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

### Shinichi Mochizuki

Kyoto University, Japan

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## 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"**.

## Cite this article

Shinichi Mochizuki, Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations. Publ. Res. Inst. Math. Sci. 57 (2021), no. 1, pp. 627–723

DOI 10.4171/PRIMS/57-1-4