Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations

Abstract

The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the log-theta-lattice, a highly noncommutative two-dimensional diagram of “miniature models of conventional scheme theory”, called ${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial $\Theta$-data. This data includes an elliptic curve $E_F$ over a number field $F$, together with a prime number $l\ge 5$. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a ${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theater related to a given ${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine – albeit from an extremely naive/non-expert point of view! – the foundational/set-theoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.