COMMENTS ON [IUTCHIV], THEOREM 1.10

Shinichi Mochizuki
October 2012

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 theABC Conjecture [i.e., [IUTchIV], Theorem A; [IUTchIV], Corollary 2.3]. That isto 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] in 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 theétale 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 thecontribution at the prime 2. In particular,
at a purely explicit level, there is no contradiction between the inequalityof [IUTchIV], Theorem 1.10, and the examples constructed in [Mss].
This was precisely my understanding when I wrote [IUTchIV].
(2.) On the other hand, it was pointed out to me by Akshay Venkatesh that theargument of [Mss] may be modiﬁed in such a way as to obtain examples for which
the contribution at the prime 2 may be bounded. This led me to reexamine the entire
theory of [IUTchI], [IUTchII], [IUTchIII], [IUTchIV] in detail. My conclusions may
be summarized as follows:(a) I continue to believe that the abstract theory of [IUTchI], [IUTchII],[IUTchIII] contains no essential errors.
(b) I continue to believe that the log-volume computations of [IUTchIV]
contain no essential errors.
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2SHINICHI MOCHIZUKI(c) On the other hand, I now see that I made a slight error in the interpretation via log-volume of the abstract theory of [IUTchI], [IUTchII],[IUTchIII], i.e., in the “bridge” between this abstract theory and the logvolume computations discussed in [IUTchIV], §1.
That is to say, at a more technical level, it appears that I made a slight error inthe deﬁnition of the constant “CΘ ” in [IUTchIII], Corollary 3.12.
(3.) The error discussed in (2.) may be explained in more detail as follows. The
essence of the abstract theory of [IUTchI], [IUTchII], [IUTchIII] lies in the computation — via anabelian geometry, the theory of Frobenioids, etc. — of an “alien”
arithmetic holomorphic structure in terms of a given initial arithmetic holomorphic
structure that is related to the “alien” structure via certain “mono-analytic” data.
This is intended to be an arithmetic analogue of the situation [i.e., in classicalcomplex Teichmüller theory] in which one considers distinct holomorphic structures
related by a single underlying real analytic structure on a topological surface. Thecurrent deﬁnition of the constant “CΘ ” amounts, in essence, to [an upper bound on]
the log-volume of the “alien” structure measured in terms of the mono-analytic[i.e., the arithmetic analogue of “underlying real analytic”] data. On the other hand,
upon further consideration, I reached the conclusion that the correct deﬁnition ofthis constant “CΘ ” is as [an upper bound on]
the log-volume of the “alien” structure measured in terms of the given
initial arithmetic holomorphic structure.
Indeed, this is in essence the content of the crucial argument given in Step (xi) of
the proof of [IUTchIII], Corollary 3.12. That is to say, in summary, my current
understanding is that
there is nothing essentially wrong with this argument/proof, but rather
that I made an error in the statement of the conclusions that one shoulddraw from this argument [i.e., in the deﬁnition of the constant “CΘ ”].
Although I am quite busy with other work, I hope to post a revised version of[IUTchIII] on my homepage [i.e., with the correct deﬁnition of the constant “CΘ ”]
in the not so distant future.
(4.) At the level of the computations of [IUTchIV], §1, the eﬀect of the change inthe deﬁnition of the constant “CΘ ” discussed in (3.) is in fact quite limited. That
is to say, in a word, there is in fact no eﬀect on the computations at archimedean
primes and at nonarchimedean primes (i.e., of the ﬁeld “K”) that are “moderatelyramiﬁed”, i.e., whose absolute ramiﬁcation index is < p − 1. At nonarchimedean
primes (i.e., of the ﬁeld “K”) that are [possibly tamely, but] not moderately ramiﬁed,
one must add a new term arising from the fact that the radius of convergence of thep-adic log/exp series is p−1/p−1 . The main contribution then occurs at the [odd!]
bad primes, i.e., where there is tame ramiﬁcation of index l, which typically ismuch larger than p. The total new contribution — say, in the case where thebase ﬁeld is the ﬁeld of rational numbers Q, the conductor of the abc-triple underCOMMENTS ON [IUTCHIV], THEOREM 1.103consideration is denoted N , and we make the assumption (which is possible in thecontext of Theorem 1.10) that l is ≤ a positive constant multiple of log(N ) — isthen roughly of the formω(N  ) · log(log(N )) − log(N  )— i.e., where “ω(−)” denotes the number of primes that divide the integer in parentheses, and we write N  for the product of p dividing N that are < l. Elementary
estimates via the prime number theorem then yield asymptotic upper bounds forthe new contribution of the form [a positive constant times]
log(N ) · (log(log(log(N ))))/log(log(N ))— i.e., which is safely out of range of the lower bound (log(N ))1/2 /log(log(N )) ofMasser’s examples. Again, although I am quite busy with other work, I hope topost a revised version of [IUTchIV] on my homepage [i.e., with the corrected versionof Theorem 1.10 and its proof] in the not so distant future.
(5.) In the context of (4.), it is of interest to note that the contribution involving
ω(N ) discussed in (4.) is [not precisely the same as, but nevertheless] strongly
reminiscent of the many reﬁnements of the ABC Conjecture considered by Baker
in his 1996 and 2004 papers on the ABC Conjecture.