Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism

Abstract

The aim of this paper is twofold. We first present a construction of the overconvergent automorphic sheaves for Siegel modular forms by generalising the perfectoid method, originally introduced by Chojecki--Hansen--Johansson for automorphic forms on compact Shimura curves over $\mathbf{Q}$. The global sections of these automorphic sheaves are precisely the overconvergent Siegel modular forms. In particular, one can compare these automorphic sheaves with the ones constructed by Andreatta--Iovita--Pilloni. Secondly, we establish an (explicit) overconvergent Eichler--Shimura morphism for Siegel modular forms, generalising the result of Andreatta--Iovita--Stevens for the elliptic modular forms.