The conjugate uniformization via 1-motives

Abstract: Given an abelian variety $A$ over a finite extension $K$ of $\mathbb{Q}_p$, Fontaine constructed an integration map from the Tate module of A to its Lie algebra. This map gives the splitting of the Hodge-Tate short exact sequence. Recent work of Iovita-Morrow-Zaharescu extends this integration map to the $\overline{K}$ points of the perfectoid universal cover of $A$. They used this result to give a uniformization of the $\mathcal{O}_{\overline K}$ points of the underlying $p$-divisible group. In this talk, we explain joint work with Sean Howe and Jackson Morrow in which we give a different perspective on this uniformization using 1-motives. We will first give some intuition from the complex uniformization of semi-abelian varieties and some background and motivation on $p$-divisible groups. Then we will explain how to construct the $p$-divisible group of a 1-motive and how this gives the desired uniformization. Finally, we will point out some interesting geometric features of this map: it embeds the rigid analytic points of a $p$-divisible group into an etale cover of a negative Banach-Colmez space.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar