Summary: GALOIS MODULES AND p-ADIC REPRESENTATIONS
Abstract. In this paper we develop a theory of class invariants associated to p-adic representations
of absolute Galois groups of number fields. Our main tool for doing this involves a new way of
describing certain Selmer groups attached to p-adic representations in terms of resolvends associated
to torsors of finite group schemes.
In this paper we shall introduce and study invariants which measure the Galois structure of
certain torsors that are constructed via p-adic Galois representations. We begin by describing the
background to the questions that we intend to discuss.
Let Y be any scheme, and suppose that G Y is a finite, flat, commutative group scheme.
Write G for the Cartier dual of G. Let ~G denote the normalisation of G, and let i : ~G G
be the natural map. Suppose that : X Y is a G-torsor, and write 0 : G Y for the trivial
G-torsor. Then OX is an OG-comodule, and so it is also an OG -module (see e.g. ). As an
OG -module, the structure sheaf OX is locally free of rank one, and so it gives a line bundle M
on G. Set
L := M M-1
Then the maps