Question

Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller theory for number fields" is certainly very beautiful, and was known to Mochizuki, along with the nature of the final estimate, already in 2000. (But let us recall, as a sane reminder of just how elusive the ABC conjecture has been, Miyaoka's flawed proof: did the idea of a Bogomolov-Miyaoka-Yau type bound involving arithmetic Chern numbers in Arakelov theory not seem equally beautiful, exciting, and promising?)

In brief, the main idea behind the IUTT-series is to construct, outside the rigid confines of algebraic geometry, a subtle object simulating a rank-1, Galois-stable quotient of $E[\ell]$. Here, $E/\mathbb{Q}$ is a (pretty much) arbitrary rational elliptic curve (and this is the main point: such a Galois-stable quotient will almost never exist!); and $\ell \geq 5$ is an auxiliary prime, generic for $E$ in a very mild sense, but otherwise free to optimize until the very final estimate. This is then applied to construct, in the non-linear discretized "Hodge-Arakelov theory," a comparison isomorphism between $(E^{\dagger}, <\ell)$ and $E[\ell]$, which is free of Gaussian poles at the bad places of $E$. For this then leads to a promising Galois-theoretic "Kodaira-Spencer map," as explained in Minhyong Kim's post, hopefully leading in the familiar way to the arithmetic Szpiro inequality for this very same elliptic curve: $\log{|\Delta_{\mathrm{min}}(E)|} \leq (6+\varepsilon) \log{N_E} + O_{\varepsilon}(1)$.

Let me, however, disagree with one point from M. Kim's post. My impression is that what Mochizuki calls an "initial $\Theta$-datum" - and which is, essentially, the pair of the rational elliptic curve $E$ (or equivalently, the $abc$-triple from the ABC-conjecture!) and, until the very final estimate in Ch. 2 of the fourth paper, the prime level $\ell$ - are fixed for good throughout the entire series of IUTT-papers. The deformation flavor of "Teichmuller theory" refers to dismantling the underlying number field, and not to the elliptic curve enhancement (indeed, in Mochizuki's dictionary with his own $p$-adic Teichmuller theory, it is the number field that corresponds to a hyperbolic curve; the elliptic curve enhancement corresponds to an "indigenous bundle" over the hyperbolic curve, and invites the anabelian philosophy via the \'etale fundamental group of the once-punctured elliptic curve). All the "Hodge theaters" associated to the initial $\Theta$-datum are isomorphic to one another, and form a vastly complicated $2$-dimensional non-commutative array - the "$\mathfrak{log}-\Theta$ lattice" - of non-ring theoretic translations between one another. What Mochizuki writes on p. 10 of IUTT-I is that the theory of $\Theta$-Hodge theaters "may be regarded as a sort of solution to the problem of constructing the global quotient $E[\ell] \twoheadrightarrow Q$" [needed for the application to arithmetic Kodaira-Spencer]. He does not seem to suggest that this is done by "moving the initial $E$ to a single elliptic curve via the intermediate case of an elliptic curve in general position," as M. Kim writes. (The term "elliptic curves in general position" indeed figures in Mochizuki's fourth paper, but it has a different, not-so-essential significance that comes through his entirely self-contained paper [GenEll], and whose purely technical purpose is to reduce the general ABC conjecture to the restricted version of Szpiro's inequality for $E$, in Thm. 1.10 of IUTT-IV, coming from the estimate in IUTT-III).

In particular, in sharp contrast to the Thue-Siegel-Roth tradition of Diophantine approximations, Mochizuki's program does not seem to compare different elliptic curves / $abc$-triples, all the way through to the key estimate $$ (*) \hspace{3cm} \log{|\Delta_{\mathrm{min}}(E)|} \leq \big(6 + \varepsilon + 200/\ell\big)\log{N_E} + 12\log(\ell\varepsilon^{-7}) $$ of IUTT-IV [asserted for all primes $\ell \geq 5$ that are generic for $E$ in a rather mild sense: essentially, $\ell$ has to be prime to the degenerate places and the $q$-parameters of $E$. Also, $\varepsilon \in (0,\epsilon_0)$ is arbitrary, with $\epsilon_0$ a numerical value. ] In this sense, Mochizuki's approach - nevermind the vast technical difficulties precipitated by the non-ring theoretic simulation of a global quotient $E[\ell] \twoheadrightarrow Q$ - is entirely direct and, consequently, effective.

So what does Mochizuki actually (claim to) prove?

Start with an $abc$-triple (co-prime rational integers with $a+b+c=0$). Since the discriminant $(\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)$ of a cubic polynomial $x^3 + \cdots$ encapsulates exactly this equation, it is a profitable, traditional idea to interpret the $abc$-datum as the giving of the rational elliptic curve $E = E_{a,b,c}$ defined by the equation $y^2 = x(x-a)(x+b)$. The (apparently weaker, but virtually as powerful) ABC conjecture $abc < K_{\varepsilon}\cdot\mathrm{rad(abc)}^{3+\varepsilon}$ then translates into Szpiro's inequality: $\log{|\Delta_{\min}(E)|} \leq (6+\varepsilon)\log{N_E} + O_{\varepsilon}(1)$ between the minimal discriminant $\Delta_{\min}(E)$ and conductor $N_E$ of $E$ (which are, essentially, $(abc)^2$ and $\mathrm{rad}(abc)$). Pick the "auxiliary prime" $\ell \geq 5$ to be generic for $E$ in the sense that, essentially: (1) $\ell \nmid abc$; (2) $\ell$ does not divide the prime exponents in $abc$; (3) for $F := \mathbb{Q}( \sqrt{-1}, E[15] )$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2 (\mathbb{Z}/\ell)$. [Conjecturally, the last condition should only exclude a finite list of primes, independent of $E$!] Then Mochizuki [IUTT-IV, Thm. 1.10] claims that (*) should hold for any $\varepsilon < \epsilon_0$.

This is the essential Diophantine estimate. Anything further than that [i.e., the deduction of the full ABC conjecture in IUTT-IV, Section 2] consists of standard, and relatively straightforward reductions [such as, e.g., the use of non-critical Belyi maps] elaborated in Mochizuki's self-contained paper [GenEll]: "Arithmetic elliptic curves in general position." Mochizuki indeed writes, in his first paper, that the auxiliary prime level $\ell \geq 5$ from the Hodge-Arakelov discretized non-linear comparison isomorphisms/correspondences $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$, will be chosen in the Diophantine application to be large, roughly on the order of the height of $E$. But this comes entirely through Theorem 3.8 in [GenEll]: there, the various non-divisibility properties are ensured by simply taking $\ell$ to exceed all the primes of bad reduction / all the $q$-parameters (also, the full Galois action is ensured unconditionally). In (*), $\ell$ could be any prime satisfying the mentioned non-divisibility conditions. (This, by the way, is what I considered highly disturbing).

My apology if I have misunderstood - and misrepresented - the points from Mochizuki's papers that I have alluded to.