Several years ago Mazur and Wiles proved a fundamental conjecture of Iwasawa which gives a precise link between the critical values of the Riemann zeta function (and, more generally, Dirichlet L-functions) and the ideal class groups of certain towers of cyclotomic fields. Probably the first hint of such a link is Kummer's well-known criterion for irregularity of primes. In Iwasawa's theory one defines for each prime p certain modules over the Iwasawa algebra A (which we will describe in Section 1). Iwasawa's conjecture then relates the structure of these A-modules to the p-adic L-functions constructed by Kubota and Leopoldt which interpolate critical values of Dirichlet L-functions. Mazur realized that, following Iwasawa's model, one could formulate a similar conjecture for an elliptic curve E (defined over Q) and for any prime p where E has good, ordinary reduction. Mazur and Swinnerton-Dyer constructed p-adic L-functions attached to E for such p (assuming Eis a Weil curve). The Iwasawa modules which Mazur's conjecture relates to these p-adic L-functions are defined in terms of Selmer groups for E, again in towers of cyclotomic fields. This time the hint of such a relationship is the Birch and Swinnerton-Dyer conjecture.�

There are now several other cases where p-adic analogues of complex L-functions have been constructed-for example, Manin's p-adic L-functions attached to classical modular forms. It seems worthwhile then to search for appropriate Iwasawa modules in a much more general context and that is our purpose in this paper. We will consider a compatible system of 1-adic representations V={Vi} over Q. Thus Vi is a finite dimensional vector space over Qi (the 1-adic numbers) of dimension d= dv on which Ga= Gal (Q/Q) acts.(For any field k, Gk denotes Gal (f/k), where le is an algebraic closure of k.) If q is any prime, then one can also consider Vi as a representation space of Gaq by choosing a place lj_ of Q over q and identifying Ga, with the decomposition group for that place. We will usually assume that pis an" ordinary" prime for Vin the follow-