nLab
Mochizuki's proof of abc

In the references (to date not formally published) listed at inter-universal Teichmüller theory, Shinichi Mochizuki has claimed in particular a proof of the abc conjecture. A few weeks after the announcement of the proof Vesselin Dimitrov observed some problems in these documents and made these public in the form of a comment to the MathOverflow forum in the thread titled “Philosophy behind Mochizuki’s work on the ABC conjecture”. This page is hopefully eventually trying to record stable insights arising from this discussion.

This page was originally created by an anonymous user. As the original post by Vesselin Dimitrov at MathOverflow is periodically revised, and this entry was created apparently without his knowledge, the contents of this entry should be treated with discretion and caution until the situation has been clarified. – Todd Trimble

The following is for the moment mostly verbatim copied from the MO discussion.

Completely rewritten. (9/26)

It seems indeed that nothing like Theorem 1.10 from Mochizuki’s IUTT-IV could hold.

Here is an infinite set of counterexamples, assuming for convenience two standard conjectures (the first being in fact a consequence of ABC), that contradict Thm. 1.10 very badly.

Assumptions:

  • A (Consequence of ABC) For all but finitely many elliptic curves over \mathbb{Q}, the conductor NN and the minimal discriminant Δ\Delta satisfy log|Δ|<(logN) 2\log{|\Delta|} \lt (\log{N})^2.

  • B (Uniform Serre Open Image conjecture) For each dd \in \mathbb{N}, there is a constant c(d)<c(d) \lt \infty such that for every number field F/F/\mathbb{Q} with [F:]d[F:\mathbb{Q}] \leq d, and every non-CM elliptic curve EE over FF, and every prime c(d)\ell \geq c(d), the Galois representation of G FG_F on E[]E[\ell] has full image GL 2(/)\mathrm{GL}_2(\mathbb{Z}/{\ell}). (In fact, it is sufficient to take the weaker version in which FF is held fixed. )

Further, as far as I can tell from the proof of Theorem 1.10 of IUTTIV, the only reason for taking FF tpd(1,E F tpd[35])F \coloneqq F_{\mathrm{tpd}}\big( \sqrt{-1}, E_{F_{\mathrm{tpd}}}[3\cdot 5] \big) — rather than simply FF tpd(1)F \coloneqq F_{\mathrm{tpd}}(\sqrt{-1}) — was to ensure that EE has semistable reduction over FF. Since I will only work in what follows with semistable elliptic curves over \mathbb{Q}, I will assume, for a mild technical convenience in the examples below, that for elliptic curves already semistable over F tpdF_{\mathrm{tpd}}, we may actually take FF tpd(1)F \coloneqq F_{\mathrm{tpd}}(\sqrt{-1}) in Theorem 1.10.

The infinite set of counterexamples. They come from Masser’s paper Masser: Note on a conjecture of Szpiro,Asterisque* 1990, as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over \mathbb{Q} whose conductor NN and minimal discriminant Δ\Delta satisfy

(1)16log|Δ|logN+logNloglogN. \frac{1}{6}\log{|\Delta|} \geq \log{N} + \frac{\sqrt{\log{N}}}{\log{\log{N}}}.

(Thus, NN in these examples may be taken arbitrarily large. ) By (A) above, taking NN big enough will ensure that

(2)log|Δ|<(logN) 2. \log{|\Delta|} \lt (\log{N})^2.

Next, the sum of the logarithms of the primes in the interval ((logN) 2,3(logN) 2)\big( (\log{N})^2, 3(\log{N})^2 \big) is 2(logN) 2+o((logN) 2)2(\log{N})^2 + o((\log{N})^2), so it is certainly >(logN) 2\gt (\log{N})^2 for N0N \gg 0 big enough. Thus, by (2), it is easy to see that the interval ((logN) 2,3(logN) 2)\big( (\log{N})^2, 3(\log{N})^2 \big) contains a prime \ell which divides neither |Δ||\Delta| nor any of the exponents α=ord p(Δ)\alpha = \mathrm{ord}_p(\Delta) in the prime factorization |Δ|=p α|\Delta| = \prod p^{\alpha} of |Δ||\Delta|.

Consider now the pair (E,)(E,\ell): it has F mod=F_{\mathrm{mod}} = \mathbb{Q}, and since EE has rational 22-torsion, F tpd=F_{\mathrm{tpd}} = \mathbb{Q} as well. Let F(1)F \coloneqq \mathbb{Q} \big( \sqrt{-1}\big). I claim that, upon taking NN big enough, the pair (E F,)(E_F,\ell) arises from an initial Θ\Theta-datum as in IUTT-I, Definition 3.1. Indeed:

  • Certainly (a), (e), (f) of IUTT-I, Def. 3.1 are satisfied (with appropriate 𝕍̲,ϵ̲\underline{\mathbb{V}}, \, \underline{\epsilon});
  • (b) of IUTT-I, Def. 3.1 is satisfied since by construction EE is semistable over \mathbb{Q};
  • (c) of IUTT-I, Def. 3.1 is satisfied, in view of (B) above and the choice of \ell, as soon as N0N \gg 0 is big enough (recall that >(logN) 2\ell \gt (\log{N})^2 by construction!), and by the observation that, for vv a place of F=(1)F = \mathbb{Q}(\sqrt{-1}), the order of the vv-adic qq-parameter of EE equals ord v(Δ)\mathrm{ord}_v (\Delta), which equals ord p(Δ)\mathrm{ord}_p(\Delta) for vp>2v \mid p \gt 2, and 2ord 2(Δ)2\cdot\mathrm{ord}_2(\Delta) for v2v \mid 2;

while 𝕍 mod bad\mathbb{V}_{\mathrm{mod}}^{\mathrm{bad}} consists of the primes dividing Δ\Delta;

  • Finally, (d) of IUTT-I, Def. 3.1 is satisfied upon excluding at most four of Masser’s examples EE. (See page 37 of IUTT-IV).

Now, take ϵ(logN) 2\epsilon \coloneqq \big( \log{N} \big)^{-2} in Theorem 1.10 of IUTT-IV; this is certainly permissible for N0N \gg 0 large enough. I claim that the conclusion of Theorem 1.10 contradicts (1) as soon as N0N \gg 0 is large enough.

For note that Mochizuki’s quantity log(𝔮)\log(\mathfrak{q}) is precisely log|Δ|\log{|\Delta|} (reference: see e.g. Szpiro’s article in the Grothendieck Festschrift, vol. 3); his log(𝔡 tpd)\log{(\mathfrak{d}^{\mathrm{tpd}})} is zero; his d modd_{\mathrm{mod}} is 11; and his log(𝔣 tpd)\log{(\mathfrak{f}^{\mathrm{tpd}})} is our logN\log{N}. By construction, our choice ϵ(logN) 2\epsilon \coloneqq \big( \log{N} \big)^{-2} then makes 1/<ϵ1/\ell \lt \epsilon and <3/ϵ\ell \lt 3/\epsilon, whence the finaly display of Theorem 1.10 would yield

16log|Δ|(1+29ϵ)logN+2log(3ϵ 8)<logN+16loglogN+32, \frac{1}{6} \log{|\Delta|} \leq (1+29\epsilon) \cdot \log{N} + 2\log{(3\epsilon^{-8})} \lt \log{N} + 16\log{\log{N}} + 32,

where we have used ϵlogN=(logN) 1<1\epsilon \log{N} = (\log{N})^{-1} \lt 1 for N>3N \gt 3, and 2log3<32\log{3} \lt 3.

The last display contradicts (1) as soon as N0N \gg 0 is big enough.

Thus Masser’s examples yield infinitely many counterexamples to Theorem 1.10 of IUTT-IV (as presently written).

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

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

(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 ϵ1/\epsilon \sim 1/\ell - which is essentially optimal - and not worrying about the best constants or the most general version):

Take the pair (E,)(E,\ell), where E/E/\mathbb{Q} is a semistable elliptic curve with (say, for the sake of simplifying) rational 22-torsion (i.e., a Frey-Hellegouarch curve) of minimal discriminant Δ\Delta and conductor NN (square-free). Assume that:

  • \ell divides neither NN nor the any of the exponents in the prime factorization of Δ\Delta;
  • The Galois representation of G (1)G_{\mathbb{Q}(\sqrt{-1})} on E[]E[\ell] has full image GL 2(/)\mathrm{GL}_2(\mathbb{Z}/\ell). (Conjecturally, this condition should only exclude a finite list of values of \ell, independent of EE. Therefore, it does not appear to be an essential condition here.)
  • The 22-adic valuation of Δ\Delta is bounded. (Assume this for simplicity in the statement below. It is not truly an essential condition either).
  • =O(logN)\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 pN,p<pM \coloneqq \prod_{p \mid N, p \lt \ell} p. Then, with the corrections outlined by Mochizuki, the revised Theorem 1.10 should essentially read (and certainly imply):

16logΔ<(1+200)logN+ω(M)loglogNlogM+O(loglogN), \frac{1}{6} \log{\Delta} \lt \Big( 1 + \frac{200}{\ell} \Big) \log{N} + \omega(M) \cdot \log{\log{N}} - \log{M} + O\big( \log{\log{N}} \big),

where ω()\omega(\cdot) denotes “number of prime factors.” If we take logN\ell \sim \log{N} and ω(M)\omega(M) bounded (i.e., restrict to conductors NN which are only divisible by a bounded number of primes <logN\lt \log{N}), then this consequence would yield (1/6)logΔ<logN+O(loglogN)(1/6) \log{\Delta} \lt \log{N} + O(\log{\log{N}}). (Must this be true for NN a large enough square-free integer such that the number of primes <logN\lt \log{N} dividing NN is bounded? A reminder: in terms of the abcabc-triple, Δ\Delta is essentially (abc) 2(abc)^2, and N=rad(abc)N = \mathrm{rad}(abc)).

A side remark: note that the inverse 1/1/\ell of the prime level from the de Rham-Etale correspondence (E ,<)E[](E^{\dagger}, \lt \ell) \leftrightarrow E[\ell] in Mochizuki’s “Hodge-Arakelov theory” ultimately figures as the ϵ\epsilon in the ABC conjecture.

(I have deleted the remainder of the 10/15 Addendum, since it is now obsolete after Mochizuki’s revised comments.)

Added on 3-13-13. Mochizuki has posted revisions of his second and third papers on Inter-Universal Teichmuller Theory. They can be found at the bottom of his webpage here.

Last revised on March 14, 2013 at 19:26:24. See the history of this page for a list of all contributions to it.