 
Summary: ON PADIC HEIGHT PAIRINGS AND
LOCALLY FREE CLASSGROUPS OF HOPF ORDERS
A. Agboola
Version of April 30th, 1996
Let E be an elliptic curve with complex multiplication by the ring of integers O
of an imaginary quadratic field K. The purpose of this paper is to describe certain
connections between the arithmetic of E on the one hand and the Galois module
structure of certain arithmetic principal homogeneous spaces arising from E on the
other. The present paper should be regarded as a complement to [AT]: we assume
that the reader is equipped with a copy of the latter paper and that he is not averse
to referring to it from time to time.
An outline of the contents of this paper is as follows. After setting up a certain
amount of notation in x0, we give an account of the algebraic padic height pairing
on E in x1. The pairing that we describe was first introduced by Bernadette Perrin
Riou (see [PR1], [PR2]). In x2 we describe the class invariant homomorphism. This
homomorphism measures information regarding the module structure of certain
principal homogeneous spaces of torsion subgroup schemes of E. In the next two
sections, we describe what we call the modified classgroup of the Hopf algebra
representing a certain torsion subgroup scheme of E, and we explain precisely how
this group is related to the height pairing on E (see Theorem 4.2). This gives a
