Mochizuki's corollary 3.12




Mochizuki’s corollary 3.12 or Mochizuki’s inequality is a conjecture by Shinichi Mochizuki in number theory and algebraic geometry that first appeared in his attemped proof of the abc conjecture from inter-universal Teichmüller theory. Mochizuki’s proof of Mochizuki’s corollary 3.12 from inter-universal Teichmüller theory was shown to be incorrect by Peter Scholze and Jakob Stix.


Let EE be an elliptic curve over a number field FF with initial theta data (F¯/F,E F,l,M̲,V̲,V mod bad,ϵ̲)(\overline{F}/F, E_F, l, \underline{M}, \underline{V}, V_{mod}^{bad}, \underline{\epsilon}). Then the following inequality holds:

deg^̲(P q)deg^̲ lgp(P hull(U Θ)) -\underline{\widehat{deg}}(P_q)\leq\underline{\widehat{deg}}_{lgp}(P_{hull(U_\Theta)})

TODO: details


If Mochizuki’s corollary 3.12 happens to be true, then Szpiro's conjecture and the abc conjecture follow as a consequence.


  • Shinichi Mochizuki, Inter-universal Teichmüller theory III, Canonical splittings of the Log-theta-lattice (2012) (pdf)

  • Peter Scholze, Jakob Stix, Why abc is still a conjecture, (pdf)

  • Taylor Dupuy, Anton Hilado?, The Statement of Mochizuki’s Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies, (arxiv:2004.13228)

  • Taylor Dupuy, Anton Hilado, Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki’s Corollary 3.12, (arxiv:2004.13108)

Lectures on Mochizuki’s inequality:

  • Taylor Dupuy, “A User’s Guide to Mochizuki’s Inequality”, Spring 2019, University of Tennesse Knoxville, Barrett Lectures (slides)

Last revised on June 21, 2021 at 16:39:18. See the history of this page for a list of all contributions to it.