Iwasawa Theory, Projective Modules, and Modular Representations. This paper concerns the structure of some modules over the group ring of a finite Galois group. The module of main interest is the Pontryagin dual of a non-primitive Selmer group for an elliptic curve E with good ordinary reduction at a prime p. Under certain hypotheses, we prove that this module is projective or of finite index in a projective module. We then use the theory of modular representations to derive consequences concerning Iwasawa theory for E. This pdf file is a new and significantly revised version of the paper and was posted here on May 16th, 2009.
Topics in Iwasawa Theory . We will post here the chapters of a monograph on Iwasawa theory as they are (more-or-less) completed. The file contains the first two chapters at present. Comments and suggestions (e.g., lack of clarity, errors, or misprints) will be appreciated.
On the Structure of Certain Galois Cohomology Groups. The Galois cohomology groups considered in this paper are associated to Galois representations over a ring R. The main theorem in this paper asserts that under certain assumptions on R and the representation,
the
Pontryagin dual of the 1st cohomology group contains no nonzero pseudo-null R-submodules.
Galois
Theory for the Selmer Group of an Abelian Variety.
Let A denote an abelian variety defined over a number field F.
Suppose that A has potentially ordinary reduction at all primes of F lying
over a fixed prime p. If K is a Galois extension of F such that Gal(K/F)
is isomorphic to a p-adic Lie group and if L is any finite extension of
F contained in K, then consider the restriction map:
We will include here the ps or pdf files for various papers and expository
articles as they are completed.
Galois properties of elliptic curves with an isogeny
. This paper studies the image of the Galois representation associated to the p-adic Tate module of an elliptic curve with an
isogeny of degree p. Under certain assumptions, the image is as large as possible.
Galois Representations with Open Image. This paper gives an algebraic number theory construction of continuous p-adic Galois representations of dimension n for a variety of pairs n and p.
SelA(L) ---> SelA(K)Gal(K/L)
The purpose of this paper is to study the kernel and cokernel of this restriction map as the field L varies. Under certain hypotheses, we show that the kernel and cokernel are always finite, and sometimes of order bounded independently of L.
The
Iwasawa Invariants of Elliptic Curves.
This is a joint paper with Nike Vatsal. Let p be an odd prime. Suppose
that E is a modular elliptic curve/Q with good ordinary reduction at p.
Let Qoo
denote the cyclotomic Zp-extension of Q. It is conjectured
that SelE(Qoo)
is a cotorsion Lambda-module and that its characteristic ideal is related
to the p-adic L-function associated to E. Under certain hypotheses
we prove that the validity of these conjectures is preserved by congruences
between the Fourier expansions of the associated modular forms.
Iwasawa
Theory-Past
and Present. This article is dedicated
to the memory of Kenkichi Iwasawa, who passed away on October 26th, 1998.
It is an historical introduction to the basic ideas of this subject going
back to the first papers of Iwasawa, various versions of the Main Conjecture,
etc..
Introduction
to Iwasawa Theory for Elliptic Curves. This
article is based on lectures given at the IAS/Park CityMathematics Institute
during the summer of 1999. The four chapters are devoted to the following
topics: 1. Mordell-Weil Groups:A general discussion of the growth
of the rank of Mordell-Weil groups in towers of number fields. 2.
Selmer Groups: The definitions of the Selmer group of an elliptic curve
and an equivalent, simpler description in the case where E has good, ordinary
reduction. 3. Lambda-Modules: The structure of modules over the Iwasawa
algebra. 4. Mazur's Control Theorem: A proof of this theorem and
several of its consequences.
Iwasawa Theory for Elliptic Curves
We study this subject by first proving that the p-primary
subgroup of the classical Selmer group for an elliptic curve with good,
ordinary reduction at a prime p has a very simple and elegant description
which involves just the Galois module of p-power torsion
points. We then prove theorems of Mazur, Schneider, and Perrin-Riou
on the basis of this description. The final section, which
is half of this long paper, contains a number of results and examples
including a thorough study of the mu-invariant.