Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

More documents

Recommendations

Info

150 SHINICHI MOCHIZUKI (ii); 4.9.2, (iv)] F l -andF⋊± l -symmetries of Propositions 4.9.1, (i); 6.8, (i), from the coarser “combinatorially real analytic” [cf. Remarks 4.9.1, (ii); 4.9.2, (iv)] S l -andS l ±-symmetries of Propositions 4.9, (ii), (iii); 6.8, (ii), (iii) — i.e., which do not admit a compatible bijection between the index sets of the capsules involved and some sort of set of [±-]label classes of cusps [cf. the discussion of Remark 4.9.2, (i)]. This relationship with a set of [±-]label classes of cusps will play a crucial role in the theory of the Hodge-Arakelov-theoretic evaluation of the étale theta function that will be developed in [IUTchII]. (iii) On the other hand, one significant feature of the additive theory of the present §6 which does not appear in the multiplicative theory of §4 is the phenomenon of “global ±-synchronization” — i.e., at a more concrete level, the various isomorphisms “ † ξ” that appear in Proposition 6.5, (i), (ii) — between the ±-indeterminacies that occur at the various v ∈ V. Note that this global ±-synchronization is a necessary “pre-condition” for the additive portion [i.e., corresponding to F l ⊆ F ⋊± l ]oftheF ⋊± l -symmetry of Proposition 6.8, (i). This “additive portion” of the F ⋊± l -symmetry plays the crucial role of allowing one to relate the zero and nonzero elements of F l [cf. the discussion of Remark 6.12.5 below]. (iv) One important property of both the “ † ζ’s” discussed in (ii) and the “ † ξ’s” discussed in (iii) is that they are constructed by means of functorial algorithms from the intrinsic structure of a D-Θ ±ell -orD-ΘNF-Hodge theater [cf. Propositions 4.7, (iii); 6.5, (i), (ii), (iii)] — i.e., not by means of comparison with some fixed reference model [cf. the discussion of [AbsTopIII], §I4], such as the objects constructed in Examples 4.3, 4.4, 4.5, 6.2, 6.3. This property will be of crucial importance when, in the theory of [IUTchIII], we combine the theory developed in the present series of papers with the theory of log-shells developed in [AbsTopIII]. Remark 6.12.5. (i) One fundamental difference between the F l -symmetry of §4 andtheF⋊± l - symmetry of the present §6 lies in the inclusion of the zero element ∈ F l in the symmetry under consideration. This inclusion of the zero element ∈ F l means, in particular, that the resulting network of comparison isomorphisms [cf. Remark 6.12.4, (i)] allows one to relate the “zero-labeled” prime-strip to the various “nonzerolabeled” prime-strips, i.e., the prime-strips labeled by nonzero elements ∈ F l [or, essentially equivalently, ∈ F l ]. Moreover, as reviewed in Remark 6.12.4, (ii), the F ⋊± l -symmetry allows one to relate the zero-labeled and non-zero-labeled prime-strips to one another in a “combinatorially holomorphic” fashion, i.e., in a fashion that is compatible with the various natural bijections [i.e., “ † ζ”] with various sets of global ±-label classes of cusps. Here, it is useful to recall that evaluation at [torsion points closely related to] the zero-labeled cusps [cf. the discussion of “evaluation points” in Example 4.4, (i)] plays an important role in the theory of normalization of the étale theta function
INTER-UNIVERSAL TEICHMÜLLER THEORY I 151 — cf. the theory of étale theta functions “of standard type”, as discussed in [EtTh], Theorem 1.10; the theory to be developed in [IUTchII]. (ii) Whereas the F ⋊± l -symmetry of the theory of the present §6 has the advantage that it allows one to relate zero-labeled and non-zero-labeled prime-strips, it has the [tautological!] disadvantage that it does not allow one to “insulate” the non-zero-labeled prime-strips from confusion with the zero-labeled prime-strip. This issue will be of substantial importance in the theory of Gaussian Frobenioids [to be developed in [IUTchII]], i.e., Frobenioids that, roughly speaking, arise from the theta values { q j2 } j v [cf. the discussion of Example 4.4, (i)] at the non-zero-labeled evaluation points. Moreover, ultimately, in [IUTchII], [IUTchIII], we shall relate these Gaussian Frobenioids to various global arithmetic line bundles on the number field F . This will require the use of both the additive and the multiplicative structures on the number field; in particular, it will require the use of the theory developed in §5. (iii) By contrast, since, in the theory of the present series of papers, we shall not be interested in analogues of the Gaussian Frobenioids that involve the zerolabeled evaluation points, we shall not require an “additive analogue” of the portion [cf. Example 5.1] of the theory developed in §5 concerning global Frobenioids. Remark 6.12.6. (i) Another fundamental difference between the F l -symmetry of §4 andthe -symmetry of the present §6 lies in the geometric nature of the “single basepoint” [cf. the discussion of Remark 6.12.4] that underlies the F ⋊± l -symmetry. That F ⋊± l is to say, the various labels ∈ T → ∼ F l that appear in a [D-]Θ ±ell -Hodge theater correspond — throughout the various portions [e.g., bridges] of the [D-]Θ ±ell -Hodge theater — to collections of cusps in a single copy [i.e., connected component] of -symmetry of the [D-]Θ ell -bridge [cf. Proposition 6.8, (i)] without permuting the collection of valuations V ± (⊆ V(K)) [cf. the discussion of Definition 6.1, (v)]. This contrasts sharply with the arithmetic nature of the “single basepoint” [cf. the discussion “D v ” at each v ∈ V; these collections of cusps are permuted by the F ⋊± l of Remark 6.12.4] that underlies the F l -symmetry of §4, i.e., in the sense that the F l -symmetry [cf. Proposition 4.9, (i)] permutes the various F l -translates of V ± = V ±un ⊆ V Bor (⊆ V(K)) [cf. Example 4.3, (i); Remark 6.1.1]. (ii) The geometric nature of the “single basepoint” of the F ⋊± l -symmetry of a[D-]Θ ±ell -Hodge theater [cf. (i)] is more suited to the theory of the Hodge-Arakelov-theoretic evaluation of the étale theta function to be developed in [IUTchII], in which the existence of a “single basepoint” corresponding to a single connected component of “D v ”forv ∈ V bad plays a central role. (iii) By contrast, the arithmetic nature of the “single basepoint” of the F l - symmetry of a [D-]ΘNF-Hodge theater [cf. (i)] is more suited to the
Page 1 and 2:
INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 3 and 4:
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
associated to † F ⊩ mod and ‡
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
completely determined by † D. INT
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
conjugation by which maps φ NF IN
Page 99 and 100: INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 129 and 130: (v) Write INTER-UNIVERSAL TEICHMÜL
Page 149: INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 157: INTER-UNIVERSAL TEICHMÜLLER THEORY
show all

Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

Create successful ePaper yourself

Delete template?

Save as template?