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

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.