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

Kazuhiro Onodera

出版者

Department of Mathematics, Tokyo Institute of Technology

雑誌

Kodai Mathematical Journal (ISSN:03865991)

巻号頁・発行日

vol.32, no.1, pp.77-90, 2009 (Released:2009-04-02)

参考文献数

9

被引用文献数

4

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It is well known that the Weierstrass product representation of the Barnes multiple gamma function Γr(z) can be calculated concretely. However, there has been no study on its explicit formulation. In this paper, its simple formulation is achieved. It is applicable to the Weierstrass product representation of the Vignéras multiple gamma function also. Moreover, the Weierstrass product representation of the Kurokawa multiple sine function Sr(z) is also formulated explicitly.

著者

Hidetoshi Komiya

出版者

Department of Mathematics, Tokyo Institute of Technology

雑誌

Kodai Mathematical Journal (ISSN:03865991)

巻号頁・発行日

vol.11, no.1, pp.5-7, 1988 (Released:2009-01-21)

参考文献数

4

被引用文献数

74

著者

Gou Nakamura

出版者

Department of Mathematics, Tokyo Institute of Technology

雑誌

Kodai Mathematical Journal (ISSN:03865991)

巻号頁・発行日

vol.35, no.1, pp.138-156, 2012 (Released:2012-03-30)

参考文献数

4

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Following the idea of P. Schmutz Schaller, we shall consider a parametrization of the Teichmüller space $\mathcal{T}$2 of compact Riemann surfaces of genus two. In the first part of this paper, we calculate the coordinates of 4 kinds of surface uniformized by Fuchsian groups whose fundamental regions can be the regular octagon. In the second part, we give a characterization of $\mathcal{T}$2 in R7.