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Journal ArticleDOI

Inter-universal Teichmüller Theory I: Construction of Hodge Theaters

04 Mar 2021-Publications of The Research Institute for Mathematical Sciences (European Mathematical Society - EMS - Publishing House GmbH)-Vol. 57, Iss: 1, pp 3-207
TL;DR: In this article, an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve is established by applying the theory of semi-graphs of anabelioids, Frobenioids and log-shells developed in earlier papers by the author.
Abstract: The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve — which we refer to as “inter-universal Teichmüller theory” — by applying the theory of semi-graphs of anabelioids, Frobenioids, the étale theta function, and log-shells developed in earlier papers by the author. We begin by fixing what we call “initial Θ-data”, which consists of an elliptic curve EF over a number field F , and a prime number l ≥ 5, as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite étale coverings to the once-punctured elliptic curve XF determined by EF . These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve. We then construct “Θ±ellNF-Hodge theaters” associated to the given Θ-data. These Θ±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±ellNF-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a “Θ-link”, which relates certain Frobenioid-theoretic portions of one Θ±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 Θ-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 Θ-link] in terms of a given ring structure [associated, say, to the codomain of the Θ-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 p-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.

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Citations
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01 Jan 2016
TL;DR: The cohomology of number fields is universally compatible with any devices to read and will help you to get the most less latency time to download any of the authors' books like this one.
Abstract: Thank you for reading cohomology of number fields. Maybe you have knowledge that, people have look numerous times for their favorite novels like this cohomology of number fields, but end up in infectious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some harmful bugs inside their computer. cohomology of number fields is available in our digital library an online access to it is set as public so you can get it instantly. Our digital library hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the cohomology of number fields is universally compatible with any devices to read.

183 citations

Book
01 Jan 1999
TL;DR: In this paper, the generalized ordinary theory and the geometrization of binary-ordinary Frobenius liftings have been studied in the context of stable bundles on a curve.
Abstract: Introduction Crys-stable bundles Torally Crys-stable bundles in positive characteristic VF-patterns Construction of examples Combinatorialization at infinity of the stack of nilcurves The stack of quasi-analytic self-isogenies The generalized ordinary theory The geometrization of binary-ordinary Frobenius liftings The geometrization of spiked Frobenius liftings Representations of the fundamental group of the curve Ordinary stable bundles on a curve Bibliography Index.

47 citations

01 Oct 1995

37 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of splitting monoids of LGP-monoids, which they called interuniversal Teichmüller theory, and proposed a multiradial algorithm for this problem.
Abstract: The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logtheta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine — albeit from an extremely naive/non-expert point of view! — the foundational/settheoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.

30 citations

Journal ArticleDOI
TL;DR: In this paper, the log-theta-lattice theory was studied in the context of the construction of log-shells, i.e., slightly adjusted forms of the image of the local units at the valuation under consideration via the local p-adic logarithm.
Abstract: The present paper constitutes the third paper in a series of four papers and may be regarded as the culmination of the abstract conceptual portion of the theory developed in the series. In the present paper, we study the theory surrounding the log-theta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters. Here, we recall that Θ±ellNF-Hodge theaters were associated, in the first paper of the series, to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Each arrow of the log-theta-lattice corresponds to a certain gluing operation between the Θ±ellNF-Hodge theaters in the domain and codomain of the arrow. The horizontal arrows of the log-theta-lattice are defined as certain versions of the “Θ-link” that was constructed, in the second paper of the series, by applying the theory of HodgeArakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the l-th root of the] theta function at l-torsion points. In the present paper, we focus on the theory surrounding the log-link between Θ±ellNFHodge theaters. The log-link is obtained, roughly speaking, by applying, at each [say, for simplicity, nonarchimedean] valuation of the number field under consideration, the local p-adic logarithm. The significance of the log-link lies in the fact that it allows one to construct log-shells, i.e., roughly speaking, slightly adjusted forms of the image of the local units at the valuation under consideration via the local p-adic logarithm. The theory of log-shells was studied extensively in a previous paper by the author. The vertical arrows of the log-theta-lattice are given by the log-link. Consideration of various properties of the log-theta-lattice leads naturally to the establishment of multiradial algorithms for constructing “splitting monoids of logarithmic Gaussian procession monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. These logarithmic Gaussian procession monoids, or LGP-monoids, for short, may be thought of as the log-shell-theoretic versions of the Gaussian monoids that were studied in the second paper of the series. Finally, by applying these multiradial algorithms for splitting monoids of LGP-monoids, we obtain estimates for the log-volume of these LGP-monoids. Explicit computations of these estimates will be applied, in the fourth paper of the series, to derive various diophantine results. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI

24 citations

References
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Book
01 Jan 1999
TL;DR: In this paper, Algebraic integral integers, Riemann-Roch theory, Abstract Class Field Theory, Local Class Field theory, Global Class Field and Zeta Functions are discussed.
Abstract: I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions and L-series.

2,824 citations

Book
01 Jan 1980
TL;DR: In this article, the authors introduce complex analysis and surface topology, and the fundamental groups of complexes of a graph and a free group of graphs are formed by knots and Braids.
Abstract: 0: Introduction and Foundations 1: Complex Analysis and Surface Topology 2: Graphs and Free Groups 3: Foundations for the Fundamental Group 4: Fundamental Groups of Complexes 5: Homology Theory and Abelianization 6: Curves on Surfaces 7: Knots and Braids 8: Three-Dimensional Manifolds 9: Unsolvable Problems

564 citations

01 Jan 2016
TL;DR: The cohomology of number fields is universally compatible with any devices to read and will help you to get the most less latency time to download any of the authors' books like this one.
Abstract: Thank you for reading cohomology of number fields. Maybe you have knowledge that, people have look numerous times for their favorite novels like this cohomology of number fields, but end up in infectious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some harmful bugs inside their computer. cohomology of number fields is available in our digital library an online access to it is set as public so you can get it instantly. Our digital library hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the cohomology of number fields is universally compatible with any devices to read.

183 citations

Journal ArticleDOI
TL;DR: In this paper, the fundamental group π 1(X) is a profinite topological group which is uniquely determined (up to inner automorphisms) by the property that the category of finite, discrete sets equipped with a continuous π1(X)-action is equivalent to the finite etale coverings of X.
Abstract: Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of finite, discrete sets equipped with a continuous π1(X)-action is equivalent to the category of finite etale coverings of X. Moreover, the assignment X → π1(X) is a functor from the category of connected schemes (and morphisms of schemes) to the category of profinite topological groups and continuous outer homomorphisms (i.e., continuous homomorphisms of topological groups, where we identify any two homomorphisms that can be obtained from one another by composition with an inner automorphism).

169 citations