Formulas for Brumer–Stark Units

Abstract

Let F be a totally real number field. There have been three p-adic formulas conjectured by Dasgupta and Dasgupta–Spieß for the Brumer–Stark units of F. These formulas are conjectured to be equal by Dasgupta–Spieß. In this thesis we first show that two of these formulas are equal in the case that F is a cubic field. This proof uses only elementary methods involving calculations of Shintani sets. We then present joint work with Dasgupta which proves that all three of the conjectural formulas are equal for any totally real field F . Finally, work of Dasgupta–Kakde has shown that one of the conjectural formulas is equal to the Brumer–Stark unit up to a root of unity. Recent work of Bullach–Burns–Daoud–Seo proves the minus part of the eTNC away from 2, for finite abelian CM extensions of totally real fields. We show that this recent work implies that the formulas hold up to a 2-power root of unity.

Date of Award	1 Aug 2022
Original language	English
Awarding Institution	King's College London
Supervisor	Mahesh Kakde (Supervisor) & David Burns (Supervisor)