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.

In this paper, we first recall the definition of $\Lambda$-submanifolds. Then we prove a Bernstein type result for 2-dimensional entire graphic $\Lambda$-submanifolds in $\mathbf{R}^{4}$. This theorem can be viewed as a generalization of Zhou's Bernstein type theorem for 2-dimensional self-shrinkers in $\mathbf{R}^{4}$.

We investigate which plane curves admit rational families of quasi-toric relations. This extends previous results of Takahashi and Tokunaga in the positive case and of the author in the negative case.

We characterize finite-dimensional thick representations over $\mathbf{C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets. Moreover, using this characterization, we give the classification of thick representations over $\mathbf{C}$ of connected complex simple Lie groups.

The associated groups of quandles play an important rôle in understanding quandles themselves. In this paper, we give several explicit descriptions of structure of associated groups by introducing certain quotient groups of them.

In this paper, we study a Howe curve $C$ in positive characteristic $p \geq 3$ which is of genus 3 and is hyperelliptic. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbf{F}_{p^2}$ without taking its $\mathbf{F}_{p^2}$-form.