ON K3 SURFACES
ALICE GARBAGNATI AND ALESSANDRA SARTI
Abstract. Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 latticeU3⊕E8(−1)2depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.
0. Introduction
An automorphism of a K3 surface is called symplectic if the induced action on the holomorphic 2-form is trivial. The finite groups acting symplectically on a K3-surface are classified by [Nik1], [Mu], [X], where also their fixed locus is described. In [Nik1] Nikulin shows that the action of a finite abelian group of symplectic automorphisms on the K3 latticeU3⊕E8(−1)2is unique (up to isometries of the lattice), i.e. it depends only on the group and not on the K3 surface. Hence one can consider a special K3 surface, compute the action, then, up to isometry, this is the same for any other K3 surface with the same finite abelian group of symplectic automorphisms. It turns out that elliptic K3 surfaces are good candidate to compute this action, in fact one can produce symplectic automorphisms by using sections of finite order.
The finite abelian group in the list of Nikulin are the following fourteen groups:
Z/nZ, 2≤n≤8, (Z/mZ)2, m= 2,3,4, Z/2Z×Z/4Z, Z/2Z×Z/6Z, (Z/2Z)i, i= 3,4.
For all the groupsGbut (Z/2Z)i, i= 3,4, there exists an elliptic K3 surface with symplectic group of automorphismsGgenerated by sections of finite order (cf. [Shim]). In general it is difficult to describe explicitly the action on the K3 lattice. An important step toward this identification is to determine the invariant sublattice, its orthogonal complement and the action ofGon this orthogonal complement.
In [Nik1] Nikulin gives only rank and discriminant of these lattices, however the discriminant are wrong if the group is not cyclic (cf. also [G2]). In the case of G=Z/2Zthe action on the K3 lattice was computed by Morrison (cf. [Mo]), and in the cases of G=Z/pZ, p= 3,5,7 we computed in [GS]
the invariant sublattice and its orthogonal in the K3 lattice. In this paper we conclude the description of these lattices for all the fourteen groups, in particular we can compute their discriminant which are not always the same as those given in [Nik1]. Its is interesting that some of the orthogonal to the invariant lattices are very well known lattices. We denote by ΩG:= (H2(X,Z)G)⊥ then
G Z/3Z (Z/2Z)2 Z/4Z (Z/2Z)4 (Z/3Z)2 ΩG K12(−2) Λ12(−1) Λ14.3(−1) Λ15(−1) K16.3(−1)
whereK12(−2) is the Coxeter-Todd lattice, the lattices Λnare laminated lattices and the latticeK16.3
is a special sublattice of the Leech-lattice (cf. [CS1] and [PP] for a description). All these give lattice packings which are very dense.
In each case one can compute the invariant lattice and its orthogonal complement by using an elliptic K3 surface. In the case ofG=Z/2Z×Z/4Z,(Z/2Z)2,Z/4Zwe use always the same elliptic fibration, with six fibers of typeI4, with symplectic automorphism group isomorphic toZ/4Z×Z/4Zgenerated
2000 Mathematics Subject Classification: 14J28, 14J50, 14J10.
Key words: K3 surfaces, symplectic automorphisms.
1
by sections, then we consider the subgroups. The K3 surface admitting this elliptic fibration is the Kummer surfaceKm(E√−1×E√−1) and the group of its automorphisms is described in [KK]. This fibration admits also symplectic automorphisms group isomorphic to (Z/2Z)i, i = 3,4, in this case some of the automorphisms come from automorphisms of the base of the fibration, i.e. of P1. Hence also for these two groups we are able to compute the invariant sublattice and its orthogonal comple- ment.
Once we know the lattices ΩG we can describe all the possible N`eron Severi groups of algebraic K3 surfaces with minimal Picard number and group of symplectic automorphisms G. Moreover we can prove that if there exists a K3 surface with one of those lattices as N´eron Severi group, then it admits a certain finite abelian groupGas group of symplectic automorphisms. These two facts are important for the classification of K3 surfaces with symplectic automorphism group (cf. [vGS], [GS]), in fact they give information on the coarse moduli space of algebraic K3 surface with symplectic automorphisms.
For example an algebraic K3 surface X has G as a symplectic automorphism group if and only if ΩG⊂N S(X) (cf. [Nik1, Theorem 4.15]) and moreoverρ(X)≥1 +rk(ΩG).
The paper is organized as follows: in the sections 1 and 2 we recall some basic results on K3 surfaces and elliptic fibrations, in section 3 we show how to find elliptic K3 surfaces with symplectic automor- phism group (Z/2Z)i,i= 3,4. The section 4 recalls some facts on the elliptic fibration described by Keum and Kondo in [KK] and contains a description of the lattices H2(X,Z)G and (H2(X,Z)G)⊥ in the cases G = Z/4Z×Z/4Z,Z/2Z×Z/4Z,Z/4Z,(Z/2Z)i, i = 2,3,4. In the section 5 we give the equations of the elliptic fibrations for the remainingGand compute the invariant lattice and its orthogonal. Finally the section 6 describe the N`eron Severi group of K3 surfaces with finite abelian symplectic automorphism group and deal with the moduli spaces. In Appendix we describe briefly the elliptic fibrations which can be used to compute the lattices ΩG for the groupG which are not analyzed in Section 4 and we give a basis for these lattices. The proofs of these results are essentially the same as the proof of Proposition 4.1 in Section 4.
We warmly thank Bert van Geemen for discussions and for his constant support during the preparation of this paper. We thank also Gabriele Nebe for comments and suggestions.
1. Basic results
LetX be a K3 surface. The second cohomology group ofX with integer coefficients,H2(X,Z), with the pairing induced by the cup product is a lattice isometric to ΛK3:=E8(−1)2⊕U3(the K3 lattice), where U is the lattice with pairing
· 0 1 1 0
¸
and E8(−1) is the lattice associated to the Dynkin diagramE8(cf. [BPV]). Letgbe an automorphism ofX. It induces an actiong∗, onH2(X,Z). This isometry induces an isometry on H2(X,C) which preserve the Hodge decomposition. In particular g∗(H2,0(X)) =H2,0(X).
Definition 1.1. An automorphism g ∈Aut(X) is symplectic if and only ifg∗|H2,0(X) =Id|H2,0(X)
(i.e. g∗(ωX) =ωX with H2,0(X) =CωX). We will say that a group of automorphisms acts sym- plectically on X if all the elements of the group are symplectic automorphisms.
Remark 1.1. [Nik1, Theorem 3.1 b)] An automorphism g of X is symplectic if and only if g∗|TX =
IdTX. ¤
In [Nik1] the finite abelian groups acting symplectically on a K3 surface are listed and many properties of this action are given. Here we recall the most important. Let Gbe a finite abelian group acting symplectically on a K3 surfaceX, then
• Gis one of the following fourteen groups
Z/nZ, 2≤n≤8, (Z/mZ)2, m= 2,3,4, Z/2Z×Z/4Z, Z/2Z×Z/6Z, (Z/2Z)i, i= 3,4;
(1)
• the desingularization of the quotientX/Gis a K3 surface;
• the action induced by the automorphisms of Gon H2(X,Z) is unique up to isometries. In particular the latticesH2(X,Z)G and (H2(X,Z)G)⊥ depend on G but they do not depend on X, up to isometry.
The last property implies that we can consider a particular K3 surface X admitting a finite abelian group Gof symplectic automorphisms to analyze the isometries induced on H2(X,Z)≃ΛK3 and to describe the lattices H2(X,Z)G and ΩG := (H2(X,Z)G)⊥. In particular sinceTX is invariant under the action ofG(by Remark 1.1),
H2(X,Z)G←֓ N S(X)G⊕TX
and the inclusion has finite index, so considering the orthogonal lattices, we obtain (H2(X,Z)G)⊥= (N S(X)G)⊥.
For a systematical approach to the problem how to construct K3 surfaces admitting certain symplectic automorphisms we will consider K3 surfaces admitting an elliptic fibration. We recall here some basic facts.
LetX be a K3 surface admitting an elliptic fibration, i.e. there exists a morphismX →P1 such that the generic fiber is a non singular genus one curve and such that there exists a section s:P1 →X, which we call the zero section. There are finitely many singular fibers, which can also be reducible. In the following we will consider only singular fibers of typeIn,n≥0,n∈N. The fibers of typeI1 are curves with a node, the fibers of type I2are reducible fibers made up of two rational curves meeting in two distinct points, the fibers of type In, n > 2 are made up of n rational curves meeting as a polygon withnedges. We will callC0the irreducible component of a reducible fiber which meets the zero section. The irreducible components of a fiber of typeIn are calledCi whereCi·Ci+1 = 1 and i∈Z/nZ. Under the assumptionC0·s= 1, these conditions identify the components completely once the component C1 is chosen, so these conditions identify the components up to the transformation Ci↔C−i for eachi∈Z/nZ. All the components of a reducible fiber of typeIn have multiplicity one, so a section can intersect a fiber of typeIn in any component.
The set of the sections of an elliptic fibration form a group (the Mordell Weil group), with the group law which is induced by the one on the fibers.
Let Red be the set Red ={v ∈P1|Fv is reducible}. Letr be the rank of the Mordell Weil group (recall that if there are no sections of infinite order then r= 0) and letρ=ρ(X) denote the Picard number of the surfaceX. Then
ρ(X) =rkN S(X) =r+ 2 + X
v∈Red
(mv−1)
(cfr. [Shio, Section 7]) where mv is the number of irreducible components of the fiberFv.
Definition 1.2. The trivial lattice T rX (or T r) of an elliptic fibration on a surface is the lattice generated by the class of the fiber, the class of the zero section and the classes of the irreducible components of the reducible fibers which do not intersect the zero section.
The lattice T r admits U as sublattice and its rank is rk(T r) = 2 +P
v∈Red(mv −1). Recall that N S(X)⊗Qis generated byT rand the sections of infinite order.
Theorem 1.1. [Shio, Theorem 1.3]The Mordell Weil group of the elliptic fibration on the surfaceX is isomorphic to the quotient N S(X)/T r:=E(K).
In Section 8 of [Shio] a pairing onE(K) is defined. The value of this pairing on a sectionP depends only on the intersection between the section P and the reducible fibers and betweenP and the zero section. Now we recall the definition and the properties of this pairing.
LetE(K)tor be the set of the torsion elements in the groupE(K).
Lemma 1.1. [Shio, Lemma 8.1, Lemma 8.2] For any P ∈E(K)there exists a unique elementφ(P) in N S(X)⊗Qsuch that:
i) φ(P)≡(P)modT r⊗Q(where(P) is the class ofP moduloT r⊗Q) ii) φ(P)⊥T r.
The map φ : E(K) → N S(X)⊗Q defined above is a group homomorphism such that Ker(φ) = E(K)tor.
Theorem 1.2. [Shio, Theorem 8.4] For any P, Q ∈ E(K) let hP, Qi = −φ(P)·φ(Q) (where · is induced on N S(X)⊗Qby the cup product). Then it defines a symmetric bilinear pairing on E(K), which induces the structure of a positive definite lattice on E(K)/E(K)tor.
In particular if P ∈E(K), thenP is a torsion section if and only if hP, Pi= 0.
For any P, Q∈E(K)the pairing h−,−i is
hP, Qi = χ+P·s+Q·s−P·Q−P
v∈Redcontrv(P, Q) hP, Pi = χ+ 2(P·s)−P
v∈Redcontrv(P)
where χ is the Euler characteristic of the surface and the rational number contrv(P, Q)are given in the table below
I2 In In∗ IV∗ III∗
contrv(P) 2/3 i(n−i)/n
½ 1 ifi= 1
1 +n/4 ifi=n−1or i=n 4/3 3/2 contrv(P, Q) 1/3 i(n−j)/n
½ 1/2 ifi= 1
2 +n/4 ifi=n−1or i=n 2/3 − (2)
where the numbering of the fibers is the one described before, P andQmeet the fiber in the component Ci and Cj andi≤j.
The pairing defined in the theorem is called height pairing. This pairing will be used to determine the intersection of the torsion sections of the elliptic fibrations with the irreducible components of the reducible fibers.
2. Elliptic fibrations and symplectic automorphisms
Using elliptic fibrations one can describe the action of a symplectic automorphism induced by a torsion section on the N´eron Severi group. Since a symplectic automorphism acts as the identity on the transcendental lattice we can describe the action of the symplectic automorphism overN S(X)⊕TX, which is a sublattice of a finite index ofH2(X,Z). Moreover, knowing the discriminant form ofN S(X) and ofTX one can explicitly find a basis for the lattice H2(X,Z), and so one can describe the action of the symplectic automorphism on the lattice H2(X,Z).
LetX be a K3 surface admitting an elliptic fibration with section, then the N´eron Severi group ofX contains the classes F ands, which are respectively the class of the fiber and of the section. Let us suppose thattis ann-torsion section of an elliptic K3 surface and letσtbe the automorphism, induced byt, which fixes the base of the fibration (so fixes each fiber) and acts on each fiber as translation by t. This automorphism is symplectic and if the sectiont is an n-torsion section (with respect to the group law of the elliptic fibration) the automorphismσthas ordern. More in general let us consider a K3 surface X admitting an elliptic fibrationEX with torsion part of the Mordell Weil group equal to a certain abelian groupG. Then the sections oftors(M W(EX)) induce symplectic automorphisms which commute. So we obtain the following:
Lemma 2.1. If X is a K3 surface with an elliptic fibration EX and tors(M W(EX)) = G, then X admits Gas abelian group of symplectic automorphisms and these automorphisms are induced by the torsion sections of EX.
Now we want to analyze the action of the symplectic automorphisms induced by torsion sections on the classes generating the N´eron Severi group of the elliptic fibration. By definition,σtfixes the class F. Since it acts as a translation on each fiber, it sends on each fiber the intersection of the fiber with the zero section, in the intersection of the fiber with the sectiont. Globallyσtsends the sectionsin
the section t. More in general the automorphismσtsends sections in sections and the section rwill be send in r+t−s, where the + and−are the operations in the Mordell Weil group. To complete the description of the action of σt on the N´eron Severi group we have to describe its action on the reducible fibers. The automorphism σtrestricted to a reducible fiber has to be an automorphism of it. This fact imposes restrictions on the existence of a torsion section and of certain reducible fibers in the same elliptic fibration.
Lemma 2.2. Lett be a torsion section and Fv a reducible fiber of the fibration.
1) If t meets the fiber Fv in a componentCi, thenσt(C0) =Ci.
2) If t meets the fiber Fv in the componentC0, thenσt(Cj) =Cj for each j.
3) If there is a fiber of type Id with component Ci and there is an n-torsion section with n6 |d, then t·C0= 1 and by 2)σt(Ci) =Ci for each i= 0, . . . , d.
4) If there is a fiber Id with component Ci and an n-torsion section t with n|d, such that t·Ci = 1, i6= 0, theni|d andd|(n·i). Moreoverσtrestricted to this fiber has orderd/i andσt(Cj) =Cj+i. Proof. 1) The sectionsmeets the reducible fiber in the componentC0, so 1 =s·C0=σt(s)·σt(C0) = t·σt(C0). Butσt(C0) has to be a component of the same fiber, becauseσtfixes the fibers, and has to be the component with a non-trivial intersection witht, soσt(C0) =Ci.
2) By 1) applied in the case i = 0 we obtain σt(C0) =C0. The group law on the fibers of type In
is C∗×Z/nZ (cf. [Mi, Section VIII.3]). Since the automorphism fixes the component C0 it acts as (ωn,0) on In, where ωn is a primitiven-th root of unity. It acts trivial on Z/nZ, so it fixes all the components Ci.
3-4) The automorphism group of the fiber of typeId isC∗×Z/dZ. The automorphismσt either acts
onId as in 2) or has order which dividesd. ¤
In the following we always use this notation: if t1 is ann-torsion section, then th corresponds to the sum, in the Mordell Weil group law, ofh timest1. Moreover Ci(j) is the i-th component of thej-th reducible fibers.
3. The cases G= (Z/2Z)3 andG= (Z/2Z)4
We have seen (cf. Lemma 2.1) that an example of a K3 surface with G as group of symplectic automorphisms is given by an elliptic fibration withGas torsion part of the Mordell Weil group. The groups which appear as torsion part of the Mordell Weil group of an elliptic fibration on a K3 surface are twelve. In particular the groupsG= (Z/2Z)3 andG= (Z/2Z)4 are groups acting symplectically on a K3 surface, but they can not be realized as the torsion part of the Mordell Weil group of an elliptic fibration on a K3 surface (cf. [Shim]). Hence to find examples of K3 surfaces with one of these groups as group of symplectic automorphisms, we have to use a different construction. One possibility is to consider a K3 surface with elliptic fibration EX with M W(EX) = (Z/2Z)2 and to find one (or two) other symplectic involutions which commute with the ones induced by torsion sections.
3.1. The group G= (Z/2Z)3 acting symplectically on an elliptic fibration. We start consid- ering an elliptic K3 surface with two 2-torsion sections. An equation of such an elliptic fibration is given by
(3) y2=x(x−p(τ))(x−q(τ)) deg(p(τ)) =deg(q(τ)) = 4, τ ∈C.
Then we consider an involution on the base of the fibration P1, which preserves the fibration. This involution fixes two points of the basis. Up to the choice of the coordinates on P1, we can suppose that the involution on the basis is σP1,a: τ 7→ −τ. So we consider on the K3 surface the involution (τ, x, y)7→(−τ, x,−y). Since this map has to be an involution of the surface, it has to fix the equation of the elliptic fibration. Moreover the involution σP1,a has to commute with the involutions induced by the torsion sections. This implies that it has to fix the curves corresponding to the torsion sections t:τ 7→(p(τ),0) andu:τ7→(q(τ),0). The equation of such an elliptic fibration is
y2=x(x−p(τ))(x−q(τ)) with
deg(p(τ)) =deg(q(τ)) = 4 andp(τ) =p(−τ) q(τ) =q(−τ).
(4)
i.e. y2=x(x−(p4τ4+p2τ2+p0))(x−(q4τ4+q2τ2+q0)), p4, p2, p0, q4, q2, q0∈C.
The involution σP1,a fixes the four curves corresponding to the sections in the torsion part of the Mordell Weil group and the two fibers over the points 0 and∞of P1. On these two fibers the auto- morphism is not the identity (because it sends yin −y). So it fixes only the eight intersection points between these four sections and the two fibers. It fixes eight isolated points, so it is a symplectic involution.
To compute the moduli of this family of surfaces we have to consider that the choice of the automor- phism onP1 corresponds to a particular choice of the coordinates, so we can not act on the equation with all the automorphisms of P1. We choose coordinates such that the involution σP1,a fixes the points (1 : 0) and (0 : 1) onP1.
The space of the automorphisms of P1 commuting with σP1,a has dimension one. Moreover we can act on the equation (4) with the transformation (x, y) 7→ (λ2x, λ3y) and divide by λ6. Since the parameters of the equation (4) are 6, we find that the number of moduli of this family is 6−2 = 4.
Collecting these results, the properties of the family satisfying the equation (4) are the following:
discriminant singularf ibers moduli p(τ)2q(τ)2(p(τ)−q(τ))2 12I2 4
Since the number of moduli of this family is four, the Picard number of the generic surface in this family is 16. The trivial lattice of this fibration has rank 14, so there are two linearly independent sections of infinite order which generate the free part of the Mordell Weil group.
3.2. The groupG= (Z/2Z)4acting symplectically on an elliptic fibration. As in the previous section we construct an involution which commutes with the three symplectic involutions σt, σu
(the involutions induced by the torsion section t and u) and σP1,a of the surfaces described by the equation (4). So we consider two commuting involutions on P1 which commute with the involutions induced by the 2-torsion sections. Up to the choice of the coordinates of P1 we can suppose that the involutions on P1 are τ 7→ −τ and τ 7→ 1/τ. We call the corresponding involution on the surface σP1,a : (τ, x, y) 7→ (−τ, x,−y) and σP1,b : (τ, x, y) 7→ (1/τ, x,−y). As before requiring that these involutions commute with the involutions induced by the torsion sections means that each of these involutions fixes the torsion sections. Elliptic K3 surfaces with the properties described have the following equation:
y2=x(x−p(τ))(x−q(τ)) with
deg(p(τ)) =deg(q(τ)) = 4 p(τ) =p(−τ) =p(1τ) q(τ) =q(−τ) =q(1τ).
(5)
i.e. y2=x(x−(p4τ4+p2τ2+p4))(x−(q4τ4+q2τ2+q4)), p4, p2, q4, q2∈C.
As before the choice of the involutions of P1 implies that the admissible transformations on the equation have to commute with σP1,a andσP1,b. The only possible transformation is the identity so no transformations of P1 can be applied to the equation (5). The only admissible transformation on that equation is (x, y)7→(λ2x, λ3y).
Collecting these results, the properties of the family satisfying the equation (5) are the following:
discriminant singularf ibers moduli p(τ)2q(τ)2(p(τ)−q(τ))2 12I2 3
As in the previous case the comparison between the number of moduli of the family and the rank of the trivial lattice implies that the free part of the Mordell Weil group is Z3.
4. An elliptic fibration with six fibers of typeI4
An equation of an elliptic K3 surface with six fibers of typeI4is the following y2=x(x−τ2σ2)
µ
x−(τ2+σ2)2 4
¶ . (6)
This equation is well known, for example it is described in [TY, Section 2.3.1], where it is constructed considering the K3 surface as double cover of a rational elliptic surface with two fibers of typeI2and two fibers of typeI4. One can find this equation also considering the equation of an elliptic K3 surface
Figure 1
with two 2-torsion sections (i.e the equation (3)) and requiring that the tangent lines to the elliptic curve defined by the equation (3) in the two rational points of the elliptic surface pass respectively through the two points of order two (p(τ),0) and (q(τ),0).
We will call X(Z/4Z)2 the elliptic K3 surface described by the equation (6). It has two 4-torsion sections t1 and u1, which induce two commuting symplectic automorphisms σt1 and σu1. We will analyze this surface to study not only the group of symplectic automorphisms G= (Z/4Z)2, but also its subgroupsZ/2Z×Z/4Z=hσt21, σu1i,Z/2Z×Z/2Z=hσ2t1, σs21i,Z/4Z=hσt1i,Z/2Z=hσ2t1iwhich act symplectically too.
It is more surprising that the equation (6) appears as a specialization also of the surfaces described in (4) and (5). So it admits also the group G= (Z/2Z)4as group of symplectic automorphisms. The automorphisms σP1,a : (τ, x, y) 7→(−τ, x,−y) andσP1,b : (τ, x, y) 7→(1/τ, x,−y) commute with the automorphism induced by the two 2-torsion sections t2 andu2.
The automorphisms group Aut(X(Z/4Z)2) of the surface X(Z/4Z)2 is described in [KK]. Keum and Kondo prove thatX(Z/4Z)2 is the Kummer surfaceKm(E√−1×E√−1). They consider forty rational curves on the surface: sixteen of them, (the ones called Gi,j), form a Kummer lattice, and the twentyfour curvesGi,j,Ei,i= 1,2,3,4,Fj,j= 1,2,3,4 form the so calleddouble Kummer, a lattice which is a sublattice of the N´eron Severi group of Km(E1×E2) for each couple of elliptic curves E1, E2. These forty curves generateN S(Km(E√−1×E√−1)) and describe five different elliptic fibrations onX(Z/4Z)2 which have six fibers of typeI4each. A complete list of the 63 elliptic fibrations present on the surfaceX(Z/4Z)2 can be found in [Nis].
In the Figure 1 some of the intersections between the forty curves introduced in [KK] are shown. The
five elliptic fibrations are associated to the classes ([KK, Proof of Lemma 3.5]) E1+E2+L1+L2, G11+G12+F1+L9, G11+G21+E1+L12,
G11+G22+D1+L8, G33+G44+D1+L7.
We identify the fibration that we will consider with one of these, say the one with fiberE1+E2+L1+L2. Then we can defines=G11,t=G13,u=G32, and soC0(1) =E1,C0(2)=L12,C0(3)=L8, C0(4) =D1, C0(5)=E1,C0(6)=L9.
We will consider the group G= (Z/2Z)4 generated by two involutions induced by torsion sections and two involutions induced by involutions on P1 (with respect to one of these elliptic fibrations).
However one can choose the last two involutions as induced by the two torsion sections of different elliptic fibrations (cf. Remark 4.2).
Proposition 4.1. Let X(Z/4Z)2 be the elliptic K3 surface with equation (6). Let t1 and u1 be two 4-torsion sections of the fibration. Thent1·C1(j)= 1ifj= 1,2,3,4,t1·C2(5) =t·C0(6)= 1,u1·C1(h)= 1 if h= 4,5,6,u1·C3(3)= 1,u1·C2(1)=u1·C0(2)= 1.
A Z-basis for the latticeN S(X(Z/4Z)2)is given byF, s, t1, u1, Ci(j), j= 1, . . . ,6,i= 1,2 andC3(j), j = 2, . . . ,5.
The trivial lattice of the fibration isU ⊕A⊕36. It has index42 in the N´eron Severi group ofX(Z/4Z)2. The latticeN S(X(Z/4Z)2)has discriminant −42 and its discriminant form is Z4(−14)⊕Z4(−14).
The transcendental lattice isTX(Z/4Z)2 =
· 4 0 0 4
¸
, and has a unique primitive embedding in the lattice ΛK3.
Proof. The singular fibers of this fibration are six fibers of type I4 (the classification of the type of the singular fibers is determined by the zero-locus of the discriminant of the equation of the surface and can be found in [Mi, IV.3.1]). Hence the trivial lattice of this elliptic fibration is U⊕A⊕36. Since it has rank 20 which is the maximal Picard number of a K3 surface, there are no sections of infinite order on this elliptic fibration. The torsion part of the Mordell Weil group is (Z/4Z)2, generated by two 4-torsion sections. We will call these sectionst1andu1. By the height formula (cf. Theorem 1.2) the intersection between a four torsion section and the six fibers of type I4 has to be of the following type: the section has to meet four fibers in the componentCiwithiodd, a fiber in the componentC2
and a fiber in the component C0. After a suitable numbering of the fibers we can suppose that the section t1 has the intersection described in the statement. The sectionu1 and the section v1 which corresponds to t1+u1 in the Mordell Weil group must intersect the fibers in a similar way (four in the componentCi with an oddi, one inC2and one inC0). Ift1·Ci(j)= 1 andu1·Ch(j)= 1, then v1
meets the fiber in the component Ci+h(j) (it is a consequence of the group law on the fibers of typeIn).
The conditions on the intersection properties of u1 and v1 imply thatu1·C1(h) = 1 if h= 3,4,5,6, u1·C2(1) = u1·C0(2) = 0, and hencev1·C1(j) = 1, i = 2,6, v1·C2(4) = 1, v1·C3(h) = 1, h = 1,5, v1·C0(3)= 1.
The torsion sections t1 and u1 can be written as a linear combination of the classes in the trivial lattice with coefficient in 14Z(because they are 4-torsion sections). Hence the trivial lattice has index 42 in the N´eron Severi group and sod(N S(X(Z/4Z)2)) = 46/44= 42. In particular
t1= 2F+s−14³P4
i=1(3C1(i)+ 2C2(i)+C3(i)) + 2C1(5)+ 4C2(5)+ 2C3(5)´ u1= 2F+s−14³P6
i=4(3C1(i)+ 2C2(i)+C3(i)) +C1(3)+ 2C2(3)+ 3C3(3)+ 2C1(1)+ 4C2(1)+ 2C3(1)´ . It is now clear that F, s, t1, u1, Ci(j), j = 1, . . . ,6, i= 1,2 and C3(j), j = 2, . . . ,5 is a Q-basis for the N´eron Severi group and since the determinant of the intersection matrix of this basis is 42, it is in fact a Z-basis. The discriminant form of the N´eron Severi lattice is generated by
d1= 14(C1(1)+ 2C2(1)+ 3C3(1)−C1(3)−2C2(3)−3C3(3)+C1(6)+ 2C2(6)+ 3C3(6)) d2= 14(C1(2)+ 2C2(2)+ 3C3(2)+C1(4)+ 2C2(4)+ 3C3(4)−C1(5)−2C2(5)−3C3(5)), (7)
hence the discriminant form ofN S(X(Z/4Z)2) isZ4(−14)⊕Z4(−14). The transcendental lattice has to be a rank 2 positive definite lattice with discriminant form Z4(14)⊕Z4(14) (the opposite of the one of the N´eron Severi group). This implies thatTX(Z/4Z)2 =
· 4 0 0 4
¸
. ¤
Up to now for simplicity we put X(Z/4Z)2=X.
Proposition 4.2. Let G4,4 be the group generated byσt1 and σu1. The invariant sublattice of the N´eron Severi group with respect to G4,4 is isometric to
· −8 8 8 0
¸ .
Its orthogonal complement (N S(X)G4,4)⊥ is Ω(Z/4Z)2 := (H2(X,Z)G4,4)⊥. It is the negative definite eighteen dimensional lattice {Z18, Q} where Q is the bilinear form obtained as the intersection form of the classes
b1=s−t1, b2=s−u1, bi+2=C1(i)−C1(i+1), i= 1, . . . ,5, bj+7=C2(j)−C2(j+1), j= 1, . . . ,5, bh+11=C1(h)−C3(h), h= 2, . . . ,5, b17=C1(1)−C2(2), b18=f+s−t1−C1(1)−C1(2)−C2(2)−C1(3). The latticeΩ(Z/4Z)2 admits a unique primitive embedding in the latticeΛK3.
The discriminant of Ω(Z/4Z)2 is28 and its discriminant group is (Z/2Z)2⊕(Z/8Z)2.
The group of isometries G4,4 acts on the discriminant groupΩ∨(Z/4Z)2/Ω(Z/4Z)2 as the identity.
The latticeH2(X,Z)G4,4, is
4 6 0 0
6 4 6 4
0 6 4 0
0 4 0 0
and it is an overlattice ofN S(X)G4,4⊕TX≃
· −8 8 8 0
¸
⊕
· 4 0 0 4
¸
of index four.
Proof. Letv1=t1+u1,w1=t2+u1,z1=t3+u1. By the definition, the automorphisms induced by the torsion sections fix the class of the fiber, soF ∈N S(X)G4,4. Moreover the groupG4,4acts on the section fixing the classs+P3
i=1ti+ui+vi+wi+zi. The action of the groupG4,4 is not trivial on the components of the reducible fibers, so the latticehF, s+P3
i=1ti+ui+vi+wi+ziiis a sublattice of N S(X)G4,4 of finite index. Since the orthogonal of a sublattice is always a primitive sublattice we have hF, s+P3
i=1ti+ui+vi+wi+zii⊥ = (N S(X)G4,4)⊥. In this way one can compute the classes generating the lattice (N S(X)G4,4)⊥. A basis for this lattice is given by the classesbi, moreover the latticeN S(X)G4,4 is isometric to ((N S(X)G4,4)⊥)⊥, and a computation shows that it is isometric to
· −8 8 8 0
¸ .
To find the lattice H2(X,Z)G4,4 we consider the orthogonal complement of (H2(X),Z)G4,4)⊥ ≃ (N S(X)G4,4)⊥ inside the latticeH2(X,Z). Since we know the generators of the discriminant form of N S(X) and ofTX we can construct a basis ofH2(X,Z). Indeed leta1 anda2 be the generators of TX, then the classesF,s,t1,u1,Ci(j)j= 1, . . . ,6,i= 1,2,C3(j),j= 2, . . . ,5,a1/4 +d1anda2/4 +d2
(di as in (7)) form aZ-basis ofH2(X,Z). The classesbi,i= 1, . . . ,18 generate (H2(X,Z)G4,4)⊥ and are expressed as a linear combination of the Z-basis of H2(X,Z) described above. The orthogonal of these classes in H2(X,Z) is the lattice H2(X,Z)G4,4. A computation with the computer shows that the action of G4,4 is trivial on the discriminant group Ω∨(Z/4Z)2/Ω(Z/4Z)2 = (Z/2Z)2⊕(Z/8Z)2. Moreover the lattice Ω(Z/4Z)2 satisfies the hypothesis of [Nik2, Theorem 1.14.4] so it admits a unique
primitive embedding in ΛK3. ¤
Proposition 4.3. 1) LetG2,4be the group generated byσt2 andσu1. The invariant sublattice of the N´eron Severi group with respect toG2,4is isometric toU(4)⊕
· −4 0 0 −4
¸
.Its orthogonal complement (N S(X)G2,4)⊥isΩZ/2Z⊕Z/4Z:= (H2(X,Z)G2,4)⊥. It is the negative definite sixteen dimensional lattice
{Z16, Q} whereQis the bilinear form obtained as the intersection form of the classes
b1=s−u1, bi=C3(i)−C1(i), i= 2, . . . ,5, bj+3=C1(j)−C1(j+1), j= 3,4,5,
bh+6=C2(h)−C2(h), h= 3,4,5, b12=C24−C13, b13=C0(3)−C1(4), b14=C1(1)+C2(1)−C2(2)−C3(2), b15=C2(2)+C3(2)−C2(3)−C3(3), b16=C2(1)−C1(2)−C1(3)+C3(3)−C2(5)+C1(6)−2t1+ 2u1.
The latticeΩZ/2Z×Z/4Z admits a unique primitive embedding in the latticeΛK3.
The discriminant of ΩZ/2Z×Z/4Z is210 and its discriminant group is (Z/2Z)2⊕(Z/4Z)4.
The group of isometries G2,4 acts on the discriminant groupΩ∨Z/2Z×Z/4Z/ΩZ/2Z×Z/4Z as the identity.
The latticeH2(X,Z)G2,4 is
4 −2 0 0 0 0
−2 0 −2 0 0 0
0 −2 −64 −4 0 0
0 0 −4 0 −4 0
0 0 0 −4 80 4
0 0 0 0 4 0
and it is an overlattice ofN S(X)G2,4⊕TX≃U(4)⊕
· −4 0 0 −4
¸
⊕
· 4 0 0 4
¸
of index two.
2) Let G2,2 be the group generated byσt2(=σ2t1)andσu2(=σ2u1).
The invariant sublattice of the N´eron Severi group with respect to G2,2 is isometric to
−4 −4 −2 −2 −4 0 0 0
−4 −4 0 2 −2 0 8 4
−2 0 −4 0 0 0 4 0
−2 2 0 0 0 0 0 0
−4 −2 0 0 −4 0 6 0
0 0 0 0 0 −4 2 0
0 8 4 0 6 2 −16 2
0 4 0 0 0 0 2 −4
.
Its orthogonal complement(N S(X)G2,2)⊥isΩZ/2Z×Z/2Z:= (H2(X,Z)G2,2)⊥. It is the negative definite twelve dimensional lattice {Z12, Q} whereQ is the bilinear form obtained as the intersection form of the classes
b1=−C1(1)−C1(2)−C1(3)+C3(3)+C1(5)+C1(6)−2t1+ 2u1, b2=−C1(1)−2C2(1)−C3(1)+F, b3=C1(2)−C3(2), b4=−C1(1)−C2(1)+C1(2)+C2(2),
b5=C1(1)+ 2C2(1)+C3(1)+C1(3)+C2(3)+C3(3)+C1(4)+C2(4)+C3(4)+C1(5)+C1(6)−3F−2s+ 2u1, b6=−C1(1)−C2(1)−C1(3)−C2(3)+F, b7=C1(4)−C3(4), b8=−C1(1)−C2(1)+C2(4)+C3(4),
b9=C1(5)−C3(5), b10=−C1(1)−C2(1)+C1(5)+C2(5), b11=C1(6)−C3(6), b12=−C1(1)−C2(1)+C1(6)+C2(6).
The latticeΩZ/2Z×Z/2Z admits a unique primitive embedding in the latticeΛK3.
The discriminant of ΩZ/2Z×Z/2Z is210 and its discriminant group is (Z/2Z)6⊕(Z/4Z)2.
The group of isometries G2,2 acts on the discriminant groupΩ∨Z/2Z×Z/2Z/ΩZ/2Z×Z/2Z as the identity.
The latticeH2(X,Z)G2,2 is
0 4 2 0 0 2 0 −2 0 0
4 0 6 −8 8 4 6 −20 8 2
2 6 0 −1 2 1 2 −6 0 2
0 −8 −1 −2 0 −1 2 1 2 0
0 8 2 0 −4 4 0 2 0 0
2 4 1 −1 4 −4 0 −1 0 −2
0 6 2 2 0 0 −4 2 0 0
−2 −20 −6 1 2 −1 2 −4 0 4
0 8 0 2 0 0 0 0 −4 0
0 2 2 0 0 −2 0 4 0 −4
and it is an overlattice ofN S(X)G2,2⊕TX of index four.
3) Let G4 be the group generated by σt1. The invariant sublattice of the N´eron Severi group with
respect toG4 is isometric toh−4i ⊕
−2 1 0 0 0
1 −2 4 0 0
0 4 4 8 4
0 0 8 0 4
0 0 4 4 0
.Its orthogonal complement(N S(X)G4)⊥ is ΩZ/4Z := (H2(X,Z)G4)⊥. It is the negative definite fourteen dimensional lattice{Z14, Q} where Q is the bilinear form obtained as the intersection form of the classes
b1=s−t1, bi+1=C1(i)−C1(i+1), i= 1,2,3, bj+3=C1(j)−C1(j), j = 2, . . . ,5, bh+7=C2(h)−C2(h+1), h= 2,3 b11=C2(2)−C1(1) b12=C2(1)−C1(2)
b13=F−C1(2)−C2(2)−C1(5)−C2(5) b14=C1(2)+C2(2)−C1(5)−C2(5). The latticeΩZ/4Z admits a unique primitive embedding in the latticeΛK3.
The discriminant of ΩZ/4Z is210 and its discriminant group is(Z/2Z)2⊕(Z/4Z)4. The group of isometries G4 acts on the discriminant group Ω∨Z/4Z/ΩZ/4Z as the identity.
The latticeH2(X,Z)G4 is
0 4 0 2 0 −1 0 0
4 0 4 4 −4 0 0 −4
0 4 0 0 0 0 0 0
2 4 0 0 0 −1 0 0
0 −4 0 0 −2 −1 0 −2
−1 0 0 −1 −1 −2 1 1
0 0 0 0 0 1 −2 0
0 −4 0 0 −2 1 0 −2
and it is an overlattice ofN S(X)G4⊕TX of index two.
Proof. The proof is similar to the proof of Proposition 4.3. ¤
Remark 4.1. The automorphismsσt1 andσu1 do not fix the other four elliptic fibrations described in [KK] (different fromE1+E2+L1+L2). The involutions (σt1)2 and (σu1)2fix the class of the fiber of those elliptic fibrations, however they are not induced by torsion sections on those fibrations, in fact they do not fix each fiber of the fibration. On the fibrations different fromE1+E2+L1+L2, the actions of (σt1)2 and (σu1)2are analogues to the ones ofσP1,a andσP1,bon the fibrationE1+E2+L1+L2.¤
Proposition 4.4. 1) Let G2,2,2 be the group generated by σt2, σu2 and σP1,a. The automorphism σP1,a acts in the following way:
t1↔v, u1↔w, Ci(1)↔Ci(2), Ci(5) ↔Ci(6), i= 0,1,2,3, C1(j)↔C3(j), j = 3,4
where w and v are respectively the section obtained as t1+u1+u1 and t1+t1+u1 with respect to the group law of the Mordell Weil group, and fixes the classes F,s andCi(j),i= 0,2,j = 3,4. The invariant sublattice of the N´eron Severi group with respect to G2,2,2 is isometric to
−4 2 0 0 0 0
2 −20 6 0 0 0
0 6 −4 −2 0 0
0 0 −2 0 −2 0
0 0 0 −2 0 −4
0 0 0 0 −4 −8
.
Its orthogonal complement(N S(X)G2,2,2)⊥ isΩ(Z/2Z)3:= (H2(X,Z)G2,2,2)⊥. It is the negative definite fourteen dimensional lattice {Z14, Q} whereQ is the bilinear form obtained as the intersection form of the classes
b1=C3(3)−C1(3), b2=C3(4)−C1(4), b3=C3(2)−C1(1), b4=C2(2)−C2(1), b5=C1(2)−C1(1), b6=C0(3)+C3(3)−C1(1)−C2(1), b7=C1(3)+C2(3)−C1(1)−C2(1), b8=C1(4)+C2(4)−C1(3)−C2(3), b9=C2(5)+C3(5)−C1(4)−C2(4), b10=C1(6)+C2(6)−C2(5)−C3(5), b11=−C1(1)+C1(5)−t1+u1, b12=C1(6)−C1(1)−t1+u1, b13=C2(5)−C2(1)+t1−u1,
b14=C1(1)+C1(2)+C1(3)+C1(4)+C1(5)+ 2C2(5)+C3(5)−2F−2s+ 2t1. The latticeΩ(Z/2Z)3 admits a unique primitive embedding in the latticeΛK3.
The discriminant of Ω(Z/2Z)3 is210 and its discriminant group is (Z/2Z)6⊕(Z/4Z)2.
The group of isometries G2,2,2 acts on the discriminant groupΩ∨(Z/2Z)3/Ω(Z/2Z)3 as the identity.
The latticeH2(X,Z)G2,2,2 is an overlattice ofN S(X)G2,2,2⊕TX of index two.
2) Let G2,2,2,2 be the group generated by σt2, σu2, σP1,a and σP1,b. The automorphism σP1,b acts in the following way:
t1↔z, u1↔w, Ci(3)↔Ci(4), Ci(5)↔C4(6)−i, i= 0,1,2,3, C1(j)↔C3(j), j= 1,2 where w is as in 1) and z is the section obtained ast1+t1+t1+u1+u1 with respect to the group law of the Mordell Weil group, and fixes the classes F,s andCi(j),i= 0,2, j = 1,2. The invariant sublattice of the N´eron Severi group with respect toG2,2,2,2 is isometric to
−20 −8 −12 −2 4
−8 8 2 2 4
−12 2 −4 0 4
−2 2 0 0 0
4 4 4 0 −8
.
Its orthogonal complement (N S(X)G2,2,2,2)⊥ is Ω(Z/2Z)4 := (H2(X,Z)G2,2,2,2)⊥. It is the negative definite fifteen dimensional lattice {Z15, Q} where Qis the bilinear form obtained as the intersection form of the classes
b1=C3(3)−C1(3), b2=C3(2)−C1(1), b3=C2(2)−C2(1), b4=C1(2)−C1(1), b5=C3(4)−C1(3), b6=C1(4)−C1(3), b7=C0(3)+C3(3)−C1(1)−C2(1), b8=C1(3)+C2(3)−C1(1)−C2(1),
b9=C1(4)+C2(4)−C1(3)−C2(3), b10=C2(5)+C3(5)−C1(4)−C2(4), b11=C1(6)+C2(6)−C2(5)−C3(5), b12=C1(6)−C1(1)−t1+u1, b13=C2(5)−C2(1)+t1−u1, b14=C1(5)−C1(1)−t1+u1, b15=−C1(1)−C1(3)−C2(3)−C1(4)+F+s−t1.
The latticeΩ(Z/2Z)4 admits a unique primitive embedding in the latticeΛK3.
The discriminant of Ω(Z/2Z)4 is−29 and its discriminant group is(Z/2Z)6⊕(Z/8Z).
The group of isometries G2,2,2,2 acts on the discriminant groupΩ∨(Z/2Z)4/Ω(Z/2Z)4 as the identity.
The latticeH2(X,Z)G2,2,2,2 is an overlattice ofN S(X)G2,2,2,2⊕TX of index two.
