Volume and symplectic structure for l-adic local systems

TL;DR

This paper introduces a notion of volume and a symplectic structure for l-adic local systems on algebraic curves, drawing analogies with 3-manifold invariants and exploring Galois actions and descent loci.

Contribution

It defines a new volume concept and symplectic form for l-adic local systems, and studies Galois actions, descent loci, and associated invariants in this arithmetic setting.

Findings

Galois group acts by conformal symplectomorphisms on the deformation space

The descent locus is characterized as the critical set of rigid analytic functions

Vanishing cycles provide additional invariants for the local systems

Abstract

We introduce a notion of volume for an l-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which extend to an l-adic analytic flow. We also prove that the locus of the deformation space over which the local system suitably descends is the critical set of a collection of rigid analytic functions. The vanishing cycles of these functions give additional invariants.