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The intrinsic Hodge theory of \(p\)-adic hyperbolic curves. (English) Zbl 1052.14024

From the introduction: A hyperbolic curve is an algebraic curve obtained by removing \(r\) points from a smooth, proper curve of genus \(g\), where \(g\) and \(r\) are nonnegative integers such that \(2g-2+r>0\). If \(X\) is a hyperbolic curve over the field of complex numbers \(\mathbb{C}\) then \(X\) gives rise in a natural way to a Riemann surface \({\mathcal X}\). As one knows from complex analysis, the most fundamental fact concerning such a Riemann surface (due to P. Köbe [Acta Math. 50, 27–157 (1927; JFM 53.0032.01)]) is that it may be uniformized by the upper half-plane, i.e., \({\mathcal X}\approx {\mathfrak H}/ \Gamma\) where \({\mathfrak H}\overset \text{def}{=} \{z\in \mathbb{C}\mid \text{Im}(z)>0\}\), and \(\Gamma\approx \pi_1 ({\mathcal X})\) (the topological fundamental group of \({\mathcal X}\) is a discontinuous group acting on \({\mathfrak H}\). Note that the action of \(\Gamma\) on \({\mathfrak H}\) defines a canonical representation \[ \rho_{\mathcal X}:\pi_1({\mathcal X})\to \text{PSL}_2(R) \overset\text{def} {=} SL_2 (R)/\{\pm \}= \operatorname{Aut}_{ \text{Holomorphic}} ({\mathcal H}). \] The goal of the present manuscript is to survey various work of the author [Publ. Res. Inst. Math. Sci. 32, No. 6, 957–1151 (1996; Zbl 0879.14009), Foundations of \(p\)–adic Teichmüller theory. AMS/IP Studies in Advanced Mathematics. 11. Providence, RI: American Mathematical Society (1997; Zbl 0969.14013), and Invent. Math. 138, No. 2, 319–423 (1999; Zbl 0935.14019), The generalized ordinary moduli of \(p\)-adic hyperbolic curves, RIMS Preprint 1051 (1995), Combinatorialization of \(p\)-adic Teichmüller theory, RIMS Preprint 1076 (1996)] devoted to generalizing Köbe uniformization to the \(p\)-adic case.

MSC:

14G20 Local ground fields in algebraic geometry
11G20 Curves over finite and local fields
14H25 Arithmetic ground fields for curves
14F30 \(p\)-adic cohomology, crystalline cohomology
14F35 Homotopy theory and fundamental groups in algebraic geometry
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
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