Spherical varieties and norm relations in Iwasawa theory

Abstract

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Spherical varieties and p-adic families of cohomology classes

“…In this paper, we consider varying the algebraic weight defining the coefficient system for our cohomology. Our main result, Theorem 5.2.2, shows that the norm-compatible families constructed in [Loe19] interpolate classical cohomology classes of many different weights (including twists by Grössencharacters of possibly non-trivial infinity-type). Several specific instances of this result are already known -see Remark 5.2.5 below -and form an important ingredient in the proofs of "explicit reciprocity laws" for Euler systems and p-adic L-functions.…”

“…Hecke actions. For general λ, the integral étale cohomology groups H * (−, V λ,O ) are Cartesian cohomology functors for G(Z p ) in the sense of [Loe19]. However, if the coefficients are non-trivial they are typically not cohomology functors for the whole of G(Q p ) since this group does not act on the lattice V λ,O .…”

“…We recall the setup considered in [Loe19], where we consider pushforward of "Eisenstein-type" cohomology classes from a group H to a larger group G. We shall fix an embedding H → G of reductive groups over Q, extending to an embedding H → G of reductive group schemes over Z p .…”

“…We set L r = { ∈ L G (Z p ) : mod p r ∈ L 0 G }. Then the "norm-compatibility machine" of [Loe19] uses the open compact subgroups U r and V r of G(Z p ) defined for r 1 by…”

“…This article is a sequel to the article [Loe19] of the first author. In the previous article, we constructed norm-compatible families of cohomology classes for Shimura varieties and other symmetric spaces, associated to pairs of reductive groups satisfying an "open orbit" condition.…”