Yamashita Go. A Proof of the ABC Conjecture after Mochizuki

Kyoto, Japan: Kyoto University, 2019. — 409 p.We give a survey of S. Mochizuki’s ingenious inter-universal Teichmüller theory and explain how it gives rise to Diophantine inequalities. The exposition was designed to be as self-contained as possible.
In fact, the main results of the preparatory papers [AbsTopIII], [EtTh], etc. are also obtained, to a substantial degree, as consequences of numerous constructions that are not so difficult. On the other hand, the discovery of the ideas and insights that underlie these constructions may be regarded as highly nontrivial in content. Examples of such ideas and insights include the "hidden endomorphisms" that play a central role in the mono-anabelian reconstruction algorithms of Section 3.2, the notions of arithmetically holomorphic structure and mono-analytic structure (cf. Section 3.5), and the distinction between étale-like and Frobenius-like objects (cf. Section 4.3). Thus, in summary, it seems to the author that, if one ignores the delicate considerations that occur in the course of interpreting and combining the main results of the preparatory papers, together with the ideas and insights that underlie the theory of these preparatory papers, then, in some sense, the only nontrivial mathematical ingredient in inter-universal Teichmüller theory is the classical result [pGC], which was already known in the last century!
A more technical introduction to the mathematical content of the main ideas of inter-universal Teichmüller theory may be found in Appendix A and the discussion at the beginning of Section 13.