transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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 $E$ be an elliptic curve over a number field $F$ with initial theta data $(\overline{F}/F, E_F, l, \underline{M}, \underline{V}, V_{mod}^{bad}, \underline{\epsilon})$. Then the following inequality holds:
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:
Claim of proof of similar statements:
Last revised on January 29, 2024 at 16:03:30. See the history of this page for a list of all contributions to it.