Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

TL;DR

This paper develops a perfectoid geometric framework for $p$-adic automorphic forms on unitary Shimura varieties, introducing an overconvergent Eichler-Shimura map that connects cohomology to automorphic forms and has implications for eigenvariety geometry and $p$-adic $L$-functions.

Contribution

It constructs a perfectoid approach to overconvergent automorphic forms and establishes a canonical $p$-adic Eichler-Shimura map linking cohomology to automorphic forms.

Findings

Constructed a perfectoid model for overconvergent automorphic forms.

Established a canonical overconvergent Eichler-Shimura map.

Applied results to eigenvariety geometry and $p$-adic $L$-functions.

Abstract

We investigate $p$-adic automorphic forms on unitary groups through the geometry of infinite-level unitary Shimura varieties and the Hodge-Tate period map. We first develop a perfectoid construction of overconvergent automorphic forms. Building on this, we establish a canonical overconvergent Eichler-Shimura map linking overconvergent cohomology to these $p$-adic automorphic forms. This map induces a comparison between the corresponding coherent sheaves on the eigenvariety, with applications to the study of its geometry and to $p$-adic $L$-functions.