REMARKS ON K3 SURFACES WITH NON-SYMPLECTIC AUTOMORPHISMS OF ORDER 7

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1 REMARKS ON K3 SURFACES WTH NON-SYMPLECTC AUTOMORPHSMS OF ORDER 7 SHNGO TAK Abstract. n this note, we treat a pair of a K3 surface and a non-symplectic automorphism of order 7m (m = 1, 3 and 6) on it. We show that if the fixed locus of a non-symplectic automorphism order 7 is special then the pair is unique up to isomorphism. And we describe fixed loci of non-symplectic automorphisms of order 21 and 42. Contents 1. ntroduction 1 2. Preliminaries 3 3. Uniqueness of K3 surfaces with a certain fixed locus 4 4. The fixed locus of a non-symplectic automorphism of order The fixed locus of a non-symplectic automorphism of order 42 7 References 9 1. ntroduction Let X be an algebraic K3 surface. n the following, we denote by S X, T X and ω X the Néron-Severi lattice, the transcendental lattice and a nowhere vanishing holomorphic 2-form on X, respectively. Let σ be an automorphism on X of finite order. t is called non-symplectic if and only if it satisfies σ ω X = ζ ω X where ζ is a primitive -th root of unity. Non-symplectic automorphisms have been studied by Nikulin who is a pioneer and several mathematicians. t is known that the dimension of a moduli space of K3 surfaces with a nonsymplectic automorphism of order is rk T X /Φ() 1 if 2 or rk T X 2 if = 2 [5, Section 11], where Φ is the Euler function. Then there exists some cases such that the dimension of a moduli space of K3 surfaces with a non-symplectic automorphism is zero. Problem 1.1. Let X be a K3 surface and σ a non-symplectic automorphism of order on X. When is a pair (X, σ ) unique up to isomorphism? Vorontsov [16] announced some answers (without proofs) for the problem. Finally these were proved by Kondo, Oguiso and Zhang. Date: December 29, Mathematics Subject Classification. Primary 14J28, 14J50; Secondary 14J10. Key words and phrases. K3 surface, non-symplectic automorphism, moduli space. 1

2 2 S. TAK Theorem 1.2. [7, Theorem] Assume that T X is unimodular and σ acts trivially on S X. f =66, 44, 42, 36, 28 or 12 and Φ() = rk T X then there exists a unique (up to isomorphism) K3 surface with σ. Here a lattice L is called unimodular if and only if L = Hom(L, Z), i.e. L is isomorphic to its dual lattice. f the transcendental lattice is not unimodular then the following theorem is important. Theorem 1.3. [12, 2, 4] Assume that T X is not unimodular and σ acts trivially on S X and Φ() = rk T X. f = 3, 5, 7, 11, 13, 19, 5 2, 3 2, 3 3 then there exists a (unique) algebraic K3 surface X with rk T X = Φ(). n some of the above cases, it seems that an assumption about the action of σ on S X is important. We can see some uniqueness theorems by changing assumptions on σ. An important assumption of Theorem 1.4 and Theorem 1.5 is the order of σ. We show uniqueness of K3 surfaces with σ from only. Theorem 1.4. [8, Main Theorem 1 and 2] Pairs (X 66, σ 66 ), (X 33, σ 33 ), (X 44, σ 44 ), (X 50, σ 50 ), (X 25, σ 25 ) and (X 40, σ 40 ) are unique up to isomorphism, respectively. Recently the following is proved. Theorem 1.5. [6] Pairs (X 21, σ 21 ) and (X 42, σ 42 ) are unique up to isomorphism, respectively. We remark that these theorems do not assume that non-symplectic automorphisms act trivially on the Néron-Severi lattice. ndeed if = 66, 44, 21 and 42 then σ acts trivially on S X. f Φ() < 12 then the uniqueness of (X, σ ) is not induced by only. An important assumption is the fixed locus of σ, hence forms of fixed loci induce uniqueness. Theorem 1.6. The followings hold by [10, Theorem 3, Theorem 4] [11, Main Theorem 4] [13, Theorem 1.5 (3)] : (1) f X σ 3 3 consists of only (smooth) rational curves and possibly some isolated points and contains at least 6 rational curves then a pair (X 3, σ 3 ) is unique up to isomorphism. (2) f X σ 2 2 consists of only (smooth) rational curves and contains at least 10 rational curves then a pair (X 2, σ 2 ) is unique up to isomorphism. (3) f X σ5 5 contains no curves of genus 2, but contains at least 3 rational curves then a pair (X 5, σ 5 ) is unique up to isomorphism. (4) Put M := {x H 2 (X 11, Z) σ 11(x) = x}. A pair (X 11, σ 11 ) is unique up to isomorphism if and only if M = U A 10. t is well known that if is prime then 19. But these theorems miss the case of = 7. Moreover Jang [6] does not determine fixed loci of automorphisms. The main purpose of this paper is to prove the following theorem: Main Theorem. (1) f X σ 7 7 consists of only (smooth) rational curves and some isolated points and contains at least 2 rational curves then a pair (, σ 7 ) is unique up to isomorphism. (2) The fixed locus of σ 21 consists of exactly 11 isolated points and one P 1. (3) The fixed locus of σ 42 consists of exactly 9 isolated points and one P 1.

3 REMARKS ON NON-SYMPLECTC AUTOMORPHSMS OF ORDER 7 3 Remark 1.7. t is easy to see that if σ holds the theorem (2) or (3) then pairs (X 21, σ 21 ) and (X 42, σ 42 ) are unique up to isomorphism by Theorem 1.5 We know further results for uniqueness. See also [15]. Throughout this article we shall denote by A m, D n, E l the negative-definite root lattice of type A m, D n, E l respectively. We denote by U the even indefinite unimodular lattice of rank 2 and U(m) the lattice whose bilinear form is the one on U multiplied by m. Acknowledgments. The author was partially supported by Grant-in-Aid for Young Scientists (B) 15K17520 from JSPS. 2. Preliminaries n this section, we collect some basic results for non-symplectic automorphisms on a K3 surface. For the details, see [9] and [2], and so on. Lemma 2.1. Let σ be a non-symplectic automorphism of order on X. Then (1) The eigen values of σ T X are the primitive -th roots of unity, hence σ T X C can be diagonalized as: ζ E q ζ n E q., ζ 1 E q where E q is the identity matrix of size q and 1 n 1 is co-prime with. (2) Let P be an isolated fixed point of σ on X. Then σ can be written as ( ) ζ i 0 0 ζ j (i + j 1 mod ) under some appropriate local coordinates around P. (3) Let C be an irreducible curve in X σ and Q a point on C. Then σ can be written as ( ) ζ under some appropriate local coordinates around Q. n particular, fixed curves are non-singular. Lemma 2.1 (1) implies that Φ() divides rk T X and Lemma 2.1 (2) and (3) imply that the fixed locus of σ is either empty or the disjoint union of non-singular curves and isolated points: X σ = {p 1,..., p M } C 1 C N, where p i is an isolated fixed point and C j is a non-singular curve. The global Torelli Theorem gives the following. Remark 2.2. [8, Lemma (1.6)] Let X be a K3 surface and g i (i = 1, 2) automorphisms of X such that g 1 S X = g 2 S X and that g 1ω X = g 2ω X. Then g 1 = g 2 in Aut (X).

4 4 S. TAK The Remark says that for study of non-symplectic automorphisms, the action on S X is important. Hence the invariant lattice S σ X := {x S X7 σ (x) = x} plays an essential role for the classification of non-symplectic automorphisms. Proposition 2.3. [2, Theorem6.3] The fixed locus X σ7 7 is of the form {p 1, p 2, p 3 } E if S σ7 = U K 7, {p 1, p 2, p 3 } if S σ 7 X σ = U(7) K 7, 7 7 = {p 1, p 2,..., p 8 } E P 1 if S σ 7 = U E 8, {p 1, p 2,..., p 8 } P 1 if S σ7 = U(7) E 8, {p 1, p 2,..., p 13 } P 1 P 1 if S σ 7 = U E 8 A 6. Here E is a non-singular( curve of genus ) 1 and K 7 is the even negative definite lattice 4 1 given by Gram matrix Uniqueness of K3 surfaces with a certain fixed locus n this section, we treat a pair (, σ 7 ) whose the fixed locus X σ7 7 consists of (smooth) rational curves and isolated points and contains at least 2 rational curves. We show that the pair (, σ 7 ) is unique up to isomorphism. Proposition 3.1. The automorphism σ 7 acts trivially on S X7. Proof. Since X σ 7 7 has at least 2 rational curves, X σ 7 7 = {p 1, p 2,..., p 13 } P 1 P 1 and S σ7 = U E 8 A 6 by Proposition 2.3. We know that rk T X7 6 by Lemma 2.1 (1) and rk S X7 16 since it contains the invariant lattice S σ7 which is of rank 16. This gives rk T X7 6 so that rk T X7 = 6 and rk S X7 = 6, hence S X7 coincides with S σ7. This implies that the action of σ 7 is trivial on the S X7. The following Corollary follows from Proposition 3.1 and Proposition 2.3. Corollary 3.2. S X7 = U E 8 A 6, T X7 = U U K 7 and the fixed locus σ 7 has 2 non-singular rational curves and 13 isolated points: X σ 7 7 = {p 1, p 2,..., p 13 } P 1 P 1 We recall that the dimension of a moduli space of K3 surfaces with a nonsymplectic automorphism of order 7 is rk T X7 /Φ(7) 1. n our case, its dimension is 0. ndeed we have the following. Theorem 3.3. A pair (, σ 7 ) is unique up to isomorphism. Proof. t follows from Proposition 3.1 and Theorem 1.3. Example 3.4. [7, (7.5)] Put X Ko : y 2 = x 3 + t 3 x + t 8, σ Ko (x, y, t) = (ζ 3 7x, ζ 7 y, ζ 2 7t). Then X Ko is a K3 surface with S XKo = U E 8 A 6 and σ Ko is a non-symplectic automorphism of order 7 acting trivially on S XKo. Example 3.5. [12, 4] Put X OZ : y 2 = x 3 + t 5 x + t 4. Then X OZ is a K3 surface with S XOZ = U E 8 A 6 and a non-symplectic automorphism of order 7. n [12], a non-symplectic automorphism of order 7 is not constructed. But φ(x, y, t) = (ζ7x, 3 ζ 7 y, ζ7t) 4 is a non-symplectic automorphism of order 7 on X OZ. Of course, it is easy to see that these examples are the same, by analysing the elliptic fibration.

5 REMARKS ON NON-SYMPLECTC AUTOMORPHSMS OF ORDER The fixed locus of a non-symplectic automorphism of order 21 We describe the fixed locus of a non-symplectic automorphism of order 21. First we recall the following. Proposition 4.1. [6, Theorem 2.1] A non-symplectic automorphism of order 21 σ 21 acts trivially on S X21. Lemma 4.2. The Euler characteristic of X σ21 21 is 3 + tr(σ 21 S X21 ) = 13. Proof. We apply the topological Lefschetz formula to the fixed locus X σ : χ(xσ ) = 2 + tr(σ21 S X21 ) + tr(σ42 T X42 ). By [9, Theorem 3.1], tr(σ21 T X21 ) = ζ 21 + ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ21 20 = ((1 + ζ ζ ζ ζ ζ ζ21) 18 + (ζ ζ21) 14 = (0 + (ζ 3 + ζ3) 2 = (0 1) = 1. Since Φ(21)=12, rk S X21 = 10. Lemma 4.3. Let P i,j and m i,j ( ) ζ i be an isolated fixed point given by the local action 0 0 ζ j the number of P i,j. Then we have m 2, m 6, m 9, , m 3, m 10, , m 4, , m 2, m 8, Proof. Since σ21(p 3 i,j 21 ) is a fixed point of σ 7, P i,j 21 is mapped to P i,j 7 (i i, j j mod 7). Thus P 2,20 21, P 6,16 21, P 9,13 σ P 2,6 7, P 3,19 21, P 5,17 21, P 10,12 σ P 3,5 7, P 4,18 21, P 11,11 σ P 4,4 7, P 7,15 21, P 8,14 21 σ 3 21 Q 7 where Q 7 is a point on fixed curves of σ 7 = σ21. 3 Since rk S X21 = 10 and Proposition 4.1, we have m 2, m 6, m 9, , (4.1) m 3, m 10, , m 4, by [2, Theorem 2.4]. Moreover σ21(p 7 i,j 21 ) is a fixed point of σ 3. f i or j 0 mod 3 then P i,j 21 is mapped to a point on a fixed curve of σ 3 = σ21 7 and if i and j 0 mod 3 then P i,j 21 is mapped to P 2,2 3. Since rk S X21 = 10 and Proposition 4.1, we have (4.2) m 2, m 8, by [1, Theorem 2.2] and [14, Proposition 3.2]

6 6 S. TAK We apply the holomorphic Lefschetz formula ([3, page 542] and [4, page 567]) to X σ : 2 M N tr(σ21 H k (X 21, O X21 )) = a(p i,j 21 ) + b(c l ), k=0 i+j=22 where a(p i,j 21 ) = 1/((1 ζi 21)(1 ζ j 21 )) and b(c l) = (1 g(c l ))/(1 ζ 21 ) ζ 21 C 2 l /(1 ζ 21 ) 2. Hence 1 + ζ = i+j=22, 2 i j m i,j 21 (1 ζ i 21 )(1 ζj 21 ) + (1 + ζ 21 )(1 g(c l )) (1 ζ 21 ) 2. Then we have m 6,16 21 = m2, m3,19 21 m5, N m 7,15 21 = 1 3m 3, N (1 g(c l)), m 8,14 21 = 1 9m2, m3, m5, N (4.3) m 9,13 21 = 1 5m 2,20 21 m 3, m 5, N m 10,12 21 = m2, m3, m5, m 4, N (1 g(c l)), m 11,11 21 = 1 3m 2,20 21 m 4, N (1 g(c l)). Proposition 4.4. The fixed locus of σ 21 consists of exactly 11 isolated points and one P 1 : X σ21 21 = {P 2,20 21, P 2,20 21, P 2,20 21, P 3,19 21, P 3,19 12, P 4,18 21, P 5,17 21, P 6,16 21, P 7,15 21, P 7,15 21, P 7,15 21 } P 1. is a non- Proof. We remark inequalities in Lemma 4.3, equations (4.3) and m i,j 21 negative integer. f m 4,18 21 < 1 then m 4,18 21 = m 11,11 21 = 0 and m 2,20 21 = 1/3 + 3 (1 g(c l )). This is a contradiction. f m 2, m6, m9,13 21 = 3 (resp. 2, 0) then m 3, m5, (1 g(c l )) = 4/3 (resp. 2/3, 2/3). These are not integer. f m 2, m 6, m 9,13 21 = 1 then m 5,17 21 = m 3, (1 g(c l )) and m 8,19 21 = 1 4 (1 g(c l )). Hence m 5,17 21 or m 8,19 21 is negative. Thus m 2, m 6, m 9,13 21 = 4. f m 3, m 10,12 21 = 2 (resp. 1) then m 4, (1 g(c l )) = 2/3 (resp. 1/3). These are not integer. Assume m 3, m 10,12 21 = 0. Since m 10,12 21 = 0, we have m 5,17 21 = 2 m 3, (1 g(c l )). This contradicts for m 3,19 21 = m 5,17 21 = 0. Hence we have m 3, m 10,12 21 = 3. f m 2, m5, m8, m11,11 21 = 2 (resp. 1, 0) then m 10,12 21 = 5 5 (1 g(c l )) or m 7,15 21 = (1 g(c l )) (resp (1 g(c l )), (1 g(c l ))) is negative. Assume m 2, m 8,14 21 = 3. Then it is easy to see M = m i,j 21 = 10 2 (1 g(c l )). n particular m 2,20 21 = 3 (1 g(c l )), m 5,17 21 = (1 g(c l )), m 8,14 21 = 4 4 (1 g(c l )) and m 11,11 21 = 2 2 (1 g(c l )). Since m 2,20 21, m5,17 21, m8,14 21 or m 11,11 21 is 0, we have (1 g(c l )) = 1 and M = 8. t follows from χ(x σ21 21 ) = M + (2 2g(C l )) and Lemma 4.2 that tr(σ21 S X21 ) = 7. This is a contradiction for Proposition 4.1, hence m 2, m5, m8, m11,11 21 = 4.

7 REMARKS ON NON-SYMPLECTC AUTOMORPHSMS OF ORDER 7 7 n conclusion inequalities in Lemma 4.3 are equations. Moreover by (4.3), we have m 2,20 21 = 3 N m 3,19 21 = 2 N m 4,18 21 = N m 5,17 21 = N m 6,16 21 = N m 7,15 21 = N m 8,14 21 = 4 4 N m 9,13 21 = 5 5 N m 10,12 21 = 5 5 N m 11,11 and M = m i,j 21 = 13 2 N 21 = 2 2 N (1 g(c l)) (1 g(c l)). f N (1 g(c l)) 1 then m 4,18 21 or m 8,14 21 are negative. Thus we have N g(c l )) = 1, M = 11 and χ(x σ ) = M + N (1 (2 2g(C l)) = = The fixed locus of a non-symplectic automorphism of order 42 We describe the fixed locus of a non-symplectic automorphism of order 42. The following is a key in this section. Proposition 5.1. [6, Corollary 2.6] A non-symplectic automorphism of order 42 σ 42 acts trivially on S X42. Lemma 5.2. The Euler characteristic of X σ is 1 + tr(σ 42 S X42 ) = 11. Proof. We apply the topological Lefschetz formula to the fixed locus X σ42 42 : χ(xσ42 ) = 4 i=0 ( 1)i tr(σ42 m H i (X 42, R)) = tr(σ42 S X42 ) + tr(σ42 T X42 ) By [9, Theorem 3.1], tr(σ42 T X42 ) = ζ 42 +ζ42 5 +ζ ζ ζ ζ ζ ζ ζ ζ ζ42 41 = ((1+ζ42 2 +ζ ζ42)+(ζ ζ42 7 +ζ42 9 +ζ ζ ζ ζ ζ ζ42)) 39 = (0+(ζ 14 +ζ14 3 +ζ14 5 +ζ14 7 +ζ14 9 +ζ ζ14)+(ζ ζ6)) 5 = (0+0+1) = 1. Since Φ(21)=12, rk S X21 = 10. Lemma 5.3. The following inequalities and equations hold: m 2, m 20, , m 3, m 19, , m 4, m 18, , m 5, m 17, , m 6, m 16, , m 7, m 15, , and m 8,35 42 = m 9,34 42 = m 10,33 42 = m 11,32 42 = m 12,31 42 = m 13,30 42 = m 14,29 42 = 0. Proof. Since σ42(p 2 i,j 42 ) is a fixed point of σ 21, P i,j 42 is mapped to P i,j 21 (i i, j j mod 21). t is easy to see these inequalities and equations by Theorem 4.4.

8 8 S. TAK Proposition 5.4. The fixed locus of σ 42 consists of exactly 9 isolated points and one P 1 : X σ42 42 = {P 2,41 42, P 2,41 42, P 2,41 42, P 3,40 42, P 3,40 42, P 4,39 42, P 5,38 42, P 6,37 42, P 7,36 42 } P 1. Proof. We apply the holomorphic Lefschetz formula ([3, page 542] and [4, page 567]) to X σ : 1 + ζ = Then we have m 15,28 42 = 0, i+j=43, 2 i j m i,j 42 (1 ζ i 42 )(1 ζj 42 ) + m 16,27 42 = 4m 2, m 3, m 5, m 6,37 42 m 7, (1 + ζ 42 )(1 g(c l )) (1 ζ 42 ) 2. (1 g(c l )), m 17,26 42 = m 2, m 3, m 5, m 6, m 7, (1 g(c l )), m 18,25 42 = m 2, m 3, m 4, m 5, m 6, m 7, m 19,24 42 = 5 58m 2, m 3, m 4, m 5, m 6, m 7, m 20,23 42 = 4 51m 2, m 3, m 4, m 5, m 6, m 7, m 21,22 42 = 2 24m 2, m 3, m 4, m 5, Moreover by Proposition 5.5, we have m 2,41 42 = N m 3,40 42 = N m 4,39 42 = N m 5,38 42 = N m 6,37 42 = N (1 g(c l)), m 7,36 42 = 1, m 16,27 42 = 4 4 N m 17,26 42 = 2 2 N m 18,25 42 = 2 2 N m 19,24 42 = 4 4 N m 20,23 42 = 2 2 N m 21,22 42 = 2 2 N (1 g(c l)). and M = m i,j 42 = 11 2 N (1 g(c l)). (1 g(c l )). (1 g(c l )), (1 g(c l )), (1 g(c l )),

9 REMARKS ON NON-SYMPLECTC AUTOMORPHSMS OF ORDER 7 9 f N (1 g(c l)) 1 then m 4,39 42 or m 16,27 42 are negative. Thus N (1 g(c l)) = 1 and M = 9. f the fixed locus X σ contains a non-singular curve then X σ also contain it. Thus X σ42 42 has at most one P 1 by Proposition 4.4. Proposition 5.5. The following equations hold: m 2, m 20,23 42 = 3, m 3, m 19,24 42 = 2, m 4, m 18,25 42 = 1, m 5, m 17,26 42 = 1, m 6, m 16,27 42 = 1, m 7, m 15,28 42 = 1. Proof. We remark inequalities in Lemma 5.3 and m i,j 42 is a non-negative integer. f m 4, m 18,25 42 = 0 (resp. m 5, m 17,26 42 = 0) then m 16,27 42 = 2/3 14m 2,41 42 /3 2m 3, m 4,39 42 /3 m6, (1 g(c l ))/3 (resp. m 16,27 42 = 1/2 2m 2,41 42 m 3,40 42 m 6, (1 g(c l ))). These are not integers, respectively. f m 6, m 16,27 42 = 0 then m 4,39 42 = 3/2 m 2, (1 g(c l )). This is not a integer. f m 3, m 19,24 42 = 0 (m 3,40 42 = m 19,24 42 = 0) then we have m 6,37 42 = 2 + 6m 5, = 1 then and m 7,36 42 = 3 8m 5, m6,37 42 or m 7,36 42 is negative. f m 3, m 19, (1 g(c l )). This is not a integer. m 6,37 42 = 3/2 + 4m 2, m 5,38 f m 7, m 15,28 42 = 3 (resp. 2, 0) then m 5,38 42 = 1/4 m 2,41 (resp. = 1/8 m 2, (1 g(c l )), = 1/8 m 2,41 are not integer (1 g(c l )) (1 g(c l ))). These f m 2, m 20,23 42 = 2 (resp. 0) then m 5,38 42 = 1/ + 2 (1 g(c l )) (resp. = 1/2 + 2 (1 g(c l ))). These are not integer. f m 2, m 20,23 42 = 1 then m 6,37 42 = (1 g(c l )) and m 18,25 42 = 1 2 (1 g(c l )). m 6,37 42 or m 18,25 42 is negative. References [1] M. Artebani, A. Sarti, Non-symplectic automorphisms of order 3 on K3 surfaces, Math. Ann. 342 (2008), [2] M. Artebani, A. Sarti, S. Taki, K3 surfaces with non-symplectic automorphisms of prime order, Math. Z. 268 (2011), [3] M.F. Atiyah, G.B. Segal, The index of elliptic operators:, Ann. of Math. 87 (1968), [4] M.F. Atiyah,.M. Singer, The index of elliptic operators:, Ann. of Math. 87 (1968), [5]. Dolgachev, S. Kondo, Moduli of K3 surfaces and complex ball quotients, Arithmetic and geometry around hypergeometric functions, , Progr. Math., 260, Birkhäuser, [6] J. Jang, A non-symplectic automorphism of order 21 of a K3 surface, preprint, arxiv: [7] S. Kondo, Automorphisms of algebraic K3 surfaces which act trivially on Picard groups, J. Math. Soc. Japan, 44 (1992), [8] N. Machida, K. Oguiso, On K3 surfaces admitting finite non-symplectic group actions, J. Math. Sci. Univ. Tokyo 5 (1998), no. 2, [9] V.V. Nikulin, Finite automorphism groups of Kählerian K3 surfaces, Trans. Moscow Math. Soc., 38 (1980), No 2, [10] K. Oguiso, D.-Q. Zhang, On the most algebraic K3 surfaces and the most extremal log Enriques surfaces, Amer. J. Math., 118 (1996), no. 6,

10 10 S. TAK [11] K. Oguiso, D.-Q. Zhang, K3 surfaces with order five automorphisms, J. Math. Kyoto Univ. 38 (1998), no. 3, [12] K. Oguiso, D.-Q. Zhang, On Vorontsov s theorem on K3 surfaces with non-symplectic group actions, Proc. Amer. Math. Soc., 128 (2000), no. 6, [13] K. Oguiso, D.-Q. Zhang, K3 surfaces with order 11 automorphisms, Pure Appl. Math. Q. 7 (2011), no. 4, [14] S. Taki, Classification of non-symplectic automorphisms of order 3 on K3 surfaces, Math. Nachr. 284 (2011), [15] S. Taki, On uniqueness of K3 surfaces with non-symplectic automorphisms, in preparation. [16] S.P. Vorontsov, Automorphisms of even lattices that arise in connection with automorphisms of algebraic K3 surfaces, Vestnik Mosk. Univ. Math. 38 (1983), Department of Mathematics, Tokai University, 4-1-1, Kitakaname, Hiratsuka, Kanagawa, , JAPAN address: taki@tsc.u-tokai.ac.jp URL: