Transcendental lattices and supersingular reduction lattices of a - PDF document

Transcendental lattices and supersingular reduction lattices of a singular K3 surface Keio, 2007 September Ichiro Shimada (Hokkaido University, Sapporo, JAPAN) • By a lattice, we mean a finitely generated free Z -module Λ equipped with a non-degenerate symmetric bilinear form Λ × Λ → Z . • A lattice Λ is said to be even if ( v, v ) ∈ 2 Z for any v ∈ Λ. • Let Λ and Λ ′ be lattices. A homomorphism Λ → Λ ′ of Z - modules is called an isometry if it preserves the symmetric bilinear forms. By definition, an isometry is injective. → Λ ′ be an isometry. We denote by • Let Λ ֒ → Λ ′ ) ⊥ (Λ ֒ the orthogonal complement of Λ in Λ ′ . 1

§ 1. (Super)singular K 3 surfaces For a K 3 surface X defined over a field k , we denote by NS( X ) the N´ eron-Severi lattice of X ⊗ ¯ k, where ¯ k is the algebraic closure of k ; that is, NS( X ) is the lattice of numerical equivalence classes of divisors on X ⊗ ¯ k with the intersection pairing. Definition. A K 3 surface X defined over a field of characteristic 0 is said to be singular if rank(NS( X )) = 20 . A K 3 surface X defined over a field of characteristic p > 0 is said to be supersingular if rank(NS( X )) = 22 . If X is singular or supersingular, then d ( X ) := disc(NS( X )) . is a negative integer. 2

Shioda and Inose showed that every singular K 3 surface is de- fined over a number field. Let X be a singular K 3 surface defined over a number field F . We denote by Z F the integer ring of F , and by π F : Spec Z F → Spec Z the natural projection. We also denote by Emb( F, C ) the set of embeddings of F into C . We consider a smooth family X → U of K 3 surfaces over a non-empty Zariski open subset U of Spec Z F such that the generic fiber X η is isomorphic to X . For a close point p of U , we denote by X p the reduction of X at p . For a prime integer p , we put S p ( X ) := { p ∈ π − 1 F ( p ) ∩ U | X p is supersingular } . 3

We investigate the following lattices of rank 2; • the transcendental lattice T ( X σ ) := (NS( X ) ֒ → H 2 ( X σ , Z )) ⊥ for each σ ∈ Emb( F, C ), where X σ is the complex K 3 surface X ⊗ F,σ C , and • the supersingular reduction lattice → NS( X p )) ⊥ L ( X , p ) := (NS( X ) ֒ for each p ∈ S p ( X ), where NS( X ) ֒ → NS( X p ) is the spe- cialization isometry. Remark. The supersingular reduction lattices and their relation with transcendental lattices was first considered in the paper T. Shioda: The elliptic K 3 surfaces with a maximal singular fibre. C. R. Math. Acad. Sci. Paris 337 (2003), 461–466, for certain elliptic K 3 surfaces. 4

§ 2. Genera of lattices Definition. Two lattices λ ′ : Λ ′ × Λ ′ → Z λ : Λ × Λ → Z and are said to be in the same genus if λ ⊗ Z p : Λ ⊗ Z p × Λ ⊗ Z p → Z p and λ ′ ⊗ Z p : Λ ′ ⊗ Z p × Λ ′ ⊗ Z p → Z p are isomorphic for any p including p = ∞ , where Z ∞ = R . We have the following: Theorem (Nikulin). Two even lattices of the same rank are in the same genus if and only if they have the same signature and their discriminant forms are isomorphic. 5

Definition. Let Λ be an even lattice. Then Λ is canonically embedded into Λ ∨ := Hom(Λ , Z ) as a subgroup of finite index, and we have a natural symmetric bilinear form Λ ∨ × Λ ∨ → Q that extends the symmetric bilinear form on Λ. The finite abelian group D Λ := Λ ∨ / Λ , together with the natural quadratic form q Λ : D Λ → Q / 2 Z is called the discriminant form of Λ. 6

§ 3. Transcendental lattices Let X be a singular K 3 surface defined over a number field F . For an embedding σ : F ֒ → C , the transcendental lattice T ( X σ ) := (NS( X ) ֒ → H 2 ( X σ , Z )) ⊥ of the complex singular K 3 surface X σ := X ⊗ F,σ C is an even positive-definite lattice of rank 2. Proposition. For σ, σ ′ ∈ Emb( F, C ), the lattices T ( X σ ) and T ( X σ ′ ) are in the same genus. This follows from Nikulin’s theorem. We have NS( X ) ∼ = NS( X σ ) ∼ = NS( X σ ′ ) . Since H 2 ( X σ , Z ) is unimodular, the discriminant form of T ( X σ ) is isomorphic to ( − 1) times the discriminant form of NS( X σ ): ( D T ( X σ ) , q T ( X σ ) ) ∼ = ( D NS( X σ ) , − q NS( X σ ) ) . The same holds for T ( X σ ′ ). Hence T ( X σ ) and T ( X σ ′ ) have the isomorphic discriminant forms. 7

For a negative integer d , we put � � � � � � 2 a b a, b, c ∈ Z , a > 0 , c > 0 , � M d := , � b 2 − 4 ac = d b 2 c on which GL 2 ( Z ) acts by M �→ t gMg , where M ∈ M d and g ∈ GL 2 ( Z ). We then denote by � L d := M d / GL 2 ( Z ) L d := M d / SL 2 ( Z ) ) (resp. the set of isomorphism classes of even, positive-definite lattices (resp. oriented lattices) of rank 2 with discriminant − d . Let S be a complex singular K 3 surface. By the Hodge decom- position T ( S ) ⊗ C = H 2 , 0 ( S ) ⊕ H 0 , 2 ( S ) , we can define a canonical orientation on T ( S ). We denote by � T ( S ) the oriented transcendental lattice of S , and by [ � T ( S )] ∈ � L d ( S ) the isomorphism class of the oriented transcendental lattice. 8

Theorem (Shioda and Inose). The map S �→ [ � T ( S )] induces a bijection from the set of iso- morphism classes of complex singular K 3 surfaces to the set � � L d d of isomorphism classes of even, positive-definite oriented lat- tices of rank 2. We have proved the following existence theorem: Theorem (S.- and Sch¨ utt). Let G ⊂ L d be a genus of even positive-definite lattices of rank 2, and let G ⊂ � � L d be the pull-back of G by the natural projection � L d → L d . Then there exists a singular K 3 surface X defined over a number field F such that the set { [ � � T ( X σ )] | σ ∈ Emb( F, C ) } ⊂ L d coincides with the oriented genus � G . 9

Corollary. Let S and S ′ be complex singular K 3 surfaces. If T ( S ) and T ( S ′ ) are in the same genus, then there exists an embedding → C of the field C into itself such that S × C ,σ C is σ : C ֒ isomorphic to S ′ as a complex surface. � � G S ⊂ L d ( S ) be the oriented genus containing Proof. Let [ � T ( S )] ∈ � L d ( S ) , and let X be the singular K 3 surface defined over a number field F such that { [ � � T ( X σ )] | σ ∈ Emb( F, C ) } = G S . By the theorem of Shioda-Inose, there exists τ ∈ Emb( F, C ) and τ ′ ∈ Emb( F, C ) such that X τ ′ ∼ X τ ∼ = S ′ . = S, → C such that σ ◦ τ = τ ′ . There exists σ : C ֒ � Corollary. Let S and S ′ be complex singular K 3 surfaces. If NS( S ) and NS( S ′ ) are in the same genus, then NS( S ) and NS( S ′ ) are iso- morphic. 10

Corollary. Let S be a complex singular K 3 surface. If S is defined over a number field L , then [ L : Q ] ≥ | � G S | , where � G S ⊂ � L d ( S ) is the oriented genus containing [ � T ( S )]. Proof. Let X be the singular K 3 surface defined over a number field F such that { [ � T ( X σ )] | σ ∈ Emb( F, C ) } = � G S . Then X σ 0 ∼ = S for some σ 0 ∈ Emb( F, C ) . Let Y be a K 3 surface defined over L such that Y τ 0 ∼ = S for some τ 0 ∈ Emb( L, C ) . Then there exists a number field M ⊂ C containing both of σ 0 ( F ) and τ 0 ( L ) such that X ⊗ M ∼ = Y ⊗ M over M . Therefore, for each σ ∈ Emb( F, C ), there exists τ ∈ Emb( L, C ) such that X σ ∼ = Y τ over C . Since there exist exactly | � G S | isomorphism classes of complex K 3 surfaces among X σ , we have | Emb( L, C ) | ≥ | � G S | . � 11

§ 4. The set S p ( X ) We fix a smooth family X → U of K 3 surfaces over an open subset U ⊂ Spec Z F such that the generic fiber X η is singular, and investigate the set S p ( X ) := { p ∈ π − 1 F ( p ) ∩ U | X p is supersingular } ) . For a supersingular K 3 surface Y in characteristic p , there is a positive integer σ ( Y ) ≤ 10, which is called the Artin invariant of Y , such that d ( Y )(:= disc(NS( Y ))) = − p 2 σ ( Y ) . Theorem. Suppose that p does not divide 2 d ( X η ) = 2 disc(NS( X η )). Let χ p : F × p → {± 1 } be the Legendre character. (1) If p ∈ S p ( X ), then the Artin invariant of X p is 1. (2) There exists a finite set N of prime integers containing the prime divisors of 2 d ( X η ) such that � ∅ if χ p ( d ( X η )) = 1, p / ∈ N ⇒ S p ( X ) = π − 1 F ( p ) if χ p ( d ( X η )) = − 1. 12

§ 5. Supersingular reduction lattices For simplicity, we assume that p � = 2. Theorem (Rudakov-Shafarevich). Let p be an odd prime, and let σ be a positive integer ≤ 10. Then there exists a lattice Λ p,σ with the following properties, and it is unique up to isomorphism: (i) even, rank 22, (ii) of signature (1 , 21), and (iii) the discriminant group is isomorphic to ( Z /p Z ) 2 σ . We call Λ p,σ the Rudakov-Shafarevich lattice . Theorem (Artin-Rudakov-Shafarevich). Let X be a supersingular K 3 surface in odd characteristic p with the Artin invariant σ . Then NS( X ) is isomorphic to Λ p,σ . 13