Abstract
In this thesis we study in detail the ‘equivariant Tamagawa Number Conjecture’ for Dirichlet L-series at s = 0 (or ‘eTNC(Gm)’, for brevity), itself an equivariant refinement of the ‘Tamagawa Number Conjecture’ of Bloch and Kato.Firstly, we prove the Iwasawa-theoretic version of a Conjecture of Mazur–Rubin and Sano in the case of elliptic units. This allows us to derive the p-part of eTNC(Gm) for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard μ-vanishing hypothesis is satisfied, also in the general case.
Secondly, by building on ideas of Coleman and Rubin, we develop a general theory of Euler systems ‘over Z’ for the multiplicative group over number fields. We show this theory has a range of concrete consequences includingthe proof of a long-standing distribution-theoretic conjecture of Coleman ‘away from 2’ and an elementary interpretation of, and thereby a more direct approach to proving, eTNC(Gm). In this way we obtain a proof of the‘minus part’ of eTNC(Gm) forCMextensions of totally real fields under a mild technical condition and an easier proof of eTNC(Gm) over Q ‘away from 2’.
We also prove that higher-rank Euler systems for a wide class of p-adic representations control the structure of Iwasawa-theoretic Selmer groups in the manner predicted by ‘main conjectures’.
| Date of Award | 1 Feb 2023 |
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| Original language | English |
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| Supervisor | David Burns (Supervisor) |