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Explicit estimates in inter-universal Teichmuller theory
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Mochizuki, Shinichi, Fesenko, Ivan, Hoshi, Yuichiro, Minamide, Arata and Porowski, Wojciech (2022) Explicit estimates in inter-universal Teichmuller theory. Kodai Mathematical Journal, 45 (2). pp. 175-236. doi:10.2996/kmj45201 ISSN 1881-5472.
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Official URL: https://doi.org/10.2996/kmj45201
Abstract
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
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Teichmüller spaces, Curves, Elliptic, Fermat's last theorem, Number theory | ||||||
| Journal or Publication Title: | Kodai Mathematical Journal | ||||||
| Publisher: | Tokyo Institute of Technology, Department of Mathematics | ||||||
| ISSN: | 1881-5472 | ||||||
| Official Date: | June 2022 | ||||||
| Dates: |
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| Volume: | 45 | ||||||
| Number: | 2 | ||||||
| Page Range: | pp. 175-236 | ||||||
| DOI: | 10.2996/kmj45201 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Reuse Statement (publisher, data, author rights): | This article has been accepted for publication in Kodai Mathematical Journal Vol. 45 No. 2. | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Copyright Holders: | Tokyo Institute of Technology, Department of Mathematics | ||||||
| Date of first compliant deposit: | 10 May 2022 | ||||||
| Date of first compliant Open Access: | 11 May 2022 | ||||||
| RIOXX Funder/Project Grant: |
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