A SHORT CLASSIFICATION OF FINITE SUBGROUPS OF SL(3,C) 0 ...

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4 YASUSHI GOMI, IKU NAKAMURA, KEN-ICHI SHINODAIn this case N = 〈S, T, R〉. For this, let σ ∈ N. Then we may assume σ(V i )=V ifor any i and σ(U 1 ) = U 1 by taking a suitable product of S j T k and σ for σ ifnecessary. Then we see σ(v i )=a i v i and a 1 = a 2 = a 3 . Hence σ = ω k for some k.Therefore σ ∈〈S, T, R〉 and N = 〈S, T, R〉.Finally we consider Case (2iv). We assume Case (iv) always occurs for any N-invariant decomposition. Then by the argument in Case (1iv), S = ω k T for somek. We may assume by the assumption of Case 2 that there is an R ∈ N whichfixes exactly one of V i , hence we assume R(V 1 )=V 2 , R(V 2 )=V 1 and R(V 3 )=V 3 .Similarly we may assume as before R(U 1 )=U 1 . Then we see as before R 2 = ω k .If k ≠ 0, then this shows ω ∈ N and we take ω 2k R for R. Hence we may assumeR 2 = id. It follows R(v 1 )=−v 2 , R(v 2 )=−v 1 , R(v 3 )=−v 3 . Let N ′ be the groupgenerated by ω, T , and R. Then N is a subgroup of N ′ . In fact, let σ ∈ N. Then wemay assume σ(V i )=V i for any i, hence σ(v i )=a i v i . By the assumption of Case 2,σ fixes at least one of U i .Ifσ(U i )=U i for any i, then σ = ω k by Sublemma. Nextassume σ(U 1 )=U 2 , σ(U 2 )=U 1 and σ(U 3 )=U 3 . Let u 3 = pv 1 + qv 2 + rv 3 . Thenpqr ≠ 0 and a 1 = a 2 = a 3 = ω k . It follows that N is a subgroup of N ′ . Howeverv 1 + v 2 + v 3 spans an N ′ -module, which contradicts that V is N-irreducible. HenceCase (iv) does not always occurs. Hence one of Cases (i)-(iii) occurs for somedecomposition U· of V . By Case (2i) N = 〈S, T, R〉 and there are exactly fourN-invariant decompositions of V . This completes the proof of the lemma.2.3. Hesse cubics. In this subsection we denote v i by x i−1 by convention. Thegroup PSL(2, Z) operates on the upper half plane H, while Γ(3) operates on theupper half plane H and the quotient curve H/Γ(3) has the universal elliptic curvewith level 3 struture on it, which is isomorphic to the family of Hesse cubics in P 2x 3 0 + x 3 1 + x 3 2 − 3µx 0 x 1 x 2 =0.It has nine 3-division points [1, −β,0], [0, 1, −β] and [−β,0, 1] (β: cube roots ofunity) if µ ≠ ∞ or µ 3 ≠ 1. The group of all 3-division points with e 0 the zero is(Z/3) ⊕2 generated by e 1 and e 2 , where we defineWe define R bye 0 =[1, −1, 0], e 1 =[1, −ω, 0], e 2 =[−1, 0, 1].R(x 0 )=−x 1 , R(x 1 )=−x 0 , R(x 2 )=−x 2Then R acts also on the family of Hesse cubics as the involution −id. of ellipticcurves which keeps the origin [1, −1, 0] invariant. The translation by e 1 or e 2 resp.is lifted to G(3) as S or T resp.S(x i )=ω i x i , T(x i )=x i+1 ,Thus G(3) acts on the family of Hesse cubics as translation by 3-division points.If µ = ∞ and µ 3 = 1, then the curve C(µ) is a union of three lines with threeordinary double points, say a 3-gon of rational curves. The j-invariant of ellipticcurves is given by the formula,j =27 µ3 (µ 3 +8) 3(µ 3 − 1) 3 .
FINITE SUBGROUS 5This is invariant under the transformations of µ:u : µ ↦→ ωµ, v : µ ↦→ µ +2µ − 1 ,which generate the Galois group PSL(2, F 3 )ofH/Γ(3) over H/ SL(2, Z) as we seebelow. These are induced from the linear transformations U and V ∈ SL(3, C)u k = U(x k )=λ(x k )(k =0, 1), u 2 = U(x 2 )=λω 2 x 2 , λ 3 = ω,v k = V (x k )= 1 √ −3(ω 2−k x 0 + ω 4−2k x 1 + x 2 ),where V 2 = R, U 3 = 1 and UT = STU. See also [Hesse, p. 83 (1844)] and[Blichfeldt17, p. 108]. Moreover we see⎧⎧⎧⎪⎨ R(e 0 )=e 0 , ⎪⎨ U(e 0 )=e 0 , ⎪⎨ V (e 0 )=e 0 ,R(e 1 )=−e 1 , U(e 1 )=e 1 ,V (e 1 )=−e 2 ,⎪⎩⎪⎩⎪⎩R(e 2 )=−e 2 , U(e 2 )=e 1 + e 2 , V (e 2 )=e 1 .We also note that U and V (as transformations of (Z/3) ⊕2 ) generate the Galoisgroup PSL(2, F 3 ). Let G largest be the largest subgroup of SL(3, C) consisting ofliftings of Aut(Hesse cubics), the projective automorphism group of Hesse cubics.Any element, say σ of G largest induces an automorphism of the family of Hessecubics. Hence it induces an element of PSL(2, F 3 ). Therefeore by modifying V andW we may assume σ maps each Hesse cubic onto itself, hence G largest maps all theinflection points of the cubics onto themselves, hence all 3-division points (in Pic 0 )onto 3-division points so that the induced map on the group of 3-divisions points(in Pic 0 ) is the identity in PSL(2, F 3 ), hence ±id. on (Z/3) ⊕2 .Ife 0 is not fixed bythe automorphism, then by modifying it by S, T and R, we may assume e 0 is fixed,hence Then by modifying it by R, we may assume it induces the identity in Pic 0 .Thus σ is a translation of the elliptic curve. Since it is an automorphism of Hessecubics, we may assume it fixes any 3-division points by modifying S and T . Then σturns out to be the identity because any Hesse cubic is an elliptic curve of level 3.This shows it is in the center of SL(3, C). Thus G largest is generated by R, S, T, U, Vand ω.2.4. Proof of Theorem 2.1. Let F i (i =1, 2, 3, 4) be singular Hesse cubics inthe family of Hesse cubics where F 1 resp. F 2 gives an N-invariant decompositionV = V 1 ⊕ V 2 ⊕ V 3 resp. V = U 1 ⊕ U 2 ⊕ U 3 . We define a homomorphism ψ : G → S 4by σ(F i )=F ψ(σ)(i) for σ ∈ G. Let N ′ =kerψ. Then N ′ be a normal subgroup of Gwhich is imprimitive. From now on we take N ′ or N. Let G be the image of G by ψ.Then G ⊂ A 4 . For this it suffices to check the permutation (3, 4) is not containedin G. Let σ ∈ G such that σ(F i )=F i (i =1, 2).Suppose R ∈ G (hence R ∈ N). Then by modifying σ by S, T and R we mayassume σ(V i )=V i for any i and σ(U 1 )=U 1 . Then σ = ω k as is easily shown.Hence ψ(σ) = 1. If R ∉ G. Then N = 〈S, T, ω〉 and φ U·(N) ≃ Z/3Z for anydecopsition {U·} of V . Then by modifying σ by S, T we may assume σ(V i )=V i forany i and σ(U 1 )=U 1 . It follows σ = ω k and ψ(σ) = 1. Thus G ⊂ A 4 . We noteA 4 ≃ PSL(2, F 3 ).
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