# Arithmetic occult period maps

## TL;DR

This paper demonstrates that certain complex configuration spaces, uniformized as arithmetic ball quotients via period maps, are inherently arithmetic, supported by an arithmetic theory of lattice-polarized K3 surfaces.

## Contribution

It proves that occult period maps are arithmetic and develops an arithmetic framework for lattice-polarized K3 surfaces.

## Key findings

- Period maps respect algebraic structures of Shimura varieties.
- Configuration spaces are uniformized as arithmetic ball quotients.
- Established an arithmetic theory for lattice-polarized K3 surfaces.

## Abstract

Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties gives the ball quotient the structure of a variety over the ring of integers of a cyclotomic field. We show that the (transcendentally-defined) period map actually respects these algebraic structures, and thus that occult period maps are arithmetic. As an intermediate tool, we develop an arithmetic theory of lattice-polarized K3 surfaces.