Invent. math. 76, 179-330 (1984) Irlverltioyle$ mathematicae ~ Springer-Verlag 1984 Class fields of abelian extensions of Q To K. Iwasawa B. Mazur and A.Wiles Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Table of contents Introduction ............................................. 179 0. Notation and preliminaries ................................... 188 1. Iwasawa theory, p-adic L-functions, and Fitting ideals .................... 191 2. Models and moduli ........................................ 225 3. A study of abelian varieties which are "good" quotients of Jz (N) .............. 261 4. The cuspidal group ........................................ 289 5. The kernel of the Eisenstein ideal ................................ 307 Appendix: Fitting ideals ....................................... 324 References .............................................. 329 Introduction w 1. General discussion Letp be an odd prime number. The object of this paper is to show that the zeroes of the p-adic L-functions of Kubota-Leopoldt are equal to the eigenvalues of certain "arithmetically-defined" operators acting on finite-dimensional Qv-vector spaces. These Qv-vector spaces were defined by Iwasawa in terms of limits of certain components of the ideal class groups of towers of cyclotomic number fields. The connection between Iwasawa's vector spaces and the p-adic L-function was conjectured by him (the "main conjecture" for the prime p over the field Q, cf. Chap. 1, w1). As the reader will see, if we take into account what is already known, - and much is known, thanks to the work of Kummer, Stickelberger, Iwasawa, Ferrero- Washington, - our problem can be reduced to the construction of enough classfields of abelian extensions of Q, while keeping close tabs on the action of Galois on the classfields. Broadly speaking, the problem we face comes under the rubric of explicit classfield theory (for abelian extensions of Q). But a wrong impression might arise from the label "explicit class field theory". First, we may ask in what sense is our construction explicit? We obtain our