February 10, 2016

August 9-13, 2007
Summer School in Iwasawa Theory
McMaster University

Organizers: Manfred Kolster, William McCallum, Romyar Sharifi

Eric Urban, Columbia University,
Eisenstein ideals and main conjectures in Iwasawa theory

Supported by
Topics of Lectures and Projects:

The goal of this series of lecture is to present the theory of the Eisenstein ideal for proving one divisibility towards the so-
called Iwasawa-Greenberg Main conjecture that establishes the equality of a p-adic L-function with the characteristic power series (or
Fitting ideal) of a certain Selmer group. We will first review the proof by Wiles of the classical Main conjecture by Iwasawa (Mazur-Wiles'theorem) and then we will move forward to explain the strategy to prove the MC for motives of unitary type. We will put the main ingredients in place and explain the basic strategy.
The topic of each lectures are the following:

  • Lecture 1: Classical Iwasawa Main conjecture over the cycltomic Zp-extension of the rationnals.
  • Lecture 2: Automorphic representations for unitary groups and Eisenstein Series.
  • Lecture 3: Galois representations attached to motifs of unitary type and statement of the Main Conjecture.
  • Lecture 4: Eisenstein Ideal and Selmer groups.

The student project will be on writting a detailed account on one of the steps of the arguments presented in the lectures. A particularly important point is to determined the relation between the type (the local conditions at primes dividing p) of the Selmer group considered in relation with the signature of the unitary group from which the Eisentein ideal is defined. In the lecture series, we will focus on the U(2,2) and U(3,1) cases. The student project would be for instance to figure out what is happening in the more general U(p,q)-situation. Another important step in this strategy is the construction of some universal lattice. I will only sketch the proof during the lecture. One project could be to write a complete proof of the construction and the main properties of this lattice.