# Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice

### Shinichi Mochizuki

Kyoto University, Japan

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## Abstract

The present paper constitutes the third paper in a series of four papers and may be regarded as the *culmination* of the *abstract conceptual* portion of the theory developed in the series. In the present paper, we study 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*. Here, we recall that $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theaters were associated, in the first paper of the series, to certain data, called *initial $\Theta$-data*, that includes an *elliptic curve $E_F$* over a *number field* $F$, together with a *prime number* $l\ge 5$. Each *arrow* of the log-theta-lattice corresponds to a certain *gluing operation* between the $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theaters in the domain and codomain of the arrow. The **horizontal arrows** of the log-theta-lattice are defined as certain versions of the *"$\Theta$-link"* that was constructed, in the second paper of the series, by applying the theory of *Hodge–Arakelov-theoretic evaluation* – i.e., evaluation in the style of the **scheme-theoretic Hodge–Arakelov theory** established by the author in previous papers – of the [reciprocal of the $l$-th root of the] **theta function** at **$l$-torsion points**. In the present paper, we focus on the theory surrounding the **$\mathfrak{log}$-link** between $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theaters. The $\mathfrak{log}$-link is obtained, roughly speaking, by applying, at each [say, for simplicity, nonarchimedean] valuation of the number field under consideration, the *local $p$-adic logarithm*. The significance of the $\mathfrak{log}$-link lies in the fact that it allows one to construct **log-shells**, i.e., roughly speaking, slightly adjusted forms of the image of the local units at the valuation under consideration via the local $p$-adic logarithm. The theory of log-shells was studied extensively in a previous paper by the author. The **vertical arrows** of the log-theta-lattice are given by the $\mathfrak{log}$-link. Consideration of various properties of the log-theta-lattice leads naturally to the establishment of **multiradial algorithms** for constructing **"splitting monoids of logarithmic Gaussian procession 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. These logarithmic Gaussian procession monoids, or **LGP-monoids**, for short, may be thought of as the log-shell-theoretic versions of the *Gaussian monoids* that were studied in the second paper of the series. Finally, by applying these multiradial algorithms for splitting monoids of LGP-monoids, we obtain **estimates** for the **log-volume** of these LGP-monoids. Explicit computations of these estimates will be applied, in the fourth paper of the series, to derive various *diophantine results*.

## Cite this article

Shinichi Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice. Publ. Res. Inst. Math. Sci. 57 (2021), no. 1, pp. 403–626

DOI 10.4171/PRIMS/57-1-3