Compactification of the moduli space of abelian varieties, Kyoto ...

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14 IKU NAKAMURA7.1. The G(K)-action and the G(K)-linearization. The G(K)-actionτ on (Z, L) is translated into G(K)-linearization{φ g ; g ∈ G(K)}(i) φ g : L→Tg ∗ (L) is a bundle isomorphism,(ii) φ gh = Th ∗φ g ◦ φ h for any g, h ∈ G(K)(T ).Then the action τ on (Z, L) is recovered from it as follows : By theisomorphismsL φ h−→ Th ∗ (L) T h−→ ∗φgTh ∗ (T g ∗ (L)) = Tgh ∗ (L),for x ∈ Z, ξ ∈ L x ,τ(h) · (z,ξ) =(T h (z),φ h (z) · ξ).We check it is the action:τ(g) · (τ(h) · (z,ξ)) = τ(g) · (T h (z),φ h (z) · ξ)=(T g (T h (z)),φ g (T h (z))φ h (z) · ξ)=(T gh (z), (Th ∗ φ g · φ h )(z) · ξ)=(T gh (z),φ gh (z) · ξ) =τ(gh) · (z,ξ).Then we define the action of G(K) onH 0 (Z, L)ρ L (g)(θ) :=Tg ∗ (φ −1 g (θ))We check that it is a homom.ρ L (gh)(θ) =Th ∗ −1 g(φ −1 gh θ)= Tg ∗ {T ∗ −1 h(T ∗ −1 h φ g · φ h θ)}= Tg ∗ {T ∗ −1 h(T ∗ −1 h φ g) · (Th ∗ φ −1 h θ)}= Tg ∗ {φ −1 g · (Th ∗ φ −1 h θ)}= ρ L (g)ρ L (h)(θ).Thus H 0 (Z, L) isaG(K)-module of weight one, that is, any a ∈ G(K)acts as a · id VH .We recallLemma 7.2. V H is a unique irreducible representation of weight one ofG(K) over Z[ζ N , 1/N ].Lemma 7.3. Assume e min (K) ≥ 3. Thenfor a level-G(K) PSQAS (Q 0 ,φ 0 ,τ 0 ), there exists a unique level-G(K)PSQAS (Q ′ 0 ,i,U H) isom. to (Q 0 ,φ 0 ,τ 0 ) such that i : Q ′ 0 ⊂ P(V H), whereU H is the Schrödinger repres. of G(K).Proof of Lemma 7.3. Let (Q 0 , L 0 )be(Z, L), and (Q 0 ,φ 0 ,τ 0 )=(Z, φ, τ).For a closed immersion φ : Z → P(V H ), we have an isomorphismφ ∗ : V H ≃ H 0 (Z, L),whence(7)ρ(φ, τ)(g)(θ) :=(φ ∗ ) −1 ρ L (g)(θ)φ ∗ .
COMPACTIFICATION OF THE MODULI SPACE OF ABELIAN VARIETIES 15Thus ρ(φ, τ) ∈ End (V H ).By Schur’s lemma, ∃ A ∈ GL(V H ) s.t.U H = A −1 ρ(φ, τ)A =(φ ∗ A) −1 ρ L (g)(θ)(φ ∗ A).Hence it suffices to choose a closed immersion ψ by ψ ∗ = φ ∗ A. Then(8)U H = ρ(ψ,τ).This proves Lemma by taking Z ′ = ψ(Z), i the inclusion of Z ′ .Remark 7.4. Suppose L to be very ample. Then the G(K)-linearizationon (Z, L) is equiv. to the G(K)-equivariant closed immersion of the pair(Z, L) into(P(V H ),O P (1)). Namely, Z is a G(K)-invariant subscheme ofP(V H ) with L = O Z (1).Let Hilb χ(n) be the Hilbert scheme parametrizing all the closed subscheme(Z, L) ofP(V H ) with χ(Z, L n )=n g√ |K| =: χ(n), and (Hilb χ(n) ) G(K)-invthe G(K)-inv. part of it.The following is an immersion of A g,K into (Hilb χ(n) ) G(K)-inv :A g,K ∋ (A 0 ,φ 0 ,τ 0 ) ↦→ (A ′ 0 ,i,U H) ∈ (Hilb χ(n) ) G(K)-inv (AV).Then we defineSQ g,K = A g,K ⊂ (Hilb χ(n) ) G(K)-invTheorem 7.5. Suppose H = ⊕ g i=1 (Z/e iZ). For any closed field k of characteristicprime to |H| = ∏ gi=1 e i,SQ g,K (k) ={(Q 0 ,i,U H ); PSQAS,i: Q 0 ⊂ P(V H )}8. RepresentabilityDefinition 8.1. The triple (X, φ, τ) or(X, L, φ, τ) isa PSQAS with level-G(K) str. if1. φ :(X, L) → (P(V ),O(1)) a closed immersionsuch that φ ∗ : V ≃ H 0 (X, L), L = φ ∗ O P(V ) (1),2. τ is a G(K)-action on the pair (X, L) so that φ is a G(K)-morphism.Define : (X, φ, τ) ≃ (X ′ ,φ ′ ,τ ′ ) isom. iff∃ (f,F):(X, L) → (X ′ ,L ′ ) G(K)-isom. such that φ = φ ′ · f.Theorem 8.2. Suppose e min (K) ≥ 3. LetN := √ |K|. The functor SQ g,Kof level-G(K) PSQASes (Q, φ, τ) over reduced base schemes is representedby the projective Z[ζ N , 1/N ]-scheme SQ g,K .SQ g,K (T )={(Q, φ, τ); PSQAS with level-G(K) str. over T }For TSQASes we proveTheorem 8.3. ([N10],[N13]) Let N := √ |K|. No restriction on e min (K).The functor SQ toricg,K of level-G(K) TSQASes (P, φ, τ) over reduced baseschemes is coarsely represented by the projective Z[ζ N , 1/N ]-scheme SQ toricg,K .
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