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http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf

https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf

Completely rewritten. (9/26)

Added on 10/15, and revised 10/19. Mochizuki has commented on the apparent contradiction between Masser's examples and Theorem 1.10:

He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. (He estimates January 2013 to be a reasonable period). He confirms the following anticipated revision of Theorem 1.10:

Let $E/\mathbb{Q}$ be a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then: $$ \frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + \mathrm{ord}_2(\Delta) + O\big( \log{(1/\epsilon)} \big), $$ where $\omega(\cdot)$ denotes "number of prime factors."

If we take $\epsilon := (\log{N})^{-1}$ and assume $\omega(N_{\epsilon})$ bounded, this would yield $(1/6)\log{|\Delta|} < \log{N} + O(\log{\log{N}})$, just as before. (Must this be true for $N$ a large enough square-free integer such that the number of primes $< \log{N}$ dividing $N$ is bounded? I cannot see this at the moment: the Masser and Erdos-Stewart-Tijdeman constructions are based on the pigeonhole principle, and do not seem to be able to exclude the small primes $< \log{N}$. A reminder: in terms of the $abc$-triple, $\Delta$ is essentially $(abc)^2$, and $N = \mathrm{rad}(abc)$).

Last revision: 10/20. (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. )

Completely rewritten. (9/26)

Added on 10/15, and revised 10/20. Mochizuki has commented on the apparent contradiction between Masser's examples and Theorem 1.10:

He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. (He estimates January 2013 to be a reasonable period). He confirms the following ["essentially"] anticipated revision of Theorem 1.10:

Let $E/\mathbb{Q}$ be a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then: $$ \frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + O\big( \log{(1/\epsilon)} \big) $$ $$ < \log{N} + \Big( \epsilon \log{N} + \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big) + o\Big( \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big), $$ where $\omega(\cdot)$ denotes "number of prime factors." The second estimate comes from the prime number theorem in the form $\pi(t) = t/\log{t} + t/(\log{t})^2 + o\big( t/(\log{t})^2 \big)$, applied to $t := \epsilon^{-1}$, and is sharp if you restrict $\epsilon$ to the range $\epsilon^{-1} < (\log{N})^{\xi}$ with $\xi < 1$, as there nothing prevents $N$ from being divisible by all primes $p < (\log{N})^{\xi}$. In particular, as the Erdos-Stewart-Tijdeman-Masser construction is based on the pigeonhole principle, which cannot preclude that $N$ be divisible by all the primes $< (\log{N})^{2/3}$, the second estimate could very well be sharp in all the Masser examples. As it is easily seen that the bracketed term exceeds the range $\sqrt{\log{N}}/(\log{\log{N}})$ of Masser's examples, this has the implication that

the Erdos-Stewart-Tijdeman-Masser method cannot disprove Mochizuki's revised inequality,

which therefore seems reasonable.

On the other hand, if we take $\epsilon := (\log{N})^{-1}$ and assume $\omega(N_{\epsilon})$ bounded, this would yield $(1/6)\log{|\Delta|} < \log{N} + O(\log{\log{N}})$, just as before. (Thus, Mochizuki predicts that this last bound must hold for $N$ a large enough square-free integer such that the number of primes $< \log{N}$ dividing $N$ is bounded. I cannot see evidence neither for nor against this at the moment: again, the Masser and Erdos-Stewart-Tijdeman constructions are based on the pigeonhole principle, and do not seem to be able to exclude the small primes $< \log{N}$. So here we have an open problem by which one could probe Mochizuki's revised inequality. A reminder: in terms of the $abc$-triple, $\Delta$ is essentially $(abc)^2$, and $N = \mathrm{rad}(abc)$).

Take $E/\mathbb{Q}$ is a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). Assume, for simplicity of the statement, that the $2$-adic valuation of $\Delta$ is bounded. For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then: $$ \frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + O\big( \log{(1/\epsilon)} \big), $$ where $\omega(\cdot)$ denotes "number of prime factors."

Let $E/\mathbb{Q}$ be a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then: $$ \frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + \mathrm{ord}_2(\Delta) + O\big( \log{(1/\epsilon)} \big), $$ where $\omega(\cdot)$ denotes "number of prime factors."