Construction of Arithmetic Teichmuller Spaces I
Kirti Joshi

TL;DR
This paper develops a p-adic and adelic version of Teichmuller spaces for algebraic varieties over number fields, extending analogies with Riemann surfaces and inspired by Mochizuki's Inter-Universal Teichmuller Theory.
Contribution
It introduces the concept of Arithmetic Teichmuller Spaces for varieties over p-adic fields and number fields, providing a new framework distinct from Mochizuki's approach.
Findings
Construction of local p-adic Arithmetic Teichmuller Space
Global adelic Arithmetic Teichmuller Space for varieties over number fields
Extension of the analogy between number fields and Riemann surfaces
Abstract
In this paper after proving (in Section 2) the Berkovich analytic space analog of the familiar fact that there exist many non-isomorphic Riemann surfaces of the fixed topological type, I introduce the precise notion of Arithmetic Holomorphic Structures. This leads, for a fixed geometrically connected, smooth quasi-projective variety over a -adic field, to the construction of a category which can be called Arithmetic Teichmuller Space of . After establishing the properties of this local i.e. -adic Arithmetic Teichmuller Space, I proceed to the global (adelic) construction, for a geometrically connected, smooth quasi-projective variety over a number field , of the Adelic Arithmetic Teichmuller Space of . A fixed number field itself has an Arithmetic Teichmuller Space--this is detailed in Constructions II(1/2) paper in this series of papers. All of these…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology
