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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."

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

(Point (4.) in those Comments was subsequently modified twice. ) He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. Taking into account the latest version of point (4.) from Mochizuki's Comments, here is the anticipated revised version of Theorem 1.10 (after taking $\epsilon \sim 1/\ell$ - which is essentially optimal - and not worrying about the best constants or the most general version):

Take the pair $(E,\ell)$, where $E/\mathbb{Q}$ 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 that:

• $\ell$ divides neither $N$ nor the any of the exponents in the prime factorization of $\Delta$;
• The Galois representation of $G_{\mathbb{Q}(\sqrt{-1})}$ on $E[\ell]$ has full image $\mathrm{GL}_2(\mathbb{Z}/\ell)$. [Conjecturally, this condition should only exclude a finite list of values of $\ell$, independent of $E$. Therefore, it does not appear to be an essential condition here. ]
• The $2$-adic valuation of $\Delta$ is bounded. (Assume this for simplicity in the statement below. It is not truly an essential condition either).
• $\ell = O(\log{N})$ (Such a choice, compliant with the three previous conditions, can be ensured; see Section 2 in IUTT-IV, or the paper [GenEll]).

Let $M := \prod_{p \mid N, p < \ell} p$. Then, with the corrections outlined by Mochizuki, the revised Theorem 1.10 should essentially read [and certainly imply]: $$ \frac{1}{6} \log{\Delta} < \Big( 1 + \frac{200}{\ell} \Big) \log{N} + \omega(M) \cdot \log{\log{N}} - \log{M} + O\big( \log{\log{N}} \big), $$ where $\omega(\cdot)$ denotes "number of prime factors." If we take $\ell \sim \log{N}$ and $\omega(M)$ bounded (i.e., restrict to conductors $N$ which are only divisible by a bounded number of primes $< \log{N}$), then this consequence would yield $(1/6) \log{\Delta} < \log{N} + O(\log{\log{N}})$. (Must this be true for $N$ a large enough square-free integer such that the number of primes $< \log{N}$ dividing $N$ is bounded? A reminder: in terms of the $abc$-triple, $\Delta$ is essentially $(abc)^2$, and $N = \mathrm{rad}(abc)$).

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:

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." 

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 $N = \mathrm{rad}(abc)$).