Global Solvably Closed Anabelian Geometry

In this paper, we study the pro-&#931; anabelian geometry of hyperbolic curves, where &#931; 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-&#931; 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 &#34;Belyi maps&#34;. 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

Explicit estimates in inter-universal Teichmuller theory

In the final paper of a series of papers concerning inter-universal Teichmuller theory, Mochizuki verified various numerically non-effective versions of the Vojta, ABC, and Szpiro Conjectures over number fields. In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results, we first establish a version of the theory of etale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime “2”. We then proceed to discuss how such a modified version of the theory of etale theta functions affects inter-universal Teichmuller theory. Finally, by applying our slightly modified version of inter-universal Teichmuller theory, together with various explicit estimates concerning heights, the j-invariants of “arithmetic” elliptic curves, and the prime number theorem, we verify the numerically effective versions of Mochizuki’s results referred to above. These numerically effective versions imply effective diophantine results such as an effective version of the ABC inequality over mono-complex number fields [i.e., the rational number field or an imaginary quadratic field] and effective versions of conjectures of Szpiro. We also obtain an explicit estimate concerning “Fermat’s Last Theorem” (FLT) — i.e., to the effect that FLT holds for prime exponents > 1.615·1014 — which is sufficient, in light of a numerical result of Coppersmith, to give an alternative proof of the first case of FLT. In the second case of FLT, if one combines the techniques of the present paper with a recent estimate due to Mihailescu and Rassias, then the lower bound “1.615·1014” can be improved to “257”. This estimate, combined with a classical result of Vandiver, yields an alternative proof of the second case of FLT. In particular, the results of the present paper, combined with the results of Vandiver, Coppersmith, and Mihailescu-Rassias, yield an unconditional new alternative proof of Fermat’s Last Theore