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Spherical varieties and p-adic families of cohomology classes

David Loeffler, Rob Rockwood, Sarah Livia Zerbes

Abstract

We prove a "twist-compatibility" result for p-adic families of cohomology classes associated to symmetric spaces. This shows that a single family of classes (lying in a finitely-generated Iwasawa module) interpolates classical cohomology classes of many different weights, including twists by Grössencharacters of possibly non-trivial infinity-type. This subsumes and generalises a number of prior results relating to Euler systems and p-adic L-functions, and we conclude with some novel applications to Euler systems for GSp(4), GSp(4) x GL(2), and GSp(4) x GL(2) x GL(2).

Spherical varieties and p-adic families of cohomology classes

Abstract

We prove a "twist-compatibility" result for p-adic families of cohomology classes associated to symmetric spaces. This shows that a single family of classes (lying in a finitely-generated Iwasawa module) interpolates classical cohomology classes of many different weights, including twists by Grössencharacters of possibly non-trivial infinity-type. This subsumes and generalises a number of prior results relating to Euler systems and p-adic L-functions, and we conclude with some novel applications to Euler systems for GSp(4), GSp(4) x GL(2), and GSp(4) x GL(2) x GL(2).

Paper Structure

This paper contains 26 sections, 27 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Control theorems for ordinary cohomology
  3. Setup
  4. Algebraic representations
  5. Integral lattices
  6. Cohomology of symmetric spaces
  7. Hecke actions
  8. Iwasawa cohomology and control theorems
  9. Proof of the control theorem
  10. Branching laws for algebraic representations
  11. Flag varieties
  12. Highest-weight representations
  13. Projection to the highest-weight subspace
  14. Norm-compatible families
  15. Subgroups
  16. ...and 11 more sections

Key Result

Lemma 2.3.4

If $\eta \in \Sigma^+$, then the endomorphism $p^{\langle \eta, \lambda\rangle} \rho_\lambda\left(\eta(p)^{-1}\right)$ of $V_\lambda$ acts on every weight space as a non-negative power of $p$, and hence restricts to an endomorphism of $\mathcal{L}$ for any admissible lattice $\mathcal{L}$. If $\eta

Theorems & Definitions (80)

  • Remark 2.1.2
  • Definition 2.3.1
  • Remark 2.3.2
  • Definition 2.3.3
  • Lemma 2.3.4
  • proof
  • Remark 2.4.1
  • Remark 2.4.2
  • Definition 2.5.1
  • Remark 2.5.2
  • ...and 70 more