Special Issue of PRIMS, Vol. 57, No. 1/2
Softcover
Price: \ 13,500.
Pages: 723
Published: March 2021
Copyright: 2021 EMS
The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmuller theory for number fields
equipped with an elliptic curve ? which we refer to as "inter-universal Teichmuller theory" ? by applying the theory of semi-graphs of anabelioids,
Frobenioids, the etale theta function, and log-shells developed in earlier papers by the author. We begin by fixing what we call "initial ΘΘ-data", which
consists of an elliptic curve EFEF over a number field FF, and a prime number l?5l?5, as well as some other technical data satisfying certain technical
properties. This data determines various hyperbolic orbicurves that are related via finite etale coverings to the once-punctured elliptic curve XFXF
determined by EFEF. These finite etale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring
Fl=Z/lZFl=Z/lZ acting on the ll-torsion points of the elliptic curve. We then construct "Θ±ellNFΘ±ellNF-Hodge theaters" associated to the given ΘΘ-
data. These Θ±ellNFΘ±ellNF-Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying
combinatorial dimensions of a number field ? which may be thought of as corresponding to the additive and multiplicative structures of a ring or,
alternatively, to the group of units and value group of a local field associated to the number field ? are, in some sense, "dismantled" or "disentangled" from
one another. All Θ±ellNFΘ±ellNF-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a "ΘΘ-link",
which relates certain Frobenioid-theoretic portions of one Θ±ellNFΘ±ellNF-Hodge theater to another in a fashion that is not compatible with the
respective conventional ring/scheme theory structures. In particular, it is a highly nontrivial problem to relate the ring structures on either side of the ΘΘ
-link to one another. This will be achieved, up to certain "relatively mild indeterminacies", in future papers in the series by applying the absolute anabelian
geometry developed in earlier papers by the author. The resulting description of an "alien ring structure" [associated, say, to the domain of the ΘΘ-link] in
terms of a given ring structure [associated, say, to the codomain of the ΘΘ-link] will be applied in the final paper of the series to obtain results in
diophantine geometry. Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered
fundamental group of a pp-adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of
independent interest.
Keywords: inter-universal, Teichmuller theory, Hodge theater, global multiplicative subspace, canonical generator, punctured elliptic curve, theta-link, etale
Contents
Preface to the Special Issue
pp. 1
Abstract | Full-Text PDF (107 KB) | Metadata
Inter-universal Teichmuller Theory I: Construction of Hodge Theaters
Shinichi Mochizuki
pp. 3?207
Abstract | Full-Text PDF (1901 KB) | Metadata
Inter-universal Teichmuller Theory II: Hodge?Arakelov-Theoretic Evaluation
Shinichi Mochizuki
pp. 209?401
Abstract | Full-Text PDF (1842 KB) | Metadata
Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-Theta-Lattice
Shinichi Mochizuki
pp. 403?626
Abstract | Full-Text PDF (1890 KB) | Metadata
Inter-universal Teichmuller Theory IV: Log-Volume Computations and Set-Theoretic Foundations
Shinichi Mochizuki
pp. 627?723
Abstract | Full-Text PDF (1075 KB) | Metadata
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