Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$

TL;DR

This paper develops explicit $p$-adic Eichler-Shimura morphisms for overconvergent Siegel modular forms of genus two, extending classical decompositions into a $p$-adic setting using advanced cohomological techniques.

Contribution

It introduces a method to interpolate the Eichler-Shimura decomposition $p$-adically for families of overconvergent Siegel modular forms, extending previous work.

Findings

Constructed explicit $p$-adic Eichler-Shimura morphisms.

Extended the $p$-adic interpolation of the Eichler-Shimura decomposition.

Utilized higher Coleman theory and pro-Kummer étale cohomology.

Abstract

We construct explicit Eichler-Shimura morphisms for families of overconvergent Siegel modular forms of genus two. These can be viewed as $p$-adic interpolations of the Eichler-Shimura decomposition of Faltings-Chai for classical Siegel modular forms. In particular, we are able to $p$-adically interpolate the entire decomposition, extending our previous work on the $H^0$-part. The key new inputs are the higher Coleman theory of Boxer-Pilloni and a theory of pro-Kummer \'etale cohomology with supports.