Euler systems and relative Satake isomorphism

TL;DR

This paper develops a unified framework linking the relative Langlands program to Euler systems via spherical varieties. By introducing motivic theta series and applying local harmonic analysis, it produces automatic tame norm relations and recovers many known Euler systems, while yielding a new split anticyclotomic Euler system in the twisted Friedberg–Jacquet setting. The approach hinges on a relative Satake machinery, generalized Cartan decompositions, and integral structures to construct test vectors and verify divisibility properties without ad hoc computations. The global constructions pass from motivic classes to Galois cohomology, yielding Euler-system-type relations for cycles and Eisenstein classes across a range of symmetric pairs, with explicit applications to Gan–Gross–Prasad, Friedberg–Jacquet, and related automorphic contexts. Overall, the work provides a computation-free pathway to generate and organize Euler-system data within the relative Langlands framework, with potential arithmetic consequences for Bloch–Kato-type conjectures.

Abstract

We explain how the unramified Plancherel formula in the relative Langlands program gives a natural way of constructing test vectors which satisfy the tame norm relations of an Euler system. This uniformly recovers many of the known Euler systems, and in the twisted Friedberg--Jacquet setting, we produce a new split anticyclotomic Euler system.

Theorems & Definitions (71)

• Theorem 1.1: Proposition \ref{['prop:cycle']}+Corollary \ref{['cor:Abstract']}
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10
• Remark 2.1
• Definition 2.2