nLab Mochizuki's corollary 3.12

Idea

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.

Statement

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:

$$-\underline{\widehat{deg}}(P_q)\leq\underline{\widehat{deg}}_{lgp}(P_{hull(U_\Theta)})$$

TODO: details

Consequences

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

Related concepts

• Szpiro's conjecture

• initial theta data

• inter-universal Teichmüller theory

References

• 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)

Claim of proof of similar statements:

• Kirti Joshi, Construction of Arithmetic Teichmuller Spaces III: A Rosetta Stone and a proof of Mochizuki’s Corollary 3.12 [arXiv:2401.13508]