August 913, 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 IwasawaGreenberg Main conjecture that establishes the equality
of a padic Lfunction 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 (MazurWiles'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:
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.