Group cohomology of the universal ordinary distribution

Abstract

Abstract. For any odd squarefree integer r, we get complete description of the Gr = Gal(Q(µr)/Q) group cohomology of the universal ordinary distribution Ur 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 * (Gr, Ur/M Ur). In particular, we explain the mysterious construction of the elements κ r ′ for r ′ |r in Rubin [10], which come exactly from a certain Z/M Z-basis of the cohomology group H 0 (Gr, Ur/M Ur) through an evaluation map.

Kronecker-Weber plus epsilon

“…The relations standing between the Main Formula, the index formulas of [Sinnott 1978], Deligne reciprocity [Deligne-Milne-Ogus-Shih 1982, Thm. 7.15, p. 91], the theory of [Fröhlich 1983], the theory of [Das 2000], the theory of the group cohomology of the universal ordinary distribution (see [Ouyang 2001] and references therein) and Stark's conjecture and its variants (see [Tate 1984]) deserve to be thoroughly investigated. We have only scratched the surface here.…”

Distributions and Euler systems in global function fields

“…Proposition 2.1.1 ( [O,Proposition 5.1]). The map u is a quasi-isomorphism between K * , * and U * , * .…”

Sign functions of imaginary quadratic fields and applications

“…He also introduces a certain double complex to compute this cohomology. The Anderson's double complex turns out to be a powerful means for other cohomological computations like those made in [15].…”

“…Let us identify G ∞ with the group J = gal(k m,s /k m ). As mentioned in the introduction we will use Anderson's method introduced in [1] and improved by Ouyang in [16] and also in [15]. The first step is to define the Anderson's resolution of U 0 s (m).…”