the results of Sands to the case where #S = r + 1. We present three in nite families of settings where our methods allow us to verify Popescu's conjecture. We formulate a conjecture that predicts when a fundamental unit of a real quadratic eld must become a square in a multiquadratic extension.

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