Online

Price, Jason

Mathematics

2009

PhD

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 "refined abelian" version of his conjecture when the L-function in question has a first order zero and is associated with an abelian extension of number fields.
In the spirit of Stark, Rubin and Popescu formulated analogous "refined abelian" conjectures for Artin L-Functions which vanish to arbitrary order r at s = 0. These conjectures are identical to Stark's own refined 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 field is minimal given minor restrictions on the S-class group of the base field. This extends the results of Sands to the case where #S = r + 1. We present three infinite 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 field must become a square in a multiquadratic extension.