RIMS - 1758 INTER - UNIVERSAL TEICHMÜLLER THEORY III : CANONICAL SPLITTINGS OF THE LOG - THETA - LATTICE By Shinichi MOCHIZUKI August 2012 R ESEARCH I NSTITUTE FOR M ATHEMATICAL

In the present paper, which is the third in a series of four papers, we study the theory surrounding the log-theta-lattice, a highly noncommutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. The horizontal arrows of the log-theta-lattice are defined as certain versions of the “Θ-link” that was constructed, in the second paper of the series, by applying the theory of Hodge-Arakelovtheoretic evaluation — i.e., evaluation in the style of the scheme-theoretic HodgeArakelov 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 study the theory surrounding the log-link between Θ±ellNF-Hodge theaters. The 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 log-link lies in the fact 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 of the author. The vertical arrows of the log-theta-lattice are given by the 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 Θ±ellNF-Hodge theater related to a given Θ±ellNF-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. These estimates will be applied to verify various diophantine results in the fourth paper of the series.