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In the following, we discuss a minor error in the theory of [IUTchIII], [IUTchIV]
concerning the precise content of the “? portion” of the ABC Conjecture. This error
is easily repaired and, moreover, has no effect on the conclusion constituted by the
ABC Conjecture [i.e., [IUTchIV], Theorem A; [IUTchIV], Corollary 2.3]. That is
to say, it only concerns the somewhat subtle content of the “? term” that appears
in these results.
(1.) In late September 2012, Vesselin Dimitrov and Akshay Venkatesh pointed out
to me, in e-mails, the possibility that the inequality of [IUTchIV], Theorem 1.10,
contradicts the examples constructed in [Mss]. In fact, I had considered this issue
when I wrote [IUTchIV] ? cf. the discussion of [IUTchIV], Remark 2.3.2, (ii).
At the time I wrote [IUTchIV], I had not studied the proof given in [Mss] detail.
However, the construction given in [Mss] is performed in such a way that there
is no apparent way to bound the contribution at the prime 2. Since the theory of
[IUTchI], [IUTchII], [IUTchIII], depends, in an essential way, on the theory of the
´etale theta function developed in [EtTh], which breaks down in an essential way
at the prime 2, the bound given in [IUTchIV], Theorem 1.10, does not involve the
contribution at the prime 2. In particular,
at a purely explicit level, there is no contradiction between the inequality
of [IUTchIV], Theorem 1.10, and the examples constructed in [Mss].
This was precisely my understanding when I wrote [IUTchIV].