Volume and symplectic structure for l-adic local systems

G. Pappas

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arXiv · 2006.03668

Volume and symplectic structure for l-adic local systems

G. Pappas

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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.

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BibTeX RIS

G. Pappas. 2020-06-05. Volume and symplectic structure for l-adic local systems. https://arxiv.org/abs/2006.03668

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