著者

Shinichi Mochizuki Ivan Fesenko Yuichiro Hoshi Arata Minamide Wojciech Porowski

出版者

Department of Mathematics, Tokyo Institute of Technology

雑誌

Kodai Mathematical Journal (ISSN:03865991)

巻号頁・発行日

vol.45, no.2, pp.175-236, 2022-06-30 (Released:2022-07-01)

参考文献数

28

被引用文献数

1

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In the final paper of a series of papers concerning inter-universal Teichmüller 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 étale 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 étale theta functions affects inter-universal Teichmüller theory. Finally, by applying our slightly modified version of inter-universal Teichmüller 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 Mihăilescu 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 Mihăilescu-Rassias, yield an unconditional new alternative proof of Fermat's Last Theorem.

著者

Yoshiya Maegawa Shinichi Mochizuki Noriko Miyamoto Yusuke Sanada Kazuo Sakurai

出版者

FCCA(Forum: Carbohydrates Coming of Age)

雑誌

Trends in Glycoscience and Glycotechnology (ISSN:09157352)

巻号頁・発行日

vol.27, no.153, pp.13-29, 2015-01-25 (Released:2015-01-23)

参考文献数

64

被引用文献数

1

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β-1,3-D-グルカンの一種であるシゾフィラン(SPG)はホモ配列のオリゴデオキシヌクレオチド(ODN)と水素結合や疎水性相互作用によってODN/SPG複合体を形成する。また、マクロファージや樹状細胞などの抗原提示細胞上には、β-1,3-D-グルカンの受容体であるデクチン-1が発現している。そのため、この複合体を用いることで、アンチセンスODN(AS-ODN)や非メチル化CpG配列を持ったODN(CpG-ODN)を抗原提示細胞に特異的に送達することが可能になると考えられる。実際、AS-ODN/SPG複合体をリポ多糖誘導型マウス肝炎モデルに投与したとき、炎症を抑えることができた。また、カニクイザルにCpG-ODN/SPG複合体をインフルエンザワクチンのアジュバントとして投与したとき、高い抗体価を促すことができた。以上より、SPGは特に抗原提示細胞を標的にした薬物送達システムのキャリアとして有用であると考えられる。

著者

Shinichi MOCHIZUKI

出版者

Faculty of Mathematics, Kyushu University

雑誌

Kyushu Journal of Mathematics (ISSN:13406116)

巻号頁・発行日

vol.62, no.2, pp.293-400, 2008 (Released:2008-09-09)

参考文献数

21

被引用文献数

2 2

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We develop the theory of Frobenioids, which may be regarded as a category-theoretic abstraction of the theory of divisors and line bundles on models of finite separable extensions of a given function field or number field. This sort of abstraction is analogous to the role of Galois categories in Galo is theory or monoids in the geometry of log schemes. This abstract category-theoretic framework preserves many o f the important features of the classical theory of divisors and line bundles on models of finite separable extensions of a function field or number field such as the global degree of an arithmetic line bundle over a number field, but also exhibits interesting new phenomena, such as a ‘Frobenius endomorphism’ of the Frobenioid associated to a number field.

著者

SHINICHI MOCHIZUKI

出版者

Mathematical Institute, Tohoku University

雑誌

Tohoku Mathematical Journal, Second Series (ISSN:00408735)

巻号頁・発行日

vol.59, no.3, pp.455-479, 2007-09-30 (Released:2010-10-21)

参考文献数

22

被引用文献数

13

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We study the “combinatorial anabelian geometry” that governs the relationship between the dual semi-graph of a pointed stable curve and various associated profinite fundamental groups of the pointed stable curve. Although many results of this type have been obtained previously in various particular situations of interest under unnecessarily strong hypotheses, the goal of the present paper is to step back from such “typical situations of interest” and instead to consider this topic in the abstract—a point of view which allows one to prove results of this type in much greater generality under very weak hypotheses.

著者

Shinichi Mochizuki

出版者

Research Institute forMathematical Sciences

雑誌

Publications of the Research Institute for Mathematical Sciences (ISSN:00345318)

巻号頁・発行日

vol.31, no.3, pp.355-441, 1995 (Released:2009-04-24)

参考文献数

33

被引用文献数

41

著者

Shinichi Mochizuki

出版者

Research Institute forMathematical Sciences

雑誌

Publications of the Research Institute for Mathematical Sciences (ISSN:00345318)

巻号頁・発行日

vol.40, no.3, pp.819-881, 2004 (Released:2009-04-24)

参考文献数

30

被引用文献数

5

著者

Shinichi MOCHIZUKI

出版者

九州大学大学院数理学研究院

雑誌

Kyushu Journal of Mathematics (ISSN:13406116)

巻号頁・発行日

vol.62, no.2, pp.401-460, 2008 (Released:2008-09-09)

参考文献数

8

被引用文献数

1 1

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We develop the theory of Frobenioids associated to non-archimedean (mixed-characteristic) and archimedean local fields. Inparticular, we show that the resulting Frobenioids satisfy the properties necessary to apply the main results of the general theory of Frobenioids. Moreover, we show that the reciprocity map in the non-archim edean case, as well as a certain archimedean analogue of this reciprocity map, admit natural Frobenioid-theoretic translations, which are, moreover, purely category-theoretic, to a substantial extent(i.e., except for the extent to which this category-theoreticity is obstructed by certain ‘Frobenius endomorphisms’ of the relevant Frobenioids). Finally, we show that certain Frobenioids which naturally encode the global arithmetic of a number field may be ‘grafted’ (i.e., glued) onto the Frobenioids associated to non-archimedean and archimedean primes of the number field to obtain ‘ poly-Frobenioids’. These poly-Frobenioids encode, in a purely category-theoretic fashion, most of the important aspects of the classical framework of the arithmetic geometry of number fields.