2020/06/05 by George J. Pappas, Pappas, G.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2006.03668
openalex publication_date 2020/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.