Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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10 SHINICHI MOCHIZUKI Classical upper half-plane Θ ±ell NF-Hodge theaters in inter-universal Teichmüller theory Additive z ↦→ z + a, F ⋊± l - symmetry z ↦→ −z + a (a ∈ R) symmetry “Functions” assoc’d to add. symm. q def = e 2πiz theta fn. evaluated at l-tors. [cf. I, 6.12.6, (ii)] Basepoint assoc’d single cusp V ± to add. symm. at infinity [cf. I, 6.1, (v)] Combinatorial prototype assoc’d cusp cusp to add. symm. Multiplicative symmetry z ↦→ z·cos(t)−sin(t) z·sin(t)+cos(t) , F l - z ↦→ z·cos(t)+sin(t) z·sin(t)−cos(t) (t ∈ R) symmetry “Functions” assoc’d to mult. symm. w def = z−i z+i elements of the number field F mod [cf. I, 6.12.6, (iii)] Basepoints assoc’d ( cos(t) −sin(t) ) ( sin(t) cos(t) , cos(t) sin(t) ) sin(t) −cos(t) F l V Bor = F l · V ±un to mult. symm. {entire boundary of H } [cf. I, 4.3, (i)] Combinatorial nodes of mod p nodes of mod p prototype assoc’d Hecke correspondence Hecke correspondence to mult. symm. [cf. II, 4.11.4, (iii), (c)] [cf. II, 4.11.4, (iii), (c)] Fig. I1.4: Comparison of F ⋊± l -, F l -symmetries with the geometry of the upper half-plane As discussed above in our explanation of the models at v ∈ V bad for F ⊢ -primestrips, by considering the 2l-th roots of the q-parameters of the elliptic curve E F
INTER-UNIVERSAL TEICHMÜLLER THEORY I 11 at v ∈ V bad , and, roughly speaking, extending to v ∈ V good in such a way as to satisfy the product formula, one may construct a natural F ⊩ -prime-strip “F ⊩ mod ” [cf. Example 3.5, (ii); Definition 5.2, (iv)]. This construction admits an abstract, algorithmic formulation that allows one to apply it to the underlying “Θ-Hodge theater” of an arbitrary Θ ±ell NF-Hodge theater † HT Θ±ell NF so as to obtain an F ⊩ - prime-strip † F ⊩ mod [cf. Definitions 3.6, (c); 5.2, (iv)]. On the other hand, by formally replacing the 2l-th roots of the q-parameters that appear in this construction by the reciprocal of the l-th root of the Frobenioid-theoretic theta function, which we shall denote “Θ v ”[forv ∈ V bad ], studied in [EtTh] [cf. also Example 3.2, (ii), of the present paper], one obtains an abstract, algorithmic formulation for the construction of an F ⊩ -prime-strip † F ⊩ tht [cf. Definitions 3.6, (c); 5.2, (iv)] from [the underlying Θ-Hodge theater of] the Θ ±ell NF-Hodge theater † HT Θ±ell NF . Now let ‡ HT Θ±ellNF be another Θ ±ell NF-Hodge theater [relative to the given initial Θ-data]. Then we shall refer to the “full poly-isomorphism” of [i.e., the collection of all isomorphisms between] F ⊩ -prime-strips † F ⊩ tht ∼ → ‡ F ⊩ mod as the Θ-link from [the underlying Θ-Hodge theater of] † HT Θ±ell NF to [the underlying Θ-Hodge theater of] ‡ HT Θ±ell NF [cf. Corollary 3.7, (i); Definition 5.2, (iv)]. One fundamental property of the Θ-link is the property that it induces a collection of isomorphisms [in fact, the full poly-isomorphism] between the F ⊢× -prime-strips † F ⊢× mod ∼ → ‡ F ⊢× mod associated to † F ⊩ mod and ‡ F ⊩ mod 4.9, (vii)]. [cf. Corollary 3.7, (ii), (iii); [IUTchII], Definition Now let { n HT Θ±ell NF } n∈Z be a collection of distinct Θ ±ell NF-Hodge theaters [relative to the given initial Θ-data] indexed by the integers. Thus, by applying the constructions just discussed, we obtain an infinite chain ... Θ −→ (n−1) HT Θ±ell NF Θ −→ n HT Θ±ell NF Θ −→ (n+1) HT Θ±ell NF Θ −→ ... of Θ-linked Θ ±ell NF-Hodge theaters [cf. Corollary 3.8], which will be referred to as the Frobenius-picture [associated to the Θ-link]. One fundamental property of this Frobenius-picture is the property that it fails to admit permutation automorphisms that switch adjacent indices n, n + 1, but leave the remaining indices ∈ Z fixed [cf. Corollary 3.8]. Roughly speaking, the Θ-link n HT Θ±ell NF Θ −→ (n+1) HT Θ±ellNF may be thought of as a formal correspondence n Θ v ↦→ (n+1) q v
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INTER-UNIVERSAL TEICHMÜLLER THEORY
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completely determined by † D. INT
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conjugation by which maps φ NF IN
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(v) Write INTER-UNIVERSAL TEICHMÜL
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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