Group cohomology of universal ordinary distribution

Abstract

For any odd squarefree integer r, we get complete description of the G_r=Gal(Q(mu_r)/Q) group cohomology of the universal ordinary distribution U_r in this paper. Moreover, if M is a fixed integer which divides l-1 for all prime factors l of r, we study the cohomology group H^{*}(G_r, U_r/MU_r). In particular, we explain the mysterious construction of the elements kappa_{r'} for r'|r in Rubin's construction of Kolyvagin elements, which come exactly from a certain Z/M Z-basis of the cohomology group H^{0}(G_r, U_r/MU_r) through an evaluation map.