TThe non-symplectic index of supersingular K3 surfaces
Junmyeong Jang
Mathematics Subject Classification : 14J20, 14J28
Let X be an algebraic complex K3 surface. The second integral singular cohomology H ( X/ Z ) is an even unimodular integral lattice of signature (3 , U ⊕ E . Here U is the unimodular hyperbolic lattice and E is the negative definite root lattice of type E . The Neron-Severi group of X is aneven lattice of signature (1 , ρ ( X ) − N S ( X ), ρ ( X ) is the Picardnumber of X . The cycle map induces a primitive embedding of lattices N S ( X ) (cid:44) → H ( X/ Z ) . The orthogonal complement of this embedding is the transcendental lattice of X whichis denoted by T ( X ). T ( X ) is an even integral lattice of signature (2 , − ρ ( X )). Bythe Hodge decomposition, we may regard the one dimensional complex space of theglobal two forms of X , H ( X, Ω X/ C ) is a direct factor of T ( X ) ⊗ C . Let us denote therepresentations of the automorphism group of X , Aut( X ) on T ( X ) and H ( X, Ω X/ C )by χ X : Aut( X ) → O ( T ( X )) and ρ X : Aut( X ) → GL ( H ( X, Ω X/ C )).Then there exists a projection p X : Im χ X → Im ρ X . It is known that p X is an iso-morphism and Im χ X (cid:39) Im ρ X is a finite cyclic group. ([13]) In particular, for any α ∈ Aut( X ), χ X ( α ) is of finite order. Moreover, if ord χ X ( α ) = n , by the Lefschetz(1,1) theorem, every eigenvalue of χ X ( α ) is a primitive n -th root of unity. We denotethe characteristic polynomial in a variable T of a linear operator L by ϕ ( L ). Let Φ n ( T )be the n -th cyclotomic polynomial. If ord χ X ( α ) = n , since ϕ ( χ X ( α )) is an integralpolynomial, ϕ ( χ X ( α )) is a power of Φ n ( T ). Therefore, when N = | Im ρ X | , the rankof T ( X ) is a multiple of φ ( N ). Here φ is the Euler φ function. We say N is thenon-symplectic index of X . Because φ ( N ) ≤ rank T ( X ) and rank T ( X ) is at most 21, φ ( N ) ≤
20. There exists a complex K3 surface of non-symplectic index 66, which isthe maximum. ([19], [10]) 1 a r X i v : . [ m a t h . AG ] M a r ssume that k is an algebraically closed filed of odd characteristic p , W is the ring ofWitt vectors of k and K is the fraction field of W .Assume X is a K3 surface defined over k . The second crystalline cohomology of X , H cris ( X/W ) is a free W module of rank 22 equipped with a canonical Frobenius semi-linear endomorphism F : H cris ( X/W ) → H cris ( X/W ). There exists a cycle map
N S ( X ) ⊗ W → H cris ( X/W )which preserves the lattice structure. Assume X is of finite height h . (1 ≤ h ≤ X is at most 22 − h . The orthogonal complement of the cyclemap is called the crystalline transcendental lattice and it is denoted by T cris ( X ). Thereexists an algebraic lifting of X over W , X /W satisfying the reduction map N S ( X ⊗ K ) → N S ( X )is an isomorphism. ([15], [11], [7]) We call such a lifting a Nerno-Severi group preservinglifting. Let us denote X K = X ⊗ K and X ¯ K = X ⊗ ¯ K . For a Neron-Severi grouppreserving lifting X /W , N S ( X ¯ K ) = N S ( X K ) = N S ( X ) and every automorphism of X ¯ K is extendable to the integral model X ⊗ o ¯ K . It follows that the reduction mapAut( X ¯ K ) → Aut( X )is injective and its image is of finite index in Aut( X ). Moreover, by the comparisontheorem, we can identity T ( X ¯ K ) with T cris ( X ). ([1], Corollary 3.7) Therefore the imageof the representation χ cris,X : Aut( X ) → O ( T cris ( X ))is finite. For a K3 surface of finite height X , H ( X, W O X ) is a direct factor of T cris ( X )and there is a projection H ( X, W O X ) (cid:16) H ( X, O X ). Considering the Serre duality,we have the canonical projection Im χ cris,X → Im ρ X and the non-symplectic index of X is finite. Assume N is the non-symplectic index of X and ρ X ( α ) generates Im ρ X for α ∈ Aut( X ). Assume ord χ cris,X ( α ) = p a · m , where m is relatively prime to p . Let β = α p a . Then ord χ cris,X ( β ) = m and ρ ( β ) also generatesIm ρ X . Since ord χ cris,X ( β ) is relatively prime to p , there is an algebraic lifting X (cid:48) /W of X such that β is liftable to X (cid:48) . ([9], Theorem 3.2) Let b : X (cid:48) → X (cid:48) be the lifting of β .The order of ρ X (cid:48) ( b ) is N and the characteristic polynomial ϕ ( χ X (cid:48) ¯ K ( b )) = ϕ ( χ cris,X ( β ))has integer coefficients and is a power of N -th cyclotomic polynomial Φ N ( T ). Therefore φ ( N ) divides rank T cris ( X ) and φ ( N ) ≤
20. The non-symplectic index of a K3 surfaceof finite height over k is at most 66.Now assume X is a supersingular K3 surface over k . The Picard number of X is 22.Let us denote the dual lattice of a lattice L by L ∗ and the discriminant group of N , N ∗ /N by l ( N ). The discriminant group l ( N S ( X )) = ( Z /p ) σ for an integer σ between2 and 10. Here σ is called the Artin invariant of X . All the supersingular K3 surfacesof Artin invariant σ form a σ − p and the Artin invariant.We denote a general supersingualr K3 surface of Artin invariant σ defined over a fieldof characteristic p by X p,σ . We also denote N S ( X p,σ ) by N p,σ . It is known that thenon-symplectic index of X p,σ is a divisor of p σ + 1. ([14]) In a previous work, we provethat the following. ([5], Theorem 3.3) Theorem If p >
3, the non-symplectic index of a supersingular K3 surface of Artininvarinat 1 is p + 1.As a direct corollary, we have the following. (Loc. cit) Corollary If φ ( p + 1) >
20, then X p, has an automorphism which can not lifted overa ring of characteristic 0.This problem of existence of a non-liftable automorphism for supersingular K3 surfacesis completely answered. ([18], [20], [3]) Theorem
Every supersingular K3 surface has a non-liftable automorphism.This result is based on the study of the Salem degree of an automorphism of a K3surface.In this paper, we generalize our previous result to obtain the non-symplecitc indexesof all supersingular K3 surfaces when the base characteristic p > In this section, we review Ogus’ classification of supersingular K3 surfaces and thecrystalline Torelli theorem in [16] and [17].Assume the characteristic of k , p > X is a supersingular K3 surface of Artininvariant σ over k . The discriminant group l ( N S ( X )) is a 2 σ dimensional vector spaceover the prime field F p equipped with the induced quadratic form. The discriminantof this quadratic form is ( − σ ∆ where ∆ is a quadratic non-residue modulo p . Hence l ( N S ( X )) does not have a σ dimensional isotropic subspace over F p . The cycle map N S ( X ) ⊗ W (cid:44) → H cris ( X/W ) gives the following chain
N S ( X ) ⊗ W ⊂ H cris ( X/W ) ⊂ N S ( X ) ∗ ⊗ W. The cokernel H cris ( X/W ) / ( N S ( X ) ⊗ W ) is a σ dimensional isotropic k subspace of( N S ( X ) ∗ ⊗ W ) / ( N S ( X ) ⊗ W ) = l ( N S ( X )) ⊗ k . We denote H cris ( X/W ) / ( N S ( X ) ⊗ W )3y K ( X ). Note that K ( X ) determines the inclusion N S ( X ) ⊗ W ⊂ H cris ( X/W ). Weset f = id ⊗ F k : l ( N S ( X )) ⊗ k → l ( N S ( X )) ⊗ k . A σ dimensional isotropic subspaceof l ( N p,σ ) ⊗ k , K is a characteristic subspace if K + f ( K ) is σ + 1 dimensional. Acharacteristic subspace K is a strictly characteristic subspace if there exists no proper F p subspace M of l ( N p,σ ) such that K ⊂ M ⊗ k . Theorem 2.1 ([16], Theorem 3.20) . For a supersingular K3 surface X , K ( X ) is astrictly characteristic subspace of l ( N S ( X )) ⊗ k . Remark 2.2.
Our definition of K ( X ) above is slightly different from one given in [16].But all the argument in this paper work for both definitions.Assume K is a strictly characteristic subspace of l ( N p,σ ) ⊗ k . Then l K = ∩ σ − i =0 f i K is aline in l ( N p,σ ) ⊗ k . Moreover we have (cid:80) σ − i =0 f − i ( l K ) = K and (cid:80) σ − i =0 f − i ( l K ) = l ( N p,σ ) ⊗ k .Let v be a non zero vector in l K and we denote v i = f − i ( v ). Then { v , · · · , v σ } is a basisof K and { v , · · · , v σ } is a basis of l ( N p,σ ) ⊗ k . Because l ( N p,σ ) is non-degenerated, v · v σ +1 (cid:54) = 0. After a suitable scalar multiplication, we may assume v · v σ +1 = 1. Here v is uniquely determined up to ( p σ + 1)-th roots of unity multiplication. Now we put a i = v · v σ +1+ i ∈ k . (ı = 1 , · · · , σ −
1) The intersection matrix of l ( N p,σ ) in terms ofthe basis v , · · · , v σ is (cid:18) AA t (cid:19) , where A = a a a · · · a σ − F − k ( a ) F − k ( a ) · · · F − k ( a σ − )0 0 1 F − k ( a ) · · · F − k ( a σ − )... ... ... ... ...0 0 0 0 · · · . (2.1)If we replace v by ξv ( ξ p σ +1 = 1), then a i is replaced by ξ p σ + i +1 a i . Hence we have amap Ψ : K (cid:55)→ ( a , · · · , a σ − ) ∈ A σ − /µ p σ +1 ( k ) . Theorem 2.3 ([16], Theorem 3.21) . The map Ψ is bijective from the set of isomorphicclasses of ( l ( N p,σ ) , K ) to A σ − /µ p σ +1 ( k ).We put ∆ X = { v ∈ N S ( X ) | v · v = − } . For any v ∈ ∆ X , let s v : w (cid:55)→ w + ( v · w ) v ∈ O ( N S ( X ))be the reflection along the line of v . The Wyle group of N S ( X ) is the subgroup of O ( N S ( X )) generated by s v ( v ∈ ∆ X ) and − id . We denote the Wyle group of X by W X .The real quadratic space N S ( X ) ⊗ R is a hyperboic space of rank (1 , N S ( X ) ⊗ R , P X = { v ∈ N S ( X ) ⊗ R | v · v > } has two connected components.Let C X be the set of connected components of P X − ∪ v ∈ ∆ X < v > ⊥ ( ⊂ N S ( X ) ⊗ R ).The ample cone of X , A X is an element of C X . The Wyle group W X acts simply and4ransitively on C X by the canonical action. The automorphism group of X , Aut( X )acts naturally on N S ( X ) and l ( N S ( X )). We denote these representations of Aut( X )by λ X : Aut( X ) → O ( N S ( X )) and ν X : Aut( X ) → O ( l ( N S ( X ))).For an autormophism α ∈ Aut( X ), λ X ( α ) preserves A X and ν ( α ) preserves K ( X ). Theorem 2.4 ([17], Theorem II, III, Crystalline Torelli theorem) . (1) If g ∈ O ( N S ( X )) preserves A X and K ( X ), then there exists a unique α ∈ Aut( X )such that λ X ( α ) = g .(2) If K is a strictly characteristic space of l ( N p,σ ), there exists a unique supersingularK3 surface X of Artin invariant σ up to isomorphism over k such that ( l ( N S ( X )) , K ( X ))is isomorphic to ( l ( N p,σ ) , K ).By the theorem, we may regard the automorphism group Aut( X ) as a subgroup of O ( N S ( X )). Two representations of Aut( X ), ρ X and ν X are isomorphic ([14], [8], [4]),so Im ρ X (cid:39) Im ν X is a finite cyclic group. Since, for any α ∈ Aut( X ), ν ( α ) is rationalover F p and preserves K ( X ), ν ( α )( l K ( X ) ) = l K ( X ) . And v , · · · , v σ are all eigen vectorsof ν ( α ). The character of Aut( X ) on H ( X, Ω X/k ) is isomorphic to the character ofAut( X ) on the 1 dimensional space < v σ +1 > . ([16]) Hence the character of Aut( X )on H ( X, O X ) is isomorphic to the character of Aut( X ) on < v > . If ν ( α )( v ) = ξv , F − σk ( ξ ) = ξ p − σ = ξ − and ν ( α )( v σ +1 ) = ξ − v σ +1 . Lemma 2.5 (c.f. [4], Corollary 3.8) . Assume g ∈ O ( l ( N S ( X ))) preserves K ( X ). Thenthere exists α ∈ Aut( X ) such that ν ( α ) = g . Proof.
Because the signature of
N S ( X ) is (1 ,
21) and the length of l ( N S ( X )) is atmost 20, the class number of N S ( X ) is 1 and the reduction map O ( N S ( X )) → O ( l ( N S ( X ))) is surjective. ([12], Theorem 1.14.2) There exists g ∈ O ( N S ( X )) suchthat g | l ( N S ( X )) = g . Let g ( A X ) = B ∈ C X . There also exists a unique ϕ ∈ W X such that ϕ ( B ) = A X . If B is in the same connected component of P X with A X , wecan write ϕ = s w ◦ s w ◦ · · · ◦ s w n for w i ∈ ∆ X . Let M be the rank 1 sub lattice of N S ( X ) generated by w ∈ ∆ X . Then l ( M ) = Z / l ( M ⊥ ) = Z / ⊕ l ( N S ( X )). Since s w | M ⊥ = id , s w induces the identity map on l ( N S ( X )). It follows that ϕ ◦ g preserves A X and K ( X ), so by the crystalline Torelli theorem, ϕ ◦ g ∈ Aut( X ). And ν ( ϕ ◦ g ) = g .Now assume B is in the other connected component of P X with A X . The signature of N S ( X ) is (1 , N S ( X ) ⊗ R of self intersection 2. If q is a primenumber different from p , N S ( X ) ⊗ Z q is even unimodular of rank 22, so it contains avector of self intersection 2. The length of the unimodular part of N S ( X ) ⊗ Z p is atleast 2, so N S ( X ) ⊗ Z p has a vector of self intersection 2. By the Hasse principle, thereexists a lattice in the genus of N S ( X ) which contains a vector of self intersection 2.But the class number of N S ( X ) is 1, so N S ( X ) contains a vector u such that u · u = 2.Let t u : w (cid:55)→ w − ( w · u ) u ∈ O ( N S ( X ))5e the reflection along the line of u . Then t u exchanges two connected components of P X and t u | l ( N S ( X )) = id . And B (cid:48) = t u ( B ) is in the same connected component of P X with A X . If ϕ ( B (cid:48) ) = A X ( ϕ ∈ W X ), ϕ is the composition of reflections along -2vectors. It follows that ϕ ◦ t u ◦ g ∈ Aut( X ) and ν ( ϕ ◦ t u ◦ g ) = g . Corollary 2.6.
The non-symplectic index of X is even. Proof.
Since − id ∈ O ( l ( N S ( X )) preserves every sub space of l ( N S ( X )) ⊗ k , − id ∈ Im ν X and the order of − id is 2. Let k be an algebraically closed field of chracteristic p > X be a supersingularK3 surface of Artin invariant σ over k . Theorem 3.1.
The non-symplectic index of X over k is as in the Table 1. σ non-symplectic index family1 p + 1 unique2 2 generic p + 1 unique3 2 generic p + 1 1 dimensional p + 1 unique4 2 generic p + 1 unique5 2 generic p + 1 2 dimensional p + 1 unique6 2 generic p + 1 1 dimensional p + 1 unique7 2 generic p + 1 3 dimensional p + 1 unique8 2 generic p + 1 unique9 2 generic p + 1 4 dimensional p + 1 1 dimensional p + 1 unique10 2 generic p + 1 2 dimensional p + 1 uniqueTable 16 roof. Let v , · · · , v σ be the basis of l ( N S ( X )) ⊗ k in the section 2 which makes theintersection matrix (2.1). For all α ∈ Aut( X ), each v i is an eigenvector of ν X ( α ).Assume ν ( α )( v ) = ξv and the order of ξ in k ∗ is n . Then ν ( α )( v i ) = F − ik ( ξ ) v i and ξ determines ν ( α ). Let m be the smallest non-negative integer such that F − mk ( ξ ) = ξ − .If m = 0, ξ is 1 or -1 and there is no restriction for a i = v · v σ +1+ i . ( i = 1 , · · · , σ − m > m is a divisor of σ and σ/m is an odd integer. The order of p in ( Z /n ) ∗ is2 m and n is a divisor of p m + 1. Becuase ν ( α ) ∈ O ( l ( N S ( X )) ⊗ k ), a i = 0 unless i is amultiple of 2 m .Conversely, assume m is a divisor of σ such that σ/m is an odd integer. Assume n is adivisor of p m +1 and ξ ∈ k ∗ is a primitive n -th root of unity. Suppose a i = 0 unless i is amultiple of 2 m . Let g be a linear operator of l ( N S ( X )) ⊗ k which sends v i to F − ik ( ξ ) v i .( i = 1 , · · · , σ ) It is clear that g preserves the subspace K ( X ). Since a i = 0 for 2 m (cid:45) i , g ∈ O ( l ( N S ( X )) ⊗ k ). g is rational over F p if and only if f − ( g ( w )) = g ( f − ( w )) forall w ∈ l ( N S ( X )) ⊗ k . For that, it is enough to check that f − ( g ( v i )) = g ( f − ( v i )) for i = 1 , · · · , σ . Because f − ( v i ) = v i +1 ( i = 1 , · · · , σ − i = 1 , · · · , σ −
1. We put v (cid:48) = f − ( v σ ) and v (cid:48) = b v + b v + · · · b σ v σ .( b i ∈ k ) By the assumption, v σ · v i = 0 unless i = σ − me for a non-negativeinteger e and v σ · v σ = 1. Since the pairing of l ( N S ( X )) ⊗ k is defined over F p , for any u, w ∈ l ( N S ( X )) ⊗ k , F − k ( u · w ) = f − ( u ) · f − ( w ). Hence v (cid:48) · v i = 0 for i (cid:54) = σ + 1 − me ( i ≥
2) and v (cid:48) · v (cid:48) = 0. It is not difficult to see that b i = 0 unless i = 1 + 2 me and g ( v (cid:48) ) = ξv (cid:48) , so f − ( g ( v σ )) = g ( f − ( v σ )). It follows that g ∈ O ( l ( N S ( X )) and, byLemma 2.5, g ∈ Im ν X . Then the non-symplectic index of X is divisible by p m + 1 andthe proof is complete. Remark 3.2.
Simon Brandhorst proves, in his dissertation, partial result of Theorem3.1. His proof is also based on the crystalline Torelli theorem.Let us say a unique supersingualr K3 surface of Artin invariant σ whose non-symplecticindex is p σ +1 is the special supersingular K3 surface of Artin invariant σ . Every specialsupersingular K3 surface has a model over a finite field. Corollary 3.3.
The maximal value of the non-symplectic index of a K3 surface definedover k is p + 1. The only K3 surface with the maximal non-symplectic index is thespecial supersingular K3 surface of Artin invariant 10. Let X be a complex algebraic K3 surface and N be the non-symplectic index of X .The rank of the transcendental lattice of X , T ( X ) is divisible by φ ( N ). If the rank of T ( X ) is equal to φ ( N ), X is a CM K3 surface and X has a model over a number field.There are many examples satisfying this condition. (See [2]) Assume the rank of T ( X )is equal to φ ( N ) and X is defined over a number field F . We choose α ∈ Aut( X ) suchthat the order of ρ X ( g ) is N . For almost all finite places υ of F , X has a good reduction X υ over υ and g is expandable to X υ . For such a υ , the height (and the Artin invariant)7f X υ is determined by the congruence class of the residue characteristic p υ modulo N . In particular, if m is the smallest positive integer such that p mυ ≡ − N ,then X υ is a supersingular of Artin invariant m . ([8], Theorem 4.7, [6], Theorem 2.3)Moreover, in this case, ν X υ ∈ O ( l ( N S ( X υ ))) has 2 σ distinct eigenvalues, so a i = 0 forall i in (2.1) and X υ is a special supersingular K3 surface.For example, an elliptic K3 surface X : y = x + t x + t has a purely non-symplectic automorphism of order 38, α : ( x, y, t ) (cid:55)→ ( ξ x, ξ y, ξ t ) . Here ξ is a primitive 38th root of unity. If a rational prime p does not divide 38, ( X, α )has a good reduction X p over Z /p . If p ≡ , , , ,
29 or 33 modulo 38 (and p > X p is special supersingular of Artin invariant 9. If p ≡
27 or 31 modulo 38, X p isspecial supersingular of Artin invariant 3. If p ≡
37 modulo 38, X p is supersingular ofArtin invariant 1.Because many complex K3 surfaces, for which the rank of transcendental lattice isequal to the phi value of the non-symplectic index, are define over Q , many specialsupersingular K3 surfaces have models over the prime field F p . We have the followingquestion naturally. Qestion.
Is every special supersingular K3 surface has a model over F p ? Acknowledgment
This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education, Scienceand Technology [2015R1D1A1A01058962].