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ABSTRACT: p-adic Lie groups occur naturally in number
theory as the image of Galois groups of local
and global fields in the automorphism group of finite
dimensional p-adic Galois representations. The
lectures will begin by using arguments from Lie algebra
cohomology to prove a rather general recent
result about the Euler characteristics of p-adic Lie groups
arising as the image of Galois groups in the
etale cohomology of an algebraic variety over a local or
global field. We will then survey the current
state of knowledge of the algebraic theory of finitely
generated modules over the Iwasawa algebra of
an arbitrary p-adic Lie group G, discussing the notion of
pseudo-nul modules recently introduced by
Venjakob in the non-commutative case. The simplest and
perhaps most intriguing non-commutative
examples of such Iwasawa modules occurring in number theory
are when G is the image of Galois in
the Tate module of an elliptic curve E, without complex
multiplication, defined over a number field;
here the relevant module for the Iwasawa algebra of G is the
dual of the Selmer group of E over the
field generated by the coordinates of the points of p-power
order on E. While little is still known in
general about these fascinating elliptic Iwasawa modules,
one can hope that some day it will be shown
that they possess as detailed an Iwasawa theory, including a
main conjecture, as those which arise in
the classical theory of cyclotomic fields. We shall discuss
our current fragmentary knowledge in support
of this belief, and indulge in a certain amount of
speculation about the future. However, the rashness of
our speculation will always be tempered by a detailed
discussion of numerical examples, especially
those which arise from the three primeval elliptic curves of
conductor 11.
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