The conjugate uniformization via 1-motives

TL;DR

This paper extends the conjugate p-adic uniformization to a broader class of p-divisible groups using 1-motives, connecting p-adic Hodge theory with geometric structures like Banach--Colmez spaces.

Contribution

It generalizes the conjugate p-adic uniformization to arbitrary p-divisible groups over p-adic rings using 1-motives, linking it to mixed Hodge theory and Fargues--Scholze spaces.

Findings

Generalization of conjugate p-adic uniformization to semi-abelian schemes

Identification of the target space as an étale cover of a Banach--Colmez space

Connection between p-adic uniformization and mixed Hodge theory

Abstract

We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an \'{e}tale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.