Summary: A GEOMETRIC DESCRIPTION OF THE
CLASS INVARIANT HOMOMORPHISM
Version of May 27, 1994
In recent years, a certain amount of work has been done on the Galois structure
of principal homogeneous spaces of finite group schemes that are constructed via
dividing points on abelian varieties. This study was begun by M. J. Taylor in [T1]
and was originally motivated by the fact that such principal homogeneous spaces
are very closely connected with certain rings of integers. The starting point of the
theory is the socalled class invariant homomorphism which was first introduced
by Waterhouse in [W]. The purpose of this note is to point out a simple geometric
description of this homomorphism (as considered in [T1]) in terms of the restric
tion of certain line bundles to torsion subgroup schemes of the abelian variety in
question. We shall also make a number of remarks concerning both old results and
new questions that arise in light of this description.
In what follows, we shall confine ourselves to the case of abelian varieties defined
over number fields. It should however be noted that there is no difficulty in carrying
out a similar analysis of the analogous situation over global function fields (cf. [A3]).
I would like to thank K. Ribet for drawing my attention to [R]. I am also grateful
to T. Chinburg, M. J. Taylor, and W. Messing for interesting conversations.