Spherical varieties and norm relations in Iwasawa theory

David Loeffler

Published 2019-09-22Version 1

Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. Firstly, the classical "modular symbols", which live in Betti cohomology of modular curves, can be used to construct $p$-adic $L$-functions for modular forms; and there are numerous generalisations of this technique used to build $p$-adic $L$-functions for automorphic forms on other reductive groups. Secondly, in the theory of Euler systems, one is interested in norm-compatibility properties for classes in etale cohomology of Shimura varieties, and these norm-compatibility relations are vital to applications to Selmer groups and the Bloch--Kato conjecture. The aim of this note is to give a systematic approach to proving "vertical" norm-compatibility relations (where the level varies at a fixed prime $p$), treating the case of Betti and \'etale 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 $U(n) \times U(n+1)$ and $U(2n)$ over the $p$-adic anticyclotomic tower.