IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS

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1 IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS SAMUEL BOISSIÈRE, CHIARA CAMERE, GIOVANNI MONGARDI, AND ALESSANDRA SARTI Abstract. We prove that there exists a holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of two points on a K3 surface that admits a non-symplectic automorphism of order 23, that is the maximal possible prime order in this deformation family. 1. Introduction The study of automorphisms on deformation families of hyperkähler manifolds is a recent and very active field of research. One of the main objective in the recent published papers concerns the classification of prime order automorphisms: fixed locus, moduli spaces and deformations. We refer for instance to [5, 13, 17] and references therein for a more complete picture. The purpose of this short note is to answer a question of [6] concerning the existence of automorphisms of order 23. Let X be an irreducible holomorphic symplectic manifold. Its second cohomology group H 2 X,Z) is an integral lattice for the Beauville Bogomolov Fujiki quadratic form [4]. Let f be a holomorphic automorphism of X of prime order p acting non-symplectically: f acts on H 2,0 X) by multiplication by a primitive p-th root of the unity. Such automorphisms can exist only when X is projective. It follows that the invariant lattice Tf) H 2 X,Z) is a primitive sublattice of the Néron Severi group NSX) and consequently the characteristic polynomial of the action of f on the transcendental lattice TransX) is a multiple k of the p-th cyclotomic polynomial Φ p. Thus kϕp) = rank Z TransX) and in particular ϕp) b 2 X) ρx), where ϕ is the Euler function and ρx) = rank Z NSX) is the Picard number of X. Assume that X is in the deformation class of the Hilbert scheme of two points on a projective K3 surface an IHS-K3 [2] for short). Since b 2 X) = 23, the maximal order for f is p = 23 and this can happen only when ρx) = 1. The main result of this paper is: Theorem 1.1. There exists an IHS-K3 [2] with a non-symplectic automorphism of order 23. We show in 3 that the Néron Severi group of X has rank one, generated by an ample line bundle of square 46 with respect to the Beauville Bogomolov Fujiki quadratic form: up to now there does not exist any geometric construction of such Date: June 26, Mathematics Subject Classification. Primary 14J50; Secondary 14C50,55T10. Key words and phrases. holomorphic symplectic manifolds, non-symplectic automorphisms, ideal lattices, cyclotomic fields. 1

2 2 S. BOISSIÈRE, C. CAMERE, G. MONGARDI, AND A. SARTI an IHS-K3 [2] see [16]). We emphasize that such an automorphism can not exist on the Hilbert scheme of two points on a K3 surface since it has Picard number two. The strategy of the proof consists in constructing an isometry of order 23 of the lattice E8 2 U 3 2 with the required properties Corollary 5.4) and then to use the surjectivity of the period map and the global Torelli theorem to construct the variety with its automorphism Theorem 6.1). Assuming that such an automorphism does exist, the invariant lattice T and its orthogonal complement S are uniquely determined up to isometry so the main step Proposition 5.3) consists in constructing an order 23 isometry on the lattice S: we obtain it by realizing S as an ideal lattice in the 23-th cyclotomic field, following results of Bayer Fluckiger [1, 2, 3]. 2. Preliminaries on lattices A lattice L is a free Z-module equipped with a nondegenerate symmetric bilinear form, L with integer values. Its dual lattice is L := Hom Z L,Z). It can be also described as follows: L = {x L Q x,v L Z v L}. ClearlyLisasublatticeofL ofthesamerank,sothediscriminant group A L := L /L is a finite abelian group whose order is denoted d L and called the discriminant of L. Inabasise i ) i ofl, forthegrammatrixm := e i,e j L ) i,j onehasd L = detm). A lattice L is called even if x,x 2Z for all x L. In this case the bilinear form induces a quadratic form q L : A L Q/2Z. Denoting by s +),s ) ) the signature of L R, the triple of invariants s +),s ),q L ) characterizes the genus of the even lattice L see [5, 9] and references therein). A lattice L is called unimodular if A L = {0}. A sublattice M L is called primitive if L/M is a free Z-module. If L is unimodular and M L is a primitive sublattice, thenm anditsorthogonalm inlhaveisomorphicdiscriminantgroups and q M = q M. Let p be a prime number. A lattice L is called p-elementary if A L = Z pz) a for some non negative integer a also called the length la L ) of A). We write Z pz α), α Q/2Z to denote that the quadratic form q L takes value α on the generator of the Z pz component of the discriminant group. Recall that an even indefinite p- elementary lattice of rank r 3 with p 3 is uniquely determined by its signature and discriminant form see [5, Theorem 2.2]). 3. Basic results on non-symplectic automorphisms From now on, we assume that X is an IHS-K3 [2] with a non-symplectic automorphism f of prime order 3 p 23. The lattice H 2 X,Z) has signature 3,19) and is isometric to L := E8 2 U 3 2, where U is the unique even unimodular hyperbolic lattice of rank two and E 8 is the negative definite lattice associated to the corresponding Dynkin diagram. We restate in this special case some results of Boissière Nieper-Wisskirchen Sarti [6]: the case p = 23 was left apart since it requires different arguments due to the fact that the ring of integers of the 23rd cyclotomic field is not a PID, but some basic facts extend easily.

3 IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS 3 The automorphism f induces an isometry g := f on H 2 X,Z). We denote by G = g the group generated by g and we put τ := g 1 Z[G], σ := 1+g + +g p 1 Z[G]. One has Tf) = kerτ) H 2 X,Z) and we define Sf) := kerσ) H 2 X,Z). Denote by Φ p Q[X] the p-th cyclotomic polynomial. Consider the cyclotomic field K = Q[X]/Φ p ) = Qζ p ) with ring of algebraic integers O K = Z[ζp ] here ζ p = X mod Φ p should not be considered as a complex number). The G-module structure of K is defined by g x = ζ p x for x K. For any fractional ideal I in K, and α I, we denote by I,α) the module I Z whose G-module structure is defined by g x,k) = ζ p x + kα,k). By a theorem of Diederichsen Reiner [8, Theorem 74.3], H 2 X,Z) is isomorphic as a Z[G]-module to a direct sum: A 1,a 1 ) A r,a r ) A r+1 A r+s Y for some r,s N, where A i are fractional ideal in K, a i A i are such that a i / ζ p 1)A i and Y is a free Z-module of finite rank on which G acts trivially. Lemma 3.1. The quotient H 2 X,Z) Tf) Sf) is a p-torsion module. Proof. First we observe that Tf) Sf) = 0 since H 2 X,Z) has no p-torsion. It is clear that Y Tf) and O K Sf). Let A = i O Kα i be a fractional ideal of K, with α i K. Clearly A Sf). In any term A,a) = A Z, denoting v := 0,1) in this decomposition, we show that pv Tf) Sf). One has τpv) = pa,0). Write a = i x iα i with x i O K. Since O K /ζ p 1) = Z/pZ, there exists z i O K such that px i = ζ p 1)z i. Hence τpv) = ζ p 1)z,0) with z := i z iα i A. Now τz,0)) = ζ p 1)z,0) hence τpv z,0)) = 0 and σz,0)) = 0 so finally pv = pv z,0))+z,0) Tf) Sf). We define a f N such that H 2 X,Z) Tf) Sf) = ) af Z. pz By definition, Sf) is a torsion-free O K -module for the action ζ p x = gx) for all x Sf), hence Sf) Q := Sf) Z Q is a K-vector space. It follows that there exists m f N such that rank Z Sf) = dim Q Sf) Q = p 1)m f. It is easy to check that Sf) is the orthogonal complement of Tf) in the lattice H 2 X,Z) see [6, Lemma 6.1]). By a similar argument as in [6, Lemma 6.5] one deduces from Lemma 3.1 that the invariant lattice Tf) has signature 1,22 p 1)m f ) and discriminant ) ) af A Tf) Z Z = 2Z pz and that Sf) has signature 2,p 1)m f 2) and discriminant A Sf) = Z pz) af. If p 3, as explained in [6, proof of Lemma 6.5] the action of G on A Sf) is trivial. Since f acts non-symplectically one has TransX) Sf) and rank Z TransX) p 1. In particular, if m f = 1 this forces TransX) = Sf) and consequently

4 4 S. BOISSIÈRE, C. CAMERE, G. MONGARDI, AND A. SARTI NSX) = Tf). Since 1 is not an eigenvalue of f Sf) the characteristic polynomial of f Sf) is Φ p. All the possible isometry classes for the lattices Tf) and Sf) have been classified in [5] when 2 p 19 only partially for p = 5) by using the previous properties for p = 2 the situation is a bit different), the Lefschetz fixed point formula and a relation between the cohomology modulo p of the fixed locus and the integers a f,m f obtained using Smith theory methods [6]. If p = 23 the only possibility is that m f = 1, Tf) has signature 1,0) and Sf) has signature 2,20). The case a f = 0 is impossible by Milnor s theorem since there exists no even unimodular lattice with signature 2,20), hence a f = 1 since Tf) has rank one, so A Tf) = Z 46Z and finally Tf) is isometric to the lattice 46. By results of Nikulin and Rudakov Shafarevich [15, 18] Sf) splits as a direct sum U W where W is hyperbolic and 23-elementary of signature 1,19), A W = Z 23Z and W is unique up to isometry. It follows that Sf) is uniquely determined and we deduce ) that Sf) is isometric to the lattice U 2 E8 2 K where K 23 :=. As a consequence, if there exists an IHS-K3 1 2 [2], say X, with a non-symplectic automorphism f of order 23, then necessarily it has ρx) = 1, NSX) = Tf) = 46 and TransX) = Sf) = E8 2 U 2 K 23. Such a variety does not belong to any of the known families: Hilbert schemes of points or moduli spaces of semi-stable sheaves on projective K3 surfaces have Picard number greater than 2, Fano varieties of lines on cubic fourfolds are polarized by a class of square 6 see [6, 5.5.2] and references therein) and similarly the degree of the polarisation is 2 for double covers of EPW sextics, it is 38 for the sums of powers of general cubics of Iliev Ranestad and it is 22 for the varieties of Debarre Voisin see [16, 0]). 4. Ideal lattices in cyclotomic fields The relation between automorphisms of lattices with given characteristic polynomials and ideals in cyclotomic fields has been studied by many authors, in particular Bayer-Fluckiger [1, 2, 3] and Gross McMullen [10]. We recall here some results that are needed in the sequel. Assume that p is an odd prime number. Recall that K = Qζ p ) denotes the cyclotomic field with ring of algebraic integers O K = Z[ζ p ]. We denote respectively by Tr K/Q and N K/Q the trace and the norm maps. The complex conjugation on K is defined as the Q-linear involution K K,x x such that ζp i = ζp p i for all i. We denote by F K the real subfield of K, that is F := {x K x = x}. Denoting µ p := ζ p +ζp 1 one has F = Qµ p ). The Q-linear pairing, ) K : K K Q,x,y) Tr K/Q xy) is non-degenerate and has determinant D K := p p 2 in the basis 1,ζ p,...,ζp p 2 ). Let S,, S ) be an integral even lattice of rank p 1, signature s +,s ) and discriminant d S. Assume that S admits a non trivial isometry ϕ of order p. Its characteristic polynomial is then Φ p so S admits a natural structure of O K -module defined by ζ p x = ϕx) for all x S. For dimensional reasons S Q := S Z Q is isomorphic to K so the inclusion S S Q = K identifies the lattice S with an O K -submodule of K a fractional ideal of K), so S becomes an ideal lattice in K.

5 IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS 5 Observe that S Q is identified with K in such a way that the isometry ϕ corresponds to the mutiplication by ζ p in K. The multiplication by ζ p is an isometry for, ) K and also for, S extended to S Q since it corresponds to the action of ϕ. It is easy to see that under the identification S Q = K there exists a unique α K such that x,y S = αx,y) K x,y S Since the bilinear form on S is symmetric one has α = α so α F. If I K is a fractional ideal and α F, we denote by I α the ideal lattice whose bilinear form is x,y α := Trαxy). Some of the main invariants of the lattice I α correspond to properties of α that we explain now. Recall that the norm of I is defined as NI) := detψ) where ψ: K K is any Q-linear automorphism such that ψo K ) = I. It follows easily that the discriminant of I α satisfies the relation: 1) Observe that since α F one has d Iα = NI) 2 N K/Q α) D K. N K/Q α) = N F/Q NK/F α) ) = N F/Q α 2 ) = N F/Q α) 2. Recall that the dual of any fractional ideal I K is defined by I := {x K TrxI) Z} this coincides with the lattice theoretical dual only when α = 1). In particular, O K is the codifferent of K. If I α is an integral lattice, for any x,y I one has Trαxy) Z so αxy O K. The integrality of I α is thus equivalent to the condition: 2) αii O K. Note that if α satisfies the above property the lattice I α is automatically even: p 1)/2 putting γ := ζp, i since γ +γ = 1 one has for any x I i=0 x,x S = Tr K/Q αxx) = Tr K/Q γ +γ)αxx) = Tr K/Q γαxx)+tr K/Q γαxx) 2Z since αxx OK by assumption. The field K admits p 1 complex embeddings defined by ζ p e 2ikπ p, 1 k p 1 that induce real embeddings of F. We denote by t the number of these real embeddings such that α is negative. One can show that the lattice I α has signature 3) p 1 2t,2t). This is a special case of [1, Proposition 2.2], we recall the argument for convenience. First observe that K is a quadratic extension of F with minimal polynomial X 2 µ p X + 1 F[X]. Denoting θ := ζp 2 + ζp 2 2 one has thus K = F θ). Each complex embedding of K induces a real embedding v: F R such that vθ) < 0. It follows that K Q R decomposes in a direct sum K Q R = ) R vθ) v: F R where the sum runs over all real embeddings of F. Each factor is isomorphic to C and the complex conjugation on K induces the usual complex conjugation on each factor C. On each factor, the form,, α computed in the R-basis 1, vθ)) is

6 6 S. BOISSIÈRE, C. CAMERE, G. MONGARDI, AND A. SARTI diag2vα), 2vα)vθ)) so it has signature 2,0) if vα) > 0 and signature 0,2) if vα) < 0. The result follows. 5. Construction of isometries of lattices We want to determine if a given integral even p-elementary lattice S of rank p 1 with fixed signature and discriminant form admits an isometry of order p whose characteristic polynomial is the cyclotomic polynomial Φ p. By the results of Section 4, one first has to find an element α F satisfying conditions 1),2),3). ) 2 1 Example 5.1. Assume that p = 5 and S = U H 5 with H 5 =. The 1 2 lattice S is 5-elementary with d S = 5, it has signature 2,2) and discriminant form A S = Z 2 ) 5Z 5. In [5, Table 2] this case is denoted by p,m,a) = 5,1,1). In order to recover this lattice as an ideal lattice, since O K is a PID we take I = βo K for some β K. Equation 1) writes: N K/Q β) 2 N F/Q α) 2 = Assuming that β = 1, by running a basic computer search program we find that α = 1 5 3µ 5 + 4) satisfies all the needed conditions: in the basis 1,ζ 5,ζ5,ζ 2 5) 3 of I the bilinear form writes and it is easy to check that this lattice has signature 2,2) and discriminant form Z 2 ) 5Z 5. As mentioned above, these invariants characterize the lattice U H5 up to isometry. By construction, the order 5 isometry of this lattice, written in this basis, is the companion matrix of Φ 5 : Example 5.2. Assume that p = 13 and that S = U 2 E 8. The lattice S is unimodular of signature 2,10). In [5, Table 5] this case is denoted by p,m,a) = 13,1,0). If S admits an order 13 isometry it induces an identification S = βo K for some β K. Equation 1) writes: N K/Q β) 2 N F/Q α) 2 = It is clear that this equation has no solution, so this lattice does not admit an isometry whose characteristic polynomial is Φ 13. This answers a question left open in [5, Theorem 7.1]: this case cannot be realized by a non-symplectic automorphism of order 13 on an IHS-K3 [2]. We assume now that p = 23 and we consider the lattice S := E8 2 U 2 K 23. It is 23-elementary with d S = 23, signature 2,20) and discriminant form A S = Z 23Z 2 23 ).

7 IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS 7 Proposition 5.3. The lattice U 2 E 2 8 K 23 admits an isometry of order 23 which acts trivially on the discriminant group A S. Proof. We apply the strategy developed above. Taking I = O K, equation 1) writes: N F/Q α) = The software MAGMA [7] provides a solution to this equation: α 0 := 1 23 µ7 23 +µ µ µ µ µ µ 23 2). By the Dirichlet unit theorem, the group of units OF is the product of the finite cyclic group of roots of unity of F with a free abelian group of rank 10. A computation with the software SAGE [19] shows that OF = Z 2Z Z10 the only roots of unity in F are ±1) where the free part is generated by the following fundamental units: ǫ 1 := µ µ ǫ 2 := µ µ µ µ ǫ 3 := µ µ µ µ 23 ǫ 4 := µ ǫ 5 := µ µ µ µ µ 23 ǫ 6 := µ µ µ µ µ µ 23 1 ǫ 7 := µ 23 ǫ 8 := µ µ µ µ µ µ µ 23 1 ǫ 9 := µ µ µ µ µ µ µ µ ǫ 10 := µ µ µ µ µ µ µ µ µ 23 1 Let us consider an element α F given as follows: α = α 0 ǫ 0 ǫ ν1 1 ǫν10 for some ν i Z with ǫ 0 { 1,1}. By running a basic computer search program we find that the choice ǫ 0 = 1, ν = 2,1,2,2,0,1,1,2,1,0) satisfies all the needed conditions: in the basis 1,ζ 23,...,ζ23) 21 of I the bilinear form is the matrix given in Appendix A. It is easy to check that this lattice has signature 2,20) and discriminant form Z 44 23Z 23). Asalreadymentioned, theseinvariants characterizethe lattice U 2 E8 2 K 23 up to isometry. By construction, the order 23 isometry of this lattice, written in this basis, is the companion matrix of the polynomial Φ 23 and a direct computation shows that that this isometry acts trivially on the discriminant group. Corollary 5.4. The lattice L := E 2 8 U 3 2 admits an order 23 isometry whose invariant lattice is isometric to T := 46 and such that the orthogonal of T in L is isometric to S := E 2 8 U 2 K 23. Proof. Denoting T = Zt with t 2 = 46, the discriminant group A T is generated by τ := t/46 T with τ 2 = 1/46. We denote by σ S a generator of A S such that

8 8 S. BOISSIÈRE, C. CAMERE, G. MONGARDI, AND A. SARTI σ 2 = 44/23. The vector 2σ +4τ S T) is isotropic in A S T so it defines an even overlattice M := S T 2σ +4τ)Z S T). Consider the quotient H := M S T A S T. One computes that H /H is generated by the class 23τ with 23τ) 2 = 3/2 Q/2Z. Since A M = H /H see [15]) we conclude that M is an even lattice of signature 3,20) and discriminant form A M = Z 3 ) 23Z 2. By [14, Theorem 2.2] these invariants characterize M up to isometry, so M is isomorphic to L. It follows directly from the construction that S is the orthogonal of T in M. Let ϕ be the isometry of order 23 on S constructed in Proposition 5.3. Since ϕ acts trivially on A S, the isometry ϕ id of S T extends to an isometry on L with the required properties. Remark 5.5. In the lattice L, denoting by e,f) a basis on one of the factors isometric to the lattice U and by δ a generator of the factor isometric to 2 it is easy to see that an explicit embedding of T in L whose orthogonal is isometric to S is given by t 2e+12f +δ. Applying Nikulin s results on primitive embeddings and decomposition of lattices see [5] and references therein) one can see that T admits up to isometry a second embedding in L whose orthogonal is isometric to E 2 8 U 2 2 K 23, an explicit embedding beeing given by t e+23f. 6. An IHS-K3 [2] with a non-symplectic automorphism of order 23 Theorem 6.1. There exists an IHS-K3 [2] with a non-symplectic automorphism of order 23. This variety X and its automorphism f have the following properties: 1) ρx) = 1, NSX) = 46 and TransX) = E 2 8 U 2 K 23 ; 2) Tf) = NSX) and Sf) = TransX). Proof. The proof is an application of the surjecivity of the period map and of the global Torelli theorem for IHS manifolds. Construction of the variety. Let M 0 L be a connected component of the moduli spaces of pairs X,η) where X is an IHS-K3 [2] and η: H 2 X,Z) L is an isometry. The period domain is Ω L := {ω PL C ) ω,ω L = 0, ω,ω L > 0}. Recall that the period map P: M 0 L Ω L defined by PX,η)) = ηh 2,0 X)) is surjective [11, Theorem 8.1]. Consider as in Corollary 5.4 the embedding of T = 46 in the lattice L whose orthogonal is S = E8 2 U 2 K 23, with the isometry ϕ of order 23 acting trivially on T. We denote by ω a generator of the one-dimensional eigenspace of S C corresponding to the eigenvalue ξ := e 2iπ 23. Recall that by construction S is identified with the ring of integers O K of the cyclotomic field K = Qζ 23 ) so that ω S C = K Q C with basis 1,ζ 23,...,ζ23). 21 In this basis, the isometry ϕ acts by the companion matrix of the 23rd cyclotomic polynomial and it is easy to check that up to a multiplicative constant one has i ω = ξ j ζ23. i i=0 j=0

9 IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS 9 Since ϕω) = ξω one has ω,ω L = 0. Denoting by Tr K/Q : K C C the C-linear extension of the trace, one has ω,ω L = ω,ω S = Tr K/Q ωαω) where α is given in the proof of Proposition 5.3. An explicit computation shows that Tr K/Q ωαω) > 0, so ω Ω L. By surjectivity of the period map, there exists X,η) M 0 L such that ηh2,0 X)) = ω. Then ηnsx)) = {λ L λ,ω L = 0} T. Let us show that ηnsx)) = T. For this, we show that there is no element λ S with the property that λ,ω S = 0. In the basis 1,ζ 23,...,ζ23), 21 denoting 1 1 Ξ := ξ 21,...,ξ,1) and J :=... one has by definition ω = JΞ. Denote 0 1 by M the matrix of the lattice S in the basis 1,ζ 23,...,ζ23) 21 see Appendix A). For any λ S, since S = O K = Z[ζ 23 ] the element λ can be identified with a column vector with integer coordinates. Then λ,ω S = λ Mω = λ MJΞ. If λ MJΞ = 0, since λ MJ has integer coordinates and since the coordinates of Ξ are linearly independent over Q it follows that λ MJ = 0. But the matrix MJ is invertible, so λ = 0. This proves that NSX) = T and in particular X is projective [11, Theorem 3.11]. Construction of the automorphism. TheisometryϕpreservesthespaceH 2,0 X) = Cω so it is a Hodge isometry. Denoting by q X the Beauville-Bogomolov-Fujiki quadratic form on H 2 X,Z), the posivite cone C X is defined as the connected component of the cone {x H 2 X,Z) q X x) > 0} that contains the Kähler cone. By Markman [12, Lemma 9.2] the group of monodromy operators of H 2 X,Z) is equal to the group of isometries of H 2 X,Z) preserving the positive cone C X. Here the generator of NSX) = T is an ample class so it lives in the Kähler cone and since NSX) is invariant by ϕ the cone C X is preserved, so ϕ is a monodromy operator that leaves invariant a Kähler class. By the Global Torelli Theorem of Markman Verbitsky [12, Theorem 1.3] there exists an automorphism f of X such that f = ϕ on H 2 X,Z). Since the natural map AutX) OH 2 X,Z)) is injective see for instance [13, Lemma 1.2] and references therein), f is an order 23 non-symplectic automorphism of X. Remark 6.2. Since NSX) = 46, it follows from [12, Theorem 2.2] that X,η) is the only Hausdorff point in the fiber P 1 ω) of the period map so this variety with its automorphism of order 23 is unique, although it belongs to a 20-dimensional family of IHS-K3 [2] polarized by a class of square 46. Remark 6.3. The same method can be used to produce order 23 automorphisms on deformations of K3 [n] with n 3, under some arithmetic conditions on n.

10 BOISSIÈRE, C. CAMERE, G. MONGARDI, AND A. SARTI 10 S. Appendix A. Matrix of the lattice with an order 23 isometry Here is the matrix of the bilinear form on the lattice U 2 E 8 2 K23, written in a basis such that the order 23 isometry of this lattice is the companion matrix of the cyclotomic polynomial Φ23:

11 IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS 11 References 1. E. Bayer-Fluckiger, Lattices and number fields, Algebraic geometry: Hirzebruch 70 Warsaw, 1998), Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp , Determinants of integral ideal lattices and automorphisms of given characteristic polynomial, J. Algebra ), no. 2, , Ideal lattices, A panorama of number theory or the view from Baker s garden Zürich, 1999), Cambridge Univ. Press, Cambridge, 2002, pp A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom ), no. 4, S. Boissière, C. Camere, and A. Sarti, Classification of automorphisms on a deformation family of hyperkähler fourfolds by p-elementary lattices, arxiv: S. Boissière, M. Nieper-Wißkirchen, and A. Sarti, Smith theory and irreducible holomorphic symplectic manifolds, J. Topol ), no. 2, W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput ), no. 3-4, , Computational algebra and number theory London, 1993). 8. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, Reprint of the 1962 original, A Wiley-Interscience Publication. 9. I. Dolgachev, Integral quadratic forms: applications to algebraic geometry after V. Nikulin), Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp B. H. Gross and C. T. McMullen, Automorphisms of even unimodular lattices and unramified Salem numbers, J. Algebra ), no. 2, D. Huybrechts, Compact hyper-kähler manifolds: basic results, Invent. Math ), no. 1, E. Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, pp G. Mongardi, On natural deformations of symplectic automorphisms of manifolds of K3 [n] type, C. R. Math. Acad. Sci. Paris ), no , D. R. Morrison, On K3 surfaces with large Picard number, Invent. Math ), no. 1, V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv ), K. G. O Grady, EPW-sextics: taxonomy, Manuscripta Math ), no. 1-2, H. Ohashi and M. Wandel, Non-natural non-symplectic involutions on symplectic manifolds of K3 [2] -type, arxiv.org/abs/ v A. N. Rudakov and I. R. Shafarevich, Surfaces of type K3 over fields of finite characteristic, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp W. A. Stein et al., Sage Mathematics Software Version 5.0.1), The Sage Development Team, 2014,

12 12 S. BOISSIÈRE, C. CAMERE, G. MONGARDI, AND A. SARTI Samuel Boissière, Laboratoire de Mathématiques et Applications, UMR CNRS 6086, Université de Poitiers, Téléport 2, Boulevard Marie et Pierre Curie, F Futuroscope Chasseneuil address: URL: sboissie/ Chiara Camere, Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten Hannover, Germany address: camere@math.uni-hannover.de URL: Giovanni Mongardi, Universitá degli studi di Milano, Dipartimento di Matematica, via Cesare Saldini Milano, Italy address: giovanni.mongardi@unimi.it Alessandra Sarti, Laboratoire de Mathématiques et Applications, UMR CNRS 6086, Université de Poitiers, Téléport 2, Boulevard Marie et Pierre Curie, F Futuroscope Chasseneuil address: alessandra.sarti@math.univ-poitiers.fr URL: sarti/