Kazhdan's Theorem on Arithmetic Varieties

TL;DR

This paper simplifies Kazhdan's proof that automorphisms of the complex numbers preserve the arithmetic nature of varieties formed from quotients of symmetric domains by arithmetic groups, avoiding complex classification theorems.

Contribution

It provides a streamlined proof of Kazhdan's theorem on the invariance of arithmetic varieties under field automorphisms, removing reliance on classification results.

Findings

Proof of Kazhdan's theorem simplified

Avoids use of classification theorems

Clarifies the algebraic nature of arithmetic varieties

Abstract

Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again arithmetic. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems. It was originally completed on March 28, 1984, and distributed in handwritten form. July 23, 2001: Fixed about 30 misprints.