Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
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Displaying 1-6 of 6 articles from this issue
  • Shinichi Mochizuki, Ivan Fesenko, Yuichiro Hoshi, Arata Minamide, Wojc ...
    2022 Volume 45 Issue 2 Pages 175-236
    Published: June 30, 2022
    Released on J-STAGE: July 01, 2022
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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.

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  • Huijuan Wang, Entao Zhao
    2022 Volume 45 Issue 2 Pages 237-249
    Published: June 30, 2022
    Released on J-STAGE: July 01, 2022
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    In this paper, we first recall the definition of Λ-submanifolds. Then we prove a Bernstein type result for 2-dimensional entire graphic Λ-submanifolds in R4. This theorem can be viewed as a generalization of Zhou's Bernstein type theorem for 2-dimensional self-shrinkers in R4.

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  • Remke Kloosterman
    2022 Volume 45 Issue 2 Pages 250-258
    Published: June 30, 2022
    Released on J-STAGE: July 01, 2022
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    We investigate which plane curves admit rational families of quasi-toric relations. This extends previous results of Takahashi and Tokunaga in the positive case and of the author in the negative case.

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  • Kazunori Nakamoto, Yasuhiro Omoda
    2022 Volume 45 Issue 2 Pages 259-269
    Published: June 30, 2022
    Released on J-STAGE: July 01, 2022
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    We characterize finite-dimensional thick representations over C of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets. Moreover, using this characterization, we give the classification of thick representations over C of connected complex simple Lie groups.

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  • Toshiyuki Akita, Aoi Hasegawa, Masayoshi Tanno
    2022 Volume 45 Issue 2 Pages 270-281
    Published: June 30, 2022
    Released on J-STAGE: July 01, 2022
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    The associated groups of quandles play an important rôle in understanding quandles themselves. In this paper, we give several explicit descriptions of structure of associated groups by introducing certain quotient groups of them.

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  • Ryo Ohashi
    2022 Volume 45 Issue 2 Pages 282-294
    Published: June 30, 2022
    Released on J-STAGE: July 01, 2022
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    In this paper, we study a Howe curve C in positive characteristic p ≥ 3 which is of genus 3 and is hyperelliptic. We will show that if C is superspecial, then its standard form is maximal or minimal over Fp2 without taking its Fp2-form.

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