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ON THE PSL (F )-INVARIANT CUBIC SEVENFOLD 2 19 ATANASILIEV,XAVIERROULLEAU 3 1 0 Abstract. IthasbeenprovedbyAdlerthatthereexistsauniquecubic 2 hypersurfaceX7 inP8 whichisinvariantundertheaction ofthesimple n group PSL2(F19). In thepresent note we study theintermediate Jaco- bianofX7andinparticularweprovethatthesubjacent85-dimensional a J torus is an Abelian variety. The symmetry group G = PSL2(F19) de- 7 fines uniquely a G-invariant abelian 9-fold A(X7), which we study in detail and describe its period lattice. ] G A . h Introduction t a Let PSL (F ) be the projective special linear group of order 2 matrices m 2 q over the finite field with q element F . There exist exactly two non trivial q [ irreducible complex representations Wq−1, W¯q−1 of PSL2(Fq) on a space of 1 2 2 v dimension q−1, each one complex conjugated to the other, see [3]. In [1], 2 2 Adler proved that for any q a power of a prime p 7, with p = 3mod 8, 4 there exists (up to a constant multiple) a unique P≥SL (F )-invariant cubic 1 2 q 1 form fq of Wq−1. We call the corresponding unique PSL2(Fq)-invariant 2 . cubic hypersurface 1 0 3 Xq−25 = fq = 0 ֒ P(Wq−1) 1 { } → 2 : the Adler cubic for q. v i In the smallest non-trivial case, q = 11 = 1 8+3, the the Adler cubic in X P4 = P(W11−1) coincides with the Klein cubic·threefold r 2 a X3 = x2x +x2x +x2x +x2x +x2x = 0 . { 1 2 2 3 3 4 4 5 5 1 } This threefold has been introduced by Felix Klein as a higher dimensional analog of the well known Klein quartic curve, the unique curve in the projective plane with symmetry group PSL (F ), see [14]. In [17], the second 2 7 author has proven that there exists a unique Abelian fivefold A(X3) with a PSL (F )-invariant principal polarization and he has explicitly described 2 11 the period lattice of A(X3). In this case, A(X3) coincide with the Griffiths intermediate Jacobian J (X3) of the cubic threefold X3, with the principal G polarization coming from the intersection of real 3-cycles on X3. 2000 Mathematics Subject Classification. 11G10, 14J50, 14J70. Keywordsandphrases. Cubicsevenfold,automorphismgroup,IntermediateJacobian. 1 2 ATANASILIEV,XAVIERROULLEAU Inthispaperwestudythenextcase-theAdlercubicforq = 19 = 2 8+3. · By using the general descriptions from Theorem 4 of [1], one can find an equation of the Adler cubic X7: f = x2x +x2x +x2x +x2x +x2x +x2x +x2x +x2x +x2x 19 1 6 6 2 2 7 7 4 4 5 5 8 8 9 9 3 3 1 2(x x x +x x x +x x x ). 1 7 8 2 3 5 4 6 9 − In the first section, we study a similar invariant principally polarized Abelian ninefold A(X7) defined uniquely by the Adler cubic sevenfold, and compute the period lattice and the first Chern class of the polarization of A(X7). In the second section, we show that on the 85-dimensional complex torus J = (H5,2 H4,3) /H (F ,Z)oftheGriffithsintermediateJacobianJ (X7) ∗ 7 19 G of X7, one⊕can introduce a structure of a PSL (F )-invariant polarization 2 19 which provides J with a structure of an Abelian variety. In the third section, we study invariant properties of the Adler-Klein pencil of cubic sevenfolds, an analog of the Dwork pencil of quintics threefolds see e.g. [9]. Notice that from the point of view of variation of Hodge structure, the cubic sevenfolds can be considered as higher dimensional analogs of Calabi-Yau threefolds, see [7] and [13]. Acknowledgements. We thank Bert van Geemen for its comments on a preliminaryversionofthispaper. Thesecondauthoracknowledgesthehospi- tality of the Seoul National University where part of this work was done. He is a member of the project Geometria Algebrica PTDC/MAT/099275/2008 and was supported by FCT grant SFRH/BPD/72719/2010. 1. The invariant Abelian 9-fold of the Adler cubic 7-fold. We begin by some notations. Let e ,...,e be a basis of a 9-dimensional 1 9 vector space V. Let τ GL(V) be the order 19 automorphism defined by ∈ τ :e ξj2e , j j → where ξ = e2iπ/19, i2 = 1, andlet beσ GL(V)the order 9automorphism − ∈ defined by σ : e e , k 6j → | | where for any integer t the notation t means the unique integer 0 u 9 | | ≤ ≤ such that t = u mod 19. We denote by µ the order 2 automorphism given ± in the basis e , ,e by the matrix: 1 9 ··· i kj µ = (ξkj ξ kj) − (cid:18)√19 (cid:18)19(cid:19) − (cid:19) 1 k,j 9 ≤ ≤ where kj is the Legendre symbol. The group generated by τ,σ and µ is 19 isomorp(cid:16)hic(cid:17)to PSL (F ) and define a representation of PSL (F ), see [3]. 2 19 2 19 ON THE PSL2(F19)-INVARIANT CUBIC SEVENFOLD 3 Let us define v = τk(e + +e ) = k 1 9 ··· = ξke +ξ4ke +ξ9ke +ξ16ke +ξ6ke +ξ17ke +ξ11ke +ξ7ke +ξ5ke 1 2 3 4 5 6 7 8 9 (thus v = v ) and k k+19 1 w = (v 5v +10v 10v +5v v ), k′ 1+2ν k − k+1 k+2− k+3 k+4− k+5 where ν = k=9ξk2 = −1+i√19. 2 Xk=1 An endomorphism h of a torus A acts on the tangent space T by its A,0 differential dh, which wecallthe analytic representation of h. For theeaseof thenotations,wewillusethesameletterforhanditsanalyticrepresentation. In the present section, we will prove the following Theorem: Theorem 1. There exist: (1) a 9-dimensional torus A = V/Λ, T = V such that the elements τ, A,0 σ of GL(V) are the analytic representations of automorphisms of A. The torus A is isomorphic to the Abelian variety E9, where E is the elliptic curve C/Z[ν], (ν = 1+i√19). − 2 (2) a unique principal polarization Θ on A which is invariant by the automorphisms σ,τ. The period lattice of A is then: Z[ν] Z[ν] Z[ν] Z[ν] 8 H (A,Z) = w + w + w + w + Z[ν]v , 1 1+2ν 0′ 1+2ν 1′ 1+2ν 2′ 1+2ν 3′ k Mk=4 and c (Θ) = 2 k=9dx dx¯ , where x ,...,x is the dual basis of the 1 √19 k=1 k ∧ k 1 9 e ’s. P j In order to prove Theorem 1, we first suppose that such a torus A = V/Λ exists and find the necessary condition for its existence. We then check immediately that these conditions are also sufficient, and that they give us the uniqueness of A. Let Λ V be a lattice such that τ and σ of GL(V) are analytic represen- ⊂ tations of automorphisms of the torus A = V/Λ. Let ℓ = ξkx +ξ4kx +ξ9kx +ξ16kx +ξ6kx +ξ17kx +ξ11kx +ξ7kx +ξ5kx . k 1 2 3 4 5 6 7 8 9 Let q be the endomorphism q = j=8σj. For z = j=9x e , we have j=0 j=1 j j q(z) = ℓ (z)v , thus the image of qPis an elliptic curve EPcontained in A. 0 0 The restriction of q τ : A E to E ֒ A is the multiplication by ν = ℓ (τv ). Thus the◦endomorp→hism group→End E of E contains the ring 0 0 Z[ν] ; since this is a maximal order, we have End E = Z[ν]. We remark that the ring Z[ν] is one of the 9 rings of integers of quadratic fields that are 4 ATANASILIEV,XAVIERROULLEAU Principal Ideal Domains, see [15]. Since Z[ν] is a PID and H (A,Z) Cv is 1 0 a rank one Z[ν]-module, there exists a constant c C such that ∩ ∗ ∈ H (A,Z) Cv = Z[ν]cv . 1 0 0 ∩ Up to normalization of the e ’s, we can suppose that c = 1. j Let Λ V be the Z-module generated by the v , k Z/19Z. The group 0 k Λ is stab⊂le under the action of τ and Λ H (A,Z) =∈Λ. 0 0 1 ⊂ Lemma 2. The Z-module Λ H (A,Z) is equal to the lattice: 0 1 ⊂ k=8 R = Z[ν]v . 0 0 Xk=0 Proof. We have νv0 = kk==91vk2 hence νv0 is an element of Λ0. This implies that the vectors νvk =Pτkνv0 are elements of Λ0 for all k, hence: R0 Λ0. ⊂ Conversely, we have: v = v +(1+ν)v 2v +(1 ν)v +(3+ν)v +( 2+ν)v (2+ν)v +2v +νv 9 0 1 2 3 4 5 6 7 8 − − − − andsimilarformulasforv ,...,v . Thisproves thatthelatticeR contains 10 17 0 the vectors v = τkv generating Λ , thus: R = Λ . (cid:3) k 0 0 0 0 Now we construct a lattice that contains H (A,Z). Let be k Z/19Z. 1 ∈ The image of z V by the endomorphism q τk : V V is ∈ ◦ → q τk(z) = ℓ (z)v . k 0 ◦ Let be λ H (A,Z). Since H (A,Z) Cv = Z[ν]v , the scalar ℓ (λ) is an 1 1 0 0 k element o∈f Z[ν]. Let ∩ Λ = z V ℓ (z) Z[ν], 0 k 8 . 8 k { ∈ | ∈ ≤ ≤ } Lemma 3. The Z-module Λ H (A,Z) is the lattice: 8 1 ⊃ k=7 Z[ν] (v v )+Z[ν]v . k k+1 0 1+2ν − Xk=0 Moreover τ stabilizes Λ . 8 Proof. Let ℓ ,...,ℓ be the basis dual to ℓ ,...,ℓ . By definition, the Z[ν]- ∗0 ∗8 0 8 module Λ is i=8Z[ν]ℓ . By expressing the ℓ in the basis v ,...,v we 8 ⊕i=0 ∗i ∗i 0 8 obtain the lattice Λ . Using the formula: 8 v v = v +(1+ν)v 2v +(1 ν)v +(3+ν)v 9 8 0 1 2 3 4 − − − +(ν 2)v (2+ν)v +2v +(ν 1)v , 5 6 7 8 − − − one can check that v v is a Z[ν]-linear combination of (2ν+1)v and the 9 8 0 − v v for k = 0,...,7. Therefore τ( 1 (v v )) = 1 (v v ) is in Λk −andk+Λ1 is stable by τ. 2ν+1 8− 7 2ν+1 9− 8 (cid:3) 8 8 ON THE PSL2(F19)-INVARIANT CUBIC SEVENFOLD 5 Wedenoteby φ: Λ Λ /Λ thequotient map. Thering Z[ν]/(1+2ν) is 8 8 0 thefinitefieldwith19el→ements. ThequotientΛ /Λ isaZ[ν]/(1+2ν)-vector 8 0 space with basis t ,...,t such that t = 1 (v v )+Λ . 1 8 i 1+2ν i−1− i 0 Let R be a lattice such that : Λ R Λ . The group φ(R) is a 0 8 ⊂ ⊂ sub-vector space of Λ /Λ and: 8 0 φ 1φ(R)= R+Λ = R. − 0 The set of such lattices R correspond bijectively to the set of sub-vector spaces of Λ /Λ . 8 0 Because the automorphism τ preserves Λ , it induces an automorphism τ on 0 the quotient Λ /Λ such that φ τ = τ φ. As τ stabilizes H (A,Z), the 8 0 1 vector space φ(H (A,Z)) is stabl◦e by τ. L◦et b 1 b j=k k b w = ( 1)k ( 1)jt , k = 0...7. 8−k − (cid:18) j (cid:19) − j+1 Xj=0 Then w = τw w for i > 1 and τw = w . The matrix of τ in the basis i 1 i i 1 1 − − w ,....,w is the size 8 8 matrix: 1 8 b × b b 1 1 0 ... 0  0 ... ... ... ...   ... ... ... ... 0     ... ... ... 1     0 ... ... 0 1    The sub-spaces stable by τ are the spaces W , 1 j 8 generated by j ≤ ≤ w ,...,w and W = 0 . Let Λ be the lattice φ 1W . It is easy to 1 j 0 j − j check that, as a lattice{in}bC9, the Λ are all isomorphic to Z[ν]9. Since j Λ= H (A,Z) is stable by τ, we have proved that : 1 Lemma 4. The torus A = V/Λ exists, it is an Abelian variety isomorphic to E9. There exists j 0,1, ,8 such that Λ = Λ . j ∈ { ··· } Let w ,...,w be the vectors defined by: 0′ 3′ 1 1 w = (v 5v +10v 10v +5v v )= (1 τ)5v 0′ 1+2ν 0− 1 2 − 3 4− 5 1+2ν − 0 and w = τkw (we have φ(w ) = w ). Let us suppose that A = V/Λ has k′ 0′ 0′ 4 moreover a principal polarization Θ that is invariant under the action of τ and σ. Then: Lemma 5. The lattice H (A,Z) is equal to Λ , and 1 4 Z[ν] Z[ν] Z[ν] Z[ν] 8 Λ = w + w + w + w + Z[ν]v . 4 1+2ν 0′ 1+2ν 1′ 1+2ν 2′ 1+2ν 3′ k Mk=4 TheHermitian matrixassociated to Θ is equal to 2 I in the basis e ,...,e √19 9 1 9 and c (Θ)= i k=9dx dx¯ . 1 √19 k=1 k ∧ k P 6 ATANASILIEV,XAVIERROULLEAU Proof. Let H be the matrix of the Hermitian form of the polarization Θ in the basis e ,...,e . Since τ preserves the polarization Θ, this implies that: 1 9 tτHτ¯= H where τ¯ is the matrix whose coefficients are conjugated of those of τ. The only Hermitian matrices that verify this equality are the diagonal matrices. By the same reasoning with σ instead of τ, we obtain that these diagonal coefficients are equal, and: 2 H = a I , 9 √19 where a is a positive real (H is a positive definite Hermitian form). As H is a polarization, the alternating form c (Θ) = m(H) takes integer values on 1 H (A,Z), hence m(tv Hv¯ ) = a is an integeℑr. 1 2 1 ℑ Let c (Θ) = m(H) = i a dx dx¯ be the alternating form of the 1 ℑ √19 k ∧ k principal polarization Θ. Let Pλ ,...,λ be a basis of a lattice Λ. By 1 18 ′ definition, the square of the Pfaffian Pf (Λ) of Λ is the determinant of the Θ ′ ′ matrix MΛ′ = (c1(Θ)(λj,λk))1 j,k 18. Since Θ is a principal polarization on A, we h≤ave≤Pf (H (A,Z)) = 1. Θ 1 For j 0,...,8 , it is easy to find a basis of the lattice Λ . We compute j ∈ { } that P2 = a18198 2j j − where P its Pfaffian of Λ . As a is positive, the only possibility that P j j j equals 1 is j = 4 and a = 1. Moreover since all coefficients of the matrix M are integers, the alternating form i dx dx¯ defines a principal Λ4 √19 k ∧ k polarization on A, see [6, Chap. 4.1]. P (cid:3) Theorem 1 follows from Lemmas 4 and 5. Remark 6. One can compute that the involution µ acts on the principally polarized Abelian variety (A,Θ), and therefore that Aut(A,Θ) contains the group PSL (F ). 2 19 Remark 7. In[5], Beauville proves that the intermediate Jacobian of the S - 6 symmetric quartic threefolds F is not a product of Jacobians of curves (see also [4]). That implies, using Clemens-Griffiths results for threefolds, the irrationality of F. Using the arguments at the end of [5], one can see that theprincipallypolarizedabelianninefold(A,Θ)isnotaproductofJacobians of curves. 2. Intermediate Jacobians of the Adler cubic sevenfold 2.1. Intermediate Jacobian. For asmooth algebraic complex manifold X of dimension n= 2k+1, let Hp,q(X) be the Hodge (p,q)-cohomology spaces Hp,q(X) = Hq(X,Ωp), andhp,q =dim Hp,q(X) betheHodgenumbers of X, 0 p+q 2n = 4k+2. The intersection of real (2k+1)-dimensional chains ≤ ≤ ON THE PSL2(F19)-INVARIANT CUBIC SEVENFOLD 7 inX defines an integer valued quadratic form Q on the integer cohomology G H (X,Z), and yields an embedding of H (X,Z) as an integer lattice 2k+1 2k+1 in the dual space of H2k+1,0(X)+H2k,1(X)+...+Hk+1,k(X). The quotient compact complex torus J (X) = (H2k+1,0(X)+H2k,1(X)+...+Hk+1,k(X)) /H (X,Z) G ∗ 2k+1 is the Griffiths intermediate Jacobian of the n-fold X, see [8, p.123]. By the Riemann-Hodge bilinear relations, the quadratic form Q is definite on any G space Hp,n p(X), and has opposite signs on Hp,n p(X) and Hp′,n p′(X) if − − − andonlyifp p odd. Moreover m(Q )takesintegralvaluesonH (X,Z). ′ G 2k+1 − ℑ We call the pair (J (X),Q ) the polarized Griffiths intermediate Jacobian. G G The quadratic form Q gives the polarized torus (J (X),Q ) a structure G G G of a polarized Abelian variety if and only if p has the same parity for all non-zero spaces Hp,n p(X), ([8, p.123]). − 2.2. Griffiths formulas. The middle Hodge groups of a smooth cubic sevenfold can be computed by the following Griffiths formulas, which we shall use below in order to determine the representation of the action of the symmetry group PSL (F ) on the middle cohomology of the Adler cubic sev- 2 19 enfold. Let X = (f(x) = 0) be a smooth hypersurface of degree m 2 in the complex projective space Pn+1(x), (x) = (x ,...,x ). Let S =≥ S be 1 n+2 d 0 d the graded polynomial ring S = C[x ,...,x ], with S being th⊕e s≥pace of 0 n+2 d polynomials of degree d. Let I = I S be the graded ideal generated by d ⊕ ⊂ the n+2 partials ∂f , j = 1,...,n+2, with I = I S . ∂xj d ∩ d Let R = S/I = R be the graded Jacobian ring of X (or the Jacobian ring d ⊕ of the polynomial f(x)), with graded components R = S /I . Then for d d d p,q p+q = n, the primitive cohomology group H (X) is isomorphic to the prim graded piece R , where R = 0 for d < 0, ([8], p. 169). For odd m(q+1) n 2 d n= 2k+1 all the mid−d−le cohomology of X are primitive, and in this case Hp,q(X) = R , p+q = 2k+1= dimX, m = degX. ∼ m(q+1) n 2 − − In particular, for a smooth cubic sevenfold X = (f(x)= 0) P8 one has ⊂ H7 q,q(X) = R , 0 q 7, − 3(q+1) 9 − ≤ ≤ which yields H7,0(X) = R = 0, H6,1(X) = R = 0, 6 3 H5,2(X) =R−= C, H4,3(X) = R −= C84. 0 ∼ 3 ∼ 2.3. Character table of PSL (F ). In Proposition 8 below, we shall use 2 19 theknowndescriptionoftheirreduciblerepresentationsoftheautomorphism group PSL (F ) Aut(X), which we state here in brief: 2 19 By [3], the grou⊂p PSL (F ) has 12 conjugacy classes: 2 19 1, w , w , x , x2 , x3 , x4 , y , y2 , y3 , y4 , y5 1 2 { } { } { } { } { } { } { } { } { } { } { } 8 ATANASILIEV,XAVIERROULLEAU 1 1 where x has order 9, y has order 10, w = , w = w2 (we observe 1 (cid:18) 0 1 (cid:19) 2 1 that w2,w3 w , w2,w3 w ). Correspondingly, the 12 irreducible represe1ntat1io∈ns{of2P}SL2(F2 )∈ar{e 1} 2 19 T ,W ,W ,W1 ,W2 ,W3 ,W4 ,W1 ,W2 ,W3 ,W4 ,W 1 9 9 18 18 18 18 20 20 20 20 19 wherethenotationsareuniquelydefinedbythecharactertableofPSL (F ) 2 19 below for which a = 2cos(2kπ), b = 2cos(kπ) and ν = 1+i√19. Note k 9 k − 5 − 2 that the representation W is described otherwise in section 1. 9 1 w w x x2 x3 x4 y y2 y3 y4 y5 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 W 9 ν ν¯ 0 0 0 0 1 -1 1 -1 1 9 W 9 ν¯ ν 0 0 0 0 1 -1 1 -1 1 9 W1 18 -1 -1 0 0 0 0 b b b b 2 18 1 2 3 4 W2 18 -1 -1 0 0 0 0 b b b b -2 18 2 4 6 8 W3 18 -1 -1 0 0 0 0 b b b b 2 18 3 6 9 12 W4 18 -1 -1 0 0 0 0 b b b b -2 18 4 8 12 16 W1 20 1 1 a a a a 0 0 0 0 0 20 1 2 3 4 W2 20 1 1 a a a a 0 0 0 0 0 20 2 4 6 8 W3 20 1 1 a a a a 0 0 0 0 0 20 3 6 9 12 W4 20 1 1 a a a a 0 0 0 0 0 20 4 8 12 16 W 19 0 0 1 1 1 1 -1 -1 -1 -1 -1 19 Figure 2.1. Character table of PSL (F ) 2 19 2.4. Periods of the Adler cubic. LetX P8 = P(W )betheAdlercubic 9 ⊂ sevenfold f = x2x +x2x +x2x +x2x +x2x +x2x +x2x +x2x +x2x { 19 1 6 6 2 2 7 7 4 4 5 5 8 8 9 9 3 3 1 2(x x x +x x x +x x x )= 0 , 1 7 8 2 3 5 4 6 9 − } and let Hp,q = Hp,q(X) be the Hodge cohomology spaces of X. Proposition 8. The representation of the group PSL (F ) on the 84- 2 19 dimensional space H4,3(X)∗ of the Adler cubic X is: H4,3(X) = W W1 W3 W W3 . ∗ 9⊕ 18⊕ 18⊕ 19⊕ 20 The group PSL (F ) acts trivially on the one-dimensional space H5,2(X) . 2 19 ∗ Remark 9. Let a group G acts on a vector space V on the left. Then the group G acts on the right on the dual space V : for ℓ V and g G, ∗ ∗ ∈ ∈ ℓ g = ℓ g. Since the traces of the action of g on V and on V are equal, ∗ · ◦ the two representations V and V are isomorphic. The representation V ∗ ∗ should not be confused with the dual representation of G defined such that the pairing between V and V is G-invariant (see [11]). ∗ ON THE PSL2(F19)-INVARIANT CUBIC SEVENFOLD 9 Proof. In order to decompose the representation of PSL (F ) Aut(X) 2 19 ⊂ on the dual cohomology space H4,3∗, we shall use the identification as representation space H4,3(X) = R = S /I = Sym3W /I ∼ 3 3 3 9 3 between H4,3(X) and the graded component of degree 3 in the quotient polynomial ring S = C[x ,...,x ] by the Jacobian ideal I spanned by the 9 1 9 partials of f , see above. 19 The space P8 containing the Adler cubic X is the projectivization of the representation space W = (S ) . We have S = Sym3(W ) = Sym3(W ). 9 1 ∗ 3 9∗ ∼ 9 Let us decompose Sym3W into irreducible representations of PSL (F ). 9 2 19 The character of the third symmetric power of a representation V is: 1 χSym3V(g) = 6(χV(g)3 +3χV(g2)χV(g)+2χV(g3)), see [16]. Therefore the traces of the action of the elements 1,w ,w etc on 1 2 Sym3W are 9 v = t(165,3 ν,3 ν¯,0,0,3,0,0,0,0,0,5). − − Using the character table of PSL (F ), we obtain: 2 19 Sym3W = T W W1 W3 W1 W2 (W3 ) 2 W4 W . 9 1⊕ 9⊕ 18⊕ 18⊕ 20⊕ 20⊕ 20 ⊕ ⊕ 20⊕ 19 ThegradedcomponentI oftheJacobianidealI ofX isgeneratedbythe9 2 derivatives df19, k =1,...,9. ThespaceI is arepresentation ofPSL (F ). dxk 2 2 19 The action of w on x is the multiplication by ξj2, where ξ = e2iπ/19, see 1 j section1. Theaction ofw onI isthen easytocompute. Byexample, since 1 2 df 19 = 2x x +x2 2x x , dx 1 6 3− 7 8 1 we get w df19 = ξ 1df19. By looking at the character table, we obtain 1 · dx1 − dx1 that the representation I is W . Therefore I = W W . Using the 2 9 3 9 9 ⊗ fact that for two representations V ,V , their characters satisfy the relation 1 2 χ = χ χ , we obtain: V1 V2 V1 V2 ⊗ I = W W = T W1 W2 W3 W4 . 3 9⊗ 9 1⊕ 20⊕ 20⊕ 20⊕ 20 Since H4,3(X)∗ ∼= (S3/I3)∗ ∼= Sym3W9/I3 then H4,3(X) = W W1 W3 W W3 . ∗ 9⊕ 18⊕ 18⊕ 19⊕ 20 The action of the group PSL (F ) Aut(X) on H5,2(X) = R = C is trivial, see the character table. 2 19 ⊂ ∗ ∼ 0∗ ∼ (cid:3) Let V be a representation of a finite group G and let W be an irreducible representation of G of character χ . We know that there exist a uniquely W determined integer a 0 and a representation V of G such that W is not a ′ ≥ 10 ATANASILIEV,XAVIERROULLEAU sub-representation ofV andV is(isomorphic to)W a V . By[11]formula ′ ⊕ ′ ⊕ (2.31) p. 23, the linear endomorphism dimW ψ = χ (g)g : V V W W G → | | gXG ∈ is the projection of V onto the factor consisting of the sum of all copies of W appearinginV i.e. istheprojectionontoW a. Forthetrivialrepresentation ⊕ T, χ = 1 and ψ is the projection of V onto the invariant space VG. T T Recall that an endomorphism h of a torus Y acts on the tangent space TY of Y by a linear endomorphism dh called the analytic representation of h. The kernel of the map End(Y) End(TY), h dh is the group of → → translations (see [6]) ; in the following we will work with endomorphisms up to translation. Suppose that for some irreducible representations W ,...,W the sum 1 k χ (g)+ +χ (g) W1 ··· Wk is an integer for every g. Then the endomorphism h = (χ (g)+ +χ (g))g End(J X) W1 ··· Wk ∈ G gXG ∈ is well defined and its analytic representation is dh = (χ (g)+ +χ (g))dg End(TJ X). W1 ··· Wk ∈ G gXG ∈ The tangent space of the image of h (translated to 0) is the image of dh. Corollary 10. The torus J X has the structure of an Abelian variety and G is isogenous to: E A A A A 9 36 19 20 × × × × where A is a k-dimensional Abelian subvariety of J X and E J X k G G an elliptic curve. The group PSL (F ) acts nontrivially on each⊂factors 2 19 A ,A ,A ,A . 9 36 19 20 Proof. Thecharacterofthetrivialrepresentationisχ = 1. Letχ ,χ ,χ ,χ 0 1 2 3 4 be respectively the characters of the representations W W , W1 W2 9⊕ 9 18⊕ 18⊕ W3 W4 , W and W1 W2 W3 W4 . By the character table, the nu1m8b⊕ers 1χ8 (g)1,9g PSL20(⊕F )20ar⊕e int2e0g⊕ers, 2t0herefore we can define k 2 19 ∈ q = χ (g)g k k g PSXL2(F19) ∈ for k = 0,...,4. The analytic representation of q is a multiple of the pro- k jection onto the subspace (H5,2) ,W ,... of TJ X. The images of q , k = ∗ 9 G k 0, ,4arethereforerespectively1,9,36,19,20-dimensional sub-toriofJ X, G sta·b··lebytheactionofthegroupringZ[PSL (F )]anddenotedrespectively 2 19 by E,A ,...,A . 9 20