Monodromy of Codimension-One Sub-Families of Universal Curves

TL;DR

This paper proves that monodromy representations of certain families of algebraic curves and abelian varieties have images of finite index in their symplectic groups, using a general non-abelian strictness theorem.

Contribution

It establishes the finiteness index of monodromy images for codimension-one families in moduli spaces, extending understanding of their algebraic and arithmetic properties.

Findings

Monodromy images have finite index in symplectic groups.

Results apply to families over divisors in moduli spaces.

Supports applications to Galois cohomology of function fields.

Abstract

Suppose that g > 2, that n > 0 and that m > 0. In this paper we show that if E is an irreducible smooth variety which dominates a divisor D in M_{g,n}[m], the moduli space of n-pointed, smooth curves of genus g with a level m structure, then the closure of the image of the monodromy representation pi_1(E,e)--> Sp_g(Zhat) has finite index in Sp_g(Zhat). A similar result is proved for codimension 1 families of the universal principally polarized abelian variety of dimension g > 2. Both results are deduced from a general "non-abelian strictness theorem". The first result is used in arXiv:1001.5008 to control the Galois cohomology of the function field of M_{g,n}[m] in degrees 1 and 2.