Apéry-Fermi pencil of K3 -surfaces and their 2 -isogenies
aa r X i v : . [ m a t h . AG ] A p r AP´ERY-FERMI PENCIL OF K -SURFACES AND THEIR -ISOGENIES MARIE JOS´E BERTIN AND ODILE LECACHEUX
Abstract.
Given a generic K Y k of the Ap´ery-Fermi pencil, we use the Kneser-Nishiyamatechnique to determine all its non isomorphic elliptic fibrations. These computations lead to deter-mine those fibrations with 2-torsion sections T. We classify the fibrations such that the translationby T gives a Shioda-Inose structure. The other fibrations correspond to a K3 surface identifiedby it transcendental lattice. The same problem is solved for a singular member Y of the familyshowing the differences with the generic case. In conclusion we put our results in the context ofrelations between 2-isogenies and isometries on the singular surfaces of the family. Introduction
The Ap´ery-Fermi pencil F is realised with the equations X + 1 X + Y + 1 Y + Z + 1 Z = k, k ∈ Z , and taking k = s + s , is seen as the Fermi threefold Z with compactification denoted ¯ Z [PS].The projection π s : ¯ Z → P ( s ) is called the Fermi fibration. In their paper [PS], Peters and Stienstraproved that for s / ∈ { , ∞ , ± , ± √ , − ± √ } the fibers of the Fermi fibration are K E ( − ⊕ E ( − ⊕ U ⊕ h− i andtranscendental lattice isometric to T = U ⊕ h i ( U denotes the hyperbolic lattice and E theunimodular lattice of rank 8). Hence this family appears as a family of M -polarized K Y k with period t ∈ H . And we deduce from a result of Dolgachev [D] the following property. Let E t = C / Z + t Z and E ′ t = C / Z + ( − t ) Z be the corresponding pair of isogenous elliptic curves.Then there exists a canonical involution τ on Y k such that Y k / ( τ ) is birationally isomorphic to theKummer surface E t × E ′ t / ( ± K K X (6) / h w , w i . Indeed it can be derived from Peters and Stienstra [PS].In [Shio], Shioda considers the problem whether every Shioda-Inose structure can be extended toa sandwich. More precisely Shioda proved a ”Kummer sandwich theorem” that is, for an elliptic K X (with a section) with two II*-fibres, there exists a unique Kummer surface S = Km ( C × C ) with two rational maps of degree 2, X → S and S → X where C and C are ellipticcurves.In van Geemen-Sarti [G], Comparin-Garbagnati [C], Koike [Ko] and Sch¨utt [Sc], sandwich Shioda-Inose structures are constructed via elliptic fibrations with 2-torsion sections.Recently Bertin and Lecacheux [BL] found all the elliptic fibrations of a singular member Y of F (i.e. of Picard number 20) and observed that many of its elliptic fibrations are endowed with 2-torsion sections. Thus a question arises: are the corresponding 2-isogenies between Y and this new K S all Morrison-Nikulin meaning that S is Kummer? Observing also that the Shioda’sKummer sandwiching between a K S and its Kummer K is in fact a 2-isogeny betweentwo elliptic fibrations of S and K , we extended the above question to the generic member Y k of thefamily F and obtained the following results. Date : April 13, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
Elliptic fibrations of K Theorem 1.1.
Suppose Y k is a generic K surface of the family with Picard number .Let π : Y k → P be an elliptic fibration with a torsion section of order which defines an involution i of Y k (Van Geemen-Sarti involution) then the quotient Y k /i is either the Kummer surface K k associated to Y k given by its Shioda-Inose structure or a surface S k with transcendental lattice T S k = h− i ⊕ h i ⊕ h i and N´eron-Severi lattice N S ( S k ) = U ⊕ E ( − ⊕ E ( − ⊕ h ( − i ⊕ h ( − i ,which is not a Kummer surface by a result of Morrison [M] . The K surface S k is the Hessian K surface of a general cubic surface with nodes studied by Dardanelli and van Geemen [DG] . Thus, π leads to an elliptic fibration either of K k or of S k . Moreover there exist some genus fibrations θ : K k → P without section such that their Jacobian variety satisfies J θ ( K k ) = S k .More precisely, among the elliptic fibrations of Y k (up to automorphisms) of them have a two-torsion section. And only of them possess a Morisson-Nikulin involution i such that Y k /i = K k . Theorem 1.2.
In the Ap´ery-Fermi pencil, the K -surface Y is singular, meaning that its Picardnumber is . Moreover Y has many more -torsion sections than the generic K surface Y k ;hence among its Van Geemen-Sarti involutions, of them are Morrison-Nikulin involutions, are symplectic automorphisms of order (self-involutions) and one exchanges two elliptic fibrationsof Y .The specializations to Y of the Morrison-Nikulin involutions of a generic member Y k are verifiedamong the Morrison-Nikulin involutions of Y , as proved in a general setting by Sch¨utt [Sc] .The specializations of the involutions between Y k and the K -surface S k are among the VanGeemen-Sarti involutions of Y which are not Morrison-Nikulin. This theorem provides an example of a Kummer surface K defined by the product of two isogenouselliptic curves (actually the same elliptic curve of j -invariant equal to 8000), having many fibrationsof genus one whose Jacobian surface is not a Kummer surface. A similar result but concerning aKummer surface defined by two non-isogenous elliptic curves has been exhibited by Keum [K].Throughout the paper we use the following result [Si]. If E denotes an elliptic fibration with a2-torsion point (0 , E : y = x + Ax + Bx, the quotient curve E/ h (0 , i has a Weierstrass equation of the form E/ h (0 , i : y = x − Ax + ( A − B ) x. The paper is organised as follows.In section 2 we recall the Kneser-Nishiyama method and use it to find all the 27 elliptic fibrations ofa generic K F . In section 3, using Elkies’s method of ”2-neighbors” [El], we exhibit anelliptic parameter giving a Weierstrass equation of the elliptic fibration. The results are summarizedin Table 2. Thus we obtain all the Weierstrass equations of the 12 elliptic fibrations with 2-torsionsections. Their 2-isogenous elliptic fibrations are computed in section 5 with their Mordell-Weilgroups and discriminants. Section 4 recalls generalities about Nikulin involutions and Shioda-Inosestructure. Section 5 is devoted to the proof of Theorem 1.1 while section 6 is concerned with theproof of Theorem 1.2.In the last section 7, using a theorem of Boissi`ere, Sarti and Veniani [BSV], we explain why Theorem1.2 cannot be generalised to the other singular K3 surfaces of the family.Computations were performed using partly the computer algebra system PARI [PA] and mostly thecomputer algebra system MAPLE and the Maple Library “Elliptic Surface Calculator” written byKuwata [Ku1]. 2. Elliptic fibrations of the family
We refer to [BL], [Sc-Shio] for definitions concerning lattices, primitive embeddings, orthogonalcomplement of a sublattice into a lattice. We recall only what is essential for understanding thissection and section 5.2.
P´ERY-FERMI PENCIL OF K Discriminant forms.
Let L be a non-degenerate lattice. The dual lattice L ∗ of L is definedby L ∗ := Hom( L, Z ) = { x ∈ L ⊗ Q / b ( x, y ) ∈ Z for all y ∈ L } and the discriminant group G L by G L := L ∗ /L. This group is finite if and only if L is non-degenerate. In the latter case, its order is equal tothe absolute value of the lattice determinant | det( G ( e )) | for any basis e of L . A lattice L is unimodular if G L is trivial.Let G L be the discriminant group of a non-degenerate lattice L . The bilinear form on L extendsnaturally to a Q -valued symmetric bilinear form on L ∗ and induces a symmetric bilinear form b L : G L × G L → Q / Z . If L is even, then b L is the symmetric bilinear form associated to the quadratic form defined by q L : G L → Q / Z q L ( x + L ) x + 2 Z . The latter means that q L ( na ) = n q L ( a ) for all n ∈ Z , a ∈ G L and b L ( a, a ′ ) = ( q L ( a + a ′ ) − q L ( a ) − q L ( a ′ )), for all a, a ′ ∈ G L , where : Q / Z → Q / Z is the natural isomorphism. The pair ( G L , b L ) (resp. ( G L , q L ) ) is called the discriminant bilinear (resp. quadratic ) form of L .The lattices A n = h a , a , . . . , a n i ( n ≥ D l = h d , d , . . . , d l i ( l ≥ E p = h e , e , . . . , e p i ( p = 6 , ,
8) defined by the following
Dynkin diagrams are called the root lattices . All thevertices a j , d k , e l are roots and two vertices a j and a ′ j are joined by a line if and only if b ( a j , a ′ j ) = 1.We use Bourbaki’s definitions [Bou]. The discriminant groups of these root lattices are given below. A n , G A n Set[1] n = n +1 P nj =1 ( n − j + 1) a j then A ∗ n = h A n , [1] n i and G A n = A ∗ n /A n ≃ Z / ( n + 1) Z .q A n ([1] n ) = − nn +1 . a a a a n D l , G D l . Set[1] D l = (cid:16)P l − i =1 id i + ( l − d l − + ld l (cid:17) [2] D l = P l − i =1 d i + ( d l − + d l )[3] D l = (cid:16)P l − i =1 id i + ld l − + ( l − d l (cid:17) then D ∗ l = h D l , [1] D l , [3] D l i ,G D l = D ∗ l /D l = h [1] D l i ≃ Z / Z if l is odd, d l d l − d l − d d l − G D l = D ∗ l /D l = h [1] D l , [2] D l i ≃ Z / Z × Z / Z if l is even .q D l ([1] D l ) = − l , q D l ([2] D l ) = − , b D l ([1] , [2]) = − . E p , G E p p = 6 , , . Set[1] E := η = − (2 e + 3 e + 4 e + 6 e + 5 e + 4 e ) and[1] E := η = − (2 e + 3 e + 4 e + 6 e + 5 e + 4 e + 3 e ),then E ∗ = h E , η i , E ∗ = h E , η i and E ∗ = E . e e e e p e G E = E ∗ /E ≃ Z / Z , G E = E ∗ /E ≃ Z / Z ,q E ( η ) = − q E ( η ) = − . Let L be a Niemeier lattice (i.e. an unimodular lattice of rank 24). Denote L root its root lattice. Weoften write L = N i ( L root ). Elements of L are defined by the glue code composed with glue vectors.Take for example L = N i ( A D E ). Its glue code is generated by the glue vector [1 , ,
1] wherethe first 1 means [1] A , the second 1 means [1] D and the third 1 means [1] E . In the glue code h [1 , (0 , , i , the notation (0 , ,
2) means any circular permutation of (0 , , M.-J. BERTIN AND ODILE LECACHEUX L root L/L root glue vectors E (0) 0 D E Z / Z h [1 , i D E ( Z / Z ) h [1 , , , [3 , , i A E Z / Z h [3 , i D Z / Z h [1] i D ( Z / Z ) h [1 , , [2 , i D ( Z / Z ) h [1 , , , [1 , , , [2 , , i A D Z / Z h [2 , i E ( Z / Z ) h [1 , (0 , , i A D E Z / Z h [1 , , i D ( Z / Z ) h even permutations of [0 , , , i A D Z / Z × Z / Z h [2 , , , [5 , , , [0 , , i A D Z / Z × Z / Z h [1 , , , , [1 , , , i Table 1.
Some Niemeier lattices and their glue codes [Co]their root lattices and glue codes used in the paper are given in Table 1 (glue codes are taken fromConway and Sloane [Co]).2.2.
Kneser-Nishiyama technique.
We use the Kneser-Nishiyama method to determine all theelliptic fibrations of Y k . For further details we refer to [Nis], [Sc-Shio], [BL], [BGL]. In [Nis], [BL],[BGL] only singular K T ( Y k ) be the transcendental lattice of Y k , that is the orthogonal complement of N S ( Y k ) in H ( Y k , Z ) with respect to the cup-product. The lattice T ( Y k ) is an even lattice of rank r = 22 −
19 =3 and signature (2 , t = r − T ( Y k )[ −
1] admits a primitive embedding into the followingindefinite unimodular lattice: T ( Y k )[ − ֒ → U ⊕ E where U denotes the hyperbolic lattice and E the unimodular lattice of rank 8. Define M as theorthogonal complement of a primitive embedding of T ( Y k )[ −
1] in U ⊕ E . Since T ( Y k )[ −
1] = − −
12 0 − , it suffices to get a primitive embedding of ( −
12) into E . From Nishiyama [Nis] we find the followingprimitive embedding: v = h e + 6 e + 12 e + 18 e + 15 e + 12 e + 8 e + 4 e i ֒ → E , giving ( v ) ⊥ E = A ⊕ D . Now the primitive embedding of T ( Y k )[ −
1] in U ⊕ E is defined by U ⊕ v ;hence M = ( U ⊕ v ) ⊥ U ⊕ E = A ⊕ D . By construction, this lattice is negative definite of rank t + 6 = 1 + 6 = r + 4 = 3 + 4 = 26 − ρ ( Y k ) = 7 with discriminant form q M = − q T ( Y k )[ − = q T ( Y k ) = − q NS ( Y k ) . Hence M takes exactly the shape required for Nishiyama’s technique.All the elliptic fibrations come from all the primitive embeddings of M = A ⊕ D into all theNiemeier lattices L . Since M is a root lattice, a primitive embedding of M into L is in fact a primitiveembedding into L root . Whenever the primitive embedding is given by a primitive embedding of A and D in two different factors of L root , or for the primitive embedding of M into E , we useNishiyama’s results [Nis]. Otherwise we have to determine the primitive embeddings of M into D l for l = 8 , , , , ,
24. This is done in the following lemma.
Lemma 2.1.
We obtain the following primitive embeddings.
P´ERY-FERMI PENCIL OF K (1) A ⊕ D = h d , d , d , d , d , d , d i ֒ → D h d , d , d , d , d , d , d i ⊥ D = h d + 4 d + 6 d + 6 d + 6 d + 6 d + 3 d + 3 d i = ( − A ⊕ D = h d , d , d , d , d , d , d i ֒ → D h d , d , d , d , d , d , d i ⊥ D = h d + d + 2 d + 2 d + 2 d + 2 d + d − d , d + 2 d + 3 d i with Gram matrix (cid:18) − − (cid:19) of determinant . (3) A ⊕ D = h d n , d n − , d n − , d n − , d n − , d n − , d n − i ֒ → D n , n ≥ h d n , d n − , d n − , d n − , d n − , d n − , d n − i ⊥ D n = h a = d n + d n − + 2( d n − + ... + d ) + d , d n − + 2 d n − + 3 d n − , d n − , ..., d i (( A ⊕ D ) ⊥ D n ) root = D n − . We have also the relation . [2] D n = a + d , a being the above root. Theorem 2.1.
There are elliptic fibrations on the generic K surface of the Ap´ery-Fermi pencil(i.e. with Picard number ). They are derived from all the non isomorphic primitive embeddingsof A ⊕ D into the various Niemeier lattices. Among them, fibrations have rank , precisely withthe type of singular fibers and torsion. A A A − torsion E D − torsion E A D − torsion E E A − torsion . The list together with the rank and torsion is given in Table 2.Proof.
The torsion groups can be computed as explained in [BL] or [BGL]. Let us recall briefly themethod.Denote φ a primitive embedding of M = A ⊕ D into a Niemeier lattice L . Define W = ( φ ( M )) ⊥ L and N = ( φ ( M )) ⊥ L root . We observe that W root = N root . Thus computing N then N root we know thetype of singular fibers. Recall also that the torsion part of the Mordell-Weil group is W root /W root ( ⊂ W/N )and can be computed in the following way [BGL]: let l + L root be a non trivial element of L/L root .If there exist k = 0 and u ∈ L root such that k ( l + u ) ∈ N root , then l + u ∈ W and the class of l is atorsion element.We use also several facts.(1) If the rank of the Mordell-Weil group is 0, then the torsion group is equal to W/N . Hence fi-brations A E E ), D E ), D A E ), A A A ) have respective torsiongroups (0), (0), Z / Z , Z / Z .(2) If there is a singular fiber of type E , then the torsion group is (0). Hence the fibrations Lemma 2.2.
Suppose A primitively embedded in A n , A = h a , a i ֒ → A n . Then for all k = 0 , k [1] A n / ∈ (( A ) ⊥ A n ) root .Proof. It follows from the fact that [1] A n is not orthogonal to a . (cid:3) (4) Using lemma 2.3 below and the shape of glue vectors we can determine the torsion forelliptic fibrations M.-J. BERTIN AND ODILE LECACHEUX L root L/L root type of Fibers Rk Tors. E (0) A ⊂ E D ⊂ E E A E A ⊕ D ⊂ E E E D E Z / Z A ⊂ E D ⊂ D E D A ⊕ D ⊂ E D Z / Z D ⊂ E A ⊂ D A D A ⊕ D ⊂ D E D D E ( Z / Z ) A ⊂ E D ⊂ D E A D Z / Z A ⊂ E D ⊂ E A A D Z / Z A ⊕ D ⊂ D E E A A Z / Z D ⊂ E A ⊂ D A D E A E Z / Z D ⊂ E A ⊂ A A A D Z / Z A ⊕ D ⊂ D D D ( Z / Z ) A ⊂ D D ⊂ D D D A ⊕ D ⊂ D D D Z / Z D ( Z / Z ) A ⊂ D D ⊂ D D A D Z / Z A ⊕ D ⊂ D D D Z / Z A D Z / Z A ⊕ D ⊂ D A Z / Z D ⊂ D A ⊂ A D A E ( Z / Z ) A ⊂ E D ⊂ E A A E E Z / Z A D E Z / Z A ⊂ E D ⊂ D A A A A A Z / Z A ⊂ A D ⊂ D A A A E A ⊂ A D ⊂ E A D D ⊂ E A ⊂ D A D Z / Z D ( Z / Z ) A ⊂ D D ⊂ D A D D Z / Z A D Z / × Z / D ⊂ D A ⊂ A A A A D Z / × Z / D ⊂ D A ⊂ D A A A A Z / Z D ⊂ D A ⊂ A D A A Table 2.
The elliptic fibrations of the Ap´ery-Fermi family
Lemma 2.3.
Suppose A primitively embedded in D l , A = h d l , d l − i ֒ → D l . Then . [2] D l ∈ (( A ) ⊥ D l ) root but there is no k satisfying k. [ i ] D l ∈ (( A ) ⊥ D l ) root , i = 1 , .Proof. It follows from Nishiyama [Nis]:( A ) ⊥ D l = h y, x , d l − , ..., d i with y = d l + 2 d l − + 2 d l − + d l − and x = d l + d l − + 2( d l − + d l − + ... + d ) + d andGram matrix L l − = − − ... − D l − . . P´ERY-FERMI PENCIL OF K Moreover (( A ) ⊥ D l ) root = h x , d l − , ..., d i . From there we compute easily the relation2 . [2] D l = x + d l − + 2( d l − + ... + d ). The last assertion follows from the fact that[ i ] D l is not orthogonal to A . (cid:3) We now give some examples showing the method in detail.2.2.1.
Fibration . It comes from a primitive embedding of A ⊕ D into D giving a primitiveembedding of A ⊕ D into N i ( A D ) with glue code h [2 , i . Since by lemma 2.1(2) N root = A ,among the elements k. [2 , . [2 ,
1] = [8 , . ∈ D ] satisfies 2 . [8 , u ] ∈ N root = A with u = 4 .
1. Hence the torsion group is Z / Z .2.2.2. Fibration . It comes from a primitive embedding of A = h e , e i into E (1)6 and D = h e , e , e , e , e i into E (2)6 giving a primitive embedding of A ⊕ D into N i ( E ). In that case N i ( E ) /E ≃ ( Z / Z ) and the glue code is h [1 , (0 , , i . Moreover ( D ) ⊥ E = 3 e + 4 e + 5 e +6 e + 4 e + 2 e = a , ( A ) ⊥ E = h e , y i ⊕ h e , e i with y = 2 e + e + 2 e + 3 e + 2 e + e . From therelation [1] E = −
13 (2 e + 3 e + 4 e + 6 e + 5 e + 4 e )we get − . [1] E = a − e − e + e + 2 e ∈ E − . [1] E = 2 y − e + e + 2 e ∈ ( A ) ⊥ E we deduce that only [1 , , , , , , , , ,
0] contribute to the torsion thus the torsion groupis Z / Z .2.2.3. Fibration . The embeddings of A = h d , d i into D and D = h e , e , e , e , e i into E (1)7 lead to a primitive embedding of A ⊕ D into N i ( D E ) satisfying N i ( D E ) / ( D E ) ≃ ( Z / Z ) with glue code h [1 , , , [3 , , i . We deduce from lemma 2.3 that no glue vector cancontribute to the torsion which is therefore (0).2.2.4. Fibration . The embeddings of A = h a , a i into A and D = h d , d , d , d , d i into D lead to a primitive embedding of A ⊕ D into N i ( A D ) satisfying N i ( A D ) / ( A D ) ≃ ( Z / Z )with glue code h [2 , i . We deduce from lemma 2.2 that no glue vector can contribute to the torsionwhich is therefore (0).2.2.5. Fibration . The primitive embeddings of A = h e , e i into E (1)7 and D = h e , e , e , e , e i into E (2)7 lead to a primitive embedding of A ⊕ D into N i ( D E ) satisfying N i ( D E ) / ( D E ) ≃ ( Z / Z ) with glue code h [1 , , , [3 , , i . From Nishiyama [Nis] we get ( A ) ⊥ E (1)7 = h e , y, e , e , e i ≃ A with y = 2 e + e + 2 e + 3 e + 2 e + e and ( D ) ⊥ E = h ( − , e + e + 2( e + e + e + e ) = ( − i .Hence N = D ⊕ A ⊕ ( − ⊕ A and W root = N root = D ⊕ A ⊕ A . Now − η = − . [1] E = 2 y − e + e + 2 e + 3 e ∈ (( A ) ⊥ E ) root and for all k = 0, k. [1] E / ∈ ( D ) ⊥ E . Hence only the generator [1 , ,
0] can contribute to the torsiongroup which is therefore Z / Z .2.2.6. Fibration . The primitive embeddings A = h d , d i into D (1)6 and D = h d , d , d , d , d i into D (2)6 give a primitive embedding of A ⊕ D into L = N i ( D ) with L/L root ≃ ( Z / Z ) andglue code h even permutations of [0 , , , i . From Nishiyama [Nis] we get ( A ) ⊥ D = h y = 2 d + d + 2 d + d , x = d + d + 2( d + d ) + d , d , d i , (( A ) ⊥ D ) root = h x , d , d i ≃ A and ( D ) ⊥ D = h x ′ i = h d + d + 2( d + d + d + d ) = ( − i . We deduce N root = A ⊕ D ⊕ D . From the relations2 . [2] D = x + d + 2 d and 2 . [3] D = y + x + d + d we deduce that the glue vectors having 1,2, 3 or 0 in the first position may belong to W . From the relation 2 . [2] D = x ′ we deduce thatonly glue vectors with 2 or 0 in the second position may belong to W . Finally only the glue vectors[0 , , , , [1 , , , , [1 , , , , [2 , , , , [2 , , , , [3 , , , , [3 , , , , [0 , , ,
0] belong to W . Since M.-J. BERTIN AND ODILE LECACHEUX y and x ′ are not roots, only glue vectors with 0 or 2 in the first position and 0 in the second positionmay contribute to torsion that is [2 , , , , [0 , , , Z / Z .2.2.7. Fibration . The primitive embeddings of A = h d , d i into D (1)5 and D into D (2)5 give a primitive embedding into L = N i ( A D ) with L/L root ≃ Z / Z × Z / Z and glue code h [1 , , , , [1 , , , i . From Nishiyama we get ( A ) ⊥ D = h y, x , d i with y = 2 d + d + 2 d + d , x = d + d + 2 d + 2 d + d and Gram matrix M = − − − − − of determinant 12. We alsodeduce N root = E A , 2 . [2] D = x + d ∈ (( A ) ⊥ D ) root . Moreover neither k. [1] D nor k. [3] D belongsto ( A ) ⊥ D . Thus only glue vectors with 2 or 0 in the third position can belong to W and eventuallycontribute to torsion, that is [2 , , , , , , , , , , , , , , , , , , , , , , , , u ∈ D satisfying 2 . (2 + u ) = 0 or 4 . (2 + u ) = 0, glue vectors withthe last component equal to 2 cannot satisfy k ( l + u ) ∈ N root with l ∈ L and u ∈ L root = A D .Hence only the glue vectors generated by h [2 , , , i contribute to torsion and the torsion group istherefore Z / Z . (cid:3) Weierstrass Equations for all the elliptic fibrations of Y k The method can be found in [BL], [El]. We follow also the same kind of computations used for Y given in [BL]. We give only explicit computations for 4 examples, Fibration
We take u = XYZ as a parameter of an elliptic fibration and with the birationaltransformation x = − u (1 + uZ )( u + Y ) , y = u (( u + Y )( uY − Z + Y ( Y + 2 u + k ) − y + ukyx + u (cid:0) u + uk + 1 (cid:1) y = x , where the point ( x = 0 , y = 0) is a 3-torsion point and the point (cid:0) − u , − u (cid:1) is of infinite order.The singular fibers are of type IV ∗ ( u = 0 , ∞ ), I (cid:0) u + uk + 1 = 0 (cid:1) and I (cid:0) u − k ( k − u + 27 = 0 (cid:1) . Moreover if k = s + s the two singular fibers of type I are above u = − s and − s .3.2. Fibration
Using the 3-neighbor method from fibration II ∗ and the parameter m = ys ( u + s ) . Then we obtain a cubic C m in w, u, with x = w ( u + s ) C m : ( s + u ) m + u (cid:0) s w + u s + w + u (cid:1) m − w s = 0 . From some component of the fiber of type I at u = − s we obtain the rational point on C m : ω m = (cid:18) u = ms − s − m , w = m ( s − ) s ( s − m ) (cid:19) which is not a flex point. The first stage is to obtain a quarticequation Qua : y = ax + bx + cx + dx + e . First we observe that ω m is on the line w = u + s , so we replace w by K with w = u + s + K and u = u + T . The transformation K = W T gives anequation of degree two in T , with constant term f W + g where f and g belong to Q ( s, m ). Withthe change variable W f + g = x we have an equation M ( x ) T + N ( x ) T + x = 0 . The discriminantof the quadratic equation in T is N ( x ) − xM ( x ) , a polynomial of degree 4 in x and constant terma square. Easily we obtain the form Qua.
P´ERY-FERMI PENCIL OF K From the quartic form, setting y = e + dx e + x X ′ , x = c X ′ − ce + d Y ′ we get Y ′ + 4 e ( dX ′ − be ) Y + 4 e (cid:0) e X ′ − ce + d (cid:1) (cid:0) X ′ − a (cid:1) = 0 . Finally the following Weierstrass equation follows from standard transformation where we replace m by t Y − X + 13 t ( s + 1)( s + 219 s − s + 1) X − t (cid:0) − s t + ( s + 14 s + 1)( s − s + 198 s − s + 1) t − s (cid:1) = 0 , with a section Φ of height 12 corresponding to (cid:0) e X ′ − ce + d (cid:1) = 0 and Y ′ = 0 . The coordinatesof Φ , too long, are omitted but we can follow the previous computation to obtain it.Writing the above form as y = x − αx + (cid:18) t + 1 t (cid:19) − β we recover the values of the j invariants of the two elliptic curves for the Shioda-Inose structure (seeparagraph 4.5.1 and 4.1 below).3.3. Fibration
Let g = XYZ . Eliminating X and writing Y = ZU we obtain an equation ofbidegree 2 in U and Z. If k = s + s there is a rational point U = − , Z = − sg on the previous curve.By standard transformations we get a Weierstrass equation y = x + 14 g (cid:0) s + 14 s + 1 (cid:1) x + s g (cid:0) g + s (cid:1) (cid:0) gs + 1 (cid:1) x and a rational point x = s ( g − (cid:0) g + s (cid:1) (cid:0) s g + 1 (cid:1) ( s − ,y = 12 s (cid:0) g − (cid:1) (cid:0) g + s (cid:1) (cid:0) s g + 1 (cid:1) (cid:0) g s + g (cid:0) s − s + 1 (cid:1) + 2 s (cid:1) ( s − . The singular fibers are of type 2
III ∗ ( ∞ , , I (cid:0) − s , − s (cid:1) , I . Fibration
Using the fibration t = xg ( g + s ) and obtain aWeierstrass equation Y = X + (cid:0) t (cid:0) t + s (cid:1) + t (cid:0) s + 14 s + 1 (cid:1)(cid:1) X + 16 s t X. The singular fibers are of type I ∗ ( ∞ , , I . Nikulin involutions and Shioda-Inose structure
Background.
Let X be a K H ( X, Z ) equipped with the cup product is an even unimodularlattice of signature (3 , T X is defined by T X = S ⊥ X ⊂ H ( X, Z )where S X is the N´eron-Severi group of X. The lattice H ( X, Z ) admits a Hodge decomposition ofweight two H ( X, C ) ≃ H , ⊕ H , ⊕ H , . Similarly, the period lattice T X has a Hodge decomposition of weight two T X ⊗ C ≃ T , ⊕ T , ⊕ T , . An isomorphism between two lattices that preserves their bilinear forms and their Hodge decompo-sition is called a Hodge isometry.An automorphism of a K X is called symplectic if it acts on H , ( X ) trivially. Suchautomorphisms were studied by Nikulin in [N1] who proved that a symplectic involution i ( Nikulin
Weierstrass Equation From Param. y + tkyx + t k ( t + 1) y = x − t ( t + 1) II ∗ ( ∞ ) , IV ∗ (0) , I ( − , I r = 0 Y ( X + Z )2( Z + Y ) XZ y = x − t (cid:0) s + 1 (cid:1) (cid:0) s + 219 ∗ s − s + 1 (cid:1) x + t ( − s t +( s +14 s +1)( s − s +198 s − s +1) t − s ) II ∗ ( ∞ , , I x P = Φ ys ( s + t ) y = x + t (cid:0) t s + (cid:0) s − s + 1 (cid:1) t + 12 (cid:1) x − t (cid:0) ts − (cid:1) x + t I ∗ ( ∞ ) , IV ∗ (0) , I r = 0 xs t y = x +( t − ( s +1 )( s +219 s − s +1 ) t + ( s − s +198 s − s +1 )( s +14 s +1 )) x +16 s xI ∗ ( ∞ ) , I x P x t y − k ( t + 1) yx + ky = x + (cid:0) t − (cid:1) x + 3 x − I ∗ ( ∞ ) , I (0) , I x P = 0 xt y = x + (cid:0) t (cid:0) s + 14 s + 1 (cid:1) + t s (cid:1) x + t s (cid:0) s + 1 (cid:1) x + t s I ∗ ( ∞ ) , II ∗ (0) , I x P x ( t + s )( ts +1 ) y = x + t (cid:0) t (cid:0) s − s + 1 (cid:1) + 8 s (cid:1) x − t s (cid:0) t − s (cid:1) xIII ∗ ( ∞ ) , I ∗ (0) , I (cid:0) s (cid:1) , I r = 0 xt y − k ( t − yx = x ( x − (cid:0) x − t (cid:1) I ∗ ( ∞ ) , I (0) , I (1) , I x P = 1 ( X + Z )( Y + Z ) XZ y = x + t (cid:0) s + 14 s + 1 (cid:1) x + t s (cid:0) t + s (cid:1) (cid:0) ts + 1 (cid:1) x III ∗ ( ∞ , , I (cid:16) − s , − s (cid:17) , I x P = s ( t − ( t + s )( ts +1 )( s − ) XYZ y + t (cid:0) s + 1 (cid:1) (cid:0) x + t s (cid:1) y = (cid:0) x − t s (cid:1) (cid:0) x + t s (cid:1) I ∗ ( ∞ ) , III ∗ (0) , I ( − , I x P = t s , x P = 0 XY ( Y + Z ) Z y + t (cid:0) st − − s (cid:1) yx − s y = x (cid:0) x + s (cid:0) t (cid:0) s − (cid:1) − s (cid:0) s + 1 (cid:1)(cid:1)(cid:1) I ( ∞ ) , I ( s ) , I x P = st, x P = − s t + s (cid:0) s + 1 (cid:1) y + sxxt y = x + t (cid:0) t s + (cid:0) s + 14 s + 1 (cid:1) + (cid:0) s + 1 (cid:1)(cid:1) x − ( t s + s ( s +14 s +1 ) t + s ( s +1 )) x + ts + s ( s +14 s +1 ) I ∗ ( ∞ ) , I x P xt + s t y = x + t (cid:0) t + (cid:0) s − s + 1 (cid:1) t + 4 s (cid:1) x +2 t s (cid:0) t − s (cid:1) x + t s I ∗ ( ∞ ) , I ∗ (0) , I x P = − ts xt Table 3.
Weierstrass equations of the elliptic fibrations of Y k involution ) has eight fixed points and that the minimal resolution Y → X/ h i i of the eight nodes isagain a K P´ERY-FERMI PENCIL OF K No Weierstrass Equation From Param. y = x + (cid:0) t (cid:0) s + 1 (cid:1) + t (cid:0) s + 14 s + 1 (cid:1) + ts (cid:1) x + s t xI ∗ ( ∞ ) , I ∗ (0) , I (cid:18) − , − s ( s − ) , .. (cid:19) x P = s (2 t +1) ( s − ) xt ( t + s )( ts +1 ) y − tx ) (cid:0) y − s tx (cid:1) = x (cid:0) x − ts (cid:1) (cid:16) x − ts ( t + 1) (cid:17) I ∗ ( ∞ ) , I ∗ (0) , I ( − , I (cid:18) (cid:16) s − s (cid:17) , .. (cid:19) x P = s t ( XY +1) ZX y = x + t (cid:0) (cid:0) t + s (cid:1) + t (cid:0) s + 14 s + 1 (cid:1)(cid:1) x + 16 s t xI ∗ ( ∞ , , I x P = − ts ( t +1) ( t + s ) xt ( t + s ) y − (cid:0) s + 14 s + 1 − s t (cid:1) yx = x (cid:0) x − s (cid:1) (cid:0) x − s (cid:1) I ( ∞ ) , I (cid:16) ± s ± s − s , .. (cid:17) x P = 4 s ; x P = s ( ts + s − ) ( ts +1 − s ) yts x y + (cid:0) − t + (cid:0) s − (cid:1) t − s (cid:1) yx + s t y = x (cid:0) x − s (cid:1) I ( ∞ ) , I ∗ (0) , I x P = 0 y − txt ( x − ts ) y + ktyx + t (cid:0) t + tk + 1 (cid:1) y = x IV ∗ ( ∞ , , I (cid:0) − s, − s (cid:1) , I x P = − t XYZ y − yx (cid:0) t − kt + 1 (cid:1) = x ( x − (cid:0) x + t − tk (cid:1) I ( ∞ ) , I (0 , k ) , I (cid:0) s, s , .. (cid:1) r = 0 X + Y + Z y = x + t (cid:0) t + 2 (cid:0) s − (cid:1) t + (cid:0) s − s + 1 (cid:1)(cid:1) x + t s (cid:0) t − (cid:0) s − (cid:1)(cid:1) x + s t I ( ∞ ) , IV ∗ (0) , I (cid:0) , − s (cid:1) , I x P = s t y − s x − ts ( t − ) x − ts ( t +1) y + (cid:0) t (cid:0) − s (cid:1) + s (cid:1) yx + t s y = x (cid:0) x − s t (cid:1) (cid:0) x + t s (1 − t ) (cid:1) I ∗ ( ∞ ) , I (0) , I x P = 1 , x P = s t Z ( XY Z + s )1+ Y Z y + (cid:0) t − tk + 1 (cid:1) yx = x (cid:0) x − t (cid:1) (cid:0) x − t (cid:1) I ∗ ( ∞ ) , I (0) , I (cid:16) k ± , .. (cid:17) x P = t , x p = ( tk − k − X + Y y + (cid:0) s + 1 (cid:1) tyx = x (cid:0) x − t s (cid:1) (cid:16) x − s t ( t + 1) (cid:17) I ∗ ( ∞ , , I ( − , I x P = t + 1; x P = t s ZY y + ( s + t ) ( ts + 1) yx + t s (cid:0) t (cid:0) s − (cid:1) + s (cid:1) y = x ( x − st ) (cid:0) x − t s ( t − s ) (cid:1) I ( ∞ ) , I (0) , I x P = ts ; x P = − t s Y − sXY + sZ y + ( ts − t − s ) xy = x (cid:0) x − t s (cid:1) I ( ∞ , , I (cid:0) s, s (cid:1) , I x P = ts Z y − (cid:0) t (cid:0) s − (cid:1) + s (cid:1) yx + t s ( t + 1) y = x (cid:0) x + t s ( t + 1) (cid:1) I ∗ ( ∞ ) , I (0) , I ( −
1) 4 I x P = 0 Z − sX + Y Table 4.
Weierstrass equations of the elliptic fibrations of Y k We have then the rational quotient map p : X → Y of degree 2. The transcendental lattices T X and T Y are related by the chain of inclusions2 T Y ⊆ p ∗ T X = T X (2) ⊆ T Y , which preserves the quadratic forms and the Hodge structures.In this paper, K τ , we considerthe symplectic involution i ( Van Geemen-Sarti involution ) given by the fiberwise translation by τ .In this situation, the rational quotient map X → Y is just an isogeny of degree 2 between ellipticcurves over C ( t ), and we have a rational map Y → X of degree 2 as the dual isogeny. Notation
We consider the fibration n of Y k with a Weierstrass equation E n : y = x + A ( t ) x + B ( t ) x and the two-torsion point T = (0 , . We will call n − i the elliptic fibration E n / h (0 , i ofthe elliptic surface Y k /i if i denotes the translation by T .4.2. Fibrations of some Kummer surfaces.
Let E l be an elliptic curve with invariant j, definedby a Weierstrass equation in the Legendre form E l : y = x ( x − x − l ) . Then l satisfies the equation j = 256 (1 − l + l ) l ( l − . For a fixed j the six values of l are given by l or l , − l, l − l , − l − , l − l .Consider the Kummer surface K given by E l × E l / ± Kx ( x − x − l ) t = x ( x − x − l ) . Following [Ku] we can construct different elliptic fibrations. In the general case we can consider thethree elliptic fibrations F i of K defined by the elliptic parameters m i , with corresponding types ofsingular fibers F : m = x x I ∗ , I F : m = ( x − l )( x − x ) l ( l − x ( x − III ∗ , I ∗ , I , I F : m = ( x − x )( l ( x − l )+( l − x )( l x − x )( x − l +( l − x ) I ∗ , I . In the special case when E = E and j = j = 8000 we obtain the following fibrations F : l = l = 3 + 2 p (2) m = x x I ∗ , I , I F : l = 3 + 2 p (2) , l = l m = ( x − l )( x − x ) l ( l − x ( x − III ∗ , I ∗ , I , I , I G : l = 3 + 2 p (2) , l = l m III ∗ , I ∗ , I F : l = l = 3 + 2 p (2) m = ( x − x )( l ( x − l )+( l − x )( l x − x )( x − l +( l − x ) I ∗ , I , I . Nikulin involutions and Kummer surfaces.Proposition 4.1.
Consider a family S a,b of K surfaces with an elliptic fibration, a two torsionsection defining an involution i and two singular fibers of type I ∗ , S a,b : Y = X + (cid:18) t + 1 t + a (cid:19) X + b X. Then the K surface S a,b /i is the Kummer surface ( E × E ) / ( ± Id ) where the j i invariants of theelliptic curves E i , i = 1 , are given by the formulae j j = 4096 (cid:0) a − b (cid:1) b ( j − j − a (2 a − − b ) b . P´ERY-FERMI PENCIL OF K Proof.
Recall that if E i , i = 1 ,
2, are two elliptic curves in the Legendre form E i : y = x ( x −
1) ( x − l i ) , the Kummer surface K K : ( E × E ) / ( ± Id )is defined by the following equation x ( x −
1) ( x − l ) t = x ( x −
1) ( x − l ) . The Kummer surface K admits an elliptic fibration with parameter u = m = x x and Weierstrassequation H u H u : Y = X ( X − u ( u −
1) ( ul − l )) ( X − u ( u − l ) ( l u − . The 2-isogenous curve S a,b / h (0 , i has the following Weierstrass equation Y = X (cid:0) X − t (cid:0) t + ( a − b ) t + 1 (cid:1)(cid:1) (cid:0) X − t (cid:0) t + ( a + 2 b ) t + 1 (cid:1)(cid:1) with two singular fibers of type I ∗ above 0 and ∞ . We easily prove that S a,b / h (0 , i and H u are isomorphic on the field Q (cid:0) √ w (cid:1) where l = w ′ w = w w , l = 1 w ′ w ′ = w w and t = w u,w , w ′ and w , w ′ being respectively the roots of polynomials t + ( a − b ) t + 1 and t +( a + 2 b ) t +1 . Recall that the modular invariant j i of the elliptic curve E i is linked to l i by the relation j i = 256 (cid:0) − l i + l i (cid:1) l i (1 − l i ) . By elimination of w and w , it follows the relations between j and j j j = 4096 (cid:0) a − b (cid:1) b ( j − j − a (2 a − − b ) b . (cid:3) In the Fermi family, the K Y k has the fibration I ∗ , a 2-torsionpoint and Weierstrass equation y = x + x t (cid:0) (cid:0) t + s (cid:1) + t (cid:0) s + 14 s + 1 (cid:1)(cid:1) + 16 t s x. Taking y = y ′ t (cid:0) √ s (cid:1) , x = x ′ t (cid:0) √ s (cid:1) and t = t ′ s, we obtain the following Weierstrass equation y ′ ´ e = x ′ + (cid:18) t ′ + 1 t ′ + 14 s + 14 s + 1 s (cid:19) + s x ′ . By the previous proposition with a = s +14 s +1 s , b = s , we derive the corollary below. Corollary 4.1.
The surface obtained with the -isogeny of kernel h (0 , i from fibration j -invariants j , j satifying j j = (cid:0) s + 1 (cid:1) (cid:0) s + 219 s − s + 1 (cid:1) s (cid:0) j − (cid:1) (cid:0) j − (cid:1) = ( s + 14 s + 1) ( s − s + 198 s − s + 1) s . Remark 4.1. If s = 1 we find j = j = 8000 . Remark 4.2. If b = 1 we obtain the family of surfaces studied by Narumiya and Shiga, [Na] .Morover if a = (resp. ) we find the two modular surfaces associated to the modular groups Γ (7) (resp. Γ (8)) . In these two cases we get j = j = − (resp. j = j = 8000) . Remark 4.3.
With the same method we can consider a family of K surfaces with Weierstrassequations E v : Y + XY − ( v + 1 v − k ) Y = X − ( v + 1 v − k ) X , singular fibers of type I ∗ , I , I and the point P v = (0 , of order . The elliptic curve E ′ v = E v / h P v i has singular fibers of type I ∗ , I . An analog computation gives E ′ v ≡ ( E × E ) / ( ± Id ) and j j = (256 k − k − (cid:0) j − (cid:1) (cid:0) j − (cid:1) = (32 k − (128 k − k − . Shioda-Inose structure.Definition 4.1. A K surface X has a Shioda-Inose structure if there is a symplectic involution i on X with rational quotient map X p → Y such that Y is a Kummer surface and p ∗ induces a Hodgeisometry T X (2) ≃ T Y .Such an involution i is called a Morrison-Nikulin involution. An equivalent criterion is that X admits a (Nikulin) involution interchanging two orthogonal copiesof E ( −
1) in
N S ( X ), where E ( −
1) is the unique unimodular even negative-definite lattice of rank8.Or even more abstractly: 2 E ֒ → N S ( X ).Applying this criterion to fibrations Proposition 4.2.
The translation by the two torsion point of fibration and endowes Y k with a Shioda-Inose structure. Fibration I at t = ∞ . The idea [G] is to use the components Θ − , Θ − , Θ ,Θ , Θ , Θ , Θ of I and the zero section to generate a lattice of type E . The two-torsion sectionintersects Θ and the translation by the two-torsion point on the fiber I transforms Θ n in Θ n +8 . The translation maps the lattice E on an another disjoint E lattice and defines a Shioda-Inosestructure.For fibration t = 0 is of type I and the section of order 2 specialises to thesingular point (0 , . Then after a blow up, it will not meet the 0-component. If we denote Θ ,i ,0 ≤ i ≤
5, the six components, then the zero section meets Θ , and the 2-torsion section meetsΘ , . The translation by the 2-torsion section induces the permutation Θ ,i → Θ ,i +3 . The fiber above t = ∞ is of type I ∗ . The simple components are denoted Θ ∞ , , Θ ∞ , and Θ ∞ , , Θ ∞ , ;the double components are denoted C i with 0 ≤ i ≤ ∞ , .C = Θ ∞ , .C = 1; Θ ∞ , .C =Θ ∞ , .C = 1 . Then the 2-torsion section intersects Θ ∞ , or Θ ∞ , and the translation by the 2-torsionsection induces the transposition C i ←→ C − i . The class of the components C , C , C , Θ ∞ , , Θ ∞ , , the zero section, Θ , and Θ , define a E ( − . The Nikulin involution defined by the two torsion section maps this E ( −
1) to another copy of E ( −
1) orthogonal to the first one; so the Nikulin involution is a Morrison-Nikulin involution.4.5.
Base change and van Geemen-Sarti involutions.
If a K X has an elliptic fi-bration with two fibers of type II ∗ , this fibration can be realised by a Weierstrass equation oftype y = x − αx + ( h + 1 /h − β ) . Moreover Shioda [Shio] deduces the “Kummer sandwiching”, K → S → K, identifying the KummerK = E × E / ± j -invariants of the two elliptic curves E , E and giving thefollowing elliptic fibration of K y = x − αx + ( t + 1 /t − β ) . P´ERY-FERMI PENCIL OF K This can be viewed as a base change of the fibration of X .4.5.1. Alternate elliptic fibration.
We shall now use an alternate elliptic fibration ([Sc-Shio] example13.6) to show that this construction is indeed a 2-isogeny between two elliptic fibrations of S andK . In the next picture we consider a divisor D of type I ∗ composed of the zero section 0 andthe components of the II ∗ fibers enclosed in dashed lines. The far double components of the II ∗ fibres can be chosen as sections of the new fibration. Take ω as the zero section. The other oneis a two-torsion point since the function h has a double pole on ω and a double zero at M. It isthe function ′ x ′ in a Weierstrass equation. More precisely with the new parameter u = x and thevariables Y = yh and X = h, we obtain the Weierstrass equation Y = X + ( u − αu − β ) X + X.M h = 0 ω − h = ∞ X (= h ) by t , we obtain an equation in W, t with Y = W t , whichis the equation for the 2-isogenous elliptic curve. Indeed the birational transformation y = 4 Y + 4 U + 2 U A, x = 2 Y + U U with inverse U = 1 / yx + A , Y = 1 / (cid:0) − y + 2 x + 4 x A + 2 xA (cid:1) y ( x + A ) transforms the curve Y = U + AU + BU in the Weierstrass form y = ( x + A )( x − B ) . This is an equation for the 2-isogenous curve of the curve Y = X + AX + BX [Si]. On the curve Y = U + AU + BU , the involution U
7→ − U means adding the two-torsion point ( x = − A, y = 0) . Using this above process with A = ( u − αu − β ) , the 2-isogenous curve E u has a Weierstrassequation Y = ( X + ( u − αu − β ))( X − I ∗ , I .The coefficients α and β can be computed using the j -invariants α = J J ; β = (1 − J )(1 − J ); j i = 1728 J i . If the elliptic curve is put in the Legendre form y ′ = x ′ ( x ′ − x ′ − l ) then j = 256 (1 − l + l ) l ( l − , so α = 16729 (1 − l + l ) (1 − l + l ) l ( l − l ( l − β = 127 (2 l − l − l − l − l + 1)( l + 1) l l ( l − l − . On the Kummer surface E × E / ± X ( X −
1) ( X − l ) Z = X ( X −
1) ( X − l )we consider an elliptic fibration (case J of [Ku]) with the parameter z = ( l X − X )( X − l + X ( l − X ( X − (in fact z = − l ( l − m − cf. 4.2) and obtain the Weierstrass equation Y =( X − l l ( l − l − X +2 l l ( l − l − ( X +4 z +4( − l l + l + l +1) z +4( l l − l l − l − l ) z +2 l l ( l − l − ) . Substituting z = w − ( − l l + l + l + 1) it follows Y = ( X − l l ( l − l − X +2 l l ( l − l − ( X +4 w − ( l − l +1 )( l − l +1 ) w + ( l − l − l − l − l +1)( l +1) ) . Up to an automorphism of this Weierstrass form we recover the equation of E u . The previous results can be used to show the following proposition
Proposition 4.3.
The translation by the two torsion point of the elliptic fibration gives to Y k a Shioda-Inose structure. Proof of Theorem . n of Y k with a two torsion section.From the Shioda-Tate formula (cf. e.g. [Shio1], Corollary 1.7]) we have the relation12 = | ∆ | Q m (1) v | Tor | where ∆ is the determinant of the height-matrix of a set of generators of the Mordell-Weil group, m (1) v the number of simple components of a singular fiber and | Tor | the order of the torsion groupof Mordell-Weil group. This formula allows us to determine generators of the Mordell-Weil groupexcept for fibration n -i. Thediscriminant is either 12 × × . Proposition 5.1.
The translation by the two torsion point of the fibration gives to Y k aShioda-Inose structure. From the previous Proposition 4.1, the translation by the two torsion point of P and the two torsion point. By computation we can see that the Mordell-Weil group of the2-isogenous curve on E ( C ( t )) is generated by p ( P ) and torsion sections. So we can compute thediscriminant of the N´eron-Severi group which is 12 ×
8. The second condition, T X (2) ≃ T Y , is thenverified. Remark 5.1.
The K surface of Picard number given with the elliptic fibration Y = X − (cid:18) t + 1 t − (cid:19) X + 116 X or y = x − / t (cid:0) t + 2 − t (cid:1) x + 1 / t x has rank . The Mordell-Weil group is generated by (0 , and P = ( x = , y = ( t − ) . Thedeterminant of the N´eron-Severi group is equal to . By computation we have p ( P ) = 2 Q with Q = (cid:0) t ( t − t − t + 1) , − t ( t − t − t + 1) (cid:1) of height . The determinant of the N´eron Severi
P´ERY-FERMI PENCIL OF K group of the -isogenous curve is then not × . So the involution induced by the two-torsionpoint is not a Nikulin-Morrison involution. Moreover the -isogenous elliptic curve is a fibration ofthe Kummer surface E × E/ ± where j ( E ) = 0 . For fibrations n -i with discriminant of the transcendental lattice 12 × n -i can be obtained by 2- or 3-neighbor method from n elliptic fibration and its 2-isogenous fibration, singular fibers and the x -coordinates of generatorsof the Mordell lattice of n -i. In the third column we give the starting fibration for the 2- or 3-neighbor method and in the last column the parameter used from the starting fibration.5.1. The K surface S k . For the remaining fibrations, (discriminant 12 × , using also the 2- or 3-neighbor method, they are proved to lie on the same surface S k . Except for the case m = yxt ( t − s ) it follows the Weierstrassequation Y + 2 (cid:0) m s − (cid:1) Y X − m s Y = (cid:0) X − m s (cid:1)(cid:0) X + 8 m s (cid:1) (cid:0) X + m (cid:0) s − s + 1 (cid:1) − (cid:1) with singular fibers I ( ∞ ) , IV ∗ (0) , I .Then the parameter m = Y ( X +8 m s ) leads to the fibration S k . First we prove that S k is the Jacobianvariety of some genus 1 fibrations of K k .Starting with the fibration y = x (cid:0) x + 4 t s (cid:1) (cid:18) x + 14 ( t − s ) ( ts − (cid:19) the new parameter m := yt ( x + ( t − s ) ( ts − ) defines an elliptic fibration of E m : Y − m (cid:0) s + 1 (cid:1) Y X = X (cid:0) X − s m (cid:1) (cid:18) X + 14 (2 m − s ) (2 m + s ) (cid:19) and singular fibers are of type 4 I (cid:0) , ± s, ∞ (cid:1) , I . Then setting as new parameter n = Xm , it follows a genus one curve in m and Y. Its equation, ofdegree 2 in Y , can be transformed in w = − n ( − n + s ) m + n ( s (8 + n ) − ns + n (1 + 4 n )) m − ns ( − n + s ) . Let us recall the formulae giving the jacobian of a genus one curve defined by the equation y = ax + bx + cx + dx + e . If c = 2 (12 ae − bd + c ) and c = 2 (72 ace − ad − b e + 9 bcd − c ),then the equation of the Jacobian curve is¯ y = ¯ x − c ¯ x − c . In our case we obtain y = x (cid:18) x + n s − n (cid:0) s − (cid:1) (cid:19)(cid:18) x + n s − (cid:0) s − s − (cid:1) (cid:0) s + 4 s − (cid:1) n + 4 ns (cid:19) , which is precisely the fibration No Weierstrass Equation From Param. y − k ( t − yx = x ( x − (cid:0) x − t (cid:1) I ∗ ( ∞ ) , I (0) , I (1) , I y = x + (cid:0) t − t k + 2 tk + 4 − k (cid:1) x + ( t − (cid:16) t + t (cid:0) − k (cid:1) + ( k − (cid:17)(cid:16) t + t (cid:0) − k (cid:1) + ( k + 2) (cid:17) xI ∗ ( ∞ ) , I (0) , I (1) , I x Q = − ( t − (cid:16) t + t (cid:0) − k (cid:1) + ( k − (cid:17) y = x + t (cid:0) (cid:0) t + s (cid:1) + t (cid:0) s + 14 s + 1 (cid:1)(cid:1) x + 16 s t x I ∗ ( ∞ , , I y = x (cid:0) x − t (cid:0) (cid:0) t + s (cid:1) + t (cid:0) s + 14 s + 1 + 8 s (cid:1)(cid:1)(cid:1)(cid:0) x − t (cid:0) (cid:0) t + s (cid:1) + t (cid:0) s + 14 s + 1 − s (cid:1)(cid:1)(cid:1) I ∗ ( ∞ , , I x Q = t (( t + s )( s ) + t ( − s +8 s +1 )) ( t +1) ( t + s ) y − (cid:0) s + 14 s + 1 − s t (cid:1) yx = x (cid:0) x − s (cid:1) (cid:0) x − s (cid:1) I ( ∞ ) , I (cid:16) ± s ± s − s , .. (cid:17) y = x (cid:0) x − (cid:0) t s − (cid:0) s + 14 s + 1 (cid:1) ± s (cid:0) s + 1 (cid:1)(cid:1)(cid:1)(cid:0) x − (cid:0) ts + s ± s − (cid:1) (cid:0) ts − s ± s + 1 (cid:1)(cid:1) I ( ∞ ) , I (cid:16) ± s ± s − s , .. (cid:17) x Q = (cid:0) ts + s − s − (cid:1) (cid:0) ts − s + 4 s + 1 (cid:1)(cid:0) t s − (cid:0) s + 14 s + 1 (cid:1) + 8 s (cid:0) s + 1 (cid:1)(cid:1) x Q = 116 ( s − ( s +1) (cid:0) ts + s + 4 s − (cid:1)(cid:0) ts − s + 4 s + 1 (cid:1) (cid:0) t s − (cid:0) s + 14 s + 1 (cid:1) − s (cid:0) s + 1 (cid:1)(cid:1) y + (cid:0) t − tk + 1 (cid:1) yx = x (cid:0) x − t (cid:1) (cid:0) x − t (cid:1) I ∗ ( ∞ ) , I (0) , I (cid:16) k ± , .. (cid:17) y = x (cid:0) x + ( t ( k − − (cid:0) t − ( k + 2) t + 1 (cid:1)(cid:1)(cid:0) x + ( t ( k + 2) − (cid:0) t − ( k − t + 1 (cid:1)(cid:1) I ∗ ( ∞ ) , I (0) , I (cid:16) k ± , .. (cid:17) x Q = − (cid:0) t − ( k − t + 1 (cid:1) ( t ( k − −
1) ; x Q = − k − k +2 ( t ( k + 2) − (cid:0) t − ( k − t + 1 (cid:1) y − y Q + k ( x − x Q ) t ( x − x Q ) y − ( t − s )( ts − xt ( x − ts ( ts − )( y − y Q ) + s ( x − x Q ) ( t +1) ( x − x Q ) y + (cid:0) s + 1 (cid:1) tyx = x (cid:0) x − t s (cid:1) (cid:16) x − s t ( t + 1) (cid:17) I ∗ ( ∞ , , I ( − , I y = x + t (cid:0) t s − t (cid:0) s − s + 1 (cid:1) + 4 s (cid:1) x + t (cid:16) t s + (cid:16) s − ( s − (cid:17) t + 4 s (cid:17)(cid:16) t s + (cid:16) s − ( s + 1) (cid:17) t + 4 s (cid:17) x I ∗ ( ∞ , , I ( − , ) x Q = (2 t s + t (cid:0) s − (cid:1) − ; x Q = − t (cid:16) t s + (cid:16) s − ( s + 1) (cid:17) t + 4 s (cid:17) y + ( ts − t − s ) xy = x (cid:0) x − t s (cid:1) I ( ∞ , , I (cid:0) s, s (cid:1) , I y = x (cid:0) x + 4 t s (cid:1) (cid:16) x + ( t − s ) ( st − (cid:17) I (cid:0) ∞ , , s, s (cid:1) , I x Q = ts ( ts − Table 5.
Fibrations with discriminant 12 × K k ) P´ERY-FERMI PENCIL OF K No Weierstrass Equation From Param. y = x + t (cid:0) t (cid:0) s − s + 1 (cid:1) + 8 s (cid:1) x − t s (cid:0) t − s (cid:1) xIII ∗ ( ∞ ) , I ∗ (0) , I (cid:0) s (cid:1) , I y = x − t (cid:0) t (cid:0) s − s + 1 (cid:1) + 8 s (cid:1) x + t (cid:16) t s + (cid:0) s − s − s − s + 1 (cid:1) t + 16 s (cid:0) s + 1 (cid:1) (cid:17) xIII ∗ ( ∞ ) , I ∗ (0) , I (cid:0) s (cid:1) , I y = x + (cid:0) s + 14 s + 1 (cid:1) t x + t s (cid:0) s + t (cid:1) (cid:0) ts + 1 (cid:1) x III ∗ ( ∞ , , I (cid:16) − s , − s (cid:17) , I y = x − (cid:0) s + 14 s + 1 (cid:1) t x − t (cid:0) t s + t (cid:0) − s + 36 s − s + 36 s − (cid:1) + 64 s (cid:1) x III ∗ ( ∞ , , I (cid:16) − s , − s (cid:17) , I x Q =
14 ( t +1) ( t s + t ( s − s +1 ) +2 s ) ( s − ) ( t − y ( ts − y = x + (cid:0) t (cid:0) s + 1 (cid:1) + t (cid:0) s + 14 s + 1 (cid:1) + ts (cid:1) x + t s xI ∗ ( ∞ ) , I ∗ (0) , I (cid:18) − , − s ( s − ) , .. (cid:19) y = x (cid:16) x − (cid:0) s + 1 (cid:1) t − (cid:0) s + 14 s + 1 (cid:1) t − ts (cid:17)(cid:16) x − (cid:0) s − (cid:1) t − (cid:0) s + 14 s + 1 (cid:1) t − ts (cid:17) I ∗ ( ∞ ) , I ∗ (0) , I (cid:18) − , − s ( s − ) , .. (cid:19) x Q = t (cid:0) s − (cid:1) (4 t + 1) t s x + t s − t ( s − ) y − tx ) (cid:0) y − s tx (cid:1) = x (cid:0) x − ts (cid:1) (cid:16) x − ts ( t + 1) (cid:17) I ∗ ( ∞ ) , I ∗ (0) , I ( − , I (cid:18) (cid:16) s − s (cid:17) , .. (cid:19) y = x (cid:0) x + t s − t ( s − (cid:1)(cid:0) x + t s − ( s − s − s + 4 s − t + 4 ts (cid:1) I ∗ ( ∞ , , I (cid:18) − , (cid:16) s − s (cid:17) , .. (cid:19) x Q = t ( s − yx ( t − s ) y − ( t s − (cid:0) s + 1 (cid:1) t + 3 s ) yx − s ( t − s ) ( ts − y = x I ( ∞ ) , I (cid:0) s, s (cid:1) , I (cid:16) , s +1 s (cid:17) , I y + (cid:0) t s − ( s + 1) t − s (cid:1) yx − s ( t − s ) ( ts − y = x I (cid:0) ∞ , s, s (cid:1) , I , I (cid:16) , s +1 s (cid:17) Table 6.
Fibrations with discriminant 12 × S k ) Remark 5.2.
Using the new parameter p = Ym ( X + (2 m − s ) (2 m + s ) ) another result can be derivedfrom E m leading to E p : Y − s (2 p −
1) (2 p + 1) Y X = X (cid:0) X + 64 s p (cid:1) ( X + (2 sp + 1) (2 sp −
1) ( s + 2 p ) ( s − p )) , with singular fibers I ∗ , I , I . From E p and the new parameter k = Xp we obtain a genus onefibration whose jacobian is -i.Starting from the fibration − i , the parameter q = xt leads to a genus one fibration whosejacobian is the fibration -i. Transcendental and N´eron-Severi lattices of the surface S k .Lemma 5.1. The five fibrations − i , − i , − i , − i , − i are fibrations of thesame K surface S k with transcendental lattice T S k = h ( − i ⊕ h i ⊕ h i and N´eron-Severi lattice N S ( S k ) = U ⊕ E ( − ⊕ E ( − ⊕ h ( − i ⊕ h ( − i . Moreover these fibrations specialise in fibrations of Y for k = 2 .Proof. These five fibrations are respectively the fibrations given in Table 6 and recalled below withthe type of their singular fibers, their rank and torsion group: − i A A D E rk 0 Z / Z − i A E rk 1 Z / Z − i A D D rk 1 Z / Z × Z / Z − i A D rk 1 Z / Z × Z / Z − i A A rk 0 Z / Z . We already know from the above results that they are fibrations of S k but we give below anotherproof of this fact. Denote S k the K Y = X + (cid:0)(cid:0) − / s + 5 s − / (cid:1) t − s t (cid:1) X (cid:18) s t + (cid:18) / s − s − / s − / s + 1 / (cid:19) t + s (cid:0) s + 1 (cid:1) t (cid:19) X with singular fibers III ∗ ( ∞ ) , I ∗ (0) , I ( s ) , I ( t , t ). t = ∞ t = 0 0 t = s T With the parameter m = Xt we obtain another fibration with singular fibers II ∗ ( ∞ ) (in blue), I ∗ (0) (in green), I ( s ( s − ) (part of it in red), I (4 s ), I ( σ ) (in yellow), where σ = − ( s − s +1 )( s +6 s +1 )( s +1 ) s . This new fibration Σ k has no torsion, rank 0, Weierstrass equation y = x + 2 m (( − s + 10 s − m + 2 s ( s + 1) ) x + (cid:0) m − s (cid:1) m (cid:0) ( s − s − s − s + 1) (cid:1) x + 256 m s ( m − s ) and N´eron-Severi group N S (Σ k ) = U ⊕ E ⊕ D ⊕ A ⊕ A . By Morrison ([M], Corollary 2.10 ii), the N´eron-Severi group of an algebraic K X with12 ≤ ρ ( X ) ≤
20 is uniquely determined by its signature and discriminant form. Thus we compute q NS ( S k ) with the help of the fibration Σ k . From D ∗ /D = h [1] D , [3] D i and q D ([1] D ) = q D ([3] D ) = − , P´ERY-FERMI PENCIL OF K we deduce the discriminant form, since b D ([1] D , [3] D ) = 0,( G NS ( S k ) , q NS ( S k ) ) = Z / Z ( −
32 ) ⊕ Z / Z ( −
32 ) ⊕ Z / Z ( −
23 ) ⊕ Z / Z ( −
12 ) mod.2 Z = Z / Z ( 12 ) ⊕ Z / Z ( −
16 ) ⊕ Z / Z ( −
12 ) . From Morrison ([M] Theorem 2.8 and Corollary 2.10) there is a unique primitive embedding of
N S ( S k ) into the K E ( − ⊕ U , whose orthogonal is by definition the transcendentallattice T S k . Now from Nikulin([Nik] Proposition 1.6.1), it follows G NS ( S k ) ≃ ( G NS ( S k ) ) ⊥ = G T Sk , q T Sk = − q NS ( s k ) . In other words the discriminant form of the transcendental lattice is( G T Sk , q T Sk ) = Z / Z ( −
12 ) ⊕ Z / Z ( 16 ) ⊕ Z / Z ( 12 ) . From this last relation we prove that T S k = h− i ⊕ h i ⊕ h i . Denoting T ′ the lattice T ′ = h− i ⊕ h i ⊕ h i , we observe that T ′ and T S k have the same signature and discriminant form. Since | det( T ′ ) | = 24 is small, there is only one equivalence class of forms in a genus, meaning that such atranscendental lattice is, up to isomorphism, uniquely determined by its signature and discriminantform ([Co] p. 395).Now computing a primitive embedding of T S k into Λ, since by Morrison ([M] Corollary 2.10 i) thisembedding is unique, its orthogonal provides N S ( S k ). Take the primitive embedding h ( − i = h e i ֒ → E , h i = h u + u i ֒ → U , h i = h u + 3 u i ֒ → U , ( u , u ) denoting a basis of U . Hence wededuce N S ( S k ) = U ⊕ E ( − ⊕ E ( − ⊕ ( − ⊕ ( − . Using their Weierstrass equations and a 2-neighbor method [El], it was proved in the previoussubsection that all the fibrations K T S k ( −
1) into U ⊕ E in the following way: ( − ⊕ ( −
6) primitively embeddedin E as in Nishiyama ([Nis] p. 334) and h i = h u + u i ֒ → U . We obtain M = ( T S k [ − ⊥ U ⊕ E = A ⊕ A ⊕ A . Now all the elliptic fibrations of S k are obtained from the primitive embeddings of M into the various Niemeier lattices, as explained in section 2.We identify some of these elliptic fibrations with fibrations N i ( D E ), given by A = h e , e , e , e , e i ֒ → E and A = h d , d i ֒ → D .Since ( A ) ⊥ E = A and ( A ) ⊥ D = A ⊕ A ⊕ D , it follows N = N root = 2 A A D E , det N = 24 × Z / Z . Hence this fibration can be identified with the ellipticfibration N i ( D E ), given by A ⊕ A = h d , d , d , d , d , d + d + 2( d + d + d + d + d ) + d , d i ֒ → D . We get ( A ⊕ A ) ⊥ D = ( − ⊕ h x i ⊕ h d i = ( − ⊕ A ⊕ A with x = d + d + 2( d + d + d + d + d + d + d ) + d and ( −
6) = 3 d + 2 d + 4 d + 3 d + 2 d + d . Cont. Cont. Cont. Cont. Cont. Cont. ht.0 + 2 F + A A A A D D , [3] , [3]] [[2] , [2] − ( d + d ) , [1]] [[2] , [1] − ( d + d ) , [2]] [[2] − ( d + d ) , [1] − ( d + d + d ) , [2]] Table 7.
Contributions and heights of the sections of 5.1.3Thus N root = A A E and the rank of the fibration is 1. Since 2[2] D = x + d and there is no otherrelation with [1] D or [3] D , among the glue vectors h [1 , , i , h [3 , , i generating N i ( D E ), only h [2 , , i contributes to torsion.Hence the torsion group is Z / Z . Moreover the 2-torsion section is2 F + 0 + [[2] , [1] , [1]]with height 4 − (1 / / / /
2) = 0. The infinite section is3 F + 0 + [( − , , N i ( D ), given by A = h d , d , d , d , d i ֒ → D (1)8 and A = h d , d i ֒ → D (2)8 . We compute ( A ) ⊥ D = ( − ⊕ h x = ( − i ⊕ h d i with x = d + d + 2( d + d + d + d + d ) + d ( A ) ⊥ D = h d i ⊕ h x = d + d + 2( d + d + d + d + d ) + d i⊕h d , d , x = d + d + 2 d + d , d i = A ⊕ A ⊕ D . We deduce N root = 4 A D D (hence the fibration has rank 1) and the relations2[2] D = x + d (1) 2([2] D − ( d + d )) = x + 2 d + 2 d + d (2) 2[3] D = x + 2 x + d + 2 d + d + d (3) 2([1] D − ( d + d )) = x + x + d + 2 d + 2 d + d (4) 2([1] D − ( d + d + d )) = 2 x + 3 d + 4 d + 3 d + 2 d + d − d . (5)Thus, among the glue vectors h [1 , , , [1 , , , [2 , , i generating the Niemeier lattice, only vectors h [0 , , , [2 , , i contribute to torsion and the torsion group is Z / Z × Z / Z .From relations (1) to (5) we deduce the various contributions and heights of the following sections(see Table 5.1.3).Hence this fibration can be identified with the fibration N i ( D ) and given by A = h d , d , d , d , d i ֒ → D (1)8 A = h d i ֒ → D (2)8 A = h d i ֒ → D (3)8 . As previously ( A ) ⊥ D = h ( − i ⊕ h x i ⊕ h d i ; we get also h d i ⊥ D = h d i ⊕ h x = d + d + 2 d + d , d , d , d , d , d i = A ⊕ D . Hence N root = 4 A D , and the rank is 1. Moreover it follows therelations 2[2] D = x + d (6) 2[2] D = x + d + 2 d + 2 d + 2 d + 2 d (7) 2([1] D − ( d + d + d + d )) = 2 x + d + 4 d + 3 d + 2 d + d − d (8) 2[3] D = 3 x + d + 2 d + 4 d + 3 d + 2 d + d ∈ A ⊕ D . (9) P´ERY-FERMI PENCIL OF K Cont. Cont. Cont. Cont. Cont. Cont. ht.0 + 2 F + A A A D A D , [3] , [3]] [[2] , [1] − ( d + d + d + d ) , [2]] [[2] , [2] , [1] − ( d + d + d + d )] [[3] , , [3]] Table 8.
Contributions and heights of the sections of 5.1.4We deduce that among the glue vectors generating
N i ( D ), only h [0 , , , [2 , , i contribute totorsion. So the torsion group is Z / Z × Z / Z . From relations (6) to (9) we deduce the variouscontributions and heights of the following sections (see Table 5.1.4).Hence this fibration can be identified with the fibration N i ( A D ) given by A ֒ → A , A ⊕ A = h d , d > i ֒ → D .Since h d , d i ⊥ D = A , we get N = N root = 3 A A ; thus the rank of the fibration is 0 and sincedet( N ) = 24 × , the torsion group is Z / Z .This fibration can be identified with the fibration (cid:3) Remark 5.3.
From fibration − i the surface S k appears to be a double cover of the rationalelliptic modular surface associated to the modular groupe Γ (6) given in Beauville’s paper [Beau]( x + y )( y + z )( z + x )( t − s )( ts −
1) = 8 sxyz. Proof of Theorem . − m ) for example refers for − m ) tonotations used in Bertin-Lecacheux [BL].Comparing to the fibrations of the family you remark more elliptic fibrations with 2-torsion sectionson Y . All the corresponding involutions are denoted τ . Some of them are specialisations for s = 1of the generic ones. Those generic which are Morrison-Nikulin still remain Morrison-Nikulin for Y by a Sch¨utt’s lemma [Sc], namely − τ , − τ , − τ , − τ , − τ , a ) − τ , − τ .Others (( − τ , − τ , − τ , < ( p, > ) − τ , b − τ , c − τ ) are specific to K and cannot be deduced from elliptic fibrations of the generic Kummer. To identify them we have touse the distinguished property of Y , that is Y is a singular K Y inherits of a Shioda-Inose structure, that is the quotient of Y by an involution is isomorphicto a Kummer surface K realized from the product of CM elliptic curves [SI], [SM] provided in thefollowing way.Since the transcendental lattice of Y is T ( Y ) = (cid:18) (cid:19) = (cid:18) a bb c (cid:19) we get b − ac = − τ = − b + √ b − ac a , τ = b + √ b − ac , hence τ = τ = τ = i √ K = E × E/ ± E = C / ( Z + τ Z ) and j ( E ) = j ( i √
2) = 8000. The fact thatthe two CM elliptic curves are equal and satisfy j ( E ) = 8000 can be obtained also by specialisationfrom the Shioda-Inose structure of the family. (see 4.1 Remark 4.1 ).The elliptic curve E can be also put in the Legendre form: E y = x ( x − x − l ) ,l satisfying the equation j = 8000 = − l + l ) l ( l − . Thus l = 3 ± √ l = − ± √ l = ±√ . Proposition 6.1.
The elliptic fibrations bis − τ and − τ are elliptic fibrations of K . L root L/L root
Fibers R Tor. E (0) − f ) A ⊂ E D ⊂ E E A E − h ) A ⊕ D ⊂ E A E E E D Z / Z − φ ) A ⊂ E D ⊂ D E D − o ) A ⊕ D ⊂ E A D Z / Z − q ) D ⊂ E A ⊂ D A A D Z / Z − δ ) A ⊕ D ⊂ D E A D E D ( Z / Z ) − β ) A ⊂ E D ⊂ D E D D Z / Z − r ) A ⊂ E D ⊂ E D A D Z / Z − ψ ) A ⊕ D ⊂ E E D Z / Z − g ) A ⊕ D ⊂ D E E A A Z / Z − e ) D ⊂ E A ⊂ D A A D E Z / Z E A Z / Z (21 − c ) A ⊕ D ⊂ E A Z / Z − n ) D ⊂ E A ⊂ A A A D Z / Z − i ) A ⊕ D ⊂ D A D D ( Z / Z ) − π ) A ⊂ D D ⊂ D A D D Z / Z − u ) A ⊕ D ⊂ D A D D Z / Z D ( Z / Z ) − p ) A ⊂ D D ⊂ D A D A D Z / − t ) A ⊕ D ⊂ D A D D Z / Z D A Z / Z − m ) A ⊕ D ⊂ D A A A A Z / Z − α ) D ⊂ D A ⊂ A D A E ( Z / Z ) − b ) A ⊂ E D ⊂ E A E E Z / Z A E D Z / Z − w ) A ⊂ E D ⊂ D A A A A Z / Z − µ ) A ⊂ A D ⊂ D A A A E − j ) A ⊕ D ⊂ D A E A Z / Z − l ) A ⊂ A D ⊂ E A D − k ) D ⊂ E A ⊂ D A A D Z / Z D ( Z / Z ) − d ) A ⊂ D D ⊂ D A D D D Z / D A Z / × Z / − v ) D ⊂ D A ⊂ A A A D A Z / × Z / − s ) D ⊂ D A ⊂ D A A A A Z / Z − a ) D ⊂ D A ⊂ A D A A Table 9.
The elliptic fibrations of Y Proof.
It follows from the 4 . F that the fibration − τ with Weierstrass equation Y = X − U ( U − X + U ( U + 1) ( U − X, singular fibers III ∗ (0), I ∗ ( ∞ ), I ( − I (4), I ( − / Z / Z -torsion is an elliptic fibrationof K . Similarly from the 4 . G , we deduce that the elliptic fibration − τ withWeierstrass equation Y = X + 2( t + 5 t ) X + t (4 t + 1)( t + 6 t + 1) X, singular fibers III ∗ ( ∞ ), I ∗ (0), 3 I ( − / , t + 6 t + 1 = 0), and Z / Z -torsion is an elliptic fibrationof K . (cid:3) To achieve the proof of Theorem 1.2 we need also the following lemma.
P´ERY-FERMI PENCIL OF K Lemma 6.1.
The Kummer K has exactly extremal elliptic fibrations given by Shimada Zhang [SZ] with the type of their singular fibers and their torsion group (1) E A A A Z / Z , (2) D A A A , Z / Z , (3) D D A , Z / Z , (4) A A A A A A , Z / Z × Z / Z . From lemma (6.1 (2)) we obtain the fibration − τ and from lemma (6.1 (3)) the fibration h ( p, i − τ .We notice also that fibrations − τ and − τ obtained by specialisation are also fibration(4) of lemma (6.1) and fibration − τ , by a 2-neighbor process of parameter m = Xk ( k +4) givesfibration (3) of lemma (6.1).Finally, by a 2-neighbor process of parameter m = Xd ( d +1) , fibrations b ) − τ and c ) − τ givesfibration − τ , hence are elliptic fibrations of K . Corollary 6.1.
As a byproduct of the proof we get Weierstrass equations for extremal fibrations oflemma (6.1) (2), (3), (4).
The table of symplectic automorphisms of order 2 (self involutions) results from an easy computation.7. 2 -isogenies and isometries
Theorem 1.2, where the 2-isogenous K Y are either its Kummer K or Y itself,cannot be generalised to all the other singular K p -isogenies ( p prime) between complex projective K X and Y define isometries between their rationaltranscendental lattices T X, Q and T Y, Q . ( These lattices are isometric if there exists M ∈ Gl( n, Q )satisfying T X, Q = M t T Y, Q M . Let us recall the part of their Theorem related to 2-isogenies. Theorem 7.1. [BSV]
Let γ : X → Y be a -isogeny between complex projective K surfaces X and Y . Then rk ( T Y, Q ) = rk ( T X, Q ) =: r and (1) If r is odd, there is no isometry between T Y, Q and T X, Q . (2) If r is even, there exists an isometry between T Y, Q and T X, Q if and only if T Y, Q is isometricto T Y, Q (2) . This property is equivalent to the following: for every prime number q congruentto or modulo , the q -adic valuation ν q (det T Y ) is even. As a Corollary we deduce the following result.
Theorem 7.2.
Among the singular K surfaces of the Ap´ery-Fermi family defined for k rationalinteger, only Y and Y possess symplectic automorphisms of order (“self -isogenies”).Proof. The singular K k rational integer are Y , Y , Y , Y , Y , Y , Y , Y . This list has been computed numerically by Boyd [Boy]. Using the notation [SZ], that is writingthe transcendental lattice T Y = (cid:18) a bb c (cid:19) as T Y = [ a b c ] we get: T Y = [4 2 4] T Y = [2 0 4] T Y = [2 0 12] . They are obtained by specialisation of fibration k = 0 , k = 0 the ellipticfibration has rank 0 and singular fibers of type I , I , 2 I . For k = 2, the transcendental lattice isalready known. For k = 6, the elliptic fibration has rank 0 and type of singular fibers I , I , 2 I .Now using Shimada-Zhang table [SZ], we derive the previous announced transcendental lattices.The transcendental lattices T Y and T Y were computed in the paper [BFFLM]. With the methodused there, we can compute the transcendental lattices of Y , Y and Y . We obtain: T Y = [2 1 8] T Y = [6 0 12] T Y = [10 0 12] No Weierstrass Equation From or to y = x + ( o − o + 2) x + xI ∗ ( ∞ ) , I (0) , I (1 , , o − o − Y = X ( X − o + 5 o )( X − o + 5 o − I ∗ ( ∞ ) , I (0) , I (1 , , o − o −
4) Spec. − i y = x + ( q + q + 2 q − x + (1 − q ) xI ∗ ( ∞ ) , I (0) , I ( ) , I ( q + 2 q + 5) Y = X − q + q + 2 q − X + Xq ( q + 2 q + 5) I ∗ ( ∞ ) , I (0) , I ( q + 2 q + 5) , I ( ) lemma(6 . y = x − r ( r − r + 2) x + r xI ∗ ( ∞ ) , I ∗ (0) , I (1) , I ( ± i ) Y = X + 2 r ( r − r + 2) X + ( r − r ( r + 4) XI ∗ ( ∞ ) , I ∗ (0) , I (1) , I ( ± i ) Spec. − i bis y = x − ( ψ + 5 ψ ) x − ψ xI ∗ ( ∞ ) , III ∗ (0) , I ( − , ψ + 6 ψ + 1) Y = X + 2(5 ψ + ψ ) X + Xψ (4 ψ + 1)( ψ + 6 ψ + 1) III ∗ ( ∞ ) , I ∗ (0) , I ( − ( ) , ψ + 6 ψ + 1) lemma(6 . y = x ( x − e ( e − x + e (2 e + 1)) I ∗ ( ∞ ) , III ∗ (0) , I ( − , − ) , I (4) Y = X + 2 e ( e − X + e ( e − e + 1) XI ∗ ( ∞ ) , III ∗ (0) , I ( − , I (4) , I ( − ) lemma(6 . < ( p, > y = x ( x − p )( x − p ( p + 1) ) I ∗ ( ∞ ) , I ∗ (0) , I ( − , I ( − Y = X − p ( p + 2 p − X − p ( p + 2) XI ∗ ( ∞ ) , I ∗ (0) , I ( − , I ( −
2) lemma(6 . y = x + t ( t + 4 t + 1) x + t x I ∗ ( ∞ , , I ( − , I ( t + 6 t + 1 = 0) Y = X − t ( t + 4 t + 1) X + t ( t + 1) ( t + 6 t + 1) X I ∗ ( ∞ , , I ( − , I ( t + 6 t + 1) Spec. − i y = x ( x + x ( ( m − −
2) + 1) I ( ∞ ) , I (0 , ± , I ( ± √ Y = X ( X − m + 2 m )( X − m + 2 m − I ( ∞ ) , I (0 , ± , I ( ± √
2) lemma(6 . − i y = x + x ( k − k + k − k ) + k xI ( ∞ ) , I ∗ (0) , I (2) , I (4 , ± i ) Y = X − ( k − k − k ) X + k ( k − k +4)( k − XI ( ∞ ) , I ∗ (0) , I (2) , I (4 , ± i ) Xk ( k +4 to lemma(6 . a ) h (0 , i b ) h ( d + d , i c ) h ( d + d , i y = x ( x − ( d + d ))( x − ( d + d ))2 I ∗ ( ∞ , , I ∗ ( − , I (1) a ) : Y = X + 2 d ( d + 1) X + d ( d − X I ∗ ( ∞ , , I ∗ ( − , I (1) b ) : Y = X + 2 d ( d + 1)( d − X + d ( d + 1) XI ∗ ( ∞ ) , I ∗ (0) , I ∗ ( − , I (1) c ) : Y = X − d ( d + 1)(2 d − X + Xd ( d + 1) a )Spec. − ib ) m = X/d ( d + 1)to − τc ) similar to b ) y = x + x ( ( s − − s ) + s x I ( ∞ , , I (1) , I ( − , I (3 ± √ Y = X ( X − ( s − + 4 s )( X − ( s − ) I (1) , I (0 , − , ∞ ) , I (3 ± √
2) Spec. − i lemma(6 . Table 10.
Morrison-Nikulin involutions of Y (fibrations of K ) T Y = [12 0 26] [ T Y = [12 0 34] . Applying Bessi`ere, Sarti and Veniani’s Theorem, we conclude that only Y and Y may have selfisogenies. By Theorem 1.2, Y has self isogenies. We shall prove that Y satisfies the same property. P´ERY-FERMI PENCIL OF K No Weierstrass Equation y = x + 2 β ( β − x + β ( β − xI ∗ ( ∞ ) , III ∗ (0) , I ∗ ( − Y = X − β ( β − X + 4 β ( β − XI ∗ ( ∞ ) , III ∗ (0) , I ∗ (1) y = x + 4 g x + g ( g + 1) x III ∗ ( ∞ , , I ( − , I (1) Y = X − g X − g ( g − X III ∗ ( ∞ , , I (1) , I ( − y = x + x π ( π − π −
2) + π (2 π + 1) xI ∗ ( ∞ ) , I ∗ (0) , I ( − / , I (4) Y = X − X π ( π − π −
2) + π ( π − XI ∗ (0) , I ∗ ( ∞ ) , I (4) , I ( − / y = x + u ( u + 4 u + 2) x + u xI ∗ ( ∞ ) , I ∗ (0) , I ( − , I ( − Y = (cid:16) X − u ( u − (cid:17) X ( X − u ) I ∗ ( ∞ ) , I ∗ (0) , I ( − , I ( − a ) < (0 , >b ) < ( p ( p + 1) , > y = x ( x − p )( x − p ( p + 1) ) I ∗ ( ∞ ) , I ∗ (0) , I ( − , I ( − a ) : Y = X ( X + 4 p + p + 4 p )( X + p ) I ∗ (0) , I ∗ ( ∞ ) , I ( − , I ( − b ) : Y = X − p (2 p + 4 p + 1) X + p XI ∗ ( ∞ ) , I ∗ (0) , I ( − , I ( − y = x − (2 − w − w ) x − ( w − xI ( ∞ ) , I (0) , I ( ± , I ( ± i √ Y = X + 2(2 − w − w ) X + w ( w + 8) XI (0) , I ( ∞ ) , I ( ± i √ , I ( ± Table 11.
Self involutions of Y Consider the following elliptic fibration of rank 0 of Y (other interesting properties of Y will bestudied in a forthcoming paper): y = x + x (9( t + 5)( t + 3) + ( t + 9) ) − xt ( t + 5) with singular fibers III ∗ ( ∞ ), I (0), I ( − I ( − I ( −
4) and 2-torsion. Its 2-isogenous curve hasa Weierstrass equation Y = X + X ( − t − t − X ( t + 4) ( t + 9) with singular fibers III ∗ ( ∞ ), I ( − I ( − I (0), I ( − Y given by x = − X , y = iY √ . (cid:3) Moreover we observe that T Y = [2 0 4] , T Y , Q = [2 0 1] ,T K = [4 0 8] , T K , Q = [2 0 1] , Similarly T Y , Q = [6 0 3] , T K , Q = [3 0 6] . Hence we suspect some relations between the transcendental lattices of K i and of S i for singular Y i .We give some examples of such relations in the following proposition. Proposition 7.1.
Even if the -isogenies from Y , Y are not isometries, the following rationaltranscendental lattices satisfy the relations (1) T K , Q = T S , Q , (2) T K , Q = T S , Q , (3) K = S .Proof. (1) For k = 0 we get two elliptic fibrations of rank 0, namely K with Weierstrass equation y = x + 2 x ( t + 1) + x ( t − ( t + t + 1) , type of singular fibers D , 3 A , A , 4-torsion and T K = [8 4 8]. On the other end thefibration S y = x ( x −
14 ( t − I )( t + I ) )( x −
14 ( t + 3 I )( t − I ) )with type of singular fibers 3 A ( ∞ , ± I ), 3 A (0 , ± I ), Z / × Z / ( t + 1) , ± ( t + 1) ). Hence, by Shimada-Zhang ’s list T S = [2 0 6]. Nowwe can easily deduce the relation (cid:18) / / − (cid:19) (cid:18) (cid:19) (cid:18) / / − (cid:19) = (cid:18) (cid:19) . (2) For k = 6 the elliptic fibration − i gives a rank 0 elliptic fibrationof S : y = x + x ( − t t − t + 18 t + 32 ) + x ( t −
16 ( t − t + 1) , with singular fibers 2 I ( t − t + 1 = 0), I ( ∞ ), I (3), 2 I (0 , Z / Z -torsion. UsingShimada-Zhang’s list [SZ], we find T S = [4 0 6]. Since T K = (cid:18) (cid:19) ∼ Q (cid:18) (cid:19) and T S = (cid:18) (cid:19) ∼ Q (cid:18) (cid:19) we get straightforward T K , Q = T S , Q . (3) Consider the elliptic fibration Y with Weierstrass equation y = x + 14 ( t − t + 15 t − t − x − t ( t − x, singular fibers I ( ∞ ), 2 I ( t − t + 1 = 0), 2 I (0 , I ( t − t + 9 = 0), rank 1 and6-torsion. The infinite section P = ( t, − t ( t − t + 3)), of height generates the freepart of the Mordell-Weil group, since det( T Y ) = 15 by the previous theorem and by theShioda-Tate formula det( T Y ) = 54 12 × × = 15 . Its 2-isogenous curve has Weierstrass equation y = x + ( − t + 3 t − t + 9 t + 32 ) x + 116 ( t − t + 9)( t − t + 1) x, singular fibers 3 I ( ∞ , t − t + 1 = 0), 2 I ( t − t + 9 = 0), 2 I (3 , Q image by the 2-isogeny of the infinite section P is an infinite section ofheight . Since neither Q nor Q + (0 ,
0) are 2-divisible, the section Q generates the freepart of the Mordell-Weil group. Hence by the Shioda-Tate formula, it followsdet( T S ) = 52 6 × = 60 = det( T K ) . We can show that K and S are the same surface. To prove this property we show that agenus one fibration is indeed an elliptic fibration. We start with the fibration of K obtained P´ERY-FERMI PENCIL OF K from m = yt ( x + ( t − s ) ( ts − ) . If k = 3 and s = s := √ we get E m . Then changing X = s x and Y = s y it follows y − myx = x (cid:0) x − m (cid:1) (cid:18) x − (cid:16)(cid:16) − √ (cid:17) m − m + 7 + 3 √ (cid:17)(cid:19) . The next fibration is obtained with the parameter n = xm . Now if w = ym it gives thefollowing quartic in w and mw − mnw + 2 (cid:16) √ − (cid:17) n ( n − m − n ( n −
1) ( n − m − (cid:16) √ (cid:17) n ( n − . Notice the point (cid:0) w = − (cid:0) √ (cid:1) n ( n + 1) ( n − , m = (cid:0) √ (cid:1) (cid:0) n − √ (cid:1)(cid:1) onthis quartic, so it is an elliptic fibration of K which is (cid:3) Remark 7.1.
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