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

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.