Arithmetic Restrictions on Geometric Monodromy

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.

arXiv comment57 pages, 1 figure. Dramatic improvements to main results. Significant improvements to exposition, several new examples, and major reorganization. Basically completely rewritten. Comments welcome!

References in corpus (3)

• Revêtements étales et groupe fondamental (SGA 1)
• p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation
• Shimura curves of genus at most two