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INTER-UNIVERSAL TEICHM ¨ ULLER THEOR Y IV: LOG-V OLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki August 2012 Abstract.  The pre sen t paper for ms the fou rth and fina l pa- per in a series of papers concerning  “inter-unive rsal Teichm¨ uller theory”. In the first three papers of the series, we introduced and studied the theory surround- ing the  log-theta-lattice, a  highly non-co mmutative  tw o-dimensional diagram of “miniature models of conventional scheme theory” , called Θ ±ell NF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called  initial Θ-data . This data includes an  elliptic curve  E F  over a number field  F , toget her with a  prime number  l  ≥ 5. Cons idera tion of v arious properties of the log-theta -latt ice led naturally to the establishment, in the third paper of the series, of  multiradial algorithms  for constructing  “split ting monoids of LGP- monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an  “ali en arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ ±ell NF-Hodge theater related to a given Θ ±ell NF-Hodge theater by means of a  non-ring/scheme-theoretic  hori zon tal 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 . Fina lly, we examine 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 thoug ht of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a  “typ e of mathe matical obje ct”. These foun- dational issues are closely related to the central role played in the present series of papers by various results from  absolute ana belian geometry, as well as to the idea of  glui ng together  distinc t mo dels of c onven tiona l scheme the ory , 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”. Contents: Introduction §0. Notations and Conventions §1. Log-volume Estimates §2. Diophantine Inequalities §3. Inter-universal Formalism: the Language of Species Typeset by  A M S -T E X 1

Inter-Universal Teichmuller Theory IV

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