ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 14F30 Shinichi Mochizuki, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan The Intrinsic Hodge Theory of p-adic Hyperbolic Curves

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 C, then X gives rise in a natural way to a Riemann surface X. As one knows from complex analysis, the most fundamental fact concerning such a Riemann surface (due to Köbe) is that it may be uniformized by the upper half-plane, i. e., X ≅ ℌ/Γ where ℌ =^{def} {z ∈ C | Im(z) > 0}, and Γ ≅ π_{1}(X) (the topological fundamental group of X) is a discontinuous group acting on ℌ. Note that the action of Γ on ℌ defines a canonical representation ρ_{X} : π_{1}(X) → P SL_{2}(R) = Aut_{Holomorphic}(ℌ). Now there are (at least) two ways to interpret the Köbe uniformization: The Physical Approach: The Köbe uniformization provides a “physical, ” geometric way to reconstruct the algebraic curve X from its topological fundamental group π_{1}(X), equipped with some canonical arithmetic information, i. e. , ρ_{X}. The Modular Approach: The representation ρ_{X} gives rise to a special kind of P^{1}-bundle with connection (P → X, ∇_{P}) on the algebraic curve X called an indigenous bundle. More generally, the moduli space of indigenous bundles forms an algebraic torsor S_{g,r} → M_{g,r} (in fact, defined over Z[\frac{1}{2}]), called the Schwarz torsor, over the moduli stack of hyperbolic curves of type (g, r). Then Teichmüller theory (i. e., in particular, the uniformization theory of hyperbolic curves and their moduli) may be formulated as the study of the canonical real analytic section of the Schwarz torsor defined by the indigenous bundle arising from the Köbe uniformization. The aim of the present talk is to discuss various generalizations to the padic case of these physical and modular aspects of Köbe’s uniformization theorem. Note that both the physical and modular approaches to the Köbe uniformization assert that there is a certain equivalence

algebraic geometry ⟺ topology endowed with an arithmetic structure

That is, on the algebraic geometry side, we have the scheme (respectively, stack) given by the curve X itself in the physical approach (respectively, its moduli M_{g,r} in the modular approach), whereas on the “topology plus arithmetic structure” side, we have the theory of the canonical representation ρ_{X} of π_{1}(X). This sort of equivalence is reminiscent of that given by classical or p-adic Hodge theory between the de Rham or Hodge cohomology of an algebraic variety (on the algebraic geometry side), and the singular or étale cohomology (equipped with Galois action) on the topology plus arithmetic side. In our case, however, instead of dealing with the cohomology of the curve, we are dealing with “the curve itself” and its moduli. It is for this reason that we refer to this theory as the intrinsic Hodge theory of the curve X. Now let X be a hyperbolic curve over a finite extension K of the field Q_{p} of p-adic numbers. Let π_{1}(X) be its algebraic fundamental group. Note that π_{1}(X) comes equipped with a natural augmentation to Γ_{K}, the absolute Galois group of K. Then one has the following p-adic analogue (cf. [4]) of the physical aspect of the Köbe uniformization: One may recover the scheme X (in the sense of the functor defined by X by considering dominant Kmorphisms to X from K-smooth schemes S of finite type) as the functor of open, continuous Γ_{K}-compatible homomorphisms π_{1}(S) → π(X). This sort of result was predicted by Grothendieck’s anabelian philosophy, and builds on earlier work of H. Nakamura and A. Tamagawa. It may also be considered as the analogue for hyperbolic curves of the Tate conjecture on homomorphisms between abelian varieties. Finally, we have a p-adic analogue of the modular aspect of the Köbe uniformization, i. e., of Teichmüller theory. Just as Teichmüller theory may be regarded as the theory of the canonical real analytic section of the Schwarz torsor, the p-adic analogue of Teichmüller theory concerns the issue of finding canonical p-adic sections of the Schwarz torsor. In characteristic p, this is achieved by considering the stack N_{g,r} of indigenous bundles with square nilpotent p-curvature. In our talk, we will discuss work on the fine structure of N_{g,r} (cf. [3]) in a “neighborhood of infinity” (i. e., near totally degenerate curves), as well as the theory of canonical p-adic liftings (cf. [1,2]) for various portions of N_{g,r}.

Bibliography

[1] Mochizuki, S., A Theory of Ordinary p-adic Curves, RIMS Preprint 1033 (September 1995); Publ. of RIMS 32, No. 6 (1996), pp. 957-1151.

[2] Mochizuki, S., The Generalized Ordinary Moduli of p-adic Hyperbolic Curves, RIMS Preprint 1051 (December 1995); 281 pp.

[3] Mochizuki, S., Combinatorialization of p-adic Teichmüller Theory, RIMS Preprint 1076 (April 1996); 32 pp.

[4] Mochizuki, S., The Local Pro-p Anabelian Geometry of Curves, RIMS Preprint 1097 (August 1996); 84 pp.