2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79349For 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.30 pages, 1 figure, 1 appendix by Prof. Greg AndersonNumber TheoryAlgebraic Topology11R34; 11RR18; 18G40text