Spherical varieties and norm relations in Iwasawa theory

Abstract

Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. The aim of this note is to give a systematic approach to proving "vertical" norm-compatibility relations for such classes (where the level varies at a fixed prime p), treating the case of Betti and étale cohomology at once, and revealing an unexpected relation to the theory of spherical varieties. This machinery can be used to construct many new examples of norm-compatible families, potentially giving rise to new constructions of both Euler systems and p-adic L-functions: examples include families of algebraic cycles on Shimura varieties for U(n) x U(n+1) and U(2n) over the p-adic anticyclotomic tower.

arXiv comment19 pages. Revised and corrected version, to appear in J Theor. Nombres Bordeaux (Iwasawa 2019 special issue)

References in corpus (9)

• Euler systems for Rankin--Selberg convolutions of modular forms
• Euler systems for GSp(4)
• Rankin--Eisenstein classes in Coleman families
• Euler systems for Hilbert modular surfaces
• Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields
• $L$-functions of $\mathrm{GL}(2n):$ $p$-adic properties and non-vanishing of twists
• Heegner points in Coleman families
• Anticyclotomic Euler systems for unitary groups
• P-adic Asai L-functions of Bianchi modular forms

Cited by in corpus (6)

• An Euler system for GU(2, 1)
• Euler Systems for ${\mathrm{GSp}}_4\times {\mathrm{GL}}_2$
• Anticyclotomic Euler systems for unitary groups
• P-adic L-functions and diagonal cycles for GSp(4) x GL(2) x GL(2)
• On some zeta-integrals for unramified representations of GSp(4)
• Special cycles on orthogonal Shimura varieties