Applications of representation theory and K-theory to problems in algebraic number theory

Abstract

In this thesis we consider three main problems: the Galois module structure of rings of integers in wildly ramified extensions of Q; Leopoldt's conjecture; and non-commutative Fitting ideals and the non-abelian Brumer-Stark conjecture. For each of these problems, which correspond to each main chapter, we will review and use tools from representation theory and algebraic K-theory.

In the first main chapter, we will prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers OK to be free over its associated order in the rational group algebra Q[G].

In the second main chapter, we will work on Leopoldt's conjecture. Let p be a rational prime and let L/K be a Galois extension of number fields with Galois group G. Under certain hypotheses, we show that Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L; a crucial tool will be the theory of norm relations in Q[G]. We also consider relations between the Leopoldt defects at p of intermediate extensions of L/K.

Finally, we will investigate new properties of (non-commutative) Fitting ideals in integral group rings, with the general idea of reducing to simpler abstract groups (such as abelian groups) that can emerge as subquotients. As an application we will provide a direct proof of the (non-abelian) Brumer-Stark conjecture in certain cases, by reducing to the abelian case as recently proved by Dasgupta and Kakde. The direct approach avoids use of technical machinery such as the equivariant Tamagawa number conjecture.