Coherent cohomology of Shimura varieties, motivic cohomology, and archimedean $L$-packets

TL;DR

This paper proposes a conjecture relating periods of automorphic forms on Shimura varieties with multiple infinity types, extending motivic action ideas and providing evidence for a unified understanding of automorphic periods.

Contribution

It formulates an analogue of the archimedean motivic action conjecture for irregular cohomological automorphic forms on Shimura varieties, addressing multiple degrees and infinity types.

Findings

Conjecture aligns with existing automorphic period conjectures.

Provides evidence for the compatibility of the new conjecture.

Suggests operations connecting different infinity types within an $L$-packet.

Abstract

We formulate an analogue of the archimedean motivic action conjecture of Prasanna--Venkatesh for irregular cohomological automorphic forms on Shimura varieties, which appear on multiple degrees of coherent cohomology of Shimura varieties. Such multiple appearances are due to many infinity types in a single $L$-packet with equal minimal $K$-types. Accordingly, we formulate the conjecture comparing periods of forms of different automorphic representations. We provide evidences for the conjecture by showing its compatibility with existing conjectures on periods of automorphic forms. The conjectures suggest the existence of certain operations which move between different infinity types in an $L$-packet.