RIMS-1854 The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmu¨ller Theory By Shinichi MOCHIZUKI July 2016 R I M S ESEARCH NSTITUTE FOR ATHEMATICAL CIENCES KYOTO UNIVERSITY, Kyoto, Japan THE MATHEMATICS OF MUTUALLY ALIEN COPIES: FROM GAUSSIAN INTEGRALS TO INTER-UNIVERSAL TEICHMU¨LLER THEORY Shinichi Mochizuki July 2016 Abstract. Inter-universal Teichmu¨ller theory may be described as a construction of certain canonical deformations of the ring structure of a number field equipped with certain auxiliary data, which includes an elliptic curve over the numberfieldandaprime number≥5. Inthepresentpaper, wesurveythistheory byfocusingontherichanalogiesbetweenthistheoryandtheclassicalcomputationof theGaussian integral. Themaincommon featuresthatunderlietheseanalogies may be summarized as follows: · the introduction of two mutually alien copies of the object of interest; · the computation of the effect — i.e., on the two mutually alien copies of the object of interest — of two-dimensional changes of coordinates by considering the effect on infinitesimals; · thepassagefromplanar cartesiantopolar coordinatesandtheresulting splitting, or decoupling, into radial — i.e., in more abstract valuation- theoretic terminology, “value group” — and angular — i.e., in more ab- stract valuation-theoretic terminology, “unit group” — portions; · thestraightforwardevaluationoftheradialportionbyapplyingthequadra- ticity of the exponent of the Gaussian distribution; · the straightforward evaluation of the angular portion by considering the metric geometry of the group of units determined by a suitable version of the natural logarithm function. [Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”.] After reviewing the classical computation of the Gaussian integral, we give a detailed survey of inter-universal Teichmu¨llertheorybyconcentratingonthecommonfeatureslistedabove. Thepaper concludes with a discussion of various historical aspects of the mathematics that appears in inter-universal Teichmu¨ller theory. Contents: Introduction §1. Review of the computation of the Gaussian integral §1.1. Inter-universal Teichmu¨ller theory via the Gaussian integral §1.2. Naive approach via changes of coordinates or partial integrations §1.3. Introduction of identical but mutually alien copies §1.4. Integrals over two-dimensional Euclidean space §1.5. The effect on infinitesimals of changes of coordinates Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI §1.6. Passage from planar cartesian to polar coordinates §1.7. Justification of naive approach up to an “error factor” §2. Changes of universe as arithmetic changes of coordinates §2.1. The issue of bounding heights: the ABC and Szpiro Conjectures §2.2. Arithmetic degrees as global integrals §2.3. Bounding heights via global multiplicative subspaces §2.4. Bounding heights via Frobenius morphisms on number fields §2.5. Fundamental example of the derivative of a Frobenius lifting §2.6. Positive characteristic model for mono-anabelian transport §2.7. The apparatus and terminology of mono-anabelian transport §2.8. Remark on the usage of certain terminology §2.9. Mono-anabelian transport and the Kodaira-Spencer morphism §2.10. Inter-universality: changes of universe as changes of coordinates §2.11. The two underlying combinatorial dimensions of a ring §2.12. Mono-anabelian transport for mixed-characteristic local fields §2.13. Mono-anabelian transport for monoids of rational functions §2.14. Finite discrete approximations of harmonic analysis §3. Multiradiality: an abstract analogue of parallel transport §3.1. The notion of multiradiality §3.2. Fundamental examples of multiradiality §3.3. The log-theta-lattice: Θ±ellNF-Hodge theaters, log-links, Θ-links §3.4. Kummer theory and multiradial decouplings/cyclotomic rigidity §3.5. Remarks on the use of Frobenioids §3.6. Galois evaluation, labels, symmetries, and log-shells §3.7. Log-volume estimates via the multiradial representation §3.8. Comparison with the Gaussian integral §3.9. Relation to scheme-theoretic Hodge-Arakelov theory §4. Historical comparisons and analogies §4.1. Numerous connections to classical theories §4.2. Contrasting aspects of class field theory and Kummer theory §4.3. Arithmetic and geometric versions of the Mordell Conjecture §4.4. Atavistic resemblance in the development of mathematics Introduction In the present paper, we survey inter-universal Teichmu¨ller theory by focusing on the rich analogies [cf. §3.8] between this theory and the classical com- putation of the Gaussian integral. Inter-universal Teichmu¨ller theory concerns the construction of canonical deformations of the ring structure of a number field equipped with certain auxiliary data. The collection of data, i.e., consisting of the number field equipped with certain auxiliary data, to which inter-universal Teichmu¨ller theory is applied is referred to as initial Θ-data [cf. §3.3, (i), for more details]. The principal components of a collection of initial Θ-data are · the given number field, · an elliptic curve over the number field, and · a prime number l ≥ 5. ALIEN COPIES, GAUSSIANS, & INTER-UNIVERSAL TEICHMU¨LLER THEORY 3 The main applications of inter-universal Teichmu¨ller theory to diophantine geometry [cf. §3.7, (iv), for more details] are obtained by applying the canonical deformation constructed for a specific collection of initial Θ-data to bound the height of the elliptic curve that appears in the initial Θ-data. Let N be a fixed natural number > 1. Then the issue of bounding a given nonnegative real number h ∈ R may be understood as the issue of showing that ≥0 N ·h is roughly equal to h, i.e., N ·h “≈” h [cf. §2.3, §2.4]. When h is the height of an elliptic curve over a number field, this issue may be understood as the issue of showing that the height of the [in fact, in most cases, fictional!] “elliptic curve” whose q-parameters are the N-th powers “qN” of the q-parameters “q” of the given elliptic curve is roughly equal to the height of the given elliptic curve, i.e., that, at least from point of view of [global] heights, qN “≈” q [cf. §2.3, §2.4]. In order to verify the approximate relation qN “≈” q, one begins by introduc- ing two distinct — i.e., two “mutually alien” — copies of the conventional scheme theory surrounding the given initial Θ-data. Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”. These two mutually alien copies of conventional scheme theory are glued together — by considering relatively weak underlying structures of the respective conven- tional scheme theories such as multiplicative monoids and profinite groups — in such a way that the “qN” in one copy of scheme theory is identified with the “q” in the other copy of scheme theory. This gluing is referred to as the Θ-link. Thus, the “qN” on the left-hand side of the Θ-link is glued to the “q” on the right-hand side of the Θ-link, i.e., qN “=” q LHS RHS [cf. §3.3, (vii), for more details]. Here, “N” is in fact taken not to be a fixed natural number, but rather a sort of symmetrized average over the values j2, where j = 1,...,l(cid:2), and we write l(cid:2) d=ef (l−1)/2. Thus, the left-hand side of the above display {qj2 } LHS j bears a striking formal resemblance to the Gaussian distribution. One then verifies the desired approximate relation qN “≈” q by computing {qj2 } LHS j — not in terms of q [which is immediate from the definitions!], but rather — LHS in terms of [the scheme theory surrounding] q RHS 4 SHINICHI MOCHIZUKI [which is a highly nontrivial matter!]. The conclusion of this computation may be summarized as follows: up to relatively mild indeterminacies — i.e., “relatively small error terms” — {qj2 } may be “confused”, or “identified”, with {qj2 } , LHS j RHS j that is to say, !! {qj2 } (cid:2) {qj2 } LHS j RHS j (“=” q ) RHS [cf. the discussion of §3.7, (i), especially, Fig. 3.19, for more details]. Once one is equipped with this “license” to confuse/identify {qj2 } with {qj2 } , the LHS j RHS j derivation of the desired approximate relation {qj2} “≈” q j and hence of the desired bounds on heights is an essentially formal matter [cf. §3.7, (ii), (iv)]. The starting point of the exposition of the present paper lies in the obser- vation [cf. §3.8 for more details] that the main features of the theory underlying the computation just discussed of {qj2 } in terms of q exhibit remarkable LHS j RHS similarities — as is perhaps foreshadowed by the striking formal resemblance ob- served above to the Gaussian distribution — to the main features of the classical computation of the Gaussian integral, namely, (1mf) the introduction of two mutually alien copies of the object of interest [cf. §3.8, (1gau), (2gau)]; (2mf) the computation of the effect — i.e., on the two mutually alien copies of the object of interest — of two-dimensional changes of coordinates by considering the effect on infinitesimals [cf. §3.8, (3gau), (4gau), (5gau), (6gau)]; (3mf) the passage from planar cartesian to polar coordinates and the re- sulting splitting, or decoupling, into radial — i.e., in more abstract valuation-theoretic terminology, “value group” — and angular — i.e., in more abstract valuation-theoretic terminology, “unit group” — por- tions [cf. §3.8, (7gau), (8gau)]; (4mf) the straightforward evaluation of the radial portion by applying the quadraticity of the exponent of the Gaussian distribution [cf. §3.8, (9gau), (11gau)]; (5mf) the straightforward evaluation of the angular portion by considering the metric geometry of the group of units determined by a suitable version of the natural logarithm function [cf. §3.8, (10gau), (11gau)]. The present paper begins, in §1, with a review of the classical computation of the Gaussian integral, by breaking down this familiar computation into steps in such a way as to facilitate the subsequent comparison with inter-universal Te- ichmu¨ller theory. We then proceed, in §2, to discuss the portion of inter-universal Teichmu¨ller theory that corresponds to (2mf). The exposition of §2 was designed so ALIEN COPIES, GAUSSIANS, & INTER-UNIVERSAL TEICHMU¨LLER THEORY 5 as to be accessible to readers familiar with well-known portions of scheme theory and the theory of the ´etale fundamental group — i.e., at the level of [Harts] and [SGA1]. The various Examples that appear in this exposition of §2 include numerous well-defined and relatively straightforward mathematical assertions often without complete proofs. In particular, the reader may think of the task of supplying a complete proof for any of these assertions as a sort of “exercise” and hence of §2 itself as a sort of workbook with exercises. At the level of papers, §2 is concerned mainly with the content of the “classi- cal” paper [Uchi] of Uchida and the “preparatory papers” [FrdI], [FrdII], [GenEll], [AbsTopI], [AbsTopII], [AbsTopIII]. By contrast, the level of exposition of §3 is substantially less elementary than that of §2. In §3, we apply the conceptual infrastructureexposedin §2tosurveythoseaspectsofinter-universalTeichmu¨ller theory that correspond to (1mf), (3mf), (4mf), and (5mf), i.e., at the level of papers, to [EtTh], [IUTchI], [IUTchII], [IUTchIII], [IUTchIV]. Finally, in §4, we reflect on various historical aspects of the theory exposed in §2 and §3. Acknowledgements: The author wishes to express his appreciation for the stimulating comments that he has received from numerous mathematicians concerning the theory exposed in the present paper and, especially, his deep gratitude to Go Yamashita, Mohamed Sa¨ıdi, Yuichiro Hoshi, and Ivan Fesenko for the very active and devoted role that theyplayedbothindiscussingthistheorywiththeauthorandindisseminatingitto others. In particular, the author would like to thank Yuichiro Hoshi for introducing the notion of mono-anabelian transport as a means of formulating a technique that is frequently applied throughout the theory. This notion plays a central role in the expository approach adopted in the present paper. Section 1: Review of the computation of the Gaussian integral §1.1. Inter-universal Teichmu¨ller theory via the Gaussian integral: The goal of the present paper is to pave the road, for the reader, from a state of complete ignorance of inter-universal Teichmu¨ller theory to a state of general appreciation of the “game plan” of inter-universal Teichmu¨ller theory by reconsidering the well-known computation of the Gaussian integral (cid:2) ∞ √ e−x2 dx = π −∞ via polar coordinates from the point of view of a hypothetical high-school student who has studied one-variable calculus and polar coordinates, but has not yet had any exposure to multi-variable calculus. That is to say, we shall begin in the present §1 by reviewing this computation of the Gaussian integral by discussing how this computation might be explained to such a hypothetical high-school stu- dent. In subsequent §’s, we then proceed to discuss how various key steps in 6 SHINICHI MOCHIZUKI such an explanation to a hypothetical high-school student may be translated into the more sophisticated language of abstract arithmetic geometry in such a way as to yield a general outline of inter-universal Teichmu¨ller theory based on the deep structural similarities between inter-universal Teichmu¨ller theory and the computation of the Gaussian integral. §1.2. Naive approach via changes of coordinates or partial integrations: In one-variable calculus, definite integrals that appear intractable at first glance are often reduced to much simpler definite integrals by performing suitable changes of coordinates or partial integrations. Thus: Step 1: Our hypothetical high-school student might initially be tempted to perform a change of coordinates e−x2 (cid:3) u and then [erroneously!] compute (cid:2) (cid:2) (cid:2) (cid:2) ∞ ∞ x=∞ 1 e−x2 dx = 2· e−x2 dx = − d(e−x2) = du = 1 −∞ 0 x=0 0 — only to realize shortly afterwards that this computation is in error, on account of the erroneous treatment of the infinitesimal “dx” when the change of coordinates was executed. Step 2: This realization might then lead the student to attempt to repair the computation of Step 1 by considering various iterated partial integra- tions (cid:2) (cid:2) (cid:2) (cid:3) (cid:4) ∞ x=∞ 1 x=∞ 1 e−x2 dx = − d(e−x2) = e−x2d = ... 2x 2x −∞ x=−∞ x=−∞ — which, of course, lead nowhere. §1.3. Introduction of identical but mutually alien copies: Atthispoint, one might suggest to the hypothetical high-school student the idea of computing the Gaussian integral by first squaring the integral and then taking the square root of the value of the square of the integral. That is to say, in effect: Step 3: One might suggest to the hypothetical high-school student that the Gaussian integral can in fact be computed by considering the product of two identical — but mutually independent! — copies of the Gaussian integral (cid:3)(cid:2) (cid:4) (cid:3)(cid:2) (cid:4) ∞ ∞ e−x2 dx · e−y2 dy −∞ −∞ — i.e., as opposed to a single copy of the Gaussian integral. Here, let us recall that our hypothetical high-school student was already in a mental stateofextreme frustrationasaresultofthe student’sintensive and heroic attempts ALIEN COPIES, GAUSSIANS, & INTER-UNIVERSAL TEICHMU¨LLER THEORY 7 in Step 2 which led only to an endless labyrinth of meaningless and increasingly complicated mathematical expressions. This experience left our hypothetical high- school student with the impression that the Gaussian integral was without question by far the most difficult integralthatthestudenthadeverencountered. Inlight ofthisexperience, thesuggestionofStep3evokedareactionofintenseindignation and distrust on the part of the student. That is to say, the idea that meaningful progress could be made in the computation of such an exceedingly difficult integral simply by considering two identical copies of the integral — i.e., as opposed to a single copy — struck the student as being utterly ludicrous. Put another way, the suggestion of Step 3 was simply not the sort of suggestion that thestudentwanted to hear. Rather,thestudentwaskeenlyinterestedinseeingsome sort of clever partial integration or change of coordinates involving “sin(−)”, “cos(−)”, “tan(−)”, “exp(−)”, “ 1 ”, etc., i.e., of the sort that the student was 1+x2 used to seeing in familiar expositions of one-variable calculus. §1.4. Integrals over two-dimensional Euclidean space: Only after quite substantial efforts at persuasion did our hypothetical high-school student reluc- tantly agree to proceed to the next step of the explanation: Step 4: If one considers the “totality”, or “total space”, of the coordi- nates that appear in the product of two copies of the Gaussian integral of Step 3, then one can regard this product of integrals as a single integral (cid:2) (cid:2) e−x2 ·e−y2 dx dy = e−(x2+y2) dx dy R2 R2 over the Euclidean plane R2. Of course, our hypothetical high-school student might have some trouble with Step 4 since it requires one to assimilate the notion of an integral over a space, i.e., the Euclidean plane R2, which is not an interval of the real line. This, however, may be explained by reviewing the essential philosophy behind the notion of the Riemann integral — a philosophy which should be familiar from one-variable calculus: Step 5: One may think of integrals over more general spaces, i.e., such as the Euclidean plane R2, as computations (cid:5) net mass = lim (infinitesimals of zero mass) of “net mass” by considering limits of sums of infinitesimals, i.e., such as “dx dy”, which one may think of as having “zero mass”. §1.5. The effect on infinitesimals of changes of coordinates: Just as in one-variable calculus, computations of integrals over more general spaces can often be simplified by performing suitable changes of coordinates. Any [say, continuously differentiable] change of coordinates results in a new factor, given by the Jacobian, in the integrand. This factor constituted by the Jacobian, i.e., the determinant of a certain matrix of partial derivatives, may appear to be somewhat mysterious to our 8 SHINICHI MOCHIZUKI hypothetical high-school student, who is only familiar with changes of coodinates in one-variable calculus. On the other hand, the appearance of the Jacobian may be justified in a computational fashion as follows: Step 6: Let U,V ⊆ R2 be open subsets of R2 and U (cid:7) (s,t) (cid:8)→ (x,y) = (f(s,t),g(s,t)) ∈ V a continuously differentiable change of coordinates such that the Ja- cobian (cid:6) (cid:7) ∂f ∂f def ∂s ∂t J = det ∂g ∂g ∂s ∂t — which may be thought of as a continuous real-valued function on U — is nonzero throughout U. Then for any continuous real-valued functions φ : U → R, ψ : V → R such that ψ(f(s,t),g(s,t)) = φ(s,t), the effect of the above change of coordinates on the integral of ψ over V may be computed as follows: (cid:2) (cid:2) ψ dx dy = φ·J ds dt. V U Step 7: In the situation of Step 6, the effect of the change of coordinates on the “infinitesimals” dx dy and ds dt may be understood as follows: First, one localizes to a sufficiently small open neighborhood of a point of U over which the various partial derivatives of f and g are roughly constant, which implies that the change of coordinates determined by f and g is roughly linear. Then the effect of such a linear transformation on areas — i.e., in the language of Step 5, “masses” — of sufficiently small parallelograms is given by multiplying by the determinant of the linear transformation. Indeed, to verify this, one observes that, after possible pre- and post-composition with a rotation [which clearly does not affect the computation of such areas], one may assume that one of the sides of the parallelogram under consideration is a line segment on the s- axis whose left-hand endpoint is equal to the origin (0,0), and, moreover, that the linear transformation may be written as a composite of toral dilations and unipotent linear transformations of the form (s,t) (cid:8)→ (a·s,b·t); (s,t) (cid:8)→ (s+c·t,t) — where a,b,c ∈ R, and ab (cid:10)= 0. On the other hand, in the case of such “upper triangular” linear transformations, the effect of the linear transformation on the area of the parallelogram under consideration is an easy computation at the level of high-school planar geometry. §1.6. Passage from planar cartesian to polar coordinates: Once the “innocuous” generalitiesofSteps5, 6, and7havebeenassimilated, onemayproceed as follows: ALIEN COPIES, GAUSSIANS, & INTER-UNIVERSAL TEICHMU¨LLER THEORY 9 Step 8: We apply Step 6 to the integral of Step 4, regarded as an inte- gral over the complement R2 \ (R × {0}) of the negative x-axis in the ≤0 Euclidean plane, to the change of coordinates R ×(−π,π) (cid:7) (r,θ) (cid:8)→ (x,y) = (rcos(θ),rsin(θ)) ∈ R2 \(R ×{0}) >0 ≤0 — where we write R for the set of positive real numbers and (−π,π) >0 for the open interval of real numbers between −π and π. Step 9: The change of coordinates of Step 8 allows one to compute as follows: (cid:3)(cid:2) (cid:4) (cid:3)(cid:2) (cid:4) (cid:2) ∞ ∞ e−x2 dx · e−y2 dy = e−x2 ·e−y2 dx dy −∞ −∞ (cid:2)R2 = e−(x2+y2) dx dy (cid:2)R2 = e−(x2+y2) dx dy R2\(R≤0×{0}) (cid:2) = e−r2 rdr dθ (cid:3)R(cid:2)>0×(−π,π) (cid:4) (cid:3)(cid:2) (cid:4) ∞ π = e−r2 ·2rdr · 1 ·dθ 2 0 −π — where we observe that the final equality is notable in that it shows that, in the computation of the integral under consideration, the radial [i.e., “r”] and angular [i.e., “θ”] coordinates may be decoupled, i.e., that the integral under consideration may be written as a product of a radial integral and an angular integral. Step 10: The radial integral of Step 9 may be evaluated (cid:2) (cid:2) (cid:2) ∞ 1 1 e−r2 ·2rdr = d(e−r2) = du = 1 0 0 0 by applying the change of coordinates e−r2 (cid:3) u that, in essence, appeared in the erroneous initial computation of Step 1! Step 11: The angular integral of Step 9 may be evaluated as follows: (cid:2) π 1 ·dθ = π 2 −π Here, we note that, if one thinks of the Euclidean plane R2 of Step 4 as the complex plane, i.e., if we write the change of coordinates of Step 8 in