 
Summary: Digital Object Identifier (DOI) 10.1007/s002080100199
Math. Ann. 320, 339365 (2001) Mathematische Annalen
On arithmetic class invariants
A. Agboola · G. Pappas
Received June 12, 2000 / Accepted August 11, 2000 /
Published online February 5, 2001 © SpringerVerlag 2001
1. Introduction
Let F be a number field with ring of integers OF , and let S be a finite set of
places of F. Assume that S contains the set S of archimedean places of F, and
write Sf for the set of finite places contained in S. Let OS (or O, when there
is no danger of confusion) denote the ring of Sf integers of F. Write Fc
for an
algebraic closure of F.
Let Y be any scheme over Spec(O). Suppose that G is a finite, flat commuta
tive group scheme over Y of exponent N, and let GD
denote the Cartier dual of
G. Let : X Y be a Gtorsor, and write 0 : G Y for the trivial Gtorsor.
Then OX is an OGcomodule, and so it is also an OGD module (see [12]). As an
OGD module, OX is locally free of rank one, and it therefore gives a line bundle
M over GD
