Counting elliptic fibrations on K3 surfaces
aa r X i v : . [ m a t h . AG ] M a r COUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES
DINO FESTI AND DAVIDE CESARE VENIANI
Abstract.
We solve the problem of counting jacobian elliptic fibrations on an arbitrary complexprojective K3 surface up to automorphisms. We then illustrate our method with several explicitexamples. Introduction An elliptic pencil on a complex projective K3 surface X is a complete linear system of divisorswhose general member is a smooth elliptic curve E . An elliptic pencil corresponds to an ellipticfibration π : X → P , which is said to be jacobian if it admits a section O . We denote by J X theset of all jacobian elliptic fibrations on X . The automorphism group Aut( X ) acts on J X with afinite number of orbits according to a result by Sterk [47, Cor. 2.7]. The aim of this paper is todetermine the number of orbits |J X / Aut( X ) | . Let π : X → P be a fixed jacobian filtration and denote by E its generic fiber. Once an arbitrarysection O is chosen as neutral element, the set of sections of π acquires the structure of a groupcalled the Mordell–Weil group of the fibration π . The classes of E and O in the Néron–Severilattice S X induce an embedding ι : U ֒ → S X , where U denotes the hyperbolic unimodular evenlattice of rank . The orthogonal complement W := ι ( U ) ⊥ ⊂ S X , is called the frame of the fibration π . The frame W encodes virtually all geometrical informationabout the fibration π . Indeed, the dual graph of the components of the reducible fibers can beinferred from its root sublattice W root , and the Mordell–Weil group is isomorphic to W/W root (see for instance the survey by Schütt and Shioda [41]). All frames have the same signature anddiscriminant form, so they belong to the same lattice genus W X , which we call the frame genus of X (Definition 2.5).Although several works have been dedicated to the classification of jacobian elliptic fibrations on agiven K3 surface [2, 4, 6, 7, 9, 12, 19, 23, 25, 31, 32, 43], in very few of them the number |J X / Aut( X ) | has been explicitly determined. The various approaches were clarified in an unpublished paper byBraun, Kimura and Watari [6], where the following distinctions were introduced: • the “ J (type) ( X ) classification” is the problem of determining the pairs ( W root , W/W root ) for W ∈ W X ; • the “ J ( X ) classification” is the problem of determining the frame genus W X ; • the “ J ( X ) classification” is the problem of determining the orbits J X / Aut( X ) . Date : March 2, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
K3 surface, elliptic fibration, transcendental lattice. • the “ J ( X ) classification” is the problem of determining J X .In general, each classification is coarser than the next one. For most geometrical applications, the“ J (type) ( X ) classification” suffices. Very few authors went as far as the “ J ( X ) classification”. Asfor the “ J ( X ) classification”, Nikulin [30, Theorem 5.1] proved that the set J X is finite if and onlyif S X belongs to a certain finite set of lattices.In a remarkable exception dating back to more than 30 years ago, Oguiso [32] proved that |J X / Aut( X ) | ∈ { , , , } when X is the Kummer surface associated to the product E × F of two non-isogenous elliptic curvesor, equivalently, when it holds T X ∼ = U (2) . Oguiso’s arguments are based on a deep understandingof the geometry of such Kummer surfaces and cannot be generalized to other K3 surfaces.In the same unpublished paper [6], Braun, Kimura and Watari also found a sufficient conditionfor |J X / Aut( X ) | = |W X | to hold (which can be seen as a direct corollary of our main theorem, seeCorollary 2.10), and listed a few cases where this condition is true. One of these cases, namely thesingular K3 surface X with transcendental lattice T X ∼ = [2] ⊕ [6] , was later completely worked outby Bertin et al. [2, 3], who found that |J X / Aut( X ) | = |W X | = 53 . The same sufficient condition was independently found by Mezzedimi [25], who also proved thata K3 surface with Néron–Severi lattice S X ∼ = U ⊕ [ − d ] satisfies |J X / Aut( X ) | = 1 if d = 1 and |J X / Aut( X ) | = 2 k − , if d ≥ , where k is the number of prime divisors of d [25, Proposition 4.4]. Moreover, he showedthat |J X | = 1 if S X belongs to a certain explicit list of lattices [25, Thm. 6.14].We are not aware of other cases in the literature where |J X / Aut( X ) | is explicitly known.Our main result, namely Theorem 2.8, is a formula for the number of jacobian fibrations up toautomorphisms with the same frame W ∈ W X , which we call the multiplicity of the frame W . Themultiplicity of W turns out to be equal to the number of certain double cosets | H \ G/K | in the orthogonal group G = O( T ♯X ) , where T ♯X is the discriminant group of the transcendentallattice T X . The subgroup H is related to the Hodge isometries of T X , while the subgroup K depends on W . Quite interestingly, this pattern is shared by many other enumerative problemssuch as counting Kummer structures on X (cf. [13]), counting Fourier–Mukai partners of X (cf. [14]),or counting Enriques surfaces covered by X (cf. [44]).The advantage of our algebraic method is that it can be implemented as soon as the transcen-dental lattice T X is known. We compute |J X / Aut( X ) | explicitly in the following cases: • K3 surfaces belonging to the Barth–Peters family (see Theorem 3.4); • Kummer surfaces associated to the product of non-isogenous elliptic curves, confirmingOguiso’s results [32] (see Theorem 3.13); • Kummer surfaces associated to Jacobian of a very general curve of genus , refining a workby Kumar [23] (see Theorem 3.18); • generic double covers of P ramified over lines, refining a work by Kloosterman [19] (seeTheorem 3.27); • K3 surfaces belonging to the Apéry–Fermi pencil, refining a work by Bertin and Lecacheux [4](see Theorem 3.30).
OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 3
In each case, all Gram matrices of the lattices in the frame genus are contained in the respectivearXiv ancillary file. Computations were carried out with GAP [10], Magma [5] and Sage [46].
Contents of the paper.
The paper is divided into two sections.The theoretical part is contained in §2: it comprises the proof of our main theorem and generalguidelines on how to implement our method.All explicit examples are contained in §3.
Acknowledgments.
We warmly thank not only Simon Brandhorst for his help with Sage, butalso Fabio Bernasconi, Alice Garbagnati and Remke Kloosterman for their insightful commentsand Edgar Costa, Noam Elkies, Klaus Hulek, Giacomo Mezzedimi, Bartosz Naskręcki, MatthiasSchütt and Evgeny Shinder for their useful remarks on an earlier draft of this paper. The secondauthor is also indebted to Christina Lienstromberg for her kind hospitality in Bonn, Germany,where this paper was finished. 2.
Main theorem
Throughout this section we let X be a complex projective K3 surface with Néron–Severi lat-tice S X and transcendental lattice T X .After fixing notation and conventions on lattices in §2.1 and recalling or proving some preliminaryresults in §2.2, we state the main theorem of the paper, namely Theorem 2.8, together with itsimmediate corollaries in §2.3. A proof of the theorem is given in §2.4. Finally, in §2.5 we providegeneral guidelines on how to compute |J X / Aut( X ) | in explicit cases applying the formula of thetheorem.2.1. Lattices. A lattice L of rank r is a free, finitely generated Z -module L ∼ = Z r endowed with asymmetric bilinear form L × L → Z , ( v, w ) v · w . The dual of L is the set L ∨ := { x ∈ L ⊗ Q | x · v ∈ Z for all v ∈ L } and the discriminant group is the finite abelian group L ♯ := L ∨ /L. If the lattice L is even , meaning that v := v · v ∈ Z for all v ∈ L , then the form on L induces afinite quadratic form L ♯ → Q / Z . We write O( L ) and O( L ♯ ) for the groups of isomorphisms of L and L ♯ respecting the corresponding bilinear or quadratic forms. There is a natural homomorphism O( L ) → O( L ♯ ) , γ γ ♯ , whose image is denoted O ♯ ( L ) .An embedding ι : M ֒ → L of lattices is called primitive if L/ι ( M ) is a free group. A vector v ∈ L is called primitive if Z v ֒ → L is a primitive embedding.A vector v in an even negative definite lattice L is called a root if v = − . The set of roots isdenoted ∆( L ) . The sublattice generated by all roots is denoted L root . A root v induces a reflection ρ v ∈ O( L ) defined by ρ v ( w ) := w + ( v · w ) v. The subgroup of O( L ) generated by all reflections ρ v , denoted W( L ) , is called the Weyl group of L .From the definition of ρ v it follows that W( L ) is always contained in the kernel of O( L ) → O( L ♯ ) . DINO FESTI AND DAVIDE CESARE VENIANI
We write L ( n ) for the lattice with the same underlying Z -module whose Gram matrix is nA ,where A is any Gram matrix of L . The standard negative definite ADE lattices are denoted A n , D n , E n .A genus is the set of isomorphism classes of all lattices of fixed signature and discriminant form.A genus is always a finite set (see for instance [20, Kapitel VII, Satz (21.3)]).2.2. Preliminary results.
Note any embedding ι : U ֒ → L is primitive, because the the lattice ( ι ( U ) ⊗ Q ) ∩ L is an overlattice of U and each overlattice of U is trivial (cf. [29, Prop. 1.4.1]).Let e, f be a fixed basis of U such that e = f = 0 and e · f = 1 . Definition 2.1.
We say that an embedding ι : U ֒ → S X is geometric if ι ( e ) is the class of an ellipticcurve E and and ι ( f − e ) is the class of a smooth rational curve O with E · O = 1 . (Such embeddingswere called “canonical” by Bertin et al. [2], but we believe this word to be slightly misleading.)Let E X denote the set of geometric embeddings of U into S X and let aut( X ) be the image of thehomomorphism Aut( X ) → O( S X ) . Lemma 2.2.
The map E X / aut( X ) → J X / Aut( X ) defined by sending a geometric embedding ι : U ֒ → S X to the fibration induced by the ellipticcurve E := ι ( e ) is a bijection.Proof. The map is clearly well defined and surjective. Consider now two geometric embeddings ι , ι such that ι ( e ) = ι ( e ) is the class of E and suppose that ι ( f − e ) , ι ( f − e ) are the classes ofthe curves O , O . Then, translation by a suitable section induces an automorphism α ∈ Aut( X ) such that α ( E ) = E and α ( O ) = O (cf. for instance [41, §7.6]). Therefore, ι and ι belong tothe same aut( X ) -orbit, so the map is also injective. (cid:3) The positive cone P X is the connected component of { x ∈ S X ⊗ R | x > } that contains one ample class. The nef cone N X is defined as N X := { x ∈ S X ⊗ R | x · C ≥ for all curves C ⊂ X } . Furthermore, we set O( S X , P X ) := { γ ∈ O( S X ) | γ ( P X ) ⊆ P X } , O( S X , N X ) := { γ ∈ O( S X ) | γ ( N X ) ⊆ N X } . Lemma 2.3. If J X = ∅ , then the Néron–Severi lattice S X is unique in its genus and the restrictionof the natural homomorphism O( S X ) → O( S ♯X ) to O( S X , N X ) is surjective.Proof. If J X = ∅ , then there exists a (geometric) embedding U ֒ → S X , hence S X ∼ = U ⊕ W forsome lattice W . From Nikulin’s [29, Thm. 1.14.2] we infer that S X is unique in its genus and thatthe natural homomorphism O( S X ) → O( S ♯X ) is surjective.The isometry γ ∈ O( S X ) defined as ( − id U , id W ) on the decomposition S X ∼ = U ⊕ W does notbelong to O( S X , P X ) . Hence, O( S X ) is generated by O( S X , P X ) and γ . Since γ is contained in thekernel of O( S X ) → O( S ♯X ) , the restriction of O( S X ) → O( S ♯X ) to O( S X , P X ) is surjective.Moreover, it holds O( S X , P X ) ∼ = W( S X ) ⋊ O( S X , N X ) (see for instance [34, Prop. 1.3]) and since W( S X ) is contained in the kernel of O( S X ) → O( S ♯X ) , the claim follows. (cid:3) OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 5
Proposition 2.4.
Let X → P be an elliptic fibration with fiber E . If D is a divisor such that D · E = 1 and D = − , then there exists a section O and an element of the Weyl group ρ ∈ W( S X ) such that ρ ( E ) = E and ρ ( D ) = O. Proof.
Imitating Kond¯o’s proof of [21, Lemma 2.1], we can show that there exist a section O and m , m , . . . , m n ∈ Z such that(1) D = O + m E + n X i =1 m i C i , where C , . . . , C n are the irreducible fiber components such that O · C i = 0 and C i = − .From D = − it follows that(2) m = − (cid:16) n X i =1 m i C i (cid:17) . As C , . . . , C n generate a negative definite lattice, it holds m ≥ . Moreover, m = 0 if and only if m = . . . = m n = 0 . We claim that whenever m > we can find ρ ′ ∈ W( S X ) such that ρ ′ ( E ) = E and ρ ′ ( D ) = O + m ′ E + n X i =1 m ′ i C i , with m ′ < m . In a finite number of steps we obtain ρ = . . . ◦ ρ ′′ ◦ ρ ′ ∈ W( S X ) with ρ ( E ) = E and ρ ( D ) = O , thus proving the theorem. From now on, we assume that m > .Without loss of generality, we can assume that C , . . . , C n are all components of the same fiber(otherwise we apply the following procedure on each fiber). Let C be the fiber component with O · C = 1 and let ρ i ∈ W( S X ) be the involution induced by the class of C i , for i = 0 , , . . . , n .Note that by applying ρ i to D , with i ∈ { , . . . , n } , only the coefficient of C i in (1) changes. Inparticular, the coefficients of O and E remain equal to and m , respectively. Since C , . . . , C n generate a negative definite lattice, there exist only a finite number of n -tuples ( m , . . . , m n ) suchthat (2) holds. If the n -tuple ( m , . . . , m j − , m ′ j , m j +1 , . . . , m n ) corresponding to ρ j ( D ) satisfies m ′ j < m j , we substitute D with ρ j ( D ) . Repeating this process in a finite number of steps, we canassume (up to substituting D with ˜ ρ ( D ) for some ˜ ρ ∈ W( S X ) contained in the subgroup generatedby ρ , . . . , ρ n ) that D satisfies the following minimality property: for each j = 1 , . . . , n it holds(3) ρ j ( D ) = O + m E + j − X i =1 m i C i + m ′ j C j + n X i = j +1 m i C i , with m ′ j ≥ m j . We claim now that the coefficient m ′ of ρ ( D ) satisfies m ′ < m , thus concluding the proof. Weneed to divide the proof according to the dual graph of C , . . . , C n .Assume first that n = 1 , so that the dual graph of C , . . . , C n is A . From the minimalityproperty (3) and the assumption m > it follows that m < . Using C = E − C and C · C = 2 we obtain ρ ( D ) = ( O + C ) + m E + m (2 C − C ) = O + m ′ E + m ′ C . with m ′ = m + 1 + 2 m < m , as wished.Assume now that the dual graph of C , . . . , C n is A n , with n ≥ . n − n DINO FESTI AND DAVIDE CESARE VENIANI
From the minimality property (3) it follows that m ≤ m , m i ≤ m i − + m i +1 , for i = 2 , . . . , n − , m n ≤ m n − . It holds im i − ≤ ( i − m i for i = 2 , . . . , n . Indeed, this is clear for i = 2 and it follows by inductionfrom im i ≤ im i − + im i +1 ≤ ( i − m i + im i +1 . From nm n − ≤ ( n − m n and m n ≤ m n − we infer that m n ≤ . It cannot be m n = 0 because itimplies m n − = . . . = m = 0 , contradicting the fact that m > . Therefore, it holds m n < and,symmetrically, m < . Using C = E − C − . . . − C n , C · C = C · C n = 1 and C · C i = 0 for i = 2 , . . . , n − we obtain ρ ( D ) = ( O + C ) + m E + m ( C − C ) + n − X i =2 m i C i + m n ( C − C n ) = O + m ′ E + n − X i =2 m ′ i C i . with m ′ = m + 1 + m + m n < m , as wished.Assume now that the dual graph of C , . . . , C n is D n ( n ≥ ). n − n From the minimality property (3) it follows that m ≤ m , m i ≤ m i − + m i +1 , for i = 2 , . . . , n − , m n − ≤ m n − + m n − + m n m n − ≤ m n − , m n ≤ m n − . It holds m n − i ≤ m n − i − for all i = 2 , . . . , n − . Indeed, this follows from the last three inequalitiesfor i = 2 and then by induction from m n − i − ≤ m n − i − + m n − i ≤ m n − i − + m n − i − . In particular m ≤ m , so from m ≤ m we infer that m ≤ . It cannot be m = 0 because itimplies m = . . . = m n = 0 , contradicting the fact that m > . Moreover, it cannot be m = − ,because (recalling that m i ∈ Z ) it implies m = · · · = m n − = − , m n − ≤ − , m n ≤ − , leadingto the contradiction − m n − ≤ m n − + m n − + m n ≤ − . Therefore, it holds m < − . Using C = E − C − C − . . . − C n − − C n − − C n , C · C = 1 and C · C i = 0 for i = 1 , , , . . . , n , we obtain ρ ( D ) = ( O + C ) + m E + m C + m ( C − C ) + n X i =3 m i C i = O + m ′ E + n X i =1 m ′ i C i . with m ′ = m + 1 + m < m , as wished.Assume now that the dual graph of C , . . . , C n is E . From the minimality property (3) it follows that m ≤ m , m ≤ m , m ≤ m + m , m ≤ m + m + m , m ≤ m + m , m ≤ m . OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 7
From the first and third inequality we obtain m ≤ m and from m ≤ m + 3 m ≤ m we get m ≤ m . Symmetrically, it holds m ≤ m . Then, from m ≤ m + 3 m + 3 m ≤ m + 4 m we infer m ≤ m . Together with m ≤ m , this implies m ≤ . It cannot be m = 0 , becauseit implies m = . . . = m n = 0 , contradicting the fact that m > . Moreover, it cannot be m = − ,because (always recalling that m i ∈ Z ) it implies m = − , m ≤ − and m ≤ − , leading to thecontradiction − m ≤ m + m + m ≤ − − − − . Therefore, it holds m < − . Using C = E − C − C − C − C − C − C , C · C = 1 and C · C i = 0 for i = 1 , , . . . , , we obtain ρ ( D ) = ( O + C ) + m E + m C + m ( C − C ) + X i =3 m i C i = O + m ′ E + X i =1 m ′ i C i . with m ′ = m + 1 + m < m , as wished.Assume now that the dual graph of C , . . . , C n is E . From the minimality property (3) it follows that m ≤ m , m ≤ m , m ≤ m + m , m ≤ m + m + m , m ≤ m + m , m ≤ m + m , m ≤ m . From the last three inequalities we obtain m ≤ m and m ≤ m . From m ≤ m + 4 m + 4 m ≤ m + 4 m + 3 m we get m ≤ m . Then, from m ≤ m + 3 m ≤ m + 4 m we infer m ≤ m . Together with m ≤ m , this implies m ≤ . It cannot be m = 0 , becauseit implies m = . . . = m = 0 , contradicting the fact that m > . Moreover, it cannot be m = − ,because (recalling that m i ∈ Z ) it implies m = − , m = − , m ≤ − , m ≤ − , and m ≤ − ,leading to the contradiction − m ≤ m + m + m ≤ − − − − . Therefore, it holds m < − . Using C = E − C − C − C − C − C − C − C , C · C = 1 and C · C i = 0 for i = 2 , . . . , , we obtain ρ ( D ) = ( O + C ) + m E + m ( C − C ) + X i =1 m i C i = O + m ′ E + X i =2 m ′ i C i . with m ′ = m + 1 + m < m , as wished.Finally, assume that the dual graph of C , . . . , C n is E . DINO FESTI AND DAVIDE CESARE VENIANI
From the minimality property (3) it follows that m ≤ m , m ≤ m , m ≤ m + m , m ≤ m + m + m , m ≤ m + m , m ≤ m + m , m ≤ m + m , m ≤ m . As in the E case, we obtain m ≤ m . From m ≤ m + 6 m + 6 m ≤ m + 4 m + 6 m we get m ≤ m . Then, from m ≤ m + 5 m ≤ m + 5 m we get m ≤ m . Similarly, we infer m ≤ m and m ≤ m . Together with m ≤ m , thisimplies m ≤ . It cannot be m = 0 , because it implies m = . . . = m = 0 , contradicting the factthat m > . Moreover, it cannot be m = − , because (recalling that m i ∈ Z ) it implies m = − , m = − , m = − , m = − , m ≤ − and m ≤ − , leading to the contradiction −
10 = 2 m ≤ m + m + m ≤ − − − − . Therefore, it holds m < − . Using C = E − C − C − C − C − C − C − C − C , C · C = 1 and C · C i = 0 for i = 1 , . . . , , we obtain ρ ( D ) = ( O + C ) + m E + X i =1 m i C i + m ( C − C ) = O + m ′ E + X i =1 m ′ i C i . with m ′ = m + 1 + m < m , as wished. (cid:3) Definition 2.5.
The frame genus of X is the genus W X of negative definite lattices W such that rk( W ) = rk( S X ) − and W ♯ ∼ = S ♯X . We define the frame map fr X as follows: fr X : E X / aut( X ) → W X , ι ι ( U ) ⊥ . The number | fr − X ( W ) | is called the multiplicity of the frame W ∈ W X . Corollary 2.6.
For each frame W ∈ W X there exists a geometric embedding ι : U ֒ → S X such that ι ( U ) ⊥ ∼ = W . In particular, the frame map fr X is surjective and it holds (4) |J X / Aut( X ) | = X W ∈W X | fr − X ( W ) | . Proof.
Fix a frame W ∈ W X . The lattices S X and U ⊕ W belong to the same genus. Since S X isunique in its genus by Lemma 2.3, it holds S X ∼ = U ⊕ W , i.e. there exists an embedding ι : U ֒ → S X with ι ( U ) ⊥ ∼ = W .Since W( S X ) acts transitively on the chambers of the positive cone, we find ρ ′ ∈ W( S X ) suchthat ρ ′ ◦ ι ( e ) ∈ N X , hence ρ ′ ◦ ι ( e ) is the class of an elliptic curve E (see [36, §3, proof of Cor. 3] or[16, Ch. 2, Prop. 3.10]). Then, D = ρ ′ ◦ ι ( f − e ) satisfies the hypothesis of Proposition 2.4, so thereexists ρ ∈ W( S X ) such that ρ ◦ ρ ′ ◦ ι is a geometric embedding. Clearly, it holds ( ρ ◦ ρ ′ ◦ ι ( U )) ⊥ ∼ = ι ( U ) ⊥ ∼ = W .By Lemma 2.2 we have |J X / Aut( X ) | = |E X / aut( X ) | , from which we infer equation (4). (cid:3) Proposition 2.7.
Let π : X → P be an elliptic fibration with fiber E and section O . If C , . . . , C n are the components of the reducible fibers not intersecting O , then there exist n , m , . . . , m n ∈ Z such that D = n O + m E + P ni =1 m i C i is an ample divisor. OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 9
Proof.
For the sake of simplicity, we assume that there is only one reducible fiber, but the sameargument works if there is more than one. Let C be the rational fiber component with O · C = 1 .The sublattice generated by C , . . . , C n is of ADE type, so we can find R := P ni =1 m i C i in thesuitable Weyl chamber in such a way that R · C i > for each i = 1 , . . . n. Fix a positive n ∈ Z such that n > − R · C . Since E is linearly equivalent to a certaincombination of C , . . . , C n with positive coefficients, taking m large enough we can suppose that D = n O + m E + R is linearly equivalent to n O + m C + X m ′ i C i with m ′ i > . Let C be a smooth rational curve on X . If C = O and C = C i for each i = 0 , . . . , n , then D · C > because C must intersect one of the components C i .If C = C i for some i ∈ { , . . . , n } , then D · C = R · C i > . If C = C , then D · C = n + R · C > . If C = O , then D · C = − n + m , which is positive if m is large enough. Finally, it holds D = − n + 2 n m + R . Therefore, if m is large enough, then D > and D · C > for every smooth rational curve on X ,proving that D is ample (cf. for instance [16, Ch. 8, Cor. 1.6]). (cid:3) Statement and corollaries.
The Hodge decomposition on H ( X, Z ) induces a Hodge decom-position on the transcendental lattice T X . The group of Hodge isometries of T X is denoted O h ( T X ) .If σ X is a generator of the subspace H , ( X ) ⊂ T X ⊗ C , we have O h ( T X ) := { η ∈ O( T X ) | η ( σ X ) ∈ C σ X } . The image of the natural homomorphism O h ( T X ) → O( T ♯X ) is denoted O ♯ h ( T X ) . We are now able to state the main theorem, whose proof is contained in §2.4. Theorem 2.8.
Let X be a complex projective K3 surface with frame genus W X (Definition 2.5).Then, the multiplicity of a frame W ∈ W X is given by | fr − X ( W ) | = | O ♯ h ( T X ) \ O( T ♯X ) / O ♯ ( W ) | . Remark 2.9.
By Definition 2.5 it holds W ♯ ∼ = S ♯X ∼ = T X ( − ♯ for each W ∈ W X , so thegroup O ♯ ( W ) can be considered as a subgroup of O( T ♯X ) thanks to the (non-canonical) isomor-phisms O( W ♯ ) ∼ = O( S ♯X ) ∼ = O( T ♯X ) , which we fix once and for all. The subgroup O ♯ ( W ) , therefore, is only well-defined up to conjugation.Still, the formula of Theorem 2.8 makes sense because the number of double cosets | H \ G/K | for any subgroups H, K of a group G only depends on the conjugacy class of H and K . Indeed, wehave the following formula due to Cauchy and Frobenius:(5) | H \ G/K | = 1 | H || K | X ( h,k ) ∈ H × K |{ g ∈ G | hgk = g }| . The following corollary providing uniform bounds on the number of orbits |J X / Aut( X ) | alreadyappears in previous literature under various forms, cf. for instance [6, Prop. C’] and [25, Thm 3.10]. Corollary 2.10.
It always holds |W X | ≤ |J X / Aut( X ) | ≤ |W X | · | O ♯ h ( T X ) \ O( T ♯X ) | . In particular, |J X / Aut( X ) | = |W X | when the map O h ( T X ) → O( T ♯X ) is surjective.Proof. The map fr X is surjective by Corollary 2.6, hence | fr − X ( W ) | ≥ for each W ∈ W X . By (4), |W X | ≤ X W ∈W X | fr − X ( W ) | = |J X / Aut( X ) | , proving the first inequality. Moreover, it holds trivially | O ♯ h ( T X ) \ O( T ♯X ) / O ♯ ( W ) | ≤ | O ♯ h ( T X ) \ O( T ♯X ) | . By Theorem 2.8 and again equation (4) we obtain therefore |J X / Aut( X ) | = X W ∈W X | fr − X ( W ) | ≤ X W ∈W X | O ♯ h ( T X ) \ O( T ♯X ) | = |W X | · | O ♯ h ( T X ) \ O( T ♯X ) | . (cid:3) Proof of Theorem 2.8.
Let G := O( T ♯X ) and H := O ♯ h ( T X ) . Fix a frame W ∈ W X anda geometric embedding ι ∈ fr − X ( W ) (Definition 2.1), which exists by Corollary 2.6. Since U isunimodular, we have a decomposition S X ∼ = ι ( U ) ⊕ ι ( U ) ⊥ ∼ = U ⊕ W. Consider the group O( S X , ι ) of isometries S X preserving this decomposition: O( S X , ι ) := { γ ∈ O( S X ) | γ ◦ ι ( U ) = ι ( U ) } . Let K be the image of O( S X , ι ) in G . Using the isomorphism S ♯X ∼ = U ♯ ⊕ W ♯ ∼ = W ♯ induced bythis decomposition, the subgroup K can be identified with O ♯ ( W ) (cf. Remark 2.9).Recall that the Torelli theorem for K3 surfaces [36] asserts that(6) aut( X ) = { γ ∈ O( S X ) | γ ∈ O( S X , N X ) and γ ♯ ∈ H } . We define a map dc : fr − X ( W ) → H \ G/K in the following way. Take a geometric embedding ι ∈ fr − X ( W ) . Recalling that S X is unique inits genus by Lemma 2.3, we see that the embedding is unique up to isometries of S X thanks toNikulin’s [29, Prop. 1.15.1]. This means that there exists γ ∈ O( S X ) such that ι = γ ◦ ι . Usingthe identification O( T ♯X ) ∼ = O( S ♯X ) , we set dc( ι ) := Hγ ♯ K. OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 11
We claim that the map dc is well defined. Indeed, take two geometric embeddings ι , ι ∈ fr − X ( W ) such that ι = α ◦ ι for some α ∈ aut( X ) . Let ι = γ ◦ ι and ι = γ ◦ ι for some γ , γ ∈ O( S X ) .The isometry κ := γ − ◦ α − ◦ γ belongs to O( S X , ι ) because κ ◦ ι = γ − ◦ α − ◦ ( γ ◦ ι ) = γ − ◦ ( α − ◦ ι ) = γ − ◦ ι = ι . In particular, it holds κ ♯ ∈ K . Since γ = α ◦ γ ◦ κ and α ♯ ∈ H by (6), the elements γ ♯ , γ ♯ ∈ G belong to the same ( H, K ) -double coset.Next we claim that the map dc is injective. Indeed, take two geometric embeddings ι = γ ◦ ι , ι = γ ◦ ι , with γ , γ ∈ O( S X ) , such that γ ♯ , γ ♯ belong to the same ( H, K ) -double coset. Bydefinition of H and K , this means that there exist η ∈ O h ( T X ) and κ ∈ O( S X , ι ) such that γ ♯ = η ♯ γ ♯ κ ♯ . We need to show that there exists α ∈ aut( X ) such that ι = α ◦ ι .We define α U := γ ◦ γ − on ι ( U ) and α W := γ ◦ κ − ◦ γ − on ι ( U ) ⊥ . As U is unimodular, α U and α W glue well together on ι ( U ) ⊕ ι ( U ) ⊥ ∼ = S X , defining an element α ∈ O( S X ) . From thedefinition, it immediately follows that ι = γ ◦ ι = α ◦ γ ◦ ι = α ◦ ι and α ♯ = ( γ ◦ κ − ◦ γ − ) ♯ = η ♯ ∈ H. For i = 1 , consider the fibrations π i : X → P corresponding to ι i . Write E i := ι i ( e ) , O i := ι i ( f − e ) and let C i , . . . , C in be the irreducible components of the reducible fibers of π i . Thanks toProposition 2.7 we know that there exist n , m , . . . , m n such that D = n O + m E + P ni =1 m i C i is an ample divisor. Set R := α ( D ) − n O − m E = n X i =1 m i α ( C i ) . As each α ( C i ) is a vector of square − in ι ( U ) ⊥ , there exist m ′ , . . . , m ′ n ∈ Z such that R = n X i =1 m ′ i C i . Since the Weyl group of the ADE lattices acts transitively on the Weyl chambers (see for in-stance [11]), we can find ρ ∈ W(( ι ( U ) ⊥ ) root ) such that ρ ( R ) · C i > for every i = 1 , . . . , n .As ρ acts trivially on the discriminant group of ( ι ( U ) ⊥ ) root , we can extend it to an isometry ˜ ρ ∈ O( ι ( U ) ⊥ ) with ˜ ρ ♯ = 1 . Therefore, up to substituting α with ˜ ρ ◦ α , we can suppose that α ( D ) is an ample divisor by the same arguments as in Proposition 2.7. In this way, α sends an ampledivisor to an ample divisor, so α ∈ O( S X , N X ) because the ample cone is the interior of N X (seefor instance [16, Ch. 8, Cor. 1.4]). The Torelli Theorem (6) implies that α ∈ aut( X ) , as wished.Finally, we claim that the map dc is surjective. Indeed, for every double coset HgK, g ∈ G , wecan assume that g = γ ♯ for some γ ∈ O( S X , N X ) thanks to Lemma 2.3. As ι ( e ) ∈ N X , it followsthat γ ◦ ι ( e ) is a primitive nef class of square , hence γ ◦ ι ( e ) is the class of some elliptic curve E .The class D = γ ◦ ι ( f − e ) then satisfies the assumptions of Proposition 2.4. Therefore, there exists ρ ∈ W( S X ) and a section O such that ρ ( D ) = O , which means that the embedding ι = ρ ◦ γ ◦ ι isgeometric and satisfies dc( ι ) = H ( ρ ◦ γ ) ♯ K = Hγ ♯ K = HgK.
Therefore, the map dc is a bijection and the theorem follows. (cid:3) Guidelines.
We now would like to explain how to compute |J X / Aut( X ) | in explicit cases.According to Theorem 2.8, the first ingredient to be computed is the image O ♯ h ( T X ) of the naturalhomomorphism O h ( T X ) → O( T ♯X ) . The following proposition summarizes the main properties ofthe group O h ( T X ) . Proposition 2.11 (see [28, Thm. 3.1] or [14, Prop. B.1]) . If T X is the transcendental lattice of aprojective K3 surface X , then the group O h ( T X ) is a finite cyclic group of even order containing − id such that ϕ ( | O h ( T X ) | ) | rk( T X ) , where ϕ denotes Euler’s totient function. The action of O h ( T X ) on C σ X is faithful. (cid:3) The last assertion means that if γ ∈ O h ( T X ) has order m , then γ ( σ X ) = ζ m σ X , where ζ m is aprimitive m th root of unity. Remark 2.12. If rk( T X ) is odd, Proposition 2.11 implies that O h ( T X ) = { ± id } . Therefore, O ♯ h ( T X ) = { ± id } . The subgroup O ♯ ( W ) always contains { ± id } , so by Theorem 2.8 we have | fr − X ( W ) | = | O( T ♯X ) / O ♯ ( W ) | = | O( T ♯X ) | / | O ♯ ( W ) | for every W ∈ W X .A direct consequence of Proposition 2.11 is that the image subgroup O ♯ h ( T X ) is also a finite cyclicsubgroup. In all explicit examples we will be interested in determining the conjugacy class of oneof its generators. Clearly, the order of a generator depends on the size of the kernel of the map O h ( T X ) → O( T ♯X ) . The following lemma was already stated without proof in [44, Rmk. 2.13]. Lemma 2.13.
The kernel of the map O h ( T X ) → O( T ♯X ) is isomorphic to the kernel of the map Aut( X ) → O( S X ) .Proof. Indeed, every element in the kernel of
Aut( X ) → O( S X ) induces a Hodge isometry of T X acting trivially on S ♯X ∼ = T ♯X and, conversely, every element in the kernel of O h ( T X ) → O( T ♯X ) canbe extended to a Hodge isometry γ of H ( X, Z ) by the identity on S X . As γ obviously acts triviallyon the nef cone, γ is the pullback of an automorphism of X by the Torelli Thm. (6). (cid:3) Lemma 2.13 will prove very useful, as the classification of the possible kernels of the map
Aut( X ) → O( S X ) was started by Vorontsov [49], followed by Kond¯o [21], Oguiso and Zhang [33],Schütt [40] and finally completed by Taki [48].The second and last ingredient to be computed is a list of the groups O ♯ ( W ) for W ∈ W X .Therefore, one first needs to determine the genus W X .Nishiyama introduced a method to compute W X using a negative definite lattice T such that rk( T ) = rk( T X ) + 4 and T ♯ ∼ = T ♯X . The method is usually known as the “Kneser–Nishiyama method” and consists in finding all primitiveembeddings of T up to isometries into the 24 Niemeier lattices N . The orthogonal complements W = T ⊥ ⊂ N run over all W ∈ W X by [29, Prop. 1.15.1]. We refer to the original paper byNishiyama [31] and to the surveys by Schütt and Shioda [41], [42, §12.3] for further details.Still, since we are interested in obtaining an explicit Gram matrix of each frame W ∈ W X , wepoint out the following algorithm. Let N be a Niemeier lattice such that there exists a primitiveembedding T ֒ → N root . • Let the primitive embedding T ֒ → N root be given by a matrix D . OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 13 • Find the isotropic subgroup H ⊂ N ♯ root corresponding to N root ֒ → N (cf. [29, Prop. 1.4.1]). • Starting from H , find a matrix C with rational coefficients such that N = CN root C ⊺ . • The embedding T ֒ → N is then given by the matrix DC − . • In this way we can compute the Gram matrix W of the orthogonal complement of T in N ,using for instance the command orthogonal_complement of the class IntegralLattice inSage.Once a Gram matrix is given, efficient methods to find a set of generators of O( W ) are wellknown (see for instance the work by Plesken and Souvignier [37]). It is then elementary to computethe image group O ♯ ( W ) . Remark 2.14.
Another efficient method to compute a genus is Kneser’s neighbor method. Ithas been implemented in several computer algebra systems (for instance, in the Magma function
GenusRepresentatives ), see the work by Scharlau and Hemkemeier [39].
Remark 2.15.
The order | O( W ) | and the mass of W X (for the definition of mass see the work byConway and Sloane [8]) can be efficiently computed using the commands number_of_automorphisms and conway_mass of the QuadraticForm class in Sage. Once a list of lattices W ∈ W X is obtained,one can verify a posteriori that the Smith–Minkowski–Siegel mass formula holds in order to checkthat the list is complete. 3. Examples
In this section we show how to apply Theorem 2.8 in order to compute |J X / Aut( X ) | in variousexplicit examples. We first consider K3 surfaces with transcendental lattice T X ∼ = U (2) in §3.1,then with T X ∼ = U (2) in §3.2, with T X ∼ = U (2) ⊕ [ − in §3.3, with T X ∼ = U (2) ⊕ [ − in §3.4and finally with T X ∼ = U ⊕ [12] in §3.5.3.1. Barth–Peters family.
Let X be a K3 surface with transcendental lattice T X ∼ = U ⊕ U (2) . Remark 3.1.
The surface X is the generic element of a -dimensional family introduced by Barthand Peters [1] in order to construct Enriques surfaces with an involution acting trivially on coho-mology. Elliptic fibrations on this family were also studied by Hulek and Schütt [15]. Lemma 3.2.
It holds O ♯ h ( T X ) = { id } .Proof. We choose a basis t , . . . , t ∈ T X so that the corresponding Gram matrix is T = . The discriminant group T ♯X is generated by the classes of t and t . The involution exchangingthese two classes generates the group O( T ♯X ) .By [28, Theorem 3.1] or [14, Proposition B.1], it holds ϕ ( | O h ( T X ) | ) | rk( T X ) = 4 , therefore | O h ( T X ) | ∈ { , , , , , } . Since | O( T ♯X ) | = 2 , we only need to check that any isometry η ∈ O h ( T X ) of order k , k > , acts trivially on T ♯X .If η has order , then η = − id by [14, Proposition B.1], so η ♯ = id . Moreover, by Lemma 2.13and Taki’s [48, Main theorem on page 18], η cannot have order . Hence, suppose that η has order . We can assume that η is represented by a matrix A with AT A ⊺ = T and A = − I , such that the generator σ = σ X of the subspace H , ( X ) ⊂ T X ⊗ C iscontained in the -dimensional, totally isotropic eigenspace V ⊂ T X ⊗ C associated with i = √− .Let τ ∈ V be linearly independent from σ and let us write σ = σ t + σ t + σ t + σ t , σ i ∈ C ,τ = τ t + τ t + τ t + τ t ∈ V, τ i ∈ C . It holds σ = 0 , otherwise t ∈ T ⊥ X , so we can rescale σ = 1 . Up to substituting τ with a linearcombination of σ and τ we can suppose τ = 0 . In addition, using the relations σ = τ = σ · τ = 0 ,up to substituting t with t we can assume that σ = − σ σ , τ = − σ τ , τ = 0 , τ = 1 . Note that ℑ ( σ ) = 0 because σ · ¯ σ > . Imposing that σ, τ ∈ V and ¯ σ, ¯ τ ∈ ¯ V , we infer that thereexist m , m ∈ Z such that A = m m + 1) / (2 m ) 00 − m m − m − m − ( m + 1) /m m , m = − ℜ ( σ ) ℑ ( σ ) , m = − ℑ ( σ ) . Therefore, A acts trivially on the classes of t and t , so η ♯ = id . (cid:3) Proposition 3.3.
The frame genus W X contains exactly isomorphism classes, listed in Table 1,whose Gram matrices are contained in the arXiv ancillary file genus_Barth_Peters.sage .Proof. We computed the frame genus W X using the command GenusRepresentatives in Magma(Remark 2.14). Alternatively, one could have applied the Kneser–Nishiyama method with T = D .The list is complete because the mass formula holds X i =1 | O( W i ) | = 50512112340763622899712000 = mass( W X ) . (cid:3) Theorem 3.4. If X is a K3 surface with transcendental lattice T X ∼ = U ⊕ U (2) , then |J X / Aut( X ) | = 7 . Proof.
By Lemma 3.2, the subgroup O ♯ h ( T X ) is trivial, so by Theorem 2.8 the multiplicity of aframe W ∈ W X is equal to the index of O ♯ ( W ) in O( T ♯X ) . A finite set of generators of O( W ) can becomputed with the command orthogonal_group of the Sage class QuadraticForm . Their imagesgenerate the subgroups O ♯ ( W ) . (cid:3) Kummer surfaces associated to a product of elliptic curves.
Let now X denote a K3surface with transcendental lattice T X ∼ = U (2) . Remark 3.5.
By results of Nikulin [27, Rmk. 2] and Morrison [26, Prop. 4.3(i)], X is the Kummersurface associated to an abelian surface A with transcendental lattice T A ∼ = U . Consequently,the Néron–Severi lattice of A is isomorphic to U , so A is isomorphic to the product of two non-isogenous elliptic curves, according to Ruppert’s criterion [38]. Therefore, the number of orbits |J X / Aut( X ) | has already been computed geometrically by Oguiso [32]. Here we verify his resultsusing our algebraic approach. OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 15
Table 1.
Lattices in the frame genus W X of a K3 surface X with transcendentallattice T X ∼ = U ⊕ U (2) . W W root
W/W root | ∆( W ) | | O( W ) | | fr − X ( W ) | W D E W D W D Z / Z
224 53271016243200 2 W A E Z / Z
256 134842259865600 1 W D D Z / Z
288 376702186291200 1 W A Z ⊕ ( Z / Z ) 240 83691159552000 1 We fix a basis t , . . . , t ∈ T X so that the corresponding Gram matrix is T := . We make the following identification:(7) O( T X ) ∼ = { A ∈ GL ( Z ) | AT A ⊺ = T } . The classes modulo T X of t , . . . , t form a basis over F for T ♯X . A computation shows that thegroup O( T ♯X ) contains elements and can be identified with the following subgroup of GL ( F ) :(8) O( T ♯X ) ∼ = * , + . Under identifications (7) and (8), the natural homomorphism O( T X ) → O( T ♯X ) is given by A ( A mod 2) . We now consider its restriction to the subgroup of Hodge isometries O h ( T X ) . Lemma 3.6.
The kernel of O h ( T X ) → O( T ♯X ) is equal to { ± id } .Proof. Clearly, { ± id } is contained in the kernel. By Lemma 2.13 and [21, Thm. 6.1], it suffices toshow that there is no automorphism of X of order acting trivially on S X . This follows from [40,Thm. 1]. (cid:3) A computation with the GAP command
ConjugacyClasses shows that the group O( T ♯X ) con-tains exactly conjugacy classes. Let us define the following elements of O( T ♯X ) : h := , h := , h := , h := . Lemma 3.7.
An element h ∈ O ♯ h ( T X ) of order belongs to the conjugacy class of h .Proof. By Proposition 2.11 and Lemma 3.6 we can suppose that h is the image of an element η ∈ O h ( T X ) represented by a matrix A ∈ GL ( Z ) of order such that AT A ⊺ = T, A = − I. Since A has finite order, it is diagonalizable over C with eigenvalues i = √− and − i .Let V ⊂ T X ⊗ C be the eigenspace of A associated with i . Then V has dimension and itscomplex conjugate ¯ V is the eigenspace of A associated with − i . As O h ( T X ) acts faithfully on C σ X ,we can assume that σ := σ X ∈ V . Let us write σ = σ t + σ t + σ t + σ t , σ i ∈ C , Certainly σ = 0 otherwise t ∈ σ ⊥ X = S X , so we are free to rescale σ = 1 . Moreover, from therelations σ = 0 and σ · ¯ σ > we obtain σ σ + σ σ ) = 0 , σ ¯ σ + σ ¯ σ + σ ¯ σ + σ ¯ σ > substituting σ = 1 and σ = − σ σ we get ℑ ( σ ) ℑ ( σ ) > (in particular, ℑ ( σ ) = 0 ).Let τ = τ t + τ t + τ t + τ t ∈ V, τ i ∈ C , be an eigenvector of A linearly independent from σ . Without loss of generality, we can substitute τ with a linear combination of σ and τ and suppose that τ = 0 Note that V is a totally isotropic subspace of T X ⊗ C , as x · y = η ( x ) · η ( y ) = (i x ) · (i y ) = − x · y, for all x, y ∈ V .
From τ = 0 it follows that τ τ = 0 . Up to exchanging t with t we can suppose that τ = 0 .From the relation σ · τ = 0 we obtain τ = − σ τ . Again without loss of generality, we can rescale τ = 1 .Imposing that σ, τ ∈ V and ¯ σ, ¯ τ ∈ ¯ V and recalling that A has integer coefficients, we infer withelementary but rather tedious computations that there exist m , m ∈ Z such that A = m − ( m + 1) /m − m − m m − m
00 ( m + 1) /m m , m := − ℜ ( σ ) ℑ ( σ ) , m := 1 ℑ ( σ ) . Note that m , m cannot be both even because h = ( A mod 2) is an invertible matrix. Looking atthe possible parities of m , m , we see that h is equal to one of the following elements: h := , , , which all belong to the same conjugacy class. (cid:3) Lemma 3.8.
An element h ∈ O ♯ h ( T X ) of order belongs to the conjugacy class of h .Proof. By Lemma 3.6, we can suppose that h is the image of an element in O h ( T X ) represented bya matrix A ∈ GL ( Z ) such that AT A ⊺ = T, A = I. Since A has finite order , it is diagonalizable over C with eigenvalues , ω, ¯ ω , where ω denotes athird root of unity. Let ζ be a primitve th root of unity such that ζ = ω and ζ = i .Let W ⊂ T X ⊗ C be the eigenspace of A associated with ω . We can suppose that σ := σ X ∈ W .We must differentiate two cases: either dim( W ) = 1 or dim( W ) = 2 .Suppose first that dim( W ) = 2 , let τ ∈ W be linearly independent from σ . Without loss ofgenerality we can suppose as in the proof Lemma 3.7 that σ = 1 , σ = − σ σ , τ = τ = 0 , τ = − σ τ , τ = 1 . OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 17
Imposing that σ, τ ∈ W and ¯ σ, ¯ τ ∈ ¯ W we find (again after some tedious computations) that A = m − ( m + m + 1) /m − m − − m m − m − m + m + 1) /m m , where m := ( ζ − ζ ) ℜ ( σ ) − ℑ ( σ )4 ℑ ( σ ) , m := 2 ζ − ζ ℑ ( σ ) . Looking at the possible parities of m , m , we see that h is equal to one of the following elements: h := , , which are conjugate to each other.In the case that dim( W ) = 1 we can assume that W is generated by σ and that the eigenspace U associated with is generated by τ and υ . We note that W and U are orthogonal to each other,as w · u = η ( w ) · η ( u ) = ωw · u, for all u ∈ U, w ∈ W . (Note, though, that U is not necessarily a totally isotropic subspace.) With similar computations wesee that the matrix A is forced to have non-integral coefficients. Thus, this case is impossible. (cid:3) Lemma 3.9.
The group O ♯ h ( T X ) contains no element of order .Proof. By inspection of the conjugacy classes of O( T ♯X ) , we see that there is only one class containingelements of order , namely the class of h := . We see, though, that h is not conjugate to h , so we conclude by Lemma 3.7. (cid:3) Proposition 3.10.
The subgroup O ♯ h ( T X ) is a cyclic group of order , , or , in which case it isgenerated by a conjugate of h , h , h or h , respectively.Proof. Let h be a generator of O ♯ h ( T X ) and let m be its order.By Proposition 2.11 it holds ϕ ( | O h ( T X ) | ) | rk( T X ) = 4 , therefore | O h ( T X ) | ∈ { , , , , , } . It follows from Lemma 3.6 that m = | O h ( T X ) | / ∈ { , , , , , } .We can exclude m = 5 because ∤ | O( T ♯X ) | = 72 and we can exclude m = 4 because of Lemma 3.9.All in all, m ∈ { , , , } as claimed.If m = 1 , then obviously h = h . If m = 2 , then h is conjugate to h by Lemma 3.7. If m = 3 ,then h is conjugate to h by Lemma 3.8. Finally, if m = 6 , then by inspection of the conjugacyclasses we see that h is conjugate to either h or h ′ := . Thanks to Lemma 3.7, we can exclude h ′ , because ( h ′ ) is not conjugate to h . (cid:3) Proposition 3.11.
The frame genus W X contains exactly isomorphism classes, listed in Table 2,whose Gram matrices are contained in the arXiv ancillary file genus_Oguiso.sage .Proof. We apply the Kneser–Nishiyama method with T = D . Thanks to [31, Lemma 41(i) andCorollary 4.6(i)] we find exatly Niemeier lattices N such that there exists a primitive embedding T ֒ → N root . All primitive embeddings are unique, except when N root = D E , in which case thereare two primitive embeddings (corresponding to W and W in Table 2). Therefore, we obtain different frames: W X = { W , . . . , W } . The list is complete because the mass formula holds (seeRemark 2.15): X i =1 | O( W i ) | = 641503671708721117016883200 = mass( W X ) . (cid:3) We introduce the following subgroups of G : K := * , + ,K := * , , + ,K := * , , + . The subgroups K , K , K contain respectively , , elements. Note that neither K nor K are normal subgroups of G , but K is, as it has index . Proposition 3.12.
The subgroup O ♯ ( W ) ⊂ O( T ♯X ) is conjugate to K if W ∈ { W , W , W , W } ,to K if W ∈ { W , W , W } , to K if W = W , and to G if W ∈ { W , W , W } .Proof. A finite set of generators of O( W ) can be computed with the command orthogonal_group of the Sage class QuadraticForm . Their images generate the subgroups O ♯ ( W ) . (cid:3) We now have all ingredients to compute the multiplicities of the frames W ∈ W X . Theorem 3.13. If X is a K3 surface with transcendental lattice T X ∼ = U (2) , then one of thefollowing cases holds: |J X / Aut( X ) | = if | O ♯ h ( T X ) | = 1 , if | O ♯ h ( T X ) | = 2 , if | O ♯ h ( T X ) | = 3 , if | O ♯ h ( T X ) | = 6 . Proof.
We apply Theorem 2.8 to compute the multiplicities of the frames W ∈ W X using eitherformula (5) or the GAP function DoubleCosets . Thanks to Proposition 3.10 and Proposition 3.12,the multiplicities are given by | H \ G/K | , where G is the group defined in (8), H is the cyclic subgroupgenerated by h , h , h or h and K ∈ { K , K , K } . Our results are listed in Table 3. (cid:3) OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 19
Table 2.
Lattices in the frame genus W X of a K3 surface X with transcendentallattice T X ∼ = U (2) , numbered according to Oguiso (cf. [32, Table 2]). W N root W root W/W root | ∆( W ) | | O( W ) | | O ♯ ( W ) | W D D W D E D D W E D E W D D D Z / Z
160 1522029035520 8 W D E A D E Z / Z
192 6421059993600 12 W D D ( Z / Z )
96 195689447424 36 W D A D ( Z / Z )
128 135895449600 8 W D E A D ( Z / Z )
192 8561413324800 72 W A D A Z ⊕ ( Z / Z ) 112 52022476800 8 W A D E A A Z ⊕ ( Z / Z ) 144 275904921600 12 W E E Z
144 773967052800 72
Table 3.
Multiplicities of the frames W ∈ W X listed in Table 2. | O ♯ h ( T X ) | W W W W W W W W W W W sum Remark 3.14.
With Theorem 3.13 we confirm independently Oguiso’s results [32]. Following hisnotation, we denote by E τ the elliptic curve with period τ . Comparing Table 3 with [32, Table B],we find a geometrical interpretation of the order of O ♯ h ( T X ) . Indeed, it holds | O ♯ h ( T X ) | = if X ∼ = Km( E ρ × E ρ ′ ) , if X ∼ = Km( E √− × E ρ ′ ) , if X ∼ = Km( E ρ × E ( − √− / ) , if X ∼ = Km( E √− × E ( − √− / ) , where E ρ , E ρ ′ are non-isogenous elliptic curves without complex multiplication.3.3. Jacobian Kummer surfaces.
Here let X denote a K3 surface with transcendental lattice T X ∼ = U (2) ⊕ [ − . Remark 3.15.
By results of Kumar [23], the Kummer surface X of the Jacobian of a curve C of genus without extra endomorphisms satisfies T X ∼ = U (2) ⊕ [ − . In the same paper, heclassified the elliptic fibrations up to the action of Aut( X ) and the permutations of the Weierstrasspoints of C , establishing that |W X | = 25 and finding all pairs ( W root , W/W root ) , W ∈ W X (cf. [23,Thm. 2]). Before Kumar’s work, some special elliptic fibrations had been found by Keum [17, 18],Shioda [45] and Kumar himself [22]. Proposition 3.16.
The frame genus W X contains exactly isomorphism classes, listed in Table 4,whose Gram matrices are contained in the arXiv ancillary file genus_Kumar.sage .Proof. We computed W X using Kneser’s neighbor method (Remark 2.14). (cid:3) Remark 3.17.
Alternatively, one could compute W X applying the Kneser–Nishiyama methodwith T = D ⊕ D ⊕ [ − . This lattice, though, is not generated by its roots, so one cannot useNishiyama’s results [31]. Theorem 3.18. If X is a K3 surface with transcendental lattice T X ∼ = U (2) ⊕ [ − , then |J X / Aut( X ) | = 491 . Proof.
It holds | O( T ♯X ) | = 1440 . Since rk( T X ) is odd, | fr − X ( W ) | is equal to the index of O ♯ ( W ) in O( T ♯X ) (Remark 2.12). A finite set of generators of O( W ) can be computed using the command orthogonal_group of the Sage class QuadraticForm . Their images generate O ♯ ( W ) . (cid:3) Double covers of the plane ramified over six lines.
Here let X denote a K3 surface withtranscendental lattice T X ∼ = U (2) ⊕ [ − . Remark 3.19.
By results of Matsumoto, Sasaki and Yoshida [24], X is the minimal resolution ofthe double cover of P ramified over lines, and, conversely, such a resolution satisfies T X ∼ = U (2) ⊕ [ − , provided that rk( S X ) = 16 . Kloosterman [19] classified the fiber types and the Mordell–Weilgroups of all jacobian elliptic fibrations on X . Here we refine Kloosterman’s classification andcompute the number of orbits |J X / Aut( X ) | .We fix a basis t , . . . , t ∈ T X so that the corresponding Gram matrix is T := − − . We make the following identification:(9) O( T X ) ∼ = { A ∈ GL ( Z ) | AT A ⊺ = T } . The classes modulo T X of t , . . . t form a basis over F for T ♯X . A computation shows that thegroup O( T ♯X ) contains · · elements and can be identified with the following subgroupof GL ( F ) :(10) O( T ♯X ) ∼ = * , , + . Under identifications (9) and (10), the natural homomorphism O( T X ) → O( T ♯X ) is given by A A mod 2 . Lemma 3.20.
The kernel of O h ( T X ) → O( T ♯X ) is equal to { ± id } . OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 21
Table 4.
Lattices in the frame genus W X of a K3 surface X with transcendentallattice T X ∼ = U (2) ⊕ [ − , numbered according to Kumar (cf. [23, §3.3]). W W root
W/W root | ∆( W ) | | O( W ) | | fr − X ( W ) | W D D W A D D W A D Z / Z
84 16307453952 15 W A D D Z / Z
108 67947724800 15 W A A E Z / Z
148 535088332800 6 W A D Z / Z
156 8561413324800 1 W A A D ( Z / Z )
84 5096079360 10 W A D Z
124 67947724800 45 W A D E Z
156 321052999680 20 W A D Z
188 2853804441600 15 W A E Z
252 64210599936000 1 W A D D Z ⊕ ( Z / Z ) 92 1698693120 60 W A D Z ⊕ ( Z / Z ) 124 31708938240 15 W A D Z ⊕ ( Z / Z )
60 452984832 15 W A A Z ⊕ ( Z / Z ) 80 743178240 45 W A Z ⊕ ( Z / Z ) 48 127401984 15 W A A Z ⊕ ( Z / Z ) 64 99532800 60 W A A A Z ⊕ ( Z / Z ) 72 743178240 15 W A A Z ⊕ ( Z / Z ) 96 2090188800 20 W D E Z
112 9555148800 15 W A Z
60 298598400 10 W D Z
80 707788800 15 W A A Z
92 3483648000 6 W D E Z
96 28665446400 1 W D Z
144 133772083200 1
Proof.
Clearly, { ± id } is contained in the kernel. By Lemma 2.13 and [21, Thm. 6.1], it sufficesto show that there is no automorphism of X of order acting trivially on S X . This follows from[48, Main theorem on p. 18], noticing that in our case it holds δ S X = 1 (cf. [48, Definition 2.2]), as ( t / = − / . (cid:3) A computation with the GAP command
ConjugacyClasses shows that the group O( T ♯X ) con-tains exactly conjugacy classes. Let us define the following elements of O( T ♯X ) : h := , h := , h := . Lemma 3.21.
An element h ∈ O ♯ h ( T X ) of order belongs to the conjugacy class of h .Proof. As in Lemma 3.7, thanks to Lemma 3.20 we can suppose that h = ( A mod 2) , where A ∈ GL ( Z ) satisfies AT A ⊺ = T and A = − I .Let V ⊂ T X ⊗ C be the eigenspace of A associated with i . Necessarily, dim( V ) = 3 and we cansuppose that σ := σ X ∈ V . Let σ, τ, υ be a basis of V over C . We write σ = σ t + σ t + σ t + σ t + σ t + σ t , σ i ∈ C ,τ = τ t + τ t + τ t + τ t + τ t + τ t , τ i ∈ C ,υ = υ t + υ t + υ t + υ t + υ t + υ t , υ i ∈ C . Again, V is a totally isotropic subspace with respect to the bilinear form induced by T X . Therefore,upon substituting t with t , or τ and υ with linear combinations of σ, τ, υ , we can make thefollowing assumptions: σ = 1 , υ = 0 σ = σ σ + σ + σ , υ = − σ υ − σ υ + σ υ + σ υ ,τ = 0 , τ = 0 , τ = − σ τ − σ τ + σ τ , υ = τ υ + i υ . Imposing that σ, τ, υ ∈ V and ¯ σ, ¯ τ , ¯ υ ∈ ¯ V and recalling that A has integer coefficients, we getwith elementary but rather tedious computations that there exist m , . . . , m ∈ Z such that A = m ∗ ∗ ∗ ∗ − m ∗ − m m m m ∗ m ∗ ∗∗ ∗ − m ∗ m m ∗ ∗ ∗ ∗ m ∗ m ∗ ∗ , where the symbol ∗ denotes other integral coefficients (not necessarily equal to or m . . . , m ).Looking at the possible parities of m , . . . , m , we see that h = ( A mod 2) is conjugate to h in allpossible cases. (cid:3) Lemma 3.22.
An element h ∈ O ♯ h ( T X ) of order belongs to the conjugacy class of h .Proof. Similarly as in Lemma 3.8, we can assume that h = ( A mod 2) for some matrix A ∈ GL ( Z ) such that AT A ⊺ = T and A = I . Again, we let U and W be the eigenspace associated with and ω , respectively. It holds dim( U ) + 2 dim( W ) = 6 . Moreover, U is orthogonal to W .Investigating all cases with similar computations, we see that it must be dim( U ) = dim( W ) = 2 ,otherwise A is forced to have non-integral coefficients.Looking at the possible parities of the entries of A , we see that h = ( A mod 2) is alwayscontained in the conjugacy class of h . (cid:3) Proposition 3.23.
The subgroup O ♯ h ( T X ) is is a cyclic group of order , or , in which case itis generated by a conjugate of h , h or h , respectively.Proof. Let h be a generator of O ♯ h ( T X ) and let m be its order.By Proposition 2.11 it holds ϕ ( | O h ( T X ) | ) | rk( T X ) = 6 , therefore | O h ( T X ) | ∈ { , , , , } . Itfollows from Lemma 3.20 that m = | O h ( T X ) | / ∈ { , , , , } .We can exclude m = 7 because ∤ | O( T ♯X ) | = 1440 and we can exclude m = 9 because wecan verify by inspection that O( T ♯X ) contains no elements of order . All in all, m ∈ { , , } asclaimed. OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 23 If m = 1 , then obviously h = h . If m = 2 , then h is conjugate to h by Lemma 3.21. If m = 3 ,then h is conjugate to h by Lemma 3.22. (cid:3) Proposition 3.24.
The frame genus W X contains exactly isomorphism classes, listed in Table 5,whose Gram matrices are contained in the arXiv ancillary file genus_Kloosterman.sage .Proof. We compute W X using Kneser’s neighbor method (see Remark 2.14). One could also haveapplied the Kneser–Nishiyama method with either T = A D or T = A D . (There is a thirdlattice in the same genus which is not generated by its roots, but this choice makes computationsmuch more difficult, as one cannot use Nishiyama’s results [31]). The list of lattices found iscomplete because the mass formula holds (see Remark 2.15): X i =1 | O( W i ) | = 130668164210599936000 = mass( W X ) . (cid:3) Remark 3.25.
As a corollary of Proposition 3.24, one can easily derive Kloosterman’s classifica-tion [19, Thm. 1.1 and Thm. 1.3] of the pairs ( W root , W/W root ) . Note that ( W , W ) and ( W , W ) correspond to the same pair ( W root , W/W root ) . In Kloosterman’s paper these cases are not distin-guished. In the language of Braun, Kimura and Watari [6], in this situation the “ J ( X ) classifica-tion” is strictly finer than the “ J (type) ( X ) classification”. For this reason we decided not to followKloosterman’s numeration.The arXiv ancillary file groups_K_Kloosterman.gap contains the definition of the subgroups K , K , K , K , K ′ , K ′′ , K , K , K , K ⊂ G. With the same proof as for Proposition 3.12 we obtain the following proposition.
Proposition 3.26.
The subgroup O ♯ ( W ) ⊂ G is conjugate to K if W = W , to K if W = W ,to K if W = W , to K if W = W , to K if W ∈ { W , W } , to K ′ if W ∈ { W , W } ,to K ′′ if W ∈ { W , W } , to K if W = W , to K if W ∈ { W , W , W } , to K if W = W , to K if W = W , and to G if W ∈ { W , W } . (cid:3) We now have all ingredients to compute the multiplicities of the frames W ∈ W X . Theorem 3.27. If X is a K3 surface with transcendental lattice T X ∼ = U (2) ⊕ [ − , then one ofthe following cases holds: |J X / Aut( X ) | = if | O ♯ h ( T X ) | = 1 , if | O ♯ h ( T X ) | = 2 , if | O ♯ h ( T X ) | = 3 . Proof.
We apply Theorem 2.8 to compute the multiplicities of the frames W ∈ W X using eitherformula (5) or the GAP function DoubleCosets . Thanks to Proposition 3.23 and Proposition 3.26,the multiplicities are given by | H \ G/K | , where G is the group defined in (10), H is the cyclicsubgroup generated by h , h or h and K ∈ { K , K , K , K , K ′ , K ′′ , K , K , K , K } .Our results are listed in Table 6. (cid:3) Table 5.
Lattices in the frame genus W X of a K3 surface X with transcendentallattice T X ∼ = U (2) ⊕ [ − . W W root
W/W root | ∆( W ) | | O( W ) | | O ♯ ( W ) | W D D W A D W A D D W A D E W A D W A E W A D Z / Z
76 2717908992 96 W A D D Z / Z
92 6794772480 96 W A D D Z / Z
92 849346560 12 W A D Z / Z
124 475634073600 1440 W A D Z / Z
124 15854469120 48 W A E Z / Z
140 89181388800 240 W A D ( Z / Z )
60 226492416 96 W A D ( Z / Z )
76 849346560 144 W A Z
60 149299200 144 W A A Z
68 185794560 96 W A A Z
92 1741824000 120 W A E Z
84 3583180800 1440
Table 6.
Multiplicities of the frames W ∈ W X listed in Table 5. | O ♯ h ( T X ) | W W W W W W W W W . . . . . . . . . W W W W W W W W W sum. . . . . . . . . Apéry–Fermi pencil.
Let X be a K3 surface with transcendental lattice T X ∼ = U ⊕ [ − . Remark 3.28.
The surface X is the generic element of a pencil of K3 surfaces studied by Peters andStienstra [35]. Elliptic fibrations on this pencil were already classified by Bertin and Lecacheux [4],who in particular determined all pairs ( W root , W/W root ) , W ∈ W X . Proposition 3.29.
The frame genus W X contains exactly isomorphism classes, listed in Table 7,whose Gram matrices are contained in the arXiv ancillary file genus_Apery_Fermi.sage . OUNTING ELLIPTIC FIBRATIONS ON K3 SURFACES 25
Proof.
The list found by Bertin and Lecacheux [4] applying the Kneser–Nishiyama method with T = A D is complete because the mass formula holds: X i =1 | O( W i ) | = 1239701105471110668726060974080000 = mass( W X ) . (cid:3) Theorem 3.30. If X is a K3 surface with transcendental lattice T X ∼ = U ⊕ [ − , then |J X / Aut( X ) | = 32 . Proof.
It holds | O( T ♯X ) | = 4 . Since rk( T X ) is odd, | fr − X ( W ) | is equal to the index of O ♯ ( W ) in O( T ♯X ) (Remark 2.12). A finite set of generators of O( W ) can be computed using the command orthogonal_group of the Sage class QuadraticForm . Their images generate O ♯ ( W ) . (cid:3) References
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148 551809843200 1 W A D E A A E Z
148 300987187200 1 W A D A D Z
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228 119859786547200 1 W D E A D Z
324 2448564210892800 1 W D E D E Z
352 14383174385664000 1 W E E Z
480 1941728542064640000 2 W D D Z
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132 73156608000 1 W A D E A D Z
156 234101145600 2 W D E A D E Z
212 7491236659200 1 W A E A A Z
212 10461394944000 1 W D A D Z ⊕ ( Z / Z ) 132 101921587200 1 W A D E A D Z ⊕ ( Z / Z ) 156 367873228800 1 W A D A Z ⊕ ( Z / Z ) 240 167382319104000 1
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