212 research outputs found

### Global Solvably Closed Anabelian Geometry

In this paper, we study the pro-Σ anabelian geometry of hyperbolic curves, where Σ is a nonempty set of prime numbers, over Galois groups of “solvably closed extensions” of number fields — i.e., infinite extensions of number fields which have no nontrivial abelian extensions. The main results of this paper are, in essence, immediate corollaries of the following three ingredients: (a) classical results concerning the structure of Galois groups of number fields; (b) an anabelian result of Uchida concerning Galois groups of solvably closed extensions of number fields; (c) a previous result of the author concerning the pro-Σ anabelian geometry of hyperbolic curves over nonarchimedean local fields.</p

### Noncritical Belyi Maps

In the present paper, we present a slightly strengthened version of a well-known theorem of Belyi on the existence of "Belyi maps".
Roughly speaking, this strengthened version asserts that there exist Belyi maps which are unramified at [cf.Theorem 2.5] - or even near [cf.Corollary 3.2] - a prescribed finite set of points.</p

### ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to conclude that the “ABC Conjecture”, or, equivalently, the so-called “Effective Mordell Conjecture”, holds for arbitrary rational points of the projective line minus three points if and only if it holds for rational points which are in “sufficiently general position” in the sense that the following properties are satisfied: (a) the rational point under
consideration is bounded away from the three points at infinity at a given finite set of primes; (b) the Galois action on the l-power torsion points of the corresponding elliptic curve determines a surjection onto
GL2(Zl), for some prime number l which is roughly of the order of the sum of the height of the elliptic curve and the logarithm of the discriminant of the minimal field of definition of the elliptic curve, but does not divide the conductor of the elliptic curve, the rational primes
that are absolutely ramified in the minimal field of definition of the elliptic curve, or the local heights [i.e., the orders of the q-parameter at primes of [bad] multiplicative reduction] of the elliptic curve.</p

### On the essential logical structure of inter-universal Teichmüller theory in terms of logical AND “∧”/logical OR “∨” relations: Report on the occasion of the publication of the four main papers on inter-universal Teichmüller theory

The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory from the point of view of the Boolean operators --such as the logical AND “∧”logical OR “∨” operators-- of propositional calculus. This essential logical structure of inter-universal Teichmüller theory may be summarized symbolically as follows: A ∧ B = A ∧ (B₁ ∨˙ B₂ ∨˙...) ⇒ A ∧ (B₁∨˙B₂∨˙...∨˙ B́₁ ∨˙ B́₂ ∨˙...) -- where · the “∨˙” denotes the Boolean operator exclusive-OR, i.e., “XOR”; · A, B, B₁, B₂, B́₁, B́₂, denote various propositions; · the logical AND “∧'s” correspond to the Θ-link of inter-universal Teichmüller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Θ-link; · the logical XOR “∨˙'s” correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of iterates of the log-link of inter-universal Teichmüller theory, which may be thought of as a device for constructing additive log-shells. This sort of concatenation of logical AND “∧'s” and logical XOR “∨˙ 's” is reminiscent of the well-known description of the “carry-addition” operation on Teichmüller representatives of the truncated Witt ring ℤ/4ℤ in terms of Boolean addition “∨˙” and Boolean multiplication “∧” in the field F₂ and may be regarded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion, the “arithmetic intertwining” between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal Teichmüller theory. One important topic in this exposition is the issue of “redundant copies”, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal Teichmüller theory impacts-- and indeed invalidates-- the essential logical structure of inter-universal Teichmüller theory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community. The exposition of the topic of “redundant copies” makes use of many interesting elementary examples from the history of mathematics

### On the combinatorial cuspidalization of hyperbolic curves

In this paper, we continue our study of the pro-Σ fundamental groups of configuration spaces associated to a hyperbolic curve, where Σ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater-Schneps. More precisely, we show that if one restricts one’s attention to outer automorphisms of such a pro-Σ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain “fiber subgroups ” [i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces] as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from n +1 to n ≥ 1 (respectively, n ≥ 2), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if n ≥ 4. The key tool in the proo

### Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves

In this paper, we prove that arbitrary hyperbolic curves over p-adic local fields admit resolution of nonsingularities [“RNS”]. This result may be regarded as a generalization of results concerning resolution of nonsingularities obtained by A. Tamagawa and E. Lepage. Moreover, by combining our RNS result with techniques from combinatorial anabelian geometry, we prove that an absolute version of the geometrically pro-Σ Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields, where Σ denotes a set of prime numbers of cardinality ≥ 2 that contains p, holds. This settles one of the major open questions in anabelian geometry. Furthermore, we prove --again by applying RNS and combinatorial anabelian geometry-- that the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature in fact coincide. As a corollary, we conclude that the metrized Grothendieck-Teichmüller group is commensurably terminal in the Grothendieck-Teichmüller group. This settles a longstanding open question in combinatorial anabelian geometry

### Development and Application of Web-Based Interface Ground Station Software (WINGS) for the Operation of SPHERE-1 EYE

The evolution of small satellites as both technological demonstrators and operational platforms has been significant. However, these projects often face constraints like limited budgets and stringent timelines, coupled with growing mission complexities. To address these challenges, the Intelligent Space System Laboratory (ISSL) at the University of Tokyo developed Web-based INterface Ground station Software (WINGS), an open-source ground station software with a modular architecture that includes front-end, back-end, and database components, as well as a telemetry and telecommand interface (WINGS-TMTC-IF). The paper highlights the application of WINGS in the SPHERE-1 EYE mission, showcasing its capability to customize and rapidly deploy satellite operations effectively, and compares its features and advantages against other ground station software. It concludes with operational insights and the lessons learned from the SPHERE-1 EYE mission, emphasizing the critical functionalities of ground station software in supporting small satellite operations

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