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 [math]\displaystyle{ c^{1-\varepsilon} }[/math] for any fixed [math]\displaystyle{ \varepsilon \gt 0 }[/math] (if a,b,c are smooth).

This shows for instance that [math]\displaystyle{ (1-\varepsilon) \log N / 3 }[/math]-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.

• Wikipedia page for the ABC conjecture
• nLab page for the ABC conjecture
• Questions about Powers of Numbers, Notices of the AMS, February 2000.
• It's As Easy As abc, Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2002.

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

Additional papers on IUT

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:

1. On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2013), Shinichi Mochizuki
2. On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2014), Shinichi Mochizuki

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:

• (HAT) The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories, Shinichi Mochizuki
• (GTKS) The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve, Shinichi Mochizuki
• (HAT-Survey-I) A Survey of the Hodge-Arakelov Theory of Elliptic Curves I, Shinichi Mochizuki
• (HAT-Survey-II) A Survey of the Hodge-Arakelov Theory of Elliptic Curves II, Shinichi Mochizuki
• (AbsTopIII) Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms, Shinichi Mochizuki, RIMS Preprint 1626 (March 2008).
• (EtTh) The Etale Theta Function and its Frobenioid-theoretic Manifestations, S. Mochizuki, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349. (See also this list of errata for the paper.)

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 [math]\displaystyle{ \ell }[/math]. In order to then deduce the true Szpiro's conjecture (which is essentially equivalent to the abc conjecture), the results from the paper

• (GenEll) Arithmetic Elliptic Curves in General Position, S. Mochizuki, Arithmetic Elliptic Curves in General Position,Math. J. Okayama Univ. 52 (2010), pp. 1-28.

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.

• The category and topos theory viewpoint is discussed at the nForum page for the abc conjecture.

Workshops

• 9.-20. March 2015: RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory: program
  • Lecture Series by Go Yamashita at Kyushu University (announcement in japanese)
  • Slides by Yuichiro Hoshi on Mono-anabelian Reconstruction of Number Fields

• 7.-11. December 2015: Clay Mathematics Institute workshop on the theory of Shinichi Mochizuki, Oxford

• Activities on the study of IUT theory of Shinichi Mochizuki

Survey articles

• Ivan Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki
• Shinichi Mochizuki, Panoramic overview of inter-universal Teichmuller theory
• Yamashita, FAQ on ‘Inter-Universality’
• Ivan Fesenko, Fukugen

Blogs

• ABC conjecture rumor, Secret Blogging Seminar, 12 June, 2012
• Mochizuki on ABC, Quomodocumque, Jordan Ellenberg, 3 Sept, 2012
• As easy as 123…, Simple City, Richard Elwes' Blog, 4 Sept, 2012
• Timothy Gowers Google+, 4 Sept, 2012
• John Baez Google+, 4 Sept 2012, see also a repost
  • John Baez Google+, 5 Sept, 2012
  • John Baez Google+, 12 Sept, 2012, by Minhyong Kim.
• Terence Tao Google+, 4 Sept, 2012
• Proof of the abc Conjecture?, Not Even Wrong, 4 Sept, 2012
• Posible demostración de la veracidad de la conjetura ABC, Gaussianos, 5 Sept, 2012.
• The abc game, bit-player, 7 Sept, 2012
• The abc Conjecture, U. Oklahoma math club, 9 Sept, 2012
• The Ax-Grothendieck Theorem According to Category Theory, The n-Category Café, 10 Sept, 2012
• touch of the galois, Oblomovka, 11 Sept, 2012
• The ABC Conjecture And Cryptography, Gödel’s Lost Letter and P=NP, 12 Sept, 2012
• Mochizuki Denial, 14 Sept 2012
• “ABC” proof opens new vistas in math, Later On, 16 Sept, 2012
• The ABC Conjecture has not been proved, Mathbabe, 14 Nov, 2012.
• in IUTeich the theta function corresponds to the gaze of the little girl into the “small house”, lieven lebruyn Google+, 27 May 2013
• MochizukiDenial, lieven lebruyn Google+, 28 May 2013
• An overview of Inter-universal Teichmüller Theory and Shinichi Mochizuki's proof of the ABC Conjecture, along with the current situation and how we can begin to understand this theory, Joseph Heavner, Quora, Aug 18 2013
• Latest on abc, Not Even Wrong, 19 Dec 2013
• Richard Elwes, Google+, 20 Dec 2013
• Peter Woit on Progress-Report 2014, 13 Jan 2015
• Alexander Kruel Google+, 8 Sept 2015
• David Roberts Google+, 8 Sept 2015
• Notes on the Oxford IUT workshop by Brian Conrad, Mathbabe, 15 Dec 2015

2013 study of Geometry of Frobenioids

• a baby Arithmetic Frobenioid, lieven lebruyn Google+, 29 May 2013
• MinuteMochizuki 2 : a quadratic arithmetic Frobenioid, lieven lebruyn Google+, 31 May 2013
• MinuteMochizuki 1, the bourbaki code, lieven's blog, 1 June 2013
• MinuteMochizuki 2, the bourbaki code, lieven's blog, 1 June 2013
• Mochizuki's menagerie of morphisms, lieven lebruyn Google+, 4 June 2013
• Mochizuki's categorical prime number sieve, lieven lebruyn Google+, 5 June 2013
• Mochizuki's Frobenioids for the Working Category Theorist, lieven lebruyn Google+, 7 June 2013
• MinuteMochizuki 3, lieven lebruyn Google+, 9 June 2013
• my problem with Mochizuki's Frobenioid1 lieven lebruyn Google+, 11 June 2013
• Mochizuki's Frobenioid reconstruction: the final bit lieven lebruyn Google+, 12 July 2013

Q & A

• What is inter-universal geometry?, Mathoverflow, 17 Oct, 2009
• Mochizuki’s proof and Siegel zeros, Mathoverflow, 4 Sept, 2012
• Philosophy behind Mochizuki's work on the ABC conjecture, Mathoverflow, 7 Sept, 2012 (see also the metapost for this question)
• Implications of proof of abc conjecture for cs theory, Theoretical Computer Science Stackexchange, 11 Sept, 2012
• Model-theoretic content of Mochizuki’s Teichmüller theory papers, Mathoverflow, 17 Sept 2012
• Groupification and perfection of a commutative monoid, Mathematics Stackexchange, 20 Sept 2012
• As of September 2014, what is the mathematical community's current understanding of Mochizuki's proof of the abc conjecture? Quora, September 2014.
• What is a Frobenioid?, Mathoverflow, 31 January 2015.
• What is an étale theta function?, Mathoverflow, 07 February 2015.

Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see this FAQ for details.

News Media

• Proof claimed for deep connection between primes, Nature News, 10 September 2012, reprinted by Scientific American
• Fiendish 'ABC proof' heralds new mathematical universe, New Scientist, 10 September 2012
• Mathematician Claims Proof of Connection between Prime Numbers, Yahoo News, 11 Sept 2012, reprinted by Huffington Post and MSNBC
• ABC Proof Could Be Mathematical Jackpot, Science, 12 Sept 2012
• A Possible Breakthrough in Explaining a Mathematical Riddle, The New York Times, 17 Sept 2012
• World's most complex mathematical theory 'cracked', The Telegraph, 19 Sept 2012, reprinted by several other news outlets
• U.-educated mathematician offers proof of pivotal number theory conjecture, The Daily Princetonian, 20 Sept 2012
• An ABC proof too tough even for mathematicians, Kevin Hartnett, 3 Nov 2012.
• The Paradox of the Proof, Caroline Chen, 10 May 2013.
• Mathematician's anger over his unread 500-page proof, Jacob Aron, 02 Jan 2015.
• Our number’s up: Machines will do maths we’ll never understand, New Scientist, 26 August 2015
• The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof, Davide Castelvecchi, 07 October 2015, Nature News
• Biggest mystery in mathematics in limbo after cryptic meeting, Davide Castelvecchi, 16 December 2015, Nature News
• Mathematicians left baffled after three-year struggle over proof, New Scientist, 16 December 2015
• Hope Rekindled for Perplexing Proof, Quanta Magazine, Kevin Hartnett, December 21, 2015
• Monumental proof to torment mathematicians for years to come, 28 July 2016, Nature News
• Mathematicians finally starting to understand epic ABC proof, New Scientist, 2 August 2016

Crowd News

• Shin Mochizuki has released his long-rumored proof of the ABC conjecture , Hacker News, 5 Sept 2012
  • Proof Claimed for Deep Connection between Prime Numbers, Hacker News, 11 Sept 212
• Possible Proof of ABC Conjecture, Slashdot, September 10, 2012
• Mathematics world abuzz with a proof of the ABC Conjecture, MetaFilter, 11 Sept 2012
• The abc conjecture, as easy as 1, 2, 3 ... or not , The Conversation, Alexandru Ghitza, 26 Nov 2012.
• A theorem in limbo shows that QED is not the last word in a mathematical proof, March 25, 2013.

See also

• inter-universal Teichmüller theory at the nLab
• Inter-universal Teichmüller theory at Wikipedia
• Frobenioid at Wikipedia