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. 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.

2 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. 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.

3 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 . 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 .

4 In the first three papers, [IUTchI],[IUTchII], and [IUTchIII], of the series, we introduced and studied the THEORY sur-rounding thelog-theta-lattice[cf. 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. [IUTchIII], TheoremA]. Here, we recall that multiradial algorithms [cf.]

5 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. [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.

6 The discussion of [IUTchIII], Introduction] interms of various well-known invariants, such asdifferents, associated to amixed-characteristic nonarchimedean local field[cf. 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. Thesecomputations yield aquite strong/explicitdiophantine inequality[cf.

7 ] 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. 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.

8 (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. Then, relative to the notation of [GenEll][reviewed in the discussion preceding Corollary of the present paper], one hasaninequality of bounded discrepancy classes ht X(D) (1 + )(log-diffX+ log-condD)of functions onUX(Q) d , the function(1 + )(log-diffX+ log-condD) ht X(D)is bounded below by aconstantonUX(Q) d[cf. [GenEll], Definition ,(ii), as well as Remark , (ii), of the present paper].Thus, Theorem A asserts aninequalityconcerning thecanonical height[ , ht X(D) ], thelogarithmic different[ , log-diffX ], and thelogarithmic conduc-tor[ , log-condD ] of points of the curveUXvalued in number fields whoseextension degree overQis d.

9 In particular, the so-calledVojta Conjectureforhyperbolic curves, theABC Conjecture,andtheSzpiro Conjecturefor ellipticcurves all follow as special cases of Theorem A. We refer to [Vjt] for a detailedexposition of these , in 3, we examine albeit from an extremelynaive/non-expertpointof view! certainfoundational issuesunderlying the THEORY of the present se-ries of papers. Typically in mathematical discussions [ , by mathematicians whoare not equipped with a detailed knowledge of the THEORY of foundations !] suchas, for instance, the THEORY developed in the present series of papers! one de-fines various types of mathematical objects [ , such as groups, topologicalspaces, or schemes], together with a notion of morphisms between two partic-ular examples of a specific type of mathematical object [ , morphisms betweengroups, between topological spaces, or between schemes].

10 Such objects and mor-phisms [typically] determine acategory. On the other hand, if one restricts one sattention to such a category, then one must keep in mind the fact that the structureof the category , which consistsonly of a collection of objects and morphismssatisfying certain properties! does not include any mention of the various setsand conditions satisfied by those sets that give rise to the type of mathematicalobject under consideration. For instance, the data consisting of the underlyingset of a group, the group multiplication law on the group, and the properties sat-isfied by this group multiplication lawcannot be recovered[at least in ana priorisense!] from the structure of the categoryof groups . Put another way, althoughthe notion of a type of mathematical object may give rise to a category of suchobjects , the notion of a type of mathematical object is muchstronger inthesense that it involves much moremathematical structure than the notion of a cat-egory.