Research Page

        We will include here the ps or pdf files for various papers and expository articles as they are completed. The most recent one is first. Here also are pdf files of recent talks: Some Lectures

On p-adic Artin L-functions II . This paper is a sequel to an earlier paper. We discuss the topic from the Selmer group point of view instead of the Galois group point of view. This results in a more transparent definition of the mu-invariant and a simpler explanation of how the main conjecture can be deduced from the work of Wiles.

Selmer groups and congruences. This is an expanded version of my talk for the ICM 2010. It is a survey concerning the question of studying the elements of order p in the Selmer group for an elliptic curve based on the structure of E[p] as a Galois module.

On elliptic curves with an isogeny of degree 7. This is a joint paper with Karl Rubin, Alice Silverberg, and Michael Stoll. We study elliptic curves with an isogeny of degree 7 defined over a field k. In the special case where k = Q, we study the image of the 7-adic Galois representations attached to such elliptic curves and prove that the image is as large as possible for all such elliptic curves, except for those which have CM.

On the structure of Selmer groups. The main results of this paper establish the nonexistence of pseudo-null submodules in the Pontryagin dual of Selmer groups. That is, the prime ideals in the support of such a module are always of height 1. As in the previous paper, we study Selmer groups defined in a very general way.

Surjectivity of the global-to-local map defining a Selmer group . Selmer groups can be attached to Galois representations over a ring $R$ under certain assumptions. A Selmer group is defined to be the kernel of a certain kind of map which we call "global-to-local." This paper studies the cokernel of that map and gives sufficient conditions for the map to be surjective.

The image of Galois representations attached to 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.

  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:

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.