Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation
04 Mar 2021-Publications of The Research Institute for Mathematical Sciences (European Mathematical Society - EMS - Publishing House GmbH)-Vol. 57, Iss: 1, pp 209-401
TL;DR: The second in a series of four papers as mentioned in this paper studies the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation of the theta function at l-torsion points.
Abstract: In the present paper, which is the second in a series of four papers, we study theKummer theory surrounding the Hodge-Arakelov-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 lth root of the] theta function at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point], for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated 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. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th root of the] theta function at primes of bad reduction of EF in one Θ ±ellNF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in another Θ±ellNF-Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this “Θ-link”. One such new version is the Θ ×μ gaulink, which is similar to the Θ-link, but involves the theta values at l-torsion points, rather than the theta function itself. One important aspect of the constructions that underlie the Θ ×μ gau-link is the study of multiradiality properties, i.e., properties of the “arithmetic holomorphic structure” — or, more concretely, the ring/scheme structure — arising from one Θ±ellNF-Hodge theater that may be formulated in such a way as to make sense from the point of the arithmetic holomorphic structure of another Θ±ellNF-Hodge theater which is related to the original Θ±ellNF-Hodge theater by means of the [non-scheme-theoretic!] Θ ×μ gau-link. For instance, certain of the various rigidity properties of the étale theta function studied in an earlier paper by the author may be intepreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the Θ ×μ gau-link is the study of “conjugate synchronization” via the F ± l -symmetry of a Θ ±ellNF-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms — which are free of any conjugacy indeterminacies! — between copies of local absolute Galois groups at the various l-torsion points at which the theta function is evaluated. Conjugate synchronization plays an important role in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study of coricity properties of [i.e., the study of objects left invariant by] the Θ ×μ gau-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia groups. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI
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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.
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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
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.
32 citations
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”.
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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.
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TL;DR: The first part of a three-part series on absolute anabelian geometry can be found in this paper, where the authors consider the problem of computing the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field.
Abstract: This paper forms the first part of a three-part series in which we treat various topics in absolute anabelian geometry from the point of view of developing abstract algorithms ,o r"software", that may be applied to abstract profinite groups that "just happen" to arise as (quotients of) ´ fundamental groups from algebraic geometry. One central theme of the present paper is the issue of understand- ing the gap between relative, "semi-absolute", and absolute anabelian geometry. We begin by studying various abstract combinatorial prop- erties of profinite groups that typically arise as absolute Galois groups or arithmetic/geometric fundamental groups in the anabelian geome- try of quite general varieties in arbitrary dimension over number fields, mixed-characteristic local fields, or finite fields. These considerations, combined with the classical theory of Albanese varieties, allow us to derive an absolute anabelian algorithm for constructing the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field in the case of quite general varieties of arbitrary dimension. Next, we take a more detailed look at certain p-adic Hodge- theoretic aspects of the absolute Galois groups of mixed-characteristic local fields. This allows us, for instance, to derive, from a certain result communicated orally to the author by A. Tamagawa, a "semi- absolute" Hom-version — whose absolute analogue is false! — of the anabelian conjecture for hyperbolic curves over mixed-characteristic lo- cal fields. Finally, we generalize to the case of varieties of arbitrary dimension over arbitrary sub-p-adic fields certain techniques devel- oped by the author in previous papers over mixed-characteristic local fields for applying relative anabelian results to obtain "semi-absolute" group-theoretic contructions of thefundamental group of one hy- perbolic curve from thefundamental group of another closely related hyperbolic curve.
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TL;DR: In this paper, the authors develop the theory ofcuspidalizations of the etale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field.
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