Quasi-projective and formal-analytic arithmetic surfaces

Abstract

This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surfaces relies crucially on the use of real-valued invariants. Some of these are intersection-theoretic, in the spirit of Arakelov intersection theory. Some other invariants involve infinite-dimensional geometry of numbers. Relating our new intersection-theoretic invariants to more classical invariants of Arakelov geometry leads us to investigate a new invariant, the Archimedean overflow, attached to an analytic map from a pointed compact Riemann surface with boundary to a Riemann surface. It is related to the characteristic functions of Nevanlinna theory. Our results on the geometry of formal-analytic arithmetic surfaces admit applications to concrete problems of arithmetic geometry. Notably we generalize the arithmetic holonomicity theorem of Calegari-Dimitrov-Tang regarding the dimension of spaces of power series with integral coefficients satisfying some convergence conditions. We also establish an arithmetic counterpart of theorems of Lefschetz and Nori by providing a bound on the index, in the \'etale fundamental group of an arithmetic surface, of the closed subgroup generated by the \'etale fundamental groups of some arithmetic curve and of some compact Riemann surfaces mapping to the arithmetic surface.