Toward Shimurian analogs of Barsotti-Tate groups

TL;DR

This paper explores the structure and reductions of Barsotti-Tate groups and their analogs in Shimura varieties, utilizing prismatic cohomology to address complex algebraic and arithmetic problems.

Contribution

It proposes new conjectures on Shimurian analogs of Barsotti-Tate groups and their reductions, connecting prismatic cohomology with algebraic stack theory.

Findings

Formulated conjectures relating prismatic cohomology to reductions of Barsotti-Tate groups.

Provided an algebraic stack framework for n-truncated Barsotti-Tate groups.

Suggested pathways for constructing Shimurian analogs with good reduction at p.

Abstract

We first recall Grothendieck's notion of n-truncated Barsotti-Tate group. Such groups form an algebraic stack over the integers. The problem is to give an illuminating description of its reductions modulo powers of p. A related problem is to construct analogs of these reductions related to general Shimura varieties with good reduction at p. We discuss some conjectures on this subject based on the theory of prismatic cohomology.