Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

More documents

Recommendations

Info

36 SHINICHI MOCHIZUKI acts on Δ E ⊗(Z/lZ) via multiplication by −1. In particular, since l is odd, it follows that the action by ι on Δ ɛ determines a decomposition into eigenspaces Δ ɛ ∼ → Δ + ɛ × Δ − ɛ — i.e., where ι acts on Δ + ɛ (respectively, Δ − ɛ ) by multiplication by +1 (respectively, −1). Moreover, the natural composite maps I ɛ ′ ↩→ Δ ɛ ↠ Δ + ɛ ; I ɛ ′′ ↩→ Δ ɛ ↠ Δ + ɛ ∼ determine isomorphisms I ɛ ′ → Δ + ɛ , I ɛ ′′ → Δ + ɛ . Since the natural action of G k on Δ ɛ clearly commutes with the action of ι, we thus conclude that the quotient Δ X ↠ Δ ɛ ↠ Δ + ɛ determines quotients ∼ Π X ↠ J X ; Π C ↠ J C — where the surjections Π X ↠ G k ,Π C ↠ G k induce natural exact sequences 1 → Δ + ɛ → J X → G k → 1, 1 → Δ + ɛ × Gal(X/C) → J C → G k → 1; we have a natural inclusion J X ↩→ J C . Next, let us consider the cusp “2ɛ”ofC — i.e., the cusp whose inverse images in X correspond to the points of E obtained by multiplying ɛ ′ , ɛ ′′ by 2, relative to the group law of the elliptic curve determined by the pair (X,ɛ 0 ). Since 2 ≠ ±1 (modl) [a consequence of our assumption that l ≥ 5], it follows that the decomposition group associated to this cusp “2ɛ” determines a section σ : G k → J C of the natural surjection J C ↠ G k . Here, we note that although, a priori, σ is only determined by 2ɛ up to composition with an inner automorphism of J C determined by an element of Δ + ɛ × Gal(X/C), in fact, since [in light of the assumption (∗)!] the natural [outer] action of G k on Δ + ɛ × Gal(X/C) istrivial, we conclude that σ is completely determined by 2ɛ, and that the subgroup Im(σ) ⊆ J C determined by the image of σ is normal in J C . Moreover, by considering the decomposition groups associated to the cusps of X lying over 2ɛ, we conclude that Im(σ) lies inside the subgroup J X ⊆ J C . Thus, the subgroups Im(σ) ⊆ J X ,Im(σ) × Gal(X/C) ⊆ J C determine [the horizontal arrows in] cartesian diagrams X−→ −→ X Π X−→ −→ Π X Δ X−→ −→ Δ X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ↓ ↓ ↓ ↓ ↓ ↓ C−→ −→ C Π C−→ −→ Π C Δ C−→ −→ Δ C of finite étale cyclic coverings of hyperbolic orbicurves and open immersions [with normal image] of profinite groups; we have Gal( −→ C /C) ∼ = Z/lZ, Gal(X/C) ∼ = Z/2Z, and Gal(X −→ /C) → ∼ Gal(X/C) × Gal( −→ C /C) ∼ = Z/2lZ. Definition 1.1. We shall refer to a hyperbolic orbicurve over k that arises, up to isomorphism, as the hyperbolic orbicurve −→ X (respectively, −→ C ) constructed above for some choice of l, ɛ as being of type (1,l-tors −−→ ) (respectively, (1,l-tors −−→ ) ±).
INTER-UNIVERSAL TEICHMÜLLER THEORY I 37 Remark 1.1.1. The arrow “→” in the notation “X −→ ”, “X −→ ”, “(1,l-tors −−→ )”, “(1,l-tors −−→ ) ±” may be thought of as denoting the “archimedean, ordered labels 1, 2,...” — i.e., determined by the choice of ɛ! —onthe{±1}-orbits of elements of the quotient “Q” that appears in the definition of a “hyperbolic orbicurve of type (1,l-tors) ± ” given in [EtTh], Definition 2.1. Remark 1.1.2. We observe that X −→ , C −→ are completely determined, up to k- isomorphism, by the data (X/k, C, ɛ). Corollary 1.2. (Characteristic Nature of Coverings) Suppose that k is an NF or an MLF. Then there exists a functorial group-theoretic algorithm [cf. [AbsTopIII], Remark 1.9.8, for more on the meaning of this terminology] to reconstruct Π X , Π C−→ , Π C (respectively, Π C ) together with the conjugacy classes of the decomposition group(s) determined by the set(s) of cusps {ɛ ′ ,ɛ ′′ }; {ɛ} (respectively, {ɛ}) fromΠ X−→ (respectively, Π C−→ ). Here, the asserted functoriality is with respect to isomorphisms of topological groups; we reconstruct Π X , Π C (respectively, Π C )asasubgroupofAut(Π X−→ ) (respectively, Aut(Π C−→ )). Proof. For simplicity, we consider the non-resp’d case; the resp’d case is entirely similar [but slightly easier]. The argument is similar to the arguments applied in [EtTh], Proposition 1.8; [EtTh], Proposition 2.4. First, we recall that Π X−→ ,Π X ,and Π C are slim [cf., e.g., [AbsTopI], Proposition 2.3, (ii)], hence embed naturally into Aut(Π X−→ ), and that one may recover the subgroup Δ X−→ ⊆ Π X−→ via the algorithms of [AbsTopI], Theorem 2.6, (v), (vi). Next, we recall that the algorithms of [AbsTopII], Corollary 3.3, (i), (ii) — which are applicable in light of [AbsTopI], Example 4.8 — allow one to reconstruct Π C [together with the natural inclusion Π X−→ ↩→ Π C ], as well as the subgroups Δ X ⊆ Δ C ⊆ Π C . In particular, l may be recovered via the formula l 2 =[Δ X :Δ X ] · [Δ X :Δ X−→ ]=[Δ X :Δ X−→ ]=[Δ C :Δ X−→ ]/2. Next, let us set H def =Ker(Δ X ↠ Δ ab X ⊗ (Z/lZ)). Then Π X ⊆ Π C may be recovered via the [easily verified] equality of subgroups Π X =Π X−→ · H. The conjugacy classes of the decomposition groups of ɛ 0 , ɛ ′ , ɛ ′′ in Π X mayberecoveredasthedecomposition groups of cusps [cf. [AbsTopI], Lemma 4.5] whose image in Gal(X −→ /X) =Π X /Π X−→ is nontrivial. Next, to reconstruct Π C ⊆ Π C , it suffices to reconstruct the splitting of the surjection Gal(X/C) =Π C /Π X ↠ Π C /Π X = Gal(X/C) determined by Gal(X/C) =Π C /Π X ; but [since l is prime to 3!] this splitting may be characterized [group-theoretically!] as the unique splitting that stabilizes the collection of conjugacy classes of subgroups of Π X determined by the decomposition groups of ɛ 0 , ɛ ′ , ɛ ′′ . Now Π C−→ ⊆ Π C may be reconstructed by applying the observation that (Z/2Z ∼ =) Gal(X −→ /C −→ ) ⊆ Gal(X −→ /C) ( ∼ = Z/2lZ) istheunique maximal subgroup of odd index. Finally, the conjugacy classes of the decomposition groups of ɛ ′ , ɛ ′′ in Π X mayberecoveredasthedecomposition groups of cusps [cf. [AbsTopI], Lemma 4.5] whose image in Gal(X −→ /X) =Π X /Π X−→ is nontrivial, but which are not fixed [up
Page 1 and 2: INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 15 and 16: associated to † F ⊩ mod and ‡
Page 35: INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 81 and 82: completely determined by † D. INT
Page 87 and 88:
INTER-UNIVERSAL TEICHMÜLLER THEORY
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:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
(v) Write INTER-UNIVERSAL TEICHMÜL
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157:
show all

Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?