FAQ about the proof of the abc-conjecture

Appendix: Additional discussion of Questions 11–14 Here are some additional pointers regarding the arithmetic case discussed in Questions 11–14. It will be useful to first fix a prime number p and understand the simplest case of X = Spec(Qp). If one takes two copies X1 = Spec(Qp) and X2 = Spec(Qp), there is no way of distinguishing them in the language of schemes as X1 ≃ X2 as Qp schemes. Let Cp be the completion of some fixed algebraic closure ¯Qp of Qp; let C♭ p be its tilt [Scholze, 2012]. The path which I found in my work to resolve this is as follows: let YC♭ p,Qp be the Fargues-Fontaine curve constructed in [Fargues and Fontaine, 2018] using input datum (C♭ p, Qp). Closed classical points of this curve parameterize untilts of C♭ p i.e. pairs (Qp ,→ K, K♭ ≃ C♭ p) where K is an algebraically closed perfectoid field, Qp ,→ K is an isometric embedding, and K♭ ≃ C♭ p is a given isometry between K♭ and C♭ p. One should think of the untilt datum (Qp ,→ K, K♭ ≃ C♭ p) as the p-adic analog of an arithmetic holomorphic structure or a p-adic analog of a Riemann surface equipped with a quasi- conformal mapping (precise definitions are given in [Joshi, 2021]). Given a pair of distinct points y1, y2 ∈ YC♭ p,Qp , by forgetting the additional untilt data, one obtains schemes X1 = Spec(Qp) and X2 = Spec(Qp). Therefore, in the language of untilts, one may solve the problem of distinguishing X1, X2 by working instead with y1, y2, just as one does in Classical Teichm¨uller Theory. How does this relate to Mochizuki’s Key Principle of Inter-Universality quoted in the answer to Question 31? Here is how: each y1 = (Qp ,→ K, K♭ ≃ C♭ p) provides the algebraically closed, perfectoid field K needed for computing one fiber functor for the tempered fundamental groups (and Galois groups) which Mochizuki refers to in his key principle. Notably, in my view, YC♭ p,Qp is the analog of the classical Teichm¨uller space of X = Spec(Qp). In [Joshi, 2021], [Joshi, 2022], [Joshi, 2023a], I show that this approach extends to quasi- projective varieties over p-adic fields and number fields, provides a satisfactory resolution of the key issues, keeping with Mochizuki’s Key Principle. Notably in [Joshi, 2023a], for each number field L (assumed to have no real embeddings for simplicity–this is also assumed by Mochizuki), I construct an Arithmetic Teichm¨uller Space YL, whose points y ∈ YL may be thought of as y = (number field L plus a Teichm¨uller datum for L), and in which one can assert y1̸ = y2 (for y1, y2 ∈ YL) even though both correspond to fixed number field L. In the parlance of [Mochizuki, 2021b, Remark 3.11.1, Page 580], y1 and y2 represent two distinct arithmetic holomorphic structures. The existence of YL (together with its list of properties, such as the global Frobenius morphism φ : YL → YL which is a precise version of Mochizuki’s log-Link) is the minimal requirement for formulating Mochizuki’s assertions (e.g. [Mochizuki, 2021b, Theorem 3.11, Page 573]) without self contradictions. The assertions of [Scholze and Stix, 2018] arose from the fact that [Mochizuki, 2021b] does not provide adequate mathematical language for asserting the existence of distinct arithmetic holomorphic structures i.e. of asserting y1̸ = y2, and I agree with them on this; however they asserted, incorrectly, that it was impossible to find distinct arithmetic holomorphic structures, and I demonstrate the existence of arithmetic holomorphic structures in my papers. Appendix 2: On Inter-Universality in [Mochizuki, 2021b] As I have noted earlier (Question 31), universes are not actually used in the main portion of the proof of the abc-conjecture asserted in [Mochizuki, 2021b]. The only reason to discuss them here is that conversations with Mochizuki reported in [Boyd, 2025] (and elsewhere) appears to have centered on this topic and this has generated some interest in it. I want to demonstrate in Theorem 60 (given below) that Mochizuki’s Key Principle of Inter- 17