Abstract
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 $> 1.615 \cdot 10^{14}$—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 "$1.615 \cdot 10^{14}$" 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.
Funding Statement
The second and fifth authors were partially supported by the ESPRC Programme Grant "Symmetries and Correspondences". The third author was supported by JSPS KAKENHI Grant Number 18K03239; the fourth author was supported by JSPS KAKENHI Grant Number 20K14285. This research was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University, as well as by the Center for Next Generation Geometry [a research center affiliated with the Research Institute for Mathematical Sciences].
Acknowledgment
Each of the co-authors of the present paper would like to thank the other co-authors for their valuable contributions to the theory exposed in the present paper. In particular, the co-authors [other than the first author] of the present paper wish to express their deep gratitude to the first author, i.e., the originator of inter-universal Teichmüller theory, for countless hours of valuable discussions related to his work. The authors are grateful to J. Sijsling for responding to our request to provide us with the computations that underlie Proposition 2.1. Moreover, the authors are grateful to P. Mihăilescu for producing a paper, co-authored with M. Rassias, based on his unpublished results on lower bounds for the second case of Fermat's Last Theorem and a new insight on lattices and an "inhomogeneous Siegel box principle".
Citation
Shinichi Mochizuki. Ivan Fesenko. Yuichiro Hoshi. Arata Minamide. Wojciech Porowski. "Explicit estimates in inter-universal Teichmüller theory." Kodai Math. J. 45 (2) 175 - 236, June 2022. https://doi.org/10.2996/kmj45201
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