Nonhomeomorphic conjugates of connected Shimura varieties

TL;DR

This paper demonstrates that automorphisms of the complex numbers can alter the topological fundamental group of certain algebraic varieties, leading to nonisomorphic fundamental groups from different embeddings of number fields.

Contribution

It reveals that conjugation by complex automorphisms can change the fundamental group of connected Shimura varieties, a novel insight into their topological and arithmetic properties.

Findings

Conjugation can change the fundamental group of locally symmetric varieties.

Different embeddings of number fields can produce varieties with nonisomorphic fundamental groups.

The result applies to a large class of algebraic varieties over number fields.

Abstract

We show that conjugation by an automorphism of the complex numbers (as an abstract field) may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups.