Volume and symplectic structure for l-adic 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. ContentsIntroduction 1 1. Preliminaries 6 2. K 2 invariants and 2-forms 10 3. Chern-Simons and volume 15 4. Representations of étale fundamental groups 25 5. Deformations and lifts 32 6. The symplectic nature of the Galois action 37 7. Appendix: Interpolation of iterates and flows 44 References 47 Notations: Throughout the paper N denotes the non-negative integers and ℓ is a prime. We denote by F ℓ the finite field of ℓ elements and by Z ℓ , resp. Q ℓ , the ℓ-adic integers, resp. ℓ-adic numbers. We fix an algebraic closure Qℓ of Q ℓ and we denote by | | ℓ (or simply | |), resp. v ℓ , the ℓ-adic absolute value, resp. ℓ-adic valuation on Qℓ , normalized so that |ℓ| ℓ = ℓ −1 , v ℓ (ℓ) = 1. We will denote by F a field of characteristic ℓ which is algebraic over the prime field F ℓ and by W (F), or simply W , the ring of Witt vectors with coefficients in F. If k is a field of characteristic = ℓ we fix an algebraic closure k. We denote by k sep the separable closure of k in k, by k(ζ ℓ ∞ ) = ∪ n≥1 k(ζ ℓ n ) the subfield of k sep generated over k by the (primitive) ℓ n -th roots of unity ζ ℓ n and by χ cycl : Gal(k sep /k) → Z × ℓ the cyclotomic character defined by σ(ζ ℓ n ) = ζ χ cycl (σ) ℓ nfor all n ≥ 1. We set G k = Gal(k sep /k). Finally, we will denote by ( ) * the Pontryagin dual, by ( ) ∨ the linear dual, and by ( ) × the units.