Difference between revisions of "ABC conjecture"

The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed $c^{1-\varepsilon}$ for any fixed $\varepsilon \gt 0$ (if a,b,c are smooth).

This shows for instance that $(1-\varepsilon) \log N / 3$-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the finding primes project.

A probabilistic heuristic justification for the ABC conjecture can be found at this blog post.

Mochizuki's proof

Papers

Mochizuki's claimed proof of the abc conjecture is conducted primarily through the following series of papers:

1. (IUTT-I) Inter-universal Teichmuller Theory I: Construction of Hodge Theaters, Shinichi Mochizuki
2. (IUTT-II) Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation, Shinichi Mochizuki
3. (IUTT-III) Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice, Shinichi Mochizuki
4. (IUTT-IV) Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations, Shinichi Mochizuki

1. A Panoramic Overview of Inter-universal Teichmuller Theory, Shinichi Mochizuki
2. Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of Inter-universal Teichmuller Theory, Shinichi Mochizuki: "Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint (perhaps even a sort of Rosetta stone!), for understanding substantial portions of inter-universal Teichmüller theory."
3. The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory.

Progress reports:

See also these earlier slides of Mochizuki on inter-universal Teichmuller theory. The answers to this MathOverflow post (and in particular Minhyong Kim's answer) describe the philosophy behind Mochizuki's proof strategy. Go Yamashita has a short FAQ on inter-universality, which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.

The argument also relies heavily on Mochizuki's previous work on the Hodge-Arakelov theory of elliptic curves, including the following references:

Anyone seeking to get a thorough "bottom-up" understanding of Mochizuki's argument will probably be best advised to start with these latter papers first. The papers (AbsTopIII), (EtTh) are directly cited heavily by the IUTT series of papers; the earlier papers (HAT), (GTKS) cover thematically related material but serve more as inspiration than as a source of mathematical results in the IUTT series.

The theory of (IUTT I-IV) is used to establish a Szpiro-type inequality, which is similar to Szpiro's conjecture but with an additional genericity hypothesis on a certain parameter $\ell$. In order to then deduce the true Szpiro's conjecture (which is essentially equivalent to the abc conjecture), the results from the paper

are used. (Note that the published version of this paper requires some small corrections, listed here.) See this MathOverflow post of Vesselin Dimitrov for more discussion.

Here are the remainder of Shinichi Mochizuki's papers, and here is the Wikipedia page for Shinichi Mochizuki.

Specific topics

• The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see this blog comment. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
• There is some discussion at this MathOverflow post as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc. In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture. Mochizuki has written comments in October 2012 to say that he hopes to post a revised version of Theorem 1.10, which were revised in 2013.
• The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to this blog post.

Q & A

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