Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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146 SHINICHI MOCHIZUKI is unique. Then by applying Propositions 4.8, 6.6, and Corollaries 5.6, 6.12, one may verify analogues of these results for such Θ ±ell NF-Hodge theaters. In a similar vein, one may glue a D-Θ ±ell -Hodge theater to a D-ΘNF-Hodge theater to obtain a “D-Θ ±ell NF-Hodge theater” [cf. Definition 6.13, (ii), below]. We leave the routine details to the reader. Remark 6.12.3. (i) One way to think of the notion of a ΘNF-Hodge theater studied in §4 isas a sort of total space of a local system of F l -torsors over a “base space” that represents a sort of “homotopy” between a number field and a Tate curve [i.e., the elliptic curve under consideration at the v ∈ V bad ]. From this point of view, the notion of a Θ ±ell -Hodge theater studied in the present §6 may be thought of as a sort of total space of a local system of F ⋊± l -torsors over a similar “base space”. Here, it is interesting to note that these F l -andF⋊± l - torsors arise, on the one hand, from the l-torsion points of the elliptic curve under consideration, hence may be thought of as discrete approximations of [the geometric portion of] this elliptic curve over a number field [cf. the point of view of scheme-theoretic Hodge-Arakelov theory discussed in [HA- SurI], §1.3.4]. On the other hand, if one thinks in terms of the tempered fundamental groups of the Tate curves that occur at v ∈ V bad , then these F l -andF⋊± l -torsors may be thought of as finite approximations of the copy of “Z” that occurs as the Galois group of a well-known tempered covering of the Tate curve [cf. the discussion of [EtTh], Remark 2.16.2]. Note, moreover, that if one works with Θ ±ell NF-Hodge theaters [cf. Remark 6.12.2, (ii)], then one is, in effect, working with both the additive and the multiplicative structures of this copy of Z — although, unlike the situation that occurs when one works with rings, i.e., in which the additive and multiplicative structures are “entangled” with one another in some sort of complicated fashion [cf. the discussion of [AbsTopIII], Remark 5.6.1], if one works with Θ ±ell NF-Hodge theaters, then each of the additive and multiplicative structures occurs in an independent fashion [i.e., in the form of Θ ±ell - and ΘNF-Hodge theaters], i.e., “extracted” from this entanglement. (ii) At this point, it is useful to recall that the idea of a distinct [i.e., from the copy of Z implicit in the “base space”] “local system-theoretic” copy of Z occurring over a “base space” that represents a number field is reminiscent not only of the discussion of [EtTh], Remark 2.16.2, but also of the Teichmüller-theoretic point of view discussed in [AbsTopIII], §I5. That is to say, relative to the analogy with p-adic Teichmüller theory, the “base space” that represents a number field corresponds to
INTER-UNIVERSAL TEICHMÜLLER THEORY I 147 a hyperbolic curve in positive characteristic, while the “local system-theoretic” copy of Z — which, as discussed in (i), also serves as a discrete approximation of the [geometric portion of the] elliptic curve under consideration — corresponds to a nilpotent ordinary indigenous bundle over the positive characteristic hyperbolic curve. -” and “F l -”] symmetries of ΘNF-, Θ±ell -Hodge theaters may be thought of as corresponding, respectively, to the two real dimensions (iii) Relative to the analogy discussed in (ii) between the “local system-theoretic” copy of Z of (i) and the indigenous bundles that occur in p-adic Teichmüller theory, it is interesting to note that the two combinatorial dimensions [cf. [AbsTopIII], Remark 5.6.1] corresponding to the additive and multiplicative [i.e., “F ⋊± l · z ↦→ z + a, z ↦→ −z + a; · z ↦→ z · cos(t) − sin(t) z · sin(t) + cos(t) , z · cos(t)+sin(t) z ↦→ z · sin(t) − cos(t) —wherea, t ∈ R; z denotes the standard coordinate on H — of transformations of the upper half-plane H, i.e., an object that is very closely related to the canonical indigenous bundles that occur in the classical complex uniformization theory of hyperbolic Riemann surfaces [cf. the discussion of Remark 4.3.3]. Here, it is also of interest to observe that the above additive symmetry of the upper half-plane is closely related to the coordinate on the upper half-plane determined by the “classical q-parameter” q def = e 2πiz — a situation that is reminiscent of the close relationship, in the theory of the present series of papers, between the F ⋊± l -symmetry and the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation of the theta function on the l-torsion points at bad primes [cf. Remark 6.12.6, (ii); the theory of [IUTchII]]. Moreover, the fixed basepoint “V ± ” [cf. Definition 6.1, (v)] with respect to which one considers l-torsion points in the context of the F ⋊± l -symmetry is reminiscent of the fact that the above additive symmetries of the upper half-plane fix the cusp at infinity. Indeed, taken as a whole, the geometry and coordinate naturally associated to this additive symmetry of the upper half-plane may be thought of, at the level of “combinatorial prototypes”, as the geometric apparatus associated to a cusp [i.e., as opposed to a node — cf. the discussion of [NodNon], Introduction]. By contrast, the “toral” multiplicative symmetry of the upper half-plane recalled above is closely related to the coordinate on the upper half-plane that determines a biholomorphic isomorphism with the unit disc w def = z−i z+i — a situation that is reminiscent of the close relationship, in the theory of the present series of papers, between the F l -symmetry and the Kummer theory surrounding the number field F mod [cf. Remark 6.12.6, (iii); the theory of §5 of the present paper]. Moreover, the action of F l on the “collection of basepoints for the l-torsion points” V Bor = F l · V ±un [cf. Example 4.3, (i)] in the context of
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associated to † F ⊩ mod and ‡
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completely determined by † D. INT
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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