Inscription, twistors, and $p$-adic periods

TL;DR

This paper develops a new theory of inscribed v-sheaves to study $p$-adic periods, constructing refined period maps and derivatives, with applications to Shimura varieties and local shtukas, revealing new structures in $p$-adic Hodge theory.

Contribution

It introduces inscribed $v$-sheaves and applies them to $p$-adic periods, constructing refined period maps and derivatives, and providing new insights into $p$-adic Hodge structures.

Findings

Construction of inscribed $v$-sheaves and period maps.

Derivatives of period maps expressed via classical $p$-adic structures.

Application to global and local Shimura varieties and local shtukas.

Abstract

We introduce the theory of inscribed $v$-sheaves, a differentiable extension of the theory of diamonds and $v$-sheaves with internal tangent bundles that are often relative inscribed Banach-Colmez spaces, then apply this theory to the study of $p$-adic periods. In particular, we construct natural inscribed versions of the Hodge and Hodge-Tate period maps and their lattice refinements for de Rham torsors, then compute the derivatives of these period maps in terms of classical structures in $p$-adic Hodge theory. These torsors include infinite level global Shimura varieties and infinite level local Shimura varieties, and for these spaces we also give another moduli-theoretic construction of the inscribed structure; the construction in the local Shimura case applies more generally to the non-minuscule moduli of mixed characterisic local shtukas with one leg. The key new ingredients in our study of inscribed structures on $p$-adic Lie group torsors over smooth rigid varieties over a $p$-adic field are the Liu-Zhu period map, a refinement of the Hodge period map whose derivative is the geometric Sen morphism/canonical Higgs field, and a closely related exact tensor functor from $\mathbb{Q}_p$-local systems to a category of twistor bundles on the relative thickened Fargues-Fontaine curve. These new structures are only visible after passing to the inscribed setting. We also discuss some possible implications of our computations in the vein of ``differential topology for diamonds."