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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 Theorem

In this article, we announce our results on the Grothendieck-Teichmüller group GT obtained in the joint work with Yuichiro Hoshi and Shinichi Mochizuki. Let n ≥ 2 be a positive integer and k an algebraically closed field of characteristic zero. Write IIn for the étale fundamental group of the n-th configuration space of P1 k/{0, 1, 1∞} and Sn+3 for the symmetric group on n + 3 letters. Then our main theorem asserts that we have a direct product decomposition Out(IIn) = GT × Sn+3. The detail of the arguments appearing in this article may be found in [2]., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In the present paper, the author introduces results concerning the indecomposability of various profinte groups related to hyperbolic curves. For instance, we discuss the indecomposability of the étale fundamental group of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero, and the pro‐l Grothendieck - Teichmüller group. The details of the results and their proofs are given in [M]., 本稿では, 双曲的曲線に関連した様々な副有限群の非分解性についての結果を紹介する. 例えば, 標数 0 の代数閉体上の双曲的曲線に付随した配置空間のエタール基本群や副l グロタンディーク・タイヒミューラー群の非分解性について議論する. 結果や証明の詳細は[M]で与えられている., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this paper, we prove that the etale fundamental group of a hyperbolic curve over an arithmetic field [e.g., a finite extension field of Q or Qp] or an algebraically closed field is indecomposable [i.e., cannot be decomposed into the direct product of nontrivial profinite groups]. Moreover, in the case of characteristic zero, we also prove that the etale fundamental group of the configuration space of a curve of the above type is indecomposable. Finally, we consider the topic of indecomposability in the context of the comparison of the absolute Galois group of Q with the Grothendieck-Teichmuller group GT and pose the question: Is GT indecomposable? We give an affirmative answer to a pro-l version of this question