Arithmetic Restrictions on Geometric Monodromy

TL;DR

This paper establishes arithmetic restrictions on geometric monodromy representations of complex algebraic varieties, showing they must be non-trivial modulo high powers of a prime, with broad implications for fundamental group properties.

Contribution

It introduces a new bound N(X, p) constraining geometric monodromy representations modulo p^N, advancing understanding of their arithmetic properties.

Findings

Existence of an integer N(X, p) for geometric monodromy representations.

Non-trivial geometric representations are non-trivial mod p^N.

Several new arithmetic properties of fundamental groups are proven.

Abstract

Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial mod p^N. The proof involves an analysis of the action of the Galois group of a finitely generated field on the etale fundamental group of X. We also prove many arithmetic statements about fundamental groups which are of independent interest, and give several applications.