INTER-UNIVERSAL TEICHMULLER THEORY IV:¨ …

1 INTER-UNIVERSAL TEICHM ULLER THEORY IV:LOG-VOLUME COMPUTATIONS ANDSET- theoretic FOUNDATIONSS hinichi MochizukiDecember present paper forms the fourth and final paper in a seriesof papers concerning INTER-UNIVERSAL Teichm uller THEORY . In the first threepapers of the series, we introduced and studied the THEORY surrounding thelog-theta-lattice,ahighly non-commutativetwo-dimensional diagram of miniaturemodels of conventional scheme THEORY , called ellNF-Hodge theaters, that wereassociated, in the first paper of the series, to certain data, calledinitial data includes anelliptic curveEFover anumber fieldF, together with aprime numberl 5.

2 Consideration of various properties of the log-theta-latticeled naturally to the establishment, in the third paper of the series, ofmultiradialalgorithmsfor constructing splitting monoids of LGP-monoids . Here, werecall that multiradial algorithms are algorithms that make sense from the pointof view of an alien arithmetic holomorphic structure , , the ring/schemestructure of a ellNF-Hodge theater related to a given ellNF-Hodge theater bymeans of anon-ring/scheme-theoretichorizontal arrow of the log-theta-lattice. Inthe present paper, estimates arising from these multiradial algorithms for splittingmonoids of LGP-monoids are applied to verify variousdiophantine resultswhichimply, for instance, the so-calledVojta Conjecturefor hyperbolic curves, theABCC onjecture, and theSzpiro Conjecturefor elliptic curves.

3 Finally, we examine albeit from an extremelynaive/non-expertpoint of view! thefoundational/set-theoreticissues surrounding theverticalandhorizontal arrowsof the log-theta-latticeby introducing and studying the basic properties of the notion of a species ,whichmay be thought of as a sort of formalization, via set- theoretic formulas, of the intuitivenotion of a type of mathematical object . These foundational issues are closelyrelated to the central role played in the present series of papers by various resultsfromabsolute anabelian geometry,aswellastotheideaofgluing togetherdistinct models of conventional scheme THEORY , , in a fashion that lies outside theframework of conventional scheme THEORY .

4 Moreover, it is precisely these foundationalissues surrounding the vertical and horizontal arrows of the log-theta-lattice that lednaturally 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 SpeciesTypeset byAMS-TEX12 SHINICHI MOCHIZUKII ntroductionThe present paper forms the fourth and final paper in a series of papers concern-ing INTER-UNIVERSAL Teichm uller THEORY . In the first three papers, [IUTchI],[IUTchII], and [IUTchIII], of the series, we introduced and studied the THEORY sur-rounding thelog-theta-lattice[cf.]

5 The discussion of [IUTchIII], Introduction],ahighly non-commutativetwo-dimensional diagram of miniature models of con-ventional scheme THEORY , called ellNF-Hodge theaters, that were associated, inthe first paper [IUTchI] of the series, to certain data, calledinitial includes anelliptic curveEFover anumber fieldF, together with aprimenumberl 5[cf. [IUTchI], I1]. Consideration of various properties of the log-theta-lattice leads naturally to the establishment ofmultiradial algorithmsforconstructing splitting monoids of LGP-monoids [cf.

6 [IUTchIII], TheoremA]. Here, we recall that multiradial algorithms [cf. the discussion of the Intro-ductions to [IUTchII], [IUTchIII]] are algorithms that make sense from the pointof view of an alien arithmetic holomorphic structure , , the ring/schemestructure of a ellNF-Hodge theater related to a given ellNF-Hodge theater bymeans of anon-ring/scheme-theoretichorizontal arrow of the log-theta-lattice. Inthe final portion of [IUTchIII], by applying these multiradial algorithms for split-ting monoids of LGP-monoids, we obtainedestimatesfor thelog-volumeof theseLGP-monoids [cf.

7 [IUTchIII], Theorem B]. In the present paper, these estimateswill be applied to verify variousdiophantine 1 of the present paper, we start by discussing variouselementary estimatesfor thelog-volumeof varioustensor productsof the modules obtained by applyingthep-adic logarithmto thelocal units , in the terminology of [IUTchIII], tensor packets of log-shells [cf. the discussion of [IUTchIII], Introduction] interms of various well-known invariants, such asdifferents, associated to amixed-characteristic nonarchimedean local field[cf.

8 Propositions , , , ]. Wethen discuss similar but technically much simpler! log-volume estimates inthe case ofcomplex archimedean local fields[cf. Proposition ]. After review-ing a certain classical estimate concerning thedistribution of prime numbers[ ], as well as some elementary general nonsense concerningweightedaverages[cf. Proposition ] and well-known elementary facts concerningellipticcurves[cf. Proposition ], we then proceed tocompute explicitly,inmoreelemen-tary language, the quantity that was estimated in [IUTchIII], Theorem B.

9 Thesecomputations yield aquite strong/explicitdiophantine inequality[cf. ] concerning elliptic curves that are in sufficiently general position ,sothat one may apply the general THEORY developed in the first three papers of 2 of the present paper, after reviewing another classical estimate concern-ing thedistribution of prime numbers[cf. Proposition , (ii)], we then proceedto apply the THEORY of [GenEll] toreducevarious diophantine results concerninganarbitraryelliptic curve over a number fieldto results of the type obtained inTheorem concerning elliptic curves that are in sufficiently general posi-tion [cf.]

10 Corollary ]. This reduction allows us to derive the following result[cf. Corollary ], which constitutes themain applicationof the inter -universalTeichm uller THEORY developed in the present series of TEICHM ULLER THEORY IV3 Theorem A. (Diophantine Inequalities)LetXbe a smooth, proper, geomet-rically connected curve over a number field;D Xa reduced divisor;UXdef=X\D;da positive integer; R>0a positive real number. Write Xfor the canon-ical sheaf onX. Suppose thatUXis ahyperbolic curve, , that the degreeofthelinebundle X(D)ispositive.