22/07/03 22:14:40.53 ufzWvOVH.net
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Kodai Mathematical Journal
J-STAGEトップ/Kodai Mathematical Journal/45 巻 (2022) 2 号/書誌
Explicit estimates in inter-universal Teichmuller theory
Shinichi Mochizuki, Ivan Fesenko, Yuichiro Hoshi, Arata Minamide, Wojciech Porowski
2022 年 45 巻 2 号 p. 175-236
抄録
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.
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