Spherical varieties and non-ordinary families of cohomology classes

TL;DR

This paper develops a method to construct non-ordinary $p$-adic families of cohomology classes in locally symmetric spaces, with applications to Galois cohomology, Euler systems, and $p$-adic $L$-functions.

Contribution

It introduces a new construction of non-ordinary $p$-adic families of classes for spherical pairs, extending previous Euler system and $p$-adic $L$-function methods.

Findings

Constructs non-ordinary $p$-adic classes in cohomology of symmetric spaces.

Maps classes into Galois cohomology in the étale case.

Generalizes $p$-part of the Lemma--Flach Euler system for Siegel modular forms.

Abstract

We give a construction of non-ordinary $p$-adic families of classes in the cohomology of locally symmetric spaces associated to spherical pairs of reductive groups. In the \'etale case, we show how to map these classes into Galois cohomology. The methods developed in this paper can be used to give new constructions of $p$-adic families of Euler systems and $p$-adic $L$-functions. As an example, we show how the constructions of this paper can be used to construct norm-compatible classes associated to non-ordinary Siegel modular forms, generalising $p$-part of the Lemma--Flach Euler system constructed by Loeffler--Skinner--Zerbes.