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 John Coates
FRS Sadleirian Professor of Pure Mathematics, University of
Cambridge
TBA
Visit: May 12 - May 16, 2008
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Series Title:
"Iwasawa Theoryn" Abstract
Lectures:
5/12
Monday @ 2 in MS 6221
5/13 Tuesday @ 2 in MS 6627
5/15 Thursday @ 2 in MS 6221
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Lecture 1 - Whyever Iwasawa theory?
Lecture 1 hopes to explain to the non-expert why the ideas of Iwasawa theory, especially the interplay between arithmetic problems, the complex world,
and the p-adic world, can be used to study a variety of theoretical and computational problems in number theory, which seem otherwise inaccessible.
Monday, May 12, 2 PM, MS 6221
Lecture 2 - Foundations of non-commutative Iwasawa theory
Lecture 2 - The Iwasawa algebra L(G) of a compact p-adic Lie group G seems to be one of the most interesting examples of a ring which is non-commutative once G is non-commutative. When G admits a closed normal subgroup H such that G/H is isomorphic to Zp (the additive group of p-adic numbers), I will explain how L(G) admits a canonical Ore set S, and how localization with respect to S enables one to give a vast generalization of the "main conjectures" of classical commutative Iwasawa theory.
Tuesday, May 13, 2 PM, MS 6627
Lecture 3 - The work of M. Kakde
Lecture 3 - In this last lecture, Kadke's proof of the first importantcase of the "main conjecture" of non-commutative Iwasawa theory.
Thursday, May 15, 2 PM, MS 6221
References:
For classical Iwasawa theory:
1. J. Coates, R. Sujatha, "Cyclotomic fields and zeta values", Springer
Monographs in Mathematics, Springer 2006.
2. R. Greenberg, "Iwasawa theory of elliptic curves", in Springer Lecture Notes
in Mathematics 1716, 1999, p.51-144.
For non-commutative Iwasawa theory:
1. J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, "The GL_2 main
conjecture dor elliptic curves without complex multiplication", Publ. Math. IHES
101 (2005), 163-208.
2. J. Coates, T. Fukaya, K. Kato, R. Sujatha, "Root numbers, Selmer groups, and
non-commutative Iwasawa theory" to appear.
3.M. Kakde, "The Main Conjecture of non-commutative Iwasawa theory for totally
real fields in certain cases", Ph.D thesis, Cambridge University, 2008."
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