A 1-dimensional formal group over the prismatization of Spf Z_p

TL;DR

This paper explores the structure of a 1-dimensional formal group over the prismatization of Spf Z_p, revealing its kernel as a Cartier dual and providing explicit descriptions of related structures.

Contribution

It introduces a new formal group over the prismatization of Spf Z_p and characterizes its properties, including its Lie algebra and explicit pullback descriptions.

Findings

Kernel of the multiplicative group map is the Cartier dual of a formal group.

Explicit description of the formal group's pullback to a quotient of the q-de Rham prism.

Description of the Lie algebra of the formal group.

Abstract

Let Sigma denote the prismatization of Spf (Z_p). The multiplicative group over Sigma maps to the prismatization of the multiplicative group over Spf (Z_p). We prove that the kernel of this map is the Cartier dual of some 1-dimensional formal group over Sigma. We obtain some results about this formal group (e.g., we describe its Lie algebra). We give a very explicit description of the pullback of the formal group to the quotient of the q-de Rham prism by the action of the multiplicative group of Z_p.