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Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms J. Math. Sci. Univ. Tokyo 22 (2015), 939–1156. Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms By Shinichi Mochizuki Abstract. In the present paper, which forms the third part of a three-part series on an algorithmic approach to absolute anabelian geometry, we apply the absolute anabelian technique of Belyi cuspi- dalization developed in the second part, together with certain ideas contained in an earlier paper of the author concerning the category- theoretic representation of holomorphic structures via either the topo- logical group SL2(R) or the use of “parallelograms, rectangles, and squares”, to develop a certain global formalism for certain hyperbolic orbicurves related to a once-punctured elliptic curve over a number field. This formalism allows one to construct certain canonical rigid integral structures, which we refer to as log-shells, that are obtained by applying the logarithm at various primes of a number field. Moreover, although each of these local logarithms is “far from being an isomor- phism”both in the sense that it fails to respect the ring structures involved and in the sense (cf. Frobenius morphisms in positive char- acteristic!) that it has the effect of exhibiting the “mass”represented by its domain as a “somewhat smaller collection of mass” than the “mass”represented by its codomain, this global formalism allows one to treat the logarithm operation as a global operation on a number field which satisfies the property of being an “isomomorphism up to an appropriate renormalization operation”, in a fashion that is rem- iniscent of the isomorphism induced on differentials by a Frobenius lifting, once one divides by p. More generally, if one thinks of number fields as corresponding to positive characteristic hyperbolic curves and of once-punctured elliptic curves on a number field as corresponding to nilpotent ordinary indigenous bundles on a positive characteristic hyperbolic curve, then many aspects of the theory developed in the present paper are reminiscent of (the positive characteristic portion of) p-adic Teichm¨uller theory. 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H25. Key words: Absolute anabelian geometry, mono-anabelian, core, Belyi cuspidal- ization, elliptic cuspidalization, arithmetic holomorphic structure, mono-analytic, log- Frobenius, log-shell, log-volume. 939 940 Shinichi Mochizuki Contents: Introduction §0. Notations and Conventions §1. Galois-theoretic Reconstruction Algorithms §2. Archimedean Reconstruction Algorithms §3. Nonarchimedean Log-Frobenius Compatibility §4. Archimedean Log-Frobenius Compatibility §5. Global Log-Frobenius Compatibility Appendix: Complements on Complex Multiplication Introduction §I1. Summary of Main Results §I2. Fundamental Naive Questions Concerning Anabelian Geometry §I3. Dismantling the Two Combinatorial Dimensions of a Ring §I4. Mono-anabelian Log-Frobenius Compatibility §I5. Analogy with p-adic Teichm¨uller Theory Acknowledgements §I1. Summary of Main Results Let k be a finite extension of the field Qp of p-adic numbers (for p a prime def number); k an algebraic closure of k; Gk = Gal(k/k). Then the starting point of the theory of the present paper lies in the elementary observation that although the p-adic logarithm × → logk : k k (normalized so that p → 0) is not a ring homomorphism, it does satisfy the important property of being Galois-equivariant (i.e., Gk-equivariant). def In a similar vein, if F is an algebraic closure of a number field F , GF = Gal(F/F), and k, k arise, respectively, as the completions of F , F at a nonarchimedean prime of F , then although the map logk does not extend, × in any natural way, to a map F → F (cf. Remark 5.4.1), it does extend to the “disjoint union of the logk’s at all the nonarchimedean primes of F ” in a fashion that is Galois-equivariant (i.e., GF -equivariant) with respect to the natural action of GF on the resulting disjoint unions of the various × k ⊆ k. Absolute Anabelian Geometry III 941 Contemplation of the elementary observations made above led the au- thor to the following point of view: The fundamental geometric framework in which the logarithm operation should be understood is not the ring-theoretic frame- work of scheme theory, but rather a geometric framework based solely on the abstract profinite groups Gk, GF (i.e., the Galois groups involved), i.e., a framework which satisfies the key prop- erty of being “immune” to the operation of applying the loga- rithm. Such a group-theoretic geometric framework is precisely what is furnished by the enhancement of absolute anabelian geometry — which we shall refer to as mono-anabelian geometry — that is developed in the present paper. This enhancement may be thought of as a natural outgrowth of the algorithm-based approach to absolute anabelian geometry, which forms the unifying theme (cf. the Introductions to [Mzk20], [Mzk21]) of the three-part series of which the present paper constitutes the third, and final, part. From the point of view of the present paper, certain portions of the theory and results developed in earlier papers of the present series — most notably, the theory of Belyi cuspidalizations developed in [Mzk21], §3 — are relevant to the theory of the present paper partly because of their logical necessity in the proofs, and partly because of their philosophical relevance (cf., especially, the discussion of “hidden endomorphisms” in the Introduction to [Mzk21]; the theory of [Mzk21], §2). Note that a ring may be thought of as a mathematical object that consists of “two combinatorial dimensions”, corresponding to its additive structure, which we shall denote by the symbol , and its multiplicative structure, which we shall denote by the symbol (cf. Remark 5.6.1, (i), for more details). One way to understand the failure of the logarithm to be compatible with the ring structures involved is as a manifestation of the fact that the logarithm has the effect of “tinkering with, or dismantling, this two-dimensional structure”. Such a dismantling operation cannot be understood within the framework of ring (or scheme) theory. That is to say, it may only be understood from the point of view of a geometric frame- work that “lies essentially outside”,or“is neutral with respect to”, this two-dimensional structure (cf. the illustration of Remark 5.10.2, (iii)). 942 Shinichi Mochizuki One important property of the p-adic logarithm logk discussed above is that the image O× ⊆ logk( k ) k — which is compact — may be thought of as defining a sort of canonical rigid integral structure on k. In the present paper, we shall refer to the “canonical rigid integral structures”obtained in this way as log-shells. Note × × O× × that the image logk(k )ofk via logk is, like logk( k ) (but unlike k !), compact. That is to say, the operation of applying the p-adic logarithm may be thought of as a sort of “compression” operation that exhibits the “mass” represented by its domain as a “somewhat smaller collection of mass” than the “mass” represented by its codomain. In this sense, the p-adic logarithm is reminiscent of the Frobenius morphism in positive characteristic (cf. Re- mark 3.6.2 for more details). In particular, this “compressing nature”of the p-adic logarithm may be thought of as being one that lies in sharp contrast with the nature of an ´etale morphism. This point of view is reminiscent of the discussion of the “fundamental dichotomy” between “Frobenius-like” and “´etale-like” structures in the Introduction of [Mzk16]. In the classical p-adic theory, the notion of a Frobenius lifting (cf. the theory of [Mzk1], [Mzk4]) may be thought of as forming a bridge between the two sides of this dichotomy (cf. the discussion of mono-theta environments in the Introduc- tion to [Mzk18]!) — that is to say, a Frobenius lifting is, on the one hand, literally a lifting of the Frobenius morphism in positive characteristic and, on the other hand, tends to satisfy the property of being ´etale in character- istic zero, i.e., of inducing an isomorphism on differentials, once one divides by p. In a word, the theory developed in the present paper may be summarized as follows: The thrust of the theory of the present paper lies in the devel- opment of a formalism, via the use of ring/scheme structures reconstructed via mono-anabelian geometry, in which the “dis- mantling/compressing nature” of the logarithm operation dis- cussed above (cf. the Frobenius morphism in positive character- istic) is “reorganized” in an abstract combinatorial fashion that exhibits the logarithm as a global operation on a number field which, moreover, is a sort of “isomomorphism up to an appro- Absolute Anabelian Geometry III 943 priate renormalization operation” (cf. the isomorphism induced on differentials by a Frobenius lifting, once one divides by p). One important aspect of this theory is the analogy between this theory and (the positive characteristic portion of) p-adic Teichm¨uller theory (cf. §I5 below), in which the “naive pull-back”of an indigenous bundle by Frobenius never yields a bundle isomorphic to the original indigenous bundle, but the “renormalized Frobenius pull-back” does, in certain cases, allow one to obtain an output bundle that is isomorphic to the original input bundle. At a more detailed level, the main results of the present paper may be summarized as follows: In §1, we develop the absolute anabelian algorithms that will be neces- sary in our theory. In particular, we obtain a semi-absolute group-theoretic reconstruction algorithm (cf. Theorem 1.9, Corollary 1.10) for hyperbolic orbicurves of strictly Belyi type (cf. [Mzk21], Definition 3.5) over sub-p- adic fields — i.e., such as number fields and nonarchimedean completions of number fields — that is functorial with respect to base-change of the base field. Moreover, we observe that the only “non-elementary” ingredi- ent of these algorithms is the technique of Belyi cuspidalization developed in [Mzk21], §3, which depends on the main results of [Mzk5] (cf.
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