Explicit Estimates in Inter-universal Teichmüller Theory

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 modifed 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 an effective version of a conjecture 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 to give an alternative proof of the first case of Fermat's Last Theorem.