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http://www.kurims.kyoto-u.ac.jp/~motizuki/Explicit%20estimates%20in%20IUTeich.pdf
EXPLICIT ESTIMATES IN INTER-UNIVERSAL
TEICHMULLER THEORY ¨
SHINICHI MOCHIZUKI, IVAN FESENKO, YUICHIRO HOSHI,
ARATA MINAMIDE, AND WOJCIECH POROWSKI

Abstract.
In the final paper of a series of papers concerning interuniversal Teichm¨uller 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
Teichm¨uller theory. Finally, by applying our slightly modified version
of inter-universal Teichm¨uller 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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