The Cayley-Oguiso automorphism of positive entropy on a K3 surface

Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit description of the automorphism.

In the second section, using results of Beauville, we give a characterisation of the K3 surfaces considered by Oguiso as determinantal quartics. Given the matrix M 0 (x) whose determinant is a defining polynomial for S 0 , Cayley indicated a method to find the corresponding matrices M 1 (y) and M 2 (z) for S 1 and S 2 which we recall in Section 3. We were not able to show that the determinants of M 1 and M 2 do not vanish identically in general. However, in the explicit example presented in Section 4, his method works and this allows us to give a convenient explicit description of the automorphism g in that case.
After we put the first version of this paper on the arXiv, Igor Dolgachev informed us that the Cayley-Oguiso automorphism was (apparently independently) also discovered by F. Schur ([S], §13, Satz I, p.30). The paper by Snyder and Sharp [SS] presents the automorphism in a way similar to our Section 3. In a series of papers, [R1],. . ., [R4], T.G. Room studies the automorphism, especially in case the surface S also contains a rational curve (in this case NS (S) has rank at least three). It is somewhat remarkable that none of these papers cites Cayley, but all refer to [S] for the automorphism.
1. The general constructions 1.1. The lattice (N, b). To describe the Néron Severi group of the K3 surfaces considered by Oguiso, we introduce a lattice (N, b). One has N ∼ = Z 2 , but to describe b and the isometries of (N, b) it is convenient to define N as the quotient ring Z[X]/(X 2 − X − 1) and let η be the class of X: N := Z[η], η 2 = 1 + η .
The norm of x ∈ N is defined as N orm(x) = xx . We define a bilinear form b = b N : N × N −→ Z, by b(x, y) = 2(x y + xy ) .
Due to the factor 4, we have b(x, y) ∈ 4Z, in particular, there are no x ∈ N with b(x, x) = ±2. The equation b(x, x) = 0 has only x = 0 as solution, since if a 2 + ab − b 2 = 0 with b = 0 then (a/b) 2 + (a/b) − 1 = 0, but this quadratic equation has no solution a/b ∈ Q.
As ηη = −1, the map N −→ N, x = a + bη −→ η 2 x = (a + b) + (a + 2b)η is an isometry of the lattice (N, b) with inverse η −2 = 2 − η. Composing this map with itself n times gives an isometry which we denote simply by η 2n . The isometries of (N, b) are given by the maps x → ±η 2k x and x → ±η 2k x with k ∈ Z. To see this, we use that an isometry is given by a 2 × 2 matrix M ∈ GL(2, Z) on Z 2 such that t M S b M = S b . Equivalently M has integer coefficients, t M S b = S b M −1 and det M = ±1. As x → x is an isometry with determinant −1, we need only consider the case det M = +1. One finds that M must be a matrix with rows (a, b), (b, a + b) and 1 = det M = a 2 + ab − b 2 . Thus the action of M is the multiplication by u = a + bη and uu = 1, so u is a unit in the ring of integers Z[η] of Q( √ 5). This group of units is well-known to be {±η m : m ∈ Z}. Thus uu = 1 implies that x = ±η 2k for some integer k.
With these definitions, Oguiso proved the following theorem, except for a refinement which we prove here.
1.2. Theorem ( [O], Theorem 4.1). There exist K3 surfaces S with NS(S) ∼ = (N, b), these form a dense subset of an 18-dimensional family of K3 surfaces. The automorphism group Aut (S) of any such surface S is isomorphic to Z. Any generator of Aut(S) is a fixed point free automorphism of positive entropy. Moreover, there is a generator g of Aut (S) such that g * = η 6 on NS(S) ⊂ H 2 (S, C) and g * = −1 on the orthogonal complement T (S) of NS (S) in H 2 (S, C).
Proof. In view of Theorem 4.1 in [O], all we need to prove is that Aut(S) ∼ = Z and that one of its two generators acts as stated in the theorem. An automorphism φ of Aut(S) is determined by its action φ * on H 2 (S, Z) ( [BHPV], VIII, Corollary 11.2). The map φ * preserves the sublattices NS(S), T (S) of H 2 (S, Z) and φ * is uniquely determined by its restriction to the sublattice (of finite index) T (S) ⊕ NS(S) of H 2 (S, Z).
Let φ * T be the restriction of φ * to T (S). Then φ * T preserves the Hodge structure on T (S) and thus it preserves the two-dimensional subspace T t := (H 2,0 The intersection form is positive, respectively negative, definite on these spaces. Thus the orthogonal groups O (T t , a discrete group, also lies in a compact group, it lies in a finite set. In particular, φ * T has finite order. If φ * T = 1, a suitable power of it will have a prime order p, let τ be such a power of φ * T . The eigenvalues of τ are thus p-th roots of unity and not all eigenvalues are equal to 1. If τ has eigenvalue 1 on T (S), then either the complexification of the sublattice T (S) τ of τ -invariants or its orthogonal complement would contain H 2,0 (S). Thus either the orthogonal complement of T (S) τ is contained in NS(S) or T (S) τ itself is contained in NS (S). Both cases contradict that T (S) is NS(S) ⊥ .
Thus φ * T has order 2 k for some integer k. In case k ≥ 2, there is thus an integer m such that the restriction of ψ := (φ * ) m to T (S) has order 4. As above, we can rule out that the restriction of ψ to T (S) has eigenvalues ±1, so ψ 2 + 1 = 0 on T (S) and also on the discriminant group. Thus ψ acts as x → ±2x on the subgroup Z/5Z of the discriminant group. The actions of ψ on T (S) and NS (S) are related through the equality ψ T (S) * /T (S) = φ NS(S) * /NS (S) under the natural isomorphism of discriminant groups T (S) * /T (S) ∼ = NS(S) * /NS(S) ∼ = N * /N . Generators of N * /N are given in [O], Proposition 3.3(2), and we checked that the generators x → x , x → −x and x → η 2 x of the isometry group all induce x → −x on the subgroup Z/5Z of the discriminant group. Therefore, S cannot have an automorphism that has order four on T (S).
Thus we must have φ * T = ±1 and, similar to the proof of Theorem 4.1 in [O], an automorphism of S induces either −1 on T (S) and η 12k+6 on NS (S) or +1 on T (S) and η 12k on NS(S) for some integer k. So φ acts as g 2k+1 or g 2k on T (S) ⊕ NS(S), where g is the fixed point free automorphism found by Oguiso. Therefore φ is a power of g.
In particular, the topological Lefschetz number of g is 0, a crucial step in the proof of Theorem 1.2. The topological Lefschetz number of g 2 is 22 + a 13 + a 11 = 344, hence g 2 does have fixed points (cf. [O], Remark 4.3). Similarly, if |n| > 1 then (g n ) * has fixed points, but it has no fixed curves since (g n ) * has no eigenvalue 1 on NS (S). In Section 4 we will present an example where g 2 has exactly 344 fixed points (see Proposition 4.10).
1.5. Ample divisors on S. Let S be a K3 surface with Picard lattice NS(S) ∼ = (N, b) as in section 1.1. We will fix the identification NS(S) ∼ = N in such a way that if D is an ample divisor class, so D 2 > 0, then D = a + bη with a > 0. As there are no elements with b(x, x) = −2 in N , any x = a + bη ∈ N with b(x, x) > 0 and a > 0 is the class of an ample divisor on S ( [BHPV], VIII, Corollary (3.9)): The isometry η 2 of NS(S) maps A onto itself: η 2 A = A. This is easily seen by observing that an isometry of N extends R-linearly to an isometry of N R = R 2 which maps the set Q defined by a 2 + ab − b 2 = 0 into itself. This set consists of two lines and the four connected components of N R − Q are thus permuted by an isometry. The isometry η 2 maps 1 ∈ A to η 2 ∈ A, so it fixes the connected component containing A, hence η 2 A = A.
1.6. Ample divisors on S are very ample. We recall the results of Saint-Donat which imply that any ample divisor on S is already very ample (cf. [O], Remark 4.2). Let L be the line bundle on S defined by an ample divisor class D. As the canonical bundle of S is trivial, Kodaira vanishing implies that h i (L) = dim H i (S, L) = 0 for i > 0. The Riemann-Roch theorem then asserts that h 0 (L) = 2 + D 2 /2. As an ample divisor D on S has D 2 > 0, we get h 0 (D) > 0 and thus we may assume that D is effective. As ∆ 2 = −2 for any divisor ∆ on S, the linear system |D| has no fixed components ( [SD], § 2.7.1, 2.7.2). By [SD], § 4.1, the map φ L defined by the global sections of L is then either of degree two or it is birational onto its image. In the first case, Theorem 5.2 of [SD] implies that S has a divisor ∆ with ∆ 2 ∈ {0, 2}, which is not the case. So φ L is birational onto its image and by Theorem 6.iii of [SD], the image of φ L is isomorphic to the image of the map θ L introduced in [SD], § 4.2. The map θ L is the contraction of all smooth rational curves ∆ in S with D · ∆ = 0. As ∆ 2 = −2 but there are no divisor classes with ∆ 2 = −2 in S, the map θ L is the identity on S and φ L : S → φ L (S) is an isomorphism, i.e. L is very ample.
1.7. Effective divisors and irreducible curves. Let D ∈ NS(S) be the class of an irreducible curve C, then by adjunction D 2 = 2p a (C) − 2 ≥ −2, and thus actually D 2 > 0. As also D · H > 0 for any ample divisor H, we conclude that D = a + bη with a > 0 and therefore any curve in S is an ample divisor. Taking linear combinations with positive coefficients of classes of curves, we conclude that any effective divisor on S is an ample divisor.
1.8. Quartic surfaces. For any integer n we define a very ample divisor class A basis of the global sections of the line bundle on S defined by D n defines a projective embedding, denoted by φ n , of S in P 3 as a quartic surface S n ⊂ P 3 : 1.9. The automorphism g. The remarkable fact that there is an automorphism of S with g * = η 6 implies that the quartic surfaces S 0 and S 3 are the same, after choosing suitable coordinates on the P 3 's. In fact, let s 0 , . . . , s 3 be a basis of H 0 (S, D 0 ). As g * D 0 = D 3 , H 0 (S, D 3 ) has basis t i := g * s i , where i = 0, . . . , 3. With a slight abuse of notation, we then get for all x ∈ S: φ 3 (x) = (t 0 (x) : . . . : t 3 (x)) = (s 0 (g(x)) : . . . : s 3 (g(x))) = φ 0 (g(x)) .

Thus, with these bases
1.10. How to find g. To give a more concrete description of g, we explain how, in principle, one can describe φ 3 in terms of φ 0 . For this we need to find H 0 (S, D 3 ), given the surface S 0 ⊂ P 3 . The zero locus of a global section t of D 3 is mapped to a curve in S 0 . This curve is not the (complete) intersection of S 0 with another surface (of degree d) in P 3 , since such an intersection has class dD 0 = d, whereas D 3 = η 6 = 5 + 8η.
The intersection of such a surface of degree d is thus the sum of two effective divisors with classes D 3 , D respectively and dD 0 = D 3 + D. As effective classes and ample classes coincide on S, the smallest possible degree d is the smallest positive integer such that Let C i := (t i = 0) be the zero divisors of a basis t i , i = 0, . . . , 3, of H 0 (S, D 3 ) and similarly, let the t j be a basis of H 0 (S, D −3 ) with zero divisor C j := (t j = 0). Then the divisor C i + C j has class D 3 + D −3 = 18D 0 and it is the zero locus of the section t i t j in H 0 (S, 18D 0 ).
Consider the exact sequence of sheaves on P 3 : where the first non-trivial map is multiplication by the equation of S 0 and where i : S 0 → P 3 is the inclusion map. As the first cohomology group Therefore, for any d, a section in H 0 (S, dD 0 ) is the restriction of a homogeneous polynomial of degree d on P 3 .
In particular, there are homogeneous polynomials R ij of degree 18 in x 0 , . . . , x 3 , such that Considering the zero loci of these sections we get: The map φ 3 is defined by the global sections t 0 , . . . , t 3 of D 3 . Since D −3 is very ample, for each x ∈ S there is an index j such that t j (x) = 0. So (with slight abuse of notation): On the open subset of S where t j = 0, we thus have: is the rational map given by the polynomials R 0j , . . . , R 3j . Hence on this open subset we get then g is just the rational map R j , for any j, and these maps glue to give an isomorphism S 0 → S 0 , which 'is' g.
To find φ 3 , given φ 0 , we thus need to find these polynomials R ij on P 3 . In practice, even if one is given that NS(S) ∼ = (N, b) this seems quite difficult. However, in Section 3, we construct explicit polynomials of degree 27 which induce the map S 0 → S 0 , corresponding to g, using a general method due to Cayley. Using these polynomials, we were able to find the degree 18 polynomials in the specific example in Section 4, see Section 4.8.

Determinantal quartic surfaces
2.1. Determinantal quartics. We now show that a result of Beauville provides an explicit description of the K3 surfaces we are interested in: the K3 surfaces S with Néron Severi group isomorphic to (N, b) are exactly the quartic determinantal surfaces with Picard number two.
More precisely, the quartic surfaces S n := φ n (S) from Section 1.8 are all determinantal. In Corollary 2.9 we show how the matrix M n which defines S n also provides explicitly the map φ n+1 φ −1 n : S n → S n+1 , once suitable bases of global sections are chosen. This is actually part of the results of Cayley in [C].
2.2. Proposition. Let S be a K3 surface with Néron Severi group NS(S) ∼ = (N, b) as in Section 1.1. Then, for any n ∈ Z, the quartic surface S n := φ n (S) is determinantal. So there is a 4 × 4 matrix M n (x), whose coefficients are linear forms in 4 variables x 0 , . . . , x 3 , such that det M n (x) = 0 is an equation for S n .
Conversely, a general determinantal quartic surface S has NS(S) ∼ = (N, b) and thus it admits a fixed point free automorphism of positive entropy.
Proof. The proposition is an easy consequence of [Be], Proposition 6.2, where Beauville proved that a smooth quartic surface X is determinantal if and only if there is a curve C ⊂ X of degree 6 and genus 3. See also [D], Section 4.2.5.
Given a K3 surface S with NS(S) ∼ = (N, b), there are smooth genus three curves C n on S with class the very ample divisor D n = η 2n , for all n ∈ Z (cf. Section 1.5). As multiplication by η −2n is an isometry of the lattice (N, b), and as one easily computes C 0 · C 1 = 6, we then get that C n · C n+1 = 6. As φ n (C n ) is a plane section of S n , the curve φ n (C n+1 ) is a genus 3 curve of degree 6 in S n . Hence S is determinantal.
For the converse, let S be a general determinantal quartic surface in P 3 . Then S is smooth ( [Be] (1.10)). Let H be the hyperplane class of S, so H 2 = 4. Let C ⊂ S be a degree 6 and genus 3 curve as in [Be] Proposition 6.2. Then H · C = 6 and the adjunction formula implies that C 2 = 4. Thus the intersection form on the sublattice ZH ⊕ ZC of NS(S) is given by the matrix This sublattice is isometric to (N, b) since the Z-basis of N given by D 0 = (1, 0) and D 1 = (1, 1) gives this intersection matrix. Thus NS(S) of a determinantal quartic K3 surface contains (N, b) as a sublattice and therefore the rank of NS(S) is at least two. In the next section we provide an example of a (smooth) determinantal quartic with rank NS(S) = 2, thus the same is true for the general determinantal quartic.
For such a quartic we thus have N ⊂ NS(S), of finite index. As | det(b)| = 20, the index can only be 1 or 2. If the index is two then D := (aH + bC)/2 ∈ NS(S) with (a, b) = (1, 0) or (0, 1) or (1, 1), but D 2 is odd in all these cases, so this is impossible. Hence NS(S) = (N, b) for a general determinantal quartic surface. The existence of a fixed point free automorphism of positive entropy now follows from Oguiso's results in [O].
2.3. Generators of NS (S). The proposition implies in particular that a general smooth determinantal surface S ⊂ P 3 has Néron Severi group of rank two. One would thus like to see a curve on S which is not a complete intersection, that is, whose class is not an integer multiple of the hyperplane class H of S ⊂ P 3 . As explained in [Be] (see also [D], Example 4.2.4), such curves, of genus 3 and degree 6, can be found as follows.
The matrix of linear forms M , whose determinant defines S, also gives a sheaf homomorphism So M defines a line bundle on S with sheaf of sections L. We will denote this line bundle by L as well. As H i (P 3 , O(−1)) = 0 for all i, we obtain an isomorphism In Proposition 2.5 we will show that L has sections whose zero locus has degree 6 and genus 3.
2.4. The cofactor matrix. We recall some well-known linear algebra. For an n × n matrix M = (m ij ), with coefficient m ij in the i-th row and j-th column, let M ij be the (n − 1) × (n − 1) matrix obtained from M by deleting the i-th row and j-th column. The cofactor matrix of M is the n × n matrix Let I be the n × n identity matrix, then we have the following matrix identities: Be], (6.7)) Let S be a smooth quartic surface defined by det M = 0. Let L be the line bundle on S defined in Section 2.3, let s j ∈ H 0 (S, L) be the global section of L which is the image of the j-th basis vector of C 4 and let C j be the zero locus of s j . Then C j ⊂ S is the divisor defined by the vanishing of the four coefficients p ij , i = 1, . . . , 4 of the cofactor matrix P of M . Moreover, the effective divisors C j have degree 6 and genus 3.
Proof. As S is smooth, for any x ∈ S at least one of the partial derivatives (∂/∂x i det M )(x) = 0. Using the expansion of the determinant of a matrix M = (m ij ), whose coefficients are variables m ij , according to the i-th row one finds that ∂/∂m ij det M = (−1) i+j det M ij . As each m ij (x) is a function of x 0 , . . . , x 3 one finds, using the chain rule, that at least one 3 × 3 minor det M ij (x) is non-zero for x ∈ S.
Notice that x ∈ C j if and only if e j (x) ∈ imM (x) where e j is the global section of the trivial bundle O ⊕4 defined by the j-th basis vector. Let P (x) be the cofactor matrix of M (x), then P (x)M (x) = 0, which implies that imM (x) ⊂ ker P (x). As at least one 3 × 3 minor of M (x) is non-zero, we have P (x) = 0 and we conclude that dim ker P (x) = 3 and so imM (x) = ker P (x). Thus e j (x) ∈ imM (x) is equivalent to P (x)e j (x) = 0 which is equivalent to the vanishing of p ij (x) for i = 1, . . . , 4.
The degree and genus of C = C j are given in [Be] Prop. 6.2, [D] Thm. 4.12.14. The genus is actually easy to compute in this case: as L ∼ = O S (C) and all effective divisors on S are ample, Kodaira vanishing and Riemann-Roch on S imply that 4 = dim H 0 (S, O S (C)) = χ(O S (C)) = p a (C) + 1.
2.6. The transposed matrix. Given a determinantal surface S with equation det M = 0, one obviously also has the (same) equation det t M = 0. However, the invertible sheaf L on S defined by the cokernel of t M is not isomorphic to L, but to O S (3) ⊗ L −1 ([Be], (6.3), [D], (4.19)).
2.7. Proposition. Let S be a determinantal surface defined by det M = 0, let L be the line bundle on S defined in Section 2.3 and let s 1 , . . . , s 4 be the basis of H 0 (S, L) defined in Proposition 2.5. The rational map coincides with the rational map given by any of the rows of the cofactor matrix P of M : for any i in {1, . . . , 4}.
Proof. In Proposition 2.5 we showed that the coefficients p ij , i = 1, . . . , 4, of the cofactor matrix P of M define the zero locus C j of the section s j of L. As t P is the cofactor matrix of t M , the coefficients p ij , j = 1, . . . , 4, of the cofactor matrix P similarly define the zero locus C i of a section t i of O S (3) ⊗ L −1 . In particular, the coefficient p ij of P is zero on both C j and C i . More precisely, we have the following identity of divisors on S: In fact, p ij = 0 is a cubic surface in P 3 and thus the left hand side is a divisor with class 3H in NS ( has rank one for x ∈ S, we get p ij (x)p 11 (x) = p i1 (x)p 1j (x) hence p ij = t i s j for any i, j and the proposition follows.
2.8. Explicitly moving from S n to S n+1 . Now we return to the quartic surfaces S n = φ n (S), where S is a K3 surface with NS(S) ∼ = (N, b) as in Section 1. Each S n is a determinantal surface, with equation det M n = 0, by Proposition 2.2. The following corollary identifies the line bundle L (up to replacing M n by t M n ), it is the line bundle defined by the divisor class D n+1 (or D n−1 ). In particular, it basically solves the problem of moving from S n to S n+1 as the following corollary shows.
2.9. Corollary. Let S be a K3 surface with Néron Severi group NS(S) ∼ = (N, b) as in Section 1. Let S n = φ n (S) ⊂ P 3 be the smooth determinantal surface defined by det M n = 0 and let P n = (p n,ij ) be the cofactor matrix of M n . Let L be the line bundle on S n defined by M n as in Section 2.3.
Then the line bundle φ * n L on S has class D n−1 (and φ * n (O(3) ⊗ L −1 ) has class D n+1 ) or it has class D n+1 (and then φ * n (O(3) ⊗ L −1 ) has class D n−1 ). In the first case, there is a basis of the global sections of the line bundle defined by D n+1 on S such that the map φ n+1 φ −1 n : S n −→ S n+1 is given by any of the columns of the cofactor matrix: x −→ (p n,1j (x) : . . . : p n,4j (x)) (j = 1, . . . , 4).
Proof. In the proof of Proposition 2.7 we found that the divisor defined by p n,ij = 0 on S n is the sum of two effective divisors φ(C n,j ), φ n (C n,i ), where C n,j is the common zero locus of the φ * n p n,kj and C n,i is the common zero locus of the φ * n p n,ik for k = 1, . . . , 4. These divisors both have genus 3, so (C n,j ) 2 = (C n,i ) 2 = 4 and C n,j + C n,i = 3D n . As these divisors are effective, they are also ample (see Section 1.7), so their classes are in A.
Now we use that D n = η 2n and that multiplication by η −2n is an isometry of N which maps A into itself. So we need to find D := a + bη, D := a + b η ∈ A with sum η −2n (3D n ) = 3D 0 . Hence a + a = 3, b + b = 0 and a, a > 0. Thus we may assume that a = 1, a = 2. As D 2 l = 4, we then have b = 0, hence also b = −b = 0, and as 0 < a 2 + ab − b 2 = 1 + b − b 2 , we get b = 1. Hence D = D 1 , D = D −1 , so (up to permutation) the class of C n,j is η 2n D = D n−1 and the class of C n,i is η 2n D = D n+1 .
As C n,j is the zero locus of a section of φ * L, the class of this line bundle is D n−1 . Therefore Proposition 2.7, applied to t M n , shows that the columns of the cofactor matrix of M n give the map defined by the global sections of the line bundle defined by D n+1 .
3. Cayley's description of the automorphism 3.1. As observed by Cayley in [C], given a quartic determinantal surface, it is easy to find two others and to find isomorphisms between them. In our setup, starting from the projective model S n of S, he produces S n+1 and S n−1 . The maps are those from Corollary 2.9. Moreover, he shows that from the matrix M n , whose determinant is the defining equation for S n , one can find the matrices M n−1 , M n+1 which define S n−1 , S n+1 respectively. The composition of the isomorphisms S 0 → S 1 → S 2 → S 3 = S 0 is basically the automorphism g we wanted to describe.

Write this matrix as
with coefficients a ijk ∈ C. The 4 3 = 64 complex numbers a ijk can be viewed as the components of a 'tritensor' in (C 4  The surprising thing is that the determinants of the M i define the quartic surfaces S i = φ i (S) respectively, provided these determinants are not identically zero, see below. Thus the tritensor, or equivalently, any one of the matrices M i , determines all the others. The maps between the S i are given by the rows and columns of the cofactor matrices of these matrices and they are thus defined by the tritensor as well.

From
where I is the 4 × 4 identity matrix. After replacing M 0 and P 0 by their transposes if necessary, and after choosing a suitable basis of the global sections of the line bundle defined by D 1 , we may assume (see Corollary 2.9) that the map φ 1 φ −1 0 : S 0 → S 1 is given by the columns of the adjoint matrix P 0 (x).
On the other hand, for x ∈ S 0 , we have det M 0 (x) = 0 and thus each column of P 0 (x), provided it is not identically zero, provides a non-trivial solution to the linear equations M 0 (x)y = 0. As S 0 is smooth, the rank of M 0 (x) is equal to three for any x ∈ S 0 and thus y = y(x) is unique up to scalar multiple.
The system of linear equations M 0 (x)y = 0 can be rewritten as: This set of four equations is equivalent to the matrix equation, For y ∈ P 3 these equations have a non-trivial solution x = x(y) if and only if det M 1 (y) = 0. In case det M 1 (y) is not identically zero, it is a quartic polynomial that vanishes on the quartic surface S 1 , and thus it is a defining equation for S 1 . We will assume that det M 1 (y) is not identically zero. Then S 1 : det M 1 (y) = 0 (⊂ P 3 ) .
3.4. From S 1 to S 2 and back to S 0 . We can repeat the procedure from section 3.3: let P 1 (y) be the cofactor matrix of M 1 (y). As S 0 consists of the points x with t xM 1 (y) = 0 and as P 1 (y)M 1 (y) = 0 for y ∈ S 1 , each row of P 1 defines the map S 1 → S 0 . Thus the columns of P 1 define the map φ 2 φ −1 1 : S 1 → S 2 = φ 2 (S) for a suitable basis of the global sections of the line bundle defined by D 2 . For y ∈ S 1 , each column of P 1 (y) is then both a solution z of P 1 (y)z = 0 and it defines a point of S 2 . Hence we have Rewriting the linear equations M 1 (y)z = 0, we get t yM 2 (z) = 0 with Now, assuming moreover that det M 2 (z) is not identically zero, we get Finally we consider the cofactor matrix P 2 (z) of M 2 (z). The columns of P 2 (z) provide us with the map φ 3 φ −1 2 : S 2 → S 3 = φ 3 (S), for a suitable basis of the global sections of the line bundle defined by D 3 . Each column is also a solution of M 2 (z)x = 0. Rewriting this system, we get t zM 0 (x) = 0, showing that S 3 is defined by det M 0 (x) = 0 since this determinant is not identically zero, being the defining equation of S 0 . Thus S 0 = S 3 (!) and the composition of the maps, each given by cubic polynomials (the minors of the matrices M i ) where we used that g = φ −1 0 φ 3 (see Section 1.9) and g is the automorphism constructed by Oguiso. To quote Cayley: "The process may be indefinitely repeated".

An explicit example
4.1. The method. In this section we give an explicit example of a determinantal K3 surface with Picard number two. The main problem is to give an upper bound for the Picard number. For this we use a method described in [vL2]. For the definition of theétale cohomology groups H í et (X, Q ) and H í et (X, Q (1)) for a scheme X, with values in Q or its Tate twist Q (1), we refer to [T] and [Mi], p. 163-165.
The following result shows that if a smooth projective surface X over a number field K has good reduction at a prime p, then the geometric Picard number of X is bounded from above by the geometric Picard number of the reduction.

Proposition.
Let K be a number field with ring of integers O, let p be a prime of O with residue field k, and let O p be the localization of O at p. Let X be a smooth projective surface over O p and set X K = X× Op K and X k = X× Op k. Let be a prime not dividing q = #k. Let F * q denote the automorphism of H 2 et (X k , Q (1)) induced by the q-th power Frobenius F q ∈ Gal(k/k). Then there are natural injections , that respect the intersection pairing and the action of Frobenius, respectively. The rank of NS(X k ) is at most the number of eigenvalues of F * q that are roots of unity, counted with multiplicity.
Proof. The Néron-Severi group modulo torsion is isomorphic to the group of divisor classes modulo numerical equivalence (see [F], 19.3.1.(ii)). Therefore, the first injection, as well as the fact that it respects the intersection pairing, follows from [F], Examples 20.3.5 and 20.3.6. The second injection is in [Mi],Remark V.3.29.(d). Each class c ∈ NS(X k ) is represented by a divisor, which is defined over some finite field; hence, some power of Frobenius fixes c. Since the Néron-Severi group NS(X k ) is finitely generated (see [F], 19.3.1.(iii)), it follows that some power of Frobenius acts as the identity on NS(X k ). This implies the last statement. See also Proposition 6.2 and Corollary 6.4 in [vL1] (which counts the eigenvalues that are a root of unity up to a factor q because it refers to the action on H 2 et (X k , Q ) without the Tate twist). 4.3. Proposition. Let X be a K3 surface over a finite field k ∼ = F q . As in Proposition 4.2, let F * q denote the automorphism of H 2 et (X k , Q (1)) induced by the q-th power Frobenius F q ∈ Gal(k/k), and for any n, let Tr((F * q ) n ) denote the trace of (F * q ) n . Then we have Furthermore, the characteristic polynomial f (t) = det(t − F * q ) ∈ Q[t] of F * q has degree 22 and satisfies the functional equation Proof. Let F X be the q-th power absolute Frobenius of X, which acts as the identity on points and by raising to the q-th power on the coordinate rings of affine open subsets of X. The geometric Frobenius ϕ = F X ×1 on X × k k = X k is an endomorphism of X k over k (cf. [Mi], proof of V.2.6 and pages 290-291). The set of fixed points of ϕ n is X(F q n ). The Weil conjectures (see [Mi], §VI.12) state that the eigenvalues of ϕ * acting on H í et (X k , Q ) have absolute value q i/2 . Since X is a K3 surface, we have dim H í et (X k , Q ) = 1, 0, 22, 0, 1 for i = 0, 1, 2, 3, 4, respectively (see [Ba], 8.4 and Theorem 10.3), so the Lefschetz trace formula for ϕ n (see [Mi], Theorems VI.12.3 and VI.12.4) yields For the remainder of this proof we restrict our attention to the middle cohomology, so H í et with i = 2. By the Weil conjectures, the characteristic polynomial f ϕ (t) = det(t − ϕ * |H 2 et (X k , Q )) is a polynomial in Z[t] satisfying the functional equation t 22 f ϕ (q 2 /t) = ±q 22 f ϕ (t) (note that the polynomial P 2 (X, t) = det(1 − ϕ * t|H 2 et (X k , Q )) of [Mi], §VI.12, is the reverse of f ϕ ). Let ϕ * (1) denote the action on H 2 et (X k , Q (1)) (with a Tate twist) induced by ϕ. Note that the fact that ϕ * (1) acts on the middle cohomology is not reflected in the notation. The eigenvalues of ϕ * (1) differ from those of ϕ * on H 2 et (X k , Q ) by a factor q (see [T]), so we have (2) Tr (ϕ * ) n |H 2 et (X k , Q ) = q · Tr ϕ * (1) n , and the characteristic polynomial f   ϕ (t). It follows that the eigenvalues, and hence the characteristic polynomials, of ϕ * (1) and ϕ * (1) −1 coincide. Finally, the product of the geometric Frobenius ϕ = F X × 1 and the Galois automorphism 1 × F q on X × k k = X k is the absolute Frobenius F X k , which acts as the identity on the cohomology groups, so the maps ϕ * (1) and F * q act as inverses of each other (see [Mi], Lemma VI.13.2 and Remark VI.13.5, and [T], §3).

An explicit determinant.
Consider the following matrix, whose entries are linear forms in the variables x 0 , . . . , x 3 with integer coefficients: The following theorem shows that any such matrix that is congruent to M 0 modulo 2 defines a determinantal quartic surface in P 3 = P 3 (C) with Picard number 2. 4.5. Theorem. Let R = Z[x 0 , x 1 , x 2 , x 3 ] and let M ∈ M 4 (R) be any matrix whose entries are homogeneous polynomials of degree 1 and such that M is congruent modulo 2 to the matrix M 0 above. Denote by S the complex surface in P 3 given by det M = 0. Then S is a K3 surface and its Picard number equals 2.
Proof. Let S denote the surface over the localization Z (2) of Z at the prime 2 given by det M = 0, and write S and S for S F 2 and S F 2 , respectively. One checks that S is smooth and S is reduced, for instance with Magma [Ma]. Since Spec Z (2) is integral and regular of dimension 1, the scheme S is integral, and the map S → Spec Z (2) is dominant, it follows from [H], Proposition III.9.7, that S is flat over Spec Z (2) . Since the fiber over the closed point is smooth, it follows from [L], Definition 4.3.35, that S is smooth over Spec Z (2) . Therefore, S = S C is smooth as well, so S and S are K3 surfaces. Let F * 2 denote the automorphism of H 2 et (S , Q (1)) induced by Frobenius F 2 ∈ Gal(F 2 /F 2 ). The divisor classes in H 2 et (S , Q (1)) defined by the hyperplane class and the curve C as in Proposition 2.5 span a two-dimensional subspace V on which F * 2 acts as the identity. We denote the linear map induced by F * 2 on the quotient W := H 2 et (S 2 , Q (1))/V by F * 2 | W , so that Tr(F * 2 ) n = Tr(F * 2 | V ) n + Tr(F * 2 | W ) n = 2 + Tr(F * 2 | W ) n for every integer n. From Proposition 4.3, we obtain We counted the number of points in S (F 2 n ) for n = 1, . . . , 10 with Magma [Ma]. The results are in the table below. If λ 1 , . . . , λ 20 denote the eigenvalues of F * 2 | W , then the trace of (F * 2 | W ) n equals Tr(F * 2 | W ) n = λ n 1 + . . . + λ n 20 , i.e., it is the n-th power sum symmetric polynomial in the eigenvalues of F * 2 | W . Let e n denote the elementary symmetric polynomial of degree n in the eigenvalues of F * 2 | W for n ≥ 0. Using Newton's identities and e 0 = 1, we compute the values of e n for n = 1, . . . , 10. They are listed in the following table.
With Magma [Ma], one checks that this polynomial is irreducible over Q, and as it is not integral, its roots are not algebraic integers, so none of its roots is a root of unity. Hence, the polynomial f F * 2 = (t − 1) 2 f F * 2 | W has exactly two roots that are a root of unity. This implies that F * 2 has only two eigenvalues (counted with multiplicity) that are roots of unity, and so, by Proposition 4.2, it follows that the rank of the Picard group NS(S) ∼ = NS(SQ) is bounded by two from above. On the other hand, by Proposition 2.2 we know that the rank is at least two, hence NS(S) has rank two. 4.6. Figure.  y 2 + y 3 y 0 + y 1 + y 2 y 0 + y 1 y 2 y 0 + y 2 y 1 + y 2 y 0 y 1 + y 2 + y 3 y 1 + y 2 y 0 + y 1 + y 3 y 1 + y 3 y 3 y 1 + y 2 + y 3 y 1 y 0 + y 2 Figure 1. The surface given by det M 0 = 0 with points in three g-orbits.
matrix P 0 (x) is (g 0 , g 1 , g 2 , g 3 ) t with Hence, the map S 0 → S 1 is given, at least on an open subset of S 0 , by sending [x 0 : x 1 : x 2 : x 3 ] to [g 0 (x) : g 1 (x) : g 2 (x) : g 3 (x)]. However, the polynomials g 0 , g 1 , g 2 , g 4 all vanish on a curve C as described in Corollary 2.9. In order to define the map S 0 → S 1 everywhere, we use the other three columns of the cofactor matrix P 0 (x). Similarly, the columns of the cofactor matrices P 1 (y) and P 2 (z) of M 1 (y) and M 2 (z) determine the maps S 1 → S 2 and S 2 → S 0 , respectively.
Cayley's method therefore gives the automorphism g explicitly as the composition S 0 → S 1 → S 2 → S 0 of three maps, each given by cubic polynomials. We can thus describe g explicitly by quadruples of coordinate functions, each of which homogeneous of degree 3 3 = 27 in x 0 , x 1 , x 2 , x 3 . Unfortunately, these quadruples are far too large to write down. 4.8. Defining polynomials of lower degree. We showed in section 1.10 that the automorphism g can in fact be given by polynomials R ij of degree 18. We now describe how we used linear algebra to find such polynomials explicitly. Unfortunately, these polynomials are still far too large to publish, but they are electronically available from the authors upon request.
We are looking for quadruples (G 0 x 3 ] 4 of homogeneous polynomials of degree 18, such that the rational map P 3 P 3 given by is equivalent with the six equalities y j G i (x) = y i G j (x) for 0 ≤ i < j ≤ 3, which are linear in the unknown coefficients of G 0 , G 1 , G 2 , G 3 . In fact, if x is defined over a field extension K of Q of degree d, then we can write the equation y j G i (x) = y i G j (x) in terms of a basis for K over Q; since we know that the G i can be chosen over Q, we can split up the equation in d independent equations, each linear in the unknown coefficients. After choosing sufficiently many points over various number fields, we obtain a large system of equations over Q that we solved with Magma [Ma]. The solution space V has dimension 2724 inside the space of quadruples (G 0 of homogeneous polynomials of degree 18. This space contains the space U of quadruples that vanish on S, which has dimension 4 · 17 3 = 2720. To verify that we used enough points, we took four quadruples (G s1 , G s2 , G s3 , G s4 ), for s = 1, 2, 3, 4, that generate the quotient V /U and checked that they indeed define the same map as g on some open subset of S; this can be done by taking, as in the previous section, a quadruple (F 0 , F 1 , F 2 , F 3 ) of homogeneous polynomials of degree 27 describing g, and checking that for each 1 ≤ s ≤ 4, and each 0 ≤ i < j ≤ 3, the polynomial F i G sj − F j G si is divisible by the defining polynomial det M 0 (x) of S = S 0 .
We also verified with Magma [Ma] that one can in fact choose three quadruples such that the 3 · 4 polynomials in them have no common base points on S, i.e., at each point on S, the automorphism g is defined by at least one of these three quadruples. This computation was done over the rational numbers and therefore holds over any field of characteristic 0.
4.9. Points of period two. As mentioned in Section 1.4, the topological Lefschetz number of g 2 is 344. Using the previous section, we can explicitly define a scheme Ξ over Z such that Ξ Q is 0-dimensional and consists of points of period 2 as follows. First we choose three quadruples (G s1 , G s2 , G s3 , G s4 ) ∈ Z[x 0 , x 1 , x 2 , x 3 ] 4 , for s = 1, 2, 3, of homogeneous polynomials of degree 18 that together describe g everywhere on S. Similarly, we compute three quadruples (H s1 , H s2 , H s3 , H s4 ) ∈ Z[x 0 , x 1 , x 2 , x 3 ] 4 , for s = 1, 2, 3, of homogeneous polynomials of degree 18 that together describe the inverse g −1 everywhere on S; as described in Section 3.4, this is done by using the rows of the cofactor matrices rather than the columns. A point x ∈ S has period 2 if and only if g(x) = g −1 (x), so if and only if H si (x)G tj (x) = H sj (x)G ti (x) for all 1 ≤ s, t ≤ 3 and all 0 ≤ i < j ≤ 3. Hence, these 54 equations of degree 36, together with the defining polynomial det M 0 for S, define a scheme Ξ over Z such that the scheme Ξ Q consists of all points of period 2.
The following proposition states that in our explicit example, none of these points is defined over a number field of degree less than 344 over Q. The proof is based on reduction modulo primes of good reduction, as we were unable to perform any significant computations with Ξ in Magma [Ma] over the rational numbers. All computations are available from the authors upon request.
4.10. Proposition. Let Π ⊂ S(Q) denote the set of all points of period 2 under the automorphism g. Then #Π = 344 and the Galois group Gal(Q/Q) acts transitively on Π.
Proof. Let K be a finite Galois extension of Q with Π ⊂ S(K) and let O K be the ring of integers of K. Note that Π consists of the K-points, or equivalently, the O K -points, of the 0-dimensional scheme Ξ constructed above. Take p ∈ {17, 101}, and let p ⊂ O K be a prime ideal above prime p. Set k(p) = O K /p and letF p be an algebraic closure of k(p). One checks with Magma [Ma] that the surface S Z ⊂ P 3 Z given by det M 0 = 0 has good reduction modulo p, i.e., the reduction S p = S Z × F p is smooth, and the same holds for the surfaces in P 3 Z given by det M i = 0 for i = 1, 2. We claim that the composition Π = Ξ(K) = Ξ(O K ) → Ξ(k(p)) → Ξ(F p ) of the reduction map and the inclusion is surjective. Indeed, one can verify with Magma [Ma] that Ξ Fp has dimension 0, degree 344, and is reduced. Since F p is perfect, it follows that ΞF p is reduced (see [G], Proposition 4.6.1). Let Ξ 0 be an irreducible component of ΞF p . Then Ξ 0 is 0-dimensional, so it is affine and its coordinate coordinate ring A(Ξ 0 ) is 0-dimensional and Noetherian, and therefore Artin (see [AM], Theorem 8.5). Hence, A(Ξ 0 ) is the product of local Artin rings ( [AM], Theorem 8.7), and since it also integral, it is local itself. From [AM], Proposition 8.6, and the fact that A(Ξ 0 ) is integral, we conclude that A(Ξ 0 ) is a field, which, being a finite extension ofF p , is isomorphic toF p . Thus, Ξ 0 is a smooth point, and Ξ Fp is smooth over F p . Hence, it follows from Hensel's Lemma that every point of Ξ Fp over some finite extension of F p lifts to some finite extension of Z p and, since every point on a 0-dimensional scheme is algebraic, to some finite extension of Q. As the topological Lefschetz number of g 2 equals 344, we find that g has at most 344 points of period 2, so #Π ≤ 344. From the claim and the equality #Ξ(F p ) = deg Ξ Fp = 344 we conclude that #Π = 344 and the reduction map r : Ξ(K) → Ξ(k(p)) = Ξ(F p ) is a bijection.
The bijection r respects the Galois action of the decomposition subgroup G p ⊂ Gal(K/Q) associated to p. Each Galois orbit C of Ξ(K) under Gal(K/Q) splits as the disjoint union of orbits under G p ; since G p naturally surjects onto Gal(k(p)/F p ), the image r(C) ⊂ Ξ(k(p)) splits as the disjoint union of orbits of Ξ(k(p)) under Gal(k(p)/F p ), or equivalently, of orbits of Ξ (F p ) under Gal (F p /F p ). This implies that the sizes of the Galois orbits of Ξ(F p ) form a partition of 344 that is a refinement of the partition corresponding to the sizes of the Galois orbits of Ξ(K) under Gal(K/Q). More precisely, if m 1 , . . . , m s are the sizes of the Galois orbits of Ξ(K) under Gal(K/Q), and n 1 , . . . , n t are the sizes of the Galois orbits of Ξ(F p ) under Gal(F p /F p ), then m 1 + · · · + m s = 344 = n 1 + · · · + n t , and there is a partition (I 1 , . . . , I s ) of the set {1, 2, . . . , t} such that m j = i∈I j n i for all 1 ≤ j ≤ s.
We computed the number of F p t -points on Ξ Fp for p = 17 and p = 101 and 1 ≤ t ≤ 150 with Magma [Ma]. For p = 17 we found that there are 4 points on Ξ Fp over F p , as well as 2 more points over F p 2 and 12 more points over F p 12 , which are not defined over a smaller field, 110 more points over F p 55 , and no other points over any field F p t with t ≤ 150. It follows that Ξ (F 17 ) has Galois orbits of sizes 1, 1, 1, 1, 2, 12, 55, 55, and as none of the remaining 216 points is defined over a field of degree less than 150 over F p , one orbit of size 216. For p = 101 we found that there are 20 points on Ξ Fp over F p 20 , which are not defined over a smaller field, 26 more points over F p 26 , and no other points over any field F p t with t ≤ 150. It follows that Ξ(F 101 ) has Galois orbits of size 20, 26, and as none of the remaining 298 points is defined over a field of degree less than 150 over F p , one orbit of size 298. The only partition of 344 of which both the partitions {1, 1, 1, 1, 2, 12, 55, 55, 216} and {20, 26, 298} are a refinement is the trivial partition {344} of one part, so we find that Ξ(K) is one orbit under Gal(K/Q), which proves the proposition.