The Cuspidal Divisor Class Group on X(N)

Abstract

In the most classical situation, the Galois group of the field Q(μ _N ) of roots of unity operates on ideal classes and units modulo cyclotomic units. Classical problems of number theory are concerned with the eigenspace decomposition of the p-primary part of these groups when N = p, and of the structures as Galois modules in general. Results include those of Kummer, Stickelberger, Herbrand, and more recently Iwasawa, Leopoldt, and Ribet. Also in recent times, such results have been the object of study in the case of complex multiplication of elliptic curves for the elliptic units as in Robert and Coates-Wiles.