SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES

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1 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES ALICE GARBAGNATI AND ALESSANDRA SARTI Abstract. We study algebrac K3 surfaces (defned over the complex number feld) wth a symplectc automorphsm of prme order. In partcular we consder the acton of the automorphsm on the second cohomology wth nteger coeffcents (by a result of Nkuln ths acton s ndependent on the choce of the K3 surface). Wth the help of ellptc fbratons we determne the nvarant sublattce and ts perpendcular complement, and show that the latter concdes wth the Coxeter-Todd lattce n the case of automorphsm of order three. 0. Introducton In the paper [N1] Nkuln studes fnte abelan groups G actng symplectcally (.e. G H 2,0 (X,C) = d H 2,0 (X,C)) on K3 surfaces (defned over C). One of hs man result s that the acton nduced by G on the cohomology group H 2 (X, Z) s unque up to sometry. In [N1] all abelan fnte groups of automorphsms of a K3 surface actng symplectcally are classfed. Later Muka n [Mu] extends the study to the non abelan case. Here we consder only abelan groups of prme order p whch, by Nkuln, are somorphc to Z/pZ for p = 2,3,5,7. In the case of p = 2 the group s generated by an nvoluton, whch s called by Morrson n [Mo, Def. 5.1] Nkuln nvoluton. Ths was very much studed n the last years, n partcular because of ts relaton wth the Shoda-Inose structure (cf. e.g. [CD], [GL], [vgt], [L], [Mo]). In [Mo] Morrson proves that the sometry nduced by a Nkuln nvoluton ι on the lattce Λ K3 U U U E 8 ( 1) E 8 ( 1), whch s sometrc to H 2 (X, Z), swtches the two copes of E 8 ( 1) and acts as the dentty on the sublattce U U U. As a consequence one sees that (H 2 (X, Z) ι ) s the lattce E 8 ( 2). Ths mples that the Pcard number ρ of an algebrac K3 surface admttng a Nkuln nvoluton s at least nne. In [vgs] van Geemen and Sart show that f ρ 9 and E 8 ( 2) NS(X) then the algebrac K3 surface X admts a Nkuln nvoluton and they classfy completely these K3 surfaces. Moreover they dscuss many examples and n partcular those surfaces admttng an ellptc fbraton wth a secton of order two. Ths secton operates by translaton on the fbers and defnes a Nkuln nvoluton on the K3 surface. The am of ths paper s to dentfy the acton of a symplectc automorphsm σ p of the remanng possble prme orders p = 3,5,7 on the K3 lattce Λ K3 and to descrbe such algebrac K3 surfaces wth mnmal possble Pcard number. Thanks to Nkuln s result ([N1, Theorem 4.7]), to fnd the acton on Λ K3, t suffces to dentfy the acton n one specal case. For ths purpose t seemed to be convenent to study algebrac K3 surfaces wth an ellptc fbraton wth a secton of order three, fve, resp. seven. Then the translaton by ths secton s a symplectc automorphsm of the surface of the same order. A The second author was partally supported by DFG Research Grant SA 1380/ Mathematcs Subject Classfcaton: 14J28, 14J10. Key words: K3 surfaces, automorphsms, modul. 1

2 2 ALICE GARBAGNATI AND ALESSANDRA SARTI concrete analyss leads us to the man result of the paper whch s the descrpton of the lattces H 2 (X, Z) σ p and Ω p = (H 2 (X, Z) σ p) gven n the Theorem 4.1. The proof of the man theorem conssts n the Propostons 4.2, 4.4, 4.6. We descrbe the lattce Ω p also as Z[ω p ]-lattces, where ω p s a prmtve p root of the unty. Ths knd of lattces are studed e.g. n [Ba], [BS] and [E]. In partcular n the case p = 3 the lattce Ω 3 s the Coxeter-Todd lattce wth the form multpled by 2, K 12 ( 2), whch s descrbed n [CT] and n [CS]. The ellptc surfaces we used to fnd the lattces Ω p do not have the mnmal possble Pcard number. We prove n Proposton 5.1 that for K3 surfaces, X, wth mnmal Pcard number and symplectc automorphsm, f L s a class n NS(X) whch s nvarant for the automorphsms, wth L 2 = 2d > 0, then ether NS(X) = ZL Ω p or the latter s a sublattce of ndex p n NS(X). Usng ths result and the one of the Proposton 5.2 we descrbe the coarse modul space of the algebrac K3 surfaces admttng a symplectc automorphsm of prme order. The structure of the paper s the followng: n secton 1 we compute the number of modul of algebrac K3 surfaces admttng a symplectc automorphsm of order p and ther mnmal Pcard number. In secton 2 we gve the defnton of Z[ω p ]-lattce and we assocate to t a module wth a blnear form, whch n some cases s a Z-lattce (we use ths constructon n secton 4 to descrbe the lattces Ω p as Z[ω p ]-lattces). In secton 3 we recall some results about ellptc fbratons and ellptc K3 surfaces (see e.g. [M1], [M2], [Shm], [Sho] for more on ellptc K3 surfaces). In partcular we ntroduce the three ellptc fbratons whch we use n secton 4 and gve also ther Weerstrass form. In secton 4 we state and proof the man result, Theorem 4.1: we dentfy the lattces Ω p and we descrbe them as Z[ω p ]-lattces. In secton 5 we descrbe the Néron-Sever group of K3 surfaces admttng a symplectc automorphsm and havng mnmal Pcard number (Proposton 5.1). In secton 5 we descrbe the coarse modul space of the algebrac K3 surfaces admttng a symplectc automorphsm and the Néron-Sever group of those havng mnmal Pcard number. We would lke to express our deep thanks to Bert van Geemen for suggestng us the problem and for hs nvaluable help durng the preparaton of ths paper. 1. Prelmnary results Defnton 1.1. A symplectc automorphsm σ p of order p on a K3 surface X s an automorphsm such that: 1. the group G generated by σ p s somorphc to Z/pZ, 2. σ p(δ) = δ, for all δ n H 2,0 (X). We recall that by [N1] an automorphsm on a K3 surface s symplectc f and only f t acts as the dentty on the transcendental lattce T X. In local coordnates at a fxed pont σ p has the form dag(ω p,ωp p 1 ) where ω p s a prmtve p-root of unty. By a result of Nkuln the only possble values for p are 2,3,5,7 see [N1, Theorem 4.5] and [N1, 5]. The automorphsm σ 3 has sx fxed ponts on X, σ 5 has four fxed ponts and σ 7 has three fxed ponts. The automorphsm σ p nduces a σp sometry on H 2 (X, Z) = Λ K3. Nkuln proved [N1, Theorem 4.7] that f σ p s symplectc, then the acton of σp s unque up to sometry of Λ K3. Let ω p be a prmtve p-root of the unty. The vector space H 2 (X, C) can be decomposed

3 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 3 n egenspaces of the egenvalues 1 and ω p: H 2 (X, C) = H 2 (X, C) σ p ( =1,...,p 1 H 2 (X, C) ω p ). We observe that the non ratonal egenvalues ω p have all the same multplcty. So we put: a p := multplcty of the egenvalue 1, b p :=multplcty of the egenvalues ω p. In the followng we fnd a p and b p by usng the Lefschetz fxed pont formula: (1) µ p = r ( 1) r trace(σ p Hr (X, C)) where µ p denotes the number of fxed ponts. For K3 surfaces we obtan µ p = trace(σ p H2 (X, C)) Proposton 1.1. Let X, σ p, a p, b p be as above, p = 3,5,7. Let ρ p be the Pcard number of X, and let m p be the dmenson of the modul space of the algebrac K3 surfaces admttng a symplectc automorphsm of order p. Then a 3 = 10 b 3 = 6 ρ 3 13 m 3 7 a 5 = 6 b 5 = 4 ρ 5 17 m 5 3 a 7 = 4 b 7 = 3 ρ 7 19 m 7 1. Proof. The proof s smlar n all the cases, here we gve the detals only n the case p = 5. A symplectc automorphsm of order fve on a K3 surface has exactly four fxed ponts. Applyng the Lefschetz fxed ponts formula (1), we have a 5 + b 5 (ω 5 + ω5 2 + ω3 5 + ω4 5 ) = 2. Snce ω p 1 p = ( p 2 =0 ω p), the equaton becomes a 5 b 5 = 2. Snce dm H 2 (X, C) = 22, a 5 and b 5 have to satsfy: { a5 b 5 = 2 (2) a 5 + 4b 5 = 22. We have dm H 2 (X, C) σ 5 = 6 = a5 and dm H 2 (X, C) ω5 = dmh 2 (X, C) ω 2 5 = dm H 2 (X, C) ω 3 5 = dm H 2 (X, C) ω 4 5 = 4 = b 5. Snce T X C H 2 (X, C) σ 5, (H 2 (X, C) σ 5 ) = H 2 (X, C) ω5 H 2 (X, C) ω 2 5 H 2 (X, C) ω 3 5 H 2 (X, C) ω 4 5 NS(X) C. We consder only algebrac K3 surfaces and so we have an ample class h on X, by takng h + σ5 h + σ 2 5 h + σ 3 5 h + σ 4 5 h we get a σ 5-nvarant class, hence n H 2 (X, C) σ 5. From here t follows that ρp =rank NS(X) = 17, whence rank T X = 5. The number of modul s at most = 3. Remark. In [N1, 10] Nkuln computes rank(h 2 (X, Z) σ p) = (p 1)b p and rank(h 2 (X, Z) σ p) = a p. In [N1, Lemma 4.2] he also proves that there are no classes wth self ntersecton 2 n the lattces (H 2 (X, Z) σ p) ; we descrbe these lattces n the sectons 4.1, 4.4, 4.6 and we fnd agan the result of Nkuln.

4 4 ALICE GARBAGNATI AND ALESSANDRA SARTI 2. The Z[ω]-lattces In the sectons 4.2, 4.5, 4.7 our purpose s to descrbe (H 2 (X, Z) σ p) as Z[ω p ]-lattce. We recall now some useful results on these lattces. Defnton 2.1. Let p be an odd prme and ω := ω p be a prmtve p-root of the unty. A Z[ω]-lattce s a free Z[ω]-module wth an hermtan form (wth values n Z[ω]). Its rank s ts rank as Z[ω]-module. Let {L,h L } be a Z[ω]-lattce of rank n. The Z[ω]-module L s also a Z-module of rank (p 1)n. In fact f e, = 1,...,n s a bass of L as Z[ω]-module, ω j e, = 1,...,n, j = 0,...,p 2 s a bass for L as Z-module (recall that ω p 1 = (ω p 2 +ω p )). The Z-module L wll be called L Z. Let Γ p := Gal(Q(ω)/Q) be the group of the automorphsms of Q(ω) whch fx Q. We recall that the group Γ p has order p 1 and ts elements are automorphsms ρ such that ρ (1) = 1, ρ (ω) = ω where = 1,...,p 1. We defne a blnear form on L Z b L (α,β) = 1 (3) ρ(h L (α,β)). p ρ Γ p Note that b L takes values n 1 p Z[ω], so n general {L Z,b L } s not a Z-lattce. We call t the assocated module (resp. lattce) of the Z[ω]-lattce L. Remark. Remark. By the defnton of the blnear form s clear that b L (α,β) = 1 p Tr Q(ω)/Q(h L (α,β)). For a precse defnton of the Trace see [E, page 128] 2.1. The Z-lattce F p. We consder a K3 surface admttng an ellptc fbraton. Let p be an odd prme number. Let I p be a semstable fber of a mnmal ellptc fbraton,.e. (cf. secton 3) I p s a fber whch s a reducble curve, whose rreducble components are the edges of a p-polygon, as descrbed n [M1, Table I.4.1], we denote the p-rreducble components by C, = 0,...,p 1, then C C j = 2 f j mod p 1 f j 1 mod p 0 otherwse. We consder now the free Z-module F p wth bass the elements of the form C C +1, = 1,...,p 1 and wth blnear form b Fp whch s the restrcton of the ntersecton form to the bass C C +1, then {F p,b Fp } s a Z-lattce The Z[ω p ]-lattce G p. Let G p be the Z[ω]-lattce G p := (1 ω) 2 Z[ω], wth the standard hermtan form: h(α,β) = α β. A bass for the Z-module G p,z s (1 ω) 2 ω, = 0,...,p 2. On Z[ω] we consder the blnear form b L defned n (3), wth values n 1 p Z, then we have b(α,β) = 1 ρ(α β), p ρ Γ p α,β Z[ω],

5 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 5 Lemma 2.1. The blnear form b restrcted to G p (denoted by b G ) has values n Z and concdes wth the ntersecton form on F p by usng the map F p G p defned by C C +1 ω (1 ω) 2, = 1,...,p 1, C p = C 0. Proof. An easy computaton shows that we have for p > 3: 6 f k h mod p, b G (ω k (1 ω) 2,ω h (1 ω) 2 4 f k h 1 mod p, ) = 1 f k h 2 mod p, 0 otherwse, and for p = 3: b G (ω k (1 ω) 2,ω h (1 ω) 2 ) = { 6 f k h mod p, 3 f k h 1 mod p. The ntersecton form on F p s easy to compute (cf. secton 2.1) and ths computaton proves that the map F p G p defned n the lemma s an sometry. In secton 4 we apply the results of ths secton and we fnd a Z[ω]-lattce {L p,h Lp } such that {L p,h Lp } contans G p as sublattce; {L p,z,b Lp } s a Z-lattce; the Z-lattce {L p,z,b Lp } s sometrc to the Z-lattce (H 2 (X, Z) σ p) for p = 3,5,7. 3. Some general facts on ellptc fbratons In the next secton we gve explct examples of K3 surfaces admttng a symplectc automorphsm σ p by usng ellptc fbratons. Here we recall some general results about these fbratons. Let X be an ellptc K3 surface, ths means that we have a morphsm f : X P 1 such that the generc fber s a (smooth) ellptc curve. We assume moreover that we have a secton s : P 1 X. The sectons of X generate the Mordell-Wel group MW f of X and we take s as zero secton. Ths group acts on X by translaton (on each fber), hence t leaves the two form nvarant. We assume that the sngular fbers of the fbraton are all of type I m, m N. Let F j be a fber of type I mj, we denote by C (j) 0 the rreducble components of the fbers meetng the zero secton. After choosng an orentaton, we denote the other rreducble components of the fbers by C (j) 1,...,C(j) m j 1. In the sequel we always consder m j a prme number, and the notaton C (j) means Z/m j Z. For each secton r we defne the number k := k j (r) by r C (k) j = 1 and r C () j = 0 f = 0,...,m j 1 k. If the secton r s a torson secton and h s the number of reducble fbers of type I mj, then by [M2, Proposton 3.1] we have h ( k j (r) 1 k ) j(r) (4) = 4. m j j=1

6 6 ALICE GARBAGNATI AND ALESSANDRA SARTI Moreover we recall the Shoda-Tate formula (cf.[sho, Corollary 5.3] or [M1, p.70]) (5) h rank(ns(x)) = 2 + (m j 1) + rank(mw f ). j=1 The rank(mw f ) s the number of generators of the free part. If there are no sectons of nfnte order then rank(mw f ) = 0. Assume that X has h fbers of type I m, m N, m > 1, and the remanng sngular fbers are of type I 1, whch are ratonal curves wth one node. Let U (A m 1 ) h denote the lattce generated by the zero secton, the generc fber and by the components of the reducble fbers not meetng s. If there are no sectons of nfnte order then t has fnte ndex n NS(X) equal to n, the order of the torson part of the group MW f. Usng ths remark we fnd that (6) det(ns(x)) = det(a m 1) h n 2 = mh n Ellptc fbratons wth a symplectc automorphsm. Now we descrbe three partcular ellptc fbratons whch admt a symplectc automorphsm σ 3, σ 5 or σ 7. Assume that we have a secton of prme order p = 3,5,7. By [Shm, No. 560, 2346, 3256] there exst ellptc fbratons wth one of the followng confguratons of components of sngular fbers I p not meetng s such that all the sngular fbers of the fbratons are semstable (.e. they are all of type I n for a certan n N) and the order of the torson subgroup of the Mordell-Wel group o(mw f ) = p : (7) p = 3 : 6A 2 o(mw f ) = 3, p = 5 : 4A 4 o(mw f ) = 5, p = 7 : 3A 6 o(mw f ) = 7. We can assume that the remanng sngular fbers are of type I 1. Snce the sum of the Euler characterstc of the fbers must add up to 24, these are sx, four, resp. three fbers. Observe that each secton of fnte order nduces a symplectc automorphsm of the same order whch corresponds to a translaton by the secton on each fber, we denote t by σ p. The nodes of the I 1 fbers are then the fxed ponts of these automorphsms, whence σ p permutes the p components of the I p fbers. For these fbratons we have rank NS(X) = 14, 18, 20 and dmensons of the modul spaces sx, two and zero, whch s one less then the maxmal possble dmenson of the modul space we have gven n the Proposton Weerstrass forms. We compute the Weerstrass form for the ellptc fbraton descrbed n (7). When X s a K3 surface then ths form s (8) y 2 = x 3 + A(t)x + B(t), t P 1 or n homogeneous coordnates (9) x 3 x 2 2 = x A(t)x 1 x B(t)x 3 3 where A(t) and B(t) are polynomals of degrees eght and twelve respectvely, x 3 = 0 s the lne at nfnty and also the tangent to the nflectonal pont (0 : 1 : 0). Fbraton wth a secton of order 3. In ths case the pont of order three must be an nflectonal pont (cf. [C, Ex. 5, p.38]), we want to determne A(t) and B(t) n the

7 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 7 equaton (8). We start by mposng to a general lne y = l(t)x + m(t) to be an nflectonal tangent so the equaton of the ellptc fbraton s y 2 = x 3 + A(t)x + B(t), t P 1, wth 2l(t)m(t) + l(t)4 A(t) = 3, B(t) = m(t)2 l(t) Snce A(t) and B(t) are of degrees eght and twelve, we have deg l(t) = 2 and deg m(t) = 6. The secton of order three s t ( ) l(t) 2 3, l(t)3 + m(t). 3 The dscrmnant = 4A B 2 of the fbraton s = (5l(t)3 + 27m(t))(l(t) 3 + 3m(t)) 3 27 hence n general t vanshes to the order three on sx values of t and to the order one on other sx values. Snce A and B n general do not vansh on these values, ths equaton parametrzes an ellptc fbraton wth sx fbers I 3 (so we have sx curves A 2 not meetng the zero secton) and sx fbers I 1 (cf. [M1, Table IV.3.1 pag.41]). Fbraton wth a secton of order 5. In the same way we can compute the Weerstrass form of the ellptc fbraton descrbed n (7) wth a secton of order fve. In [BM] a geometrcal condton for the exstence of a pont of order fve on an ellptc curve s gven. For fxed t let the cubc curve be n the form (9) then take two arbtrary dstnct lnes through O whch meet the cubc n two other dstnct ponts each. Call 1, 4 the ponts on the frst lne and 2, 3 the ponts on the second lne, then 1 (or any of the other pont) has order fve f: -the tangent through 1 meets the cubc n 3, -the tangent through 4 meets the cubc n 2, -the tangent through 3 meets the cubc n 4, -the tangent through 2 meets the cubc n 1. These condtons gve the Weerstrass form: y 2 = x 3 + A(t)x + B(t), t P 1, wth A(t) = ( b(t)4 + b(t) 2 a(t) 2 a(t) 4 3a(t)b(t) 3 + 3a(t) 3 b(t)) 3, B(t) = (b(t)2 + a(t) 2 )(19b(t) 4 34b(t) 2 a(t) a(t) a(t)b(t) 3 18a(t) 3 b(t)) 108 where deg a(t) = 2, deg b(t) = 2. The secton of order fve s t ((2b(t) 2 a(t) 2 )2 : 3(a(t) + b(t))(a(t) b(t)) 2 : 6) and the dscrmnant s = 1 16 (b(t)2 a(t) 2 ) 5 (11(b(t) 2 a(t) 2 ) + 4a(t)b(t)). By a careful analyss of the zeros of the dscrmnant we can see that the fbraton has four fbers I 5 and four fbers I 1 (cf. [M1, Table IV.3.1 pag.41]). Fbraton wth a secton of order 7. To fnd the Weerstrass form we use also n ths case the results of [BM]. We explan brefly the dea to fnd a set of ponts of order seven on an ellptc curve. One takes ponts 0,3,4 and 1,2,4 on two lnes n the plane. Then the ntersectons of a lne through 3 dfferent from the lnes {3, 2}, {3, 4}, {3, 1} wth the lnes {1,0} and {2,0} gve two new ponts 1 and 2. By usng the condtons that the tangent

8 8 ALICE GARBAGNATI AND ALESSANDRA SARTI through 1 goes through 2 and the tangent through 2 goes through 3 one can determne a cubc havng a pont of order seven whch s e.g. 1. By usng these condtons one can fnd the equaton, but snce the computatons are qute nvolved, we recall the Weerstrass form gven n [T, p.195] y 2 + (1 + t t 2 )xy + (t 2 t 3 )y = x 3 + (t 2 t 3 )x 2. By a drect check one sees that the pont of order seven s (0(t),0(t)). Ths ellptc fbraton has three fbers I 7 and three fbers I Ellptc K3 surfaces wth an automorphsm of prme order In ths secton we prove the man theorem: Theorem 4.1. For any K3 surface X wth a symplectc automorphsm σ p of order p = 2,3,5,7 the acton on H 2 (X, Z) decomposes n the followng way: p = 2 : H 2 (X, Z) σ 2 = E8 ( 2) U U U, (H 2 (X, Z) σ 2 ) = E 8 ( 2). p = 3 : H 2 (X, Z) σ 3 = U U(3) U(3) A2 A 2 x x j mod (1 ω 3 ), (H 2 (X, Z) σ 3 ) = (x 1,...,x 6 ) (Z[ω 3 ]) 6 : 6 =1 x 0 mod (1 ω 3 ) 2 = K 12( 2) wth hermtan form h(α,β) = 6 =1 (α β ). p = 5 : H 2 (X, Z) σ 5 = U U(5) U(5) (H 2 (X, Z) σ 5 ) = (x 1,...,x 4 ) (Z[ω 5 ]) 4 : x 1 x 2 2x 3 2x 4 mod (1 ω 5 ), (3 ω 5 )(x 1 + x 2 ) + x 3 + x 4 0 mod (1 ω 5 ) 2 wth hermtan form h(α,β) = 2 =1 α β + 4 j=3 fα jfβ j where f = 1 (ω5 2 + ω3 5 ). ( ) p = 7 : H 2 (X, Z) σ = U(7) 1 2 x 1 x 2 6x 3 mod (1 ω 7 ), (H 2 (X, Z) σ 7 ) = (x 1,x 2,x 3 ) (Z[ω 7 ]) 3 : (1 + 5ω 7 )x 1 + 3x 2 + 2x 3 0 mod (1 ω 7 ) 2 wth hermtan form h(α,β) = α 1 β 1 + f 1 α 2 f 1 β 2 + f 2 α 3 f 2 β 3 where f 1 = 3 + 2(ω 7 + ω 6 7 ) + (ω2 7 + ω5 7 ) and f 2 = 2 + (ω 7 + ω 6 7 ). In the case p = 3, K 12 ( 2) denotes the Coxeter-Todd lattce wth the blnear form multpled by 2. Ths theorem gves a complete descrpton of the nvarant sublattce H 2 (X, Z) σ p and ts orthogonal complement n H 2 (X, Z) for the symplectc automorphsms σ p of all possble prme order p = 2,3,5,7 actng on a K3 surface. The results about the order two automorphsm s proven by Morrson n [Mo, Theorem 5.7]. We descrbe the lattces of the theorem and ther hermtan forms n the sectons from 4.1 to 4.7. The proof s the followng: we dentfy the acton of σ p on H 2 (X, Z) n the case of X an ellptc K3 surface, ths s done n several propostons n these sectons, then we apply [N1, Theorem 4.7] whch assure the unqueness of ths acton.

9 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES A secton of order three. Let X be a K3 surface wth an ellptc fbraton whch admts a secton of order three descrbed n (7) of secton 3. We recall that X has sx reducble fbres of type I 3 and sx sngular rreducble fbres of type I 1. In the precedng secton we have seen that the rank of the Néron-Sever group s 14. We determne now NS(X) and T X. Let t 1 denote the secton of order three and t 2 = t 1 + t 1. Let σ 3 be the automorphsm of X whch corresponds to the translaton by t 1. It leaves each fber nvarant and σ 3 (s) = t 1, σ 3 (t 1) = t 2, σ 3 (t 2) = s. Denoted by C () 0,C() 1,C() 2 the components of the th reducble fber ( = 1,...,6), we can assume that C () 1 t 1 = C () 2 t 2 = C () 0 s = 1. Proposton 4.1. A Z-bass for the lattce NS(X) s gven by s,t 1,t 2,F,C (1) 1,C(1) 2,C(2) 1,C(2) 2,C(3) 1,C(3) 2,C(4) 1,C(4) 2,C(5) 1,C(5) 2. Let U A 6 2 be the lattce generated by the secton, the fber and the rreducble components of the sx fbers I 3 whch do not ntersect the zero secton s. It has ndex three n the Néron- Sever group of X, NS(X). The lattce NS(X) has dscrmnant 3 4 and ts dscrmnant form s The transcendental lattce T X s Z 3 ( 2 3 ) Z 3( 2 3 ) Z 3( 2 3 ) Z 3( 2 3 ). T X = U U(3) A 2 A 2 and has a unque prmtve embeddng n the lattce Λ K3. Proof. It s clear that a Q-bass for NS(X) s gven by s,f,c () 1,C() 2, = 1,...,6. Ths bass generates the lattce U A 6 2. It has dscrmnant d(u A6 2 ) = 36. We denote by c = 2C () 1 + C () 2, C = c, d = C () 1 + 2C () 2, D = d. Snce we know that t 1 NS(X) we can wrte t 1 = αs + βf + γ C () 1 + δ C () 2, α,β,γ,δ Q. Then by usng the fact that t 1 s = t 1 C () 2 = 0 and t 1 C () 1 = t 1 F = 1 one obtans that α = 1, β = 2 and γ 1 = 2/3, δ 1 = 1/3 hence 1 3C NS(X). A smlar computaton wth t 2 shows that 1 3D NS(X). So one obtans that (10) and so t 1 = s + 2F 1 3 C NS(X), t 2 = s + 2F 1 3 D NS(X). 3(t 2 t 1 ) = 6 =1 (C () 1 C () 2 ) = C D. We consder now the Q-bass for the Néron-Sever group s,t 1,t 2,F,C (1) 1,C(1) 2,C(2) 1,C(2) 2,C(3) 1,C(3) 2,C(4) 1,C(4) 2,C(5) 1,C(5) 2. By computng the matrx of the ntersecton form respect to ths bass one fnds that the determnant s 3 4. By the Shoda-Tate formula we have det(ns(x)) = 3 4. Hence ths s a Z-bass for the Néron-Sever group. We add to the classes whch generate U A 6 2 the

10 10 ALICE GARBAGNATI AND ALESSANDRA SARTI classes t 1 and σ3 (t 1) = t 2 gven n the formula (10). Snce d(u A 6 2 ) = 36 and d(ns(x)) = 3 4 the ndex of U A 6 2 n NS(X) s 3. Observe that ths s also a consequence of a general result gven at the end of secton 3. The classes v = C() 1 C () 2 (C (5) 1 C (5) 3 2 ), = 1,...,4 generate the dscrmnant group, whch s NS(X) /NS(X) = (Z/3Z) 4. These classes are not orthogonal to each other wth respect to the blnear form, so we take w 1 = v 1 v 2, w 2 = v 3 v 4, w 3 = v 1 + v 2 + v 3 + v 4, w 4 = v 1 + v 2 (v 3 + v 4 ) whch form an orthogonal bass wth respect to the blnear form wth values n Q/Z. And t s easy to compute that w1 2 = w2 2 = w2 3 = 2/3, w2 4 = 2/3. The dscrmnant form of the lattce NS(X) s then (11) Z 3 ( 2 3 ) Z 3( 2 3 ) Z 3( 2 3 ) Z 3( 2 3 ). The transcendental lattce T X orthogonal to NS(X) has rank eght. Snce NS(X) has sgnature (1, 13), the transcendental lattce has sgnature (2, 6). The dscrmnant form of the transcendental lattce s the opposte of the dscrmnant form of the Néron-Sever lattce. So the transcendental lattce has sgnature (2,6), dscrmnant 3 4, dscrmnant group TX /T X = (Z/3Z) 4 and dscrmnant form Z 3 ( 2 3 ) Z 3( 2 3 ) Z( 2 3 ) Z 3( 2 3 ). By [N2, Cor ] we have T = U T where T has rank sx, sgnature (1,5) and T has dscrmnant form as before. These data dentfy T unquely ([N2, Corollary ]). Hence t s somorphc to U(3) A 2 A 2 wth generators for the dscrmnant form (e f)/3, (e + f)/3, (A B)/3, (A B )/3, where e,f,a,b,a,b are the usual bases of the lattces. The transcendental lattce T X = U U(3) A 2 A 2 has a unque embeddng n the lattce Λ K3 by [N2, Theorem ] or [Mo, Corollary 2.10] The nvarant lattce and ts orthogonal complement. Proposton 4.2. The nvarant sublattce of the Néron-Sever group s sometrc to U(3) and t s generated by the classes F and s + t 1 + t 2. The nvarant sublattce H 2 (X, Z) σ 3 s sometrc to U U(3) U(3) A2 A 2. Its orthogonal complement Ω 3 := (H 2 (X, Z) σ 3 ) s the negatve defnte twelve dmensonal

11 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 11 lattce {Z 12,M} where M s the blnear form and t s equal to the lattce (NS(X) σ 3 ). The lattce Ω 3 admts a unque prmtve embeddng n the lattce Λ K3. The dscrmnant of Ω 3 s 3 6 and ts dscrmnant form s (Z 3 ( 2 3 )) 6. The sometry σ 3 acts on the dscrmnant group Ω 3 /Ω 3 as the dentty. Proof. It s clear that the sometry σ 3 fxes the classes F and s + t 1 + t 2. These generate a lattce U(3) (wth bass F and F + s + t 1 + t 2 ). The nvarant sublattce H 2 (X, Z) σ 3 contans TX and the nvarant sublattce of the Néron- Sever group. So (H 2 (X, Z) σ 3 ) = (NS(X) σ 3 ), ths lattce has sgnature (0,12) and by [N1, p. 133] the dscrmnant group s (Z/3Z) 6. Hence by [N2, Theorem ] there s a unque prmtve embeddng of (H 2 (X, Z) σ 3 ) n the K3-lattce. By usng the orthogonalty condtons one fnds the followng bass of Ω 3 = (NS(X) σ 3 ) : b 1 = t 2 t 1, b 2 = s t 2, b 3 = F 3C (5) 2, b 2(+1) = C () 1 C () 2, = 1,...,5 b 2j+3 = C (j) 1 C (j+1) 1, j = 1,...,4. An easy computaton shows that the Gram matrx of ths bass s exactly the matrx M whch ndeed has determnant 3 6. Snce H 2 (X, Z) σ 3 TX NS(X) σ 3 = U U(3) U(3) A2 A 2 and these lattces have the same rank, to prove that the ncluson s an equalty we compare ther dscrmnants. The lattce (H 2 (X, Z) σ 3 ) has determnant 3 6. So the lattce (H 2 (X, Z) σ 3 ) has determnant 3 6 (because these are prmtve sublattces of H 2 (X, Z)). The lattce U U(3) U(3) A 2 A 2 has determnant exactly 3 6, so H 2 (X, Z) σ 3 = U U(3) U(3) A2 A 2. Snce NS(X) /NS(X) Ω 3 /Ω 3 the generators of the dscrmnant form of the lattce Ω 3 are classes w 1,...,w 6 wth w 1,...,w 4 the classes whch generate the dscrmnant form of NS(X) (cf. the proof of the Proposton 4.1) and w 5 = 1 3 (b 1 +2b 2 ) = 1 3 (2s t 1 t 2 ) w 6 = 1 3 (b 1 +2b 2 2b 3 ) = 1 3 (2s t 1 t 2 2F +6C (5) 2 ). These sx classes are orthogonal, wth respect to the blnear form takng values n Q/Z, and generate the dscrmnant form. Ther squares are w1 2 = w2 2 = w2 3 = w mod 2Z, w4 2 = w mod 2Z. By replacng w 4, w 6 by w 4 w 6, w 4 + w 6 we obtan the dscrmnant form (Z 3 ( 2 3 )) 6. By computng the mage of w, = 1,...,6 under σ3 one fnds that σ 3 (w ) w Ω 3. For

12 12 ALICE GARBAGNATI AND ALESSANDRA SARTI example: σ3 (w 5) w 5 = 1 3 (2t 1 t 2 s) 1 3 (2s t 1 t 2 ) = t 1 s whch s an element of Ω 3 (n fact t s orthogonal to F and to s + t 1 + t 2 ). Hence the acton of σ3 s trval on Ω 3 /Ω 3 as clamed. In the next two subsectons we apply the results of secton 2 about the Z[ω]-lattces to descrbe the lattce {Ω 3,M} and to prove that Ω 3 s somorphc to the lattce K 12 ( 2), where K 12 s the Coxeter-Todd lattce (cf. e.g. [CT], [CS] for a descrpton of ths lattce) The lattce Ω 3. Let ω 3 be a prmtve thrd root of the unty. In ths secton we prove the followng result (we use the same notatons of secton 2): Theorem 4.2. The lattce Ω 3 s sometrc to the Z-lattce assocated to the Z[ω 3 ]-lattce {L 3, h L3 } where x x j mod (1 ω 3 ), L 3 = (x 1,...,x 6 ) (Z[ω 3 ]) 6 : 6 =1 x 0 mod (1 ω 3 ) 2 and h L3 s the restrcton of the standard hermtan form on Z[ω 3 ] 6. Proof. Let F = F 6 3 be the Z-sublattce of NS(X) generated by C (j) C (j) +1, = 0,1,2, j = 1,...,6 wth blnear form nduced by the ntersecton form on NS(X). Let G = G 6 3 denote the Z[ω 3]-lattce (1 ω 3 ) 2 Z[ω 3 ] 6 wth the standard hermtan form. Ths s a sublattce of Z[ω 3 ] 6. Applyng to each component of G the Lemma 2.1 we know that {G Z,b G } s a Z-lattce sometrc to the lattce F. The explct sometry s gven by C (1) C (1) +1 (1 ω 3) 2 (ω3 1,0,0,0,0,0) C (2) C (2) +1 (1 ω 3) 2 (0,ω3 1,0,0,0,0). C (6) C (6) +1 (1 ω 3) 2 (0,0,0,0,0,ω 1 3 ). The multplcaton by ω 3 of an element (1 ω 3 ) 2 e j (where e j s the canoncal bass) corresponds to a translaton by t 1 on a sngular fber, whch sends the curve C (j) to the curve C (j) +1. Hence we have a commutatve dagram: F G σ 3 ω 3 F G. The elements C (j) C (j) k,,k = 0,1,2, j = 1,...,6 are all contaned n the lattce Ω 3 = (NS(X) σ 3 ), but they do not generate ths lattce. A set of generators for Ω 3 s s t 1, t 1 t 2, C (j) From the formula (10) we obtan that s t 1 = C (k) h,h = 0,1,2, j,k = 1,...,6. 6 [ 1 3 (C(j) 1 C (j) 2 ) σ 3(C (j) 1 C (j) 2 )]. j=1

13 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 13 After the dentfcaton of F wth G Z we have s t 1 = (1 ω 3 ) 2 ( 1 3 (1 + ω 3))(1,1,1,1,1,1) = (1,1,1,1,1,1). The dvsor t 1 t 2, whch s the mage of s t 1 under the acton of σ3, corresponds to the vector (ω 3,ω 3,ω 3,ω 3,ω 3,ω 3 ). Smlarly one can see that the element C (1) 1 C (2) 1 corresponds to the vector (1 ω 3 )(1, 1,0,0,0,0,0) and more n general C (j) C (k) wth j k corresponds to the vector (1 ω 3 )(ω3 1 e j ω3 1 e k ) where e s the standard bass. The lattce L 3 generated by the vectors of G Z and by ω 3 (1,1,1,1,1,1) s thus sometrc to Ω 3. In concluson a bass for L 3 s (1 ω 3)ω 1 3 ( e j + ω 3 e k ) = 0,1,2, j,k = 1,... 6, l 1 = ω 3 (1,1,1,1,1,1) l 2 = ω3 2(1,1,1,1,1,1) = (1 + ω 3)(1,1,1,1,1,1) l 3 = (1 ω 3 ) 2 (0,0,0,0,1 ω 3,0) l 4 = (1 ω 3 ) 2 (1,0,0,0,0,0) l 5 = (1 ω 3 )(1, 1,0,0,0,0) l 6 = (1 ω 3 ) 2 (0,1,0,0,0,0) l 7 = (1 ω 3 )(0,1, 1,0,0,0) l 8 = (1 ω 3 ) 2 (0,0,1,0,0,0) l 9 = (1 ω 3 )(0,0,1, 1,0,0) l 10 = (1 ω 3 ) 2 (0,0,0,1,0,0) l 11 = (1 ω 3 )(0,0,0,1, 1,0) l 12 = (1 ω 3 ) 2 (0,0,0,0,1,0). The dentfcaton between Ω 3 and L 3 s gven by the map b l. After ths dentfcaton the ntersecton form on Ω 3 s exactly the form b L3 on L 3. The bass l of L 3 satsfes the condton gven n the theorem, and so L 3 {(x 1,...,x 6 ) (Z[ω 3 ]) 6 : x x j mod (1 ω 3 ), 6 =1 x 0 mod (1 ω 3 ) 2 }. Snce the vectors (1 ω 3 ) 2 e j, (1 ω 3 )(e e j ) and (1,1,1,1,1,1) generate the Z[ω 3 ]-lattce {(x 1,...,x 6 ) (Z[ω 3 ]) 6 : x x j mod (1 ω 3 ), 6 =1 x 0 mod (1 ω 3 ) 2 } and snce they are all vectors contaned n L 3, the equalty holds The Coxeter-Todd lattce K 12. Theorem 4.3. The lattce Ω 3 s sometrc to the lattce K 12 ( 2). Proof. The lattce K 12 s descrbed by Coxeter and Todd n [CT] and by Conway and Sloane n [CS]. The lattce K 12 s the twelve dmensonal Z-module assocated to a sx dmensonal Z[ω 3 ]-lattce Λ ω 3 6. The Z[ω 3 ]-lattce Λ ω 3 6 s descrbed n [CS] n four dfferent ways. We recall one of them denoted by Λ (3) n [CS, Defnton 2.3], whch s convenent for us. Let θ = ω 3 ω 3, then Λ ω 3 6 s the Z[ω 3 ]-lattce Λ ω 3 6 = {(x 1,...,x 6 ) : x Z[ω 3 ],x x j mod θ, 6 x 0 mod 3} wth hermtan form 3t 1 xȳ. We observe that θ = ω3 (1 ω 3 ). The element ω 3 s a unt n Z[ω 3 ] so the congruence modulo θ s the same as the congruence modulo (1 ω 3 ). Observng that 3 = θ 2 t s then clear that the Z[ω 3 ]-module Λ ω 3 6 s the Z[ω 3 ]-module L 3. The Z-modules K 12 and L 3,Z are somorphc snce they are the twelve dmensonal =1

14 14 ALICE GARBAGNATI AND ALESSANDRA SARTI Z-lattces assocated to the same Z[ω 3 ]-lattce. The blnear form on the Z-module K 12 s gven by and the blnear form on L 3,Z s gven by b K12 (x,y) = 1 3 xȳ = 1 6 Tr(xȳ) b L3 (x,y) = 1 3 Tr(xȳ). So the Z-lattce {L 3,Z,b L3 } s sometrc to K 12 ( 2). Remark. 1) In [CT] Coxeter and Todd gve an explct bass of the Z-lattce K 12. By a drect computaton one can fnd the change of bass between the bass descrbed n [CT] and the bass {b } gven n the proof of Proposton ) The lattce Ω 3 does not contan vectors of norm 2 (cf. [N1, Lemma 4.2]), but has 756 vectors of norm 4, 4032 of norm 6 and of norm 8. Snce these propertes defne the lattce K 12 ( 2), (cf. [CS, Theorem 1]), ths s another way to prove the equalty between Ω 3 and K 12 ( 2). 3) The lattce K 12 ( 2) s generated by vectors of norm 4, [PP, Secton 3] Secton of order fve. Let X be a K3 surface wth an ellptc fbraton whch admts a secton of order fve as descrbed n secton 3. We recall that X has four reducble fbres of type I 5 and four sngular rreducble fbres of type I 1. We have seen that the rank of the Néron-Sever group s 18. We determne now NS(X) and T X. We label the fbers and ther components as descrbed n the secton 3. Let t 1 denote the secton of order fve whch meets the frst sngular fber n C (1) 1. By the formula (4) of secton 3 up to permutaton of the fbers only the followng stuatons are possble: t 1 C (1) 1 = t 1 C (2) 1 = t 1 C (3) 2 = t 1 C (4) 2 = 1 and t 1 C (j) = 0 otherwse; or t 1 C (1) 1 = t 1 C (2) 4 = t 1 C (3) 2 = t 1 C (4) 3 = 1 and t 1 C (j) = 0 otherwse. Observe that these two cases descrbe the same stuaton f we change the orentaton on the last two fbers, so we assume to be n the frst case. Let σ 5 be the automorphsm of order fve whch leaves each fber nvarant and s the translaton by t 1, so σ 5 (s) = t 1, σ 5 (t 1) = t 2, σ 5 (t 2) = t 3, σ 5 (t 3) = t 4, σ 5 (t 4) = s. Proposton 4.3. A Z-bass for the lattce NS(X) s gven by s,t 1,t 2,t 3,t 4,F,C (1) 1,C(1) 2,C(1) 3,C(1) 4,C(2) 1,C(2) 2,C(2) 3,C(2) 4,C(3) 1,C(3) 2,C(3) 3,C(3) 4. Let U A 4 4 be the lattce generated by the secton, the fber and the rreducble components of the four fbers I 5 whch do not ntersect the zero secton s. It has ndex fve n the Néron-Sever group of X, NS(X). The lattce NS(X) has dscrmnant 5 2 and ts dscrmnant form s The transcendental lattce s Z 5 ( 2 5 ) Z 5( 2 5 ). T X = U U(5) and has a unque prmtve embeddng n the lattce Λ K3.

15 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 15 Proof. The proof s smlar to the proof of Proposton 4.1. So we sketch t brefly. The classes s,f,c (j), = 1,...,4, j = 1,...,4, generate U A 4 4. By usng the ntersecton form, or by the result of [M2, p. 299], we fnd [ t 1 = s + 2F =1 (4C() 1 + 3C () 2 + 2C () 3 + C () 4 (12) )+ + ] 4 j=3 (3C(j) 1 + 6C (j) 2 + 4C (j) 3 + 2C (j) 4 ). A Z-bass s s,t 1,t 2,t 3,t 4,F,C (1) 1,C(1) 2,C(1) 3,C(1) 4,C(2) 1,C(2) 2, C(2) 3, C(2),C(3) 1,C(3) 2,C(3) 3,C(3) 4. Snce d(ns(x)) = 5 2 and d(u A 4 4 ) = 54, the ndex of U A 4 4 n NS(X) s fve. Let w 1 and w 2 be w 1 = 1 5 (2C(1) 1 + 4C (1) 2 + C (1) 3 + 3C (1) 4 + 4C (3) 1 + 3C (3) 2 + 2C (3) 3 + C (3) 4 ); w 2 = 1 5 (3C(2) 1 + C (2) 2 + 4C (2) 3 + 2C (2) 4 + C (3) 1 + 2C (3) 2 + 3C (3) 3 + 4C (3) 4 ). The classes v 1 = w 1 w 2, v 2 = w 1 +w 2 are orthogonal classes and generate the dscrmnant group of NS(X), the dscrmnant form s Z 5 ( 2 5 ) Z 5( 2 5 ). The transcendental lattce T X has rank four, sgnature (2,2) and dscrmnant form Z 5 ( 2 5 ) Z 5( 2 5 ). Snce n ths case T X s unquely determned by sgnature and dscrmnant form (cf. [N2, Corollary ]) ths s the lattce T X = U U(5). The transcendental lattce has a unque embeddng n the lattce Λ K3 by [N2, Theorem ] or [Mo, Corollary 2.10] The nvarant lattce and ts orthogonal complement. Proposton 4.4. The nvarant sublattce of the Néron-Sever lattce s sometrc to the lattce U(5) and t s generated by the classes F and s + t 1 + t 2 + t 3 + t 4. The nvarant lattce H 2 (X, Z) σ 5 s sometrc to U U(5) U(5) and ts orthogonal complement Ω 5 = (H 2 (X, Z) σ 5 ) s the negatve defnte sxteen dmensonal lattce {Z 16,M} where M s the blnear form

16 16 ALICE GARBAGNATI AND ALESSANDRA SARTI and t s equal to the lattce (NS(X) σ 5 ). The lattce Ω 5 admts a unque prmtve embeddng n the lattce Λ K3. The dscrmnant of Ω 5 s 5 4 and ts dscrmnant form s (Z 5 ( 2 5 )) 4. The sometry σ 5 acts on the dscrmnant group Ω 5 /Ω 5 as the dentty. Proof. As n the case of an ellptc fbraton wth a secton of order three, t s clear that σ 5 fxes the classes F and s + t 1 + t 2 + t 3 + t 4. These classes generate the lattce U(5), and so H 2 (X, Z) σ 5 U(5) TX = U(5) U(5) U. Usng Nkuln s result n [N1, p. 133] we fnd that the lattce H 2 (X, Z) σ 5 has determnant 5 4, whch s exactly the determnant of U(5) U(5) U. Snce these have the same rank, we conclude that H 2 (X, Z) σ 5 = U(5) U(5) U. The orthogonal complement (H 2 (X, Z) σ 5 ) s equal to (NS(X) σ 5 ) as n Proposton 4.2. It has sgnature (0,16) and by [N1, p. 133] the dscrmnant group s (Z/5Z) 4. Hence by [N2, Theorem ] there s a unque prmtve embeddng of (H 2 (X, Z) σ 5 ) n the K3-lattce. By usng the orthogonalty condtons one fnds the followng bass of Ω 5 = (NS(X) σ 5 ) : b 1 = s t 1, b 2 = t 1 t 2, b 3 = t 2 t 3, b 4 = t 3 t 4, b 5 = F 5C (3) 4, b = C (1) 5 C(1) 4, = 6,7,8, b 9 = C (1) 1 C (2) 1, b = C (2) 9 C(2) 8, = 10,11,12, b 13 = C (2) 1 C (3) 1, b = C (3) 13 C(3) 12, = 14,15,16. The Gram matrx of ths bass s exactly the matrx M. The generators of the dscrmnant group of Ω 5 are the classes v 1, v 2 of the dscrmnant form of NS(X) and the classes v 3 = 1 5 (b 3 + 2b 1 + 3b 4 + 4b 2 ), v 4 = 1 5 (b 3 + 2b 1 + 3b 4 + 4b 2 b 5 ). These have v3 2 = 2/5 mod 2Z, v2 4 = 2/5 mod 2Z. The generators v 1, 2v 2 4v 3 v 4, 2v 3, v 4 are orthogonal to each other and have self-ntersecton 2/ The lattce Ω 5. Let ω 5 be a prmtve ffth root of the unty. In ths secton we prove the followng result Theorem 4.4. The lattce Ω 5 s sometrc to the Z-lattce assocated to the Z[ω 5 ]-lattce {L 5, h L5 } where x 1 x 2 2x 3 2x 4 mod (1 ω 5 ) L 5 = (x 1,...,x 4 ) (Z[ω 5 ]) 4 : (3 ω 5 )x 1 + (3 ω 5 )x 2 + x 3 + x 4 0 mod (1 ω 5 ) 2 wth the hermtan form (13) h L5 (α,β) = 2 α β + =1 4 fα j fβ j = j=3 2 α β + τ =1 4 α j β j, where α,β L 5 Z[ω 5 ] 4, f = 1 (ω ω3 5 ) and τ = ff = 2 3(ω2 5 + ω3 5 ). Proof. The strategy of the proof s the same as n the case wth an automorphsm of order three, but the stuaton s more complcated because the secton t 1 does not meet all the fbers I 5 n the same component. For ths reason the hermtan form of the Z[ω 5 ]-lattce L 5 s not the standard hermtan form on all the components. It s possble to repeat the j=3

17 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 17 constructon used n the case of order three, but wth the hermtan form (13). We explan now how we fnd ths hermtan form. Let F := F5 4 be the lattce generated by the elements C(j) C (j) +1, = 0,...,4, j = 1,...,4. Ths s a sublattce of (NS(X) σ5 ). A bass s d 1+ = (σ5 ) (C (1) 1 C (1) 2 ), d 5+ = (σ5 ) (C (2) 1 C (2) 2 ), d 9+ = (σ5 ) (C (3) 1 C (3) 2 ), d 13+ = (σ5 ) (C (4) 1 C (4) ), = 0,...,3 2 and the blnear form s thus the dagonal block matrx Q = dag(a,a,b,b) A = , B = We want to dentfy the multplcaton by ω 5 n the lattce G wth the acton of the sometry σ 5 on the lattce F. We consder the Z[ω 5]-module G = (1 ω) 2 Z[ω] 4. Now we consder the Z-module G Z. The map φ : (σ 5 ) (C (1) 1 C (1) 2 ) (1 ω 5) 2 ω 5 (1,0,0,0) (σ 5 ) (C (2) 1 C (2) 2 ) (1 ω 5) 2 ω 5 (0,1,0,0) (σ 5 ) (C (3) 1 C (3) 2 ) (1 ω 5) 2 ω 5 (0,0,1,0) (σ 5 ) (C (4) 1 C (4) 2 ) (1 ω 5) 2 ω 5 (0,0,0,1) s an somorphsm between the Z-modules G Z and F. Now we have to fnd a blnear form b G on G such that {G Z,b G } s sometrc to {F,Q}. On the frst and second fber the acton of σ5 s σ 5 (C(j) ) = C (j) +1, j = 1,2, = 0,... 4, so (σ5 ) (C (j) 1 C (j) 2 ) = C(j) +1 C(j) +2. Hence the map φ operates on the frst two fbers n the followng way: φ : C (1) +1 C(1) +2 (1 ω 5 ) 2 ω5 (1,0,0,0) C (2) +1 C(2) +2 (1 ω 5 ) 2 ω5 (0,1,0,0). Ths dentfcaton s exactly the dentfcaton descrbed n Lemma 2.1, so on these generators of the lattces F and G we can choose exactly the form descrbed n the lemma. On the thrd and fourth fber the acton of σ5 s dfferent (because σ 5 s the translaton by t 1 and t meets the frst and second fber n the component C 1 and the thrd and fourth fber n the component C 2 ). In fact (σ5 ) (C (j) 1 C (j) 2 ) = C(j) 2+1 C(j) 2+2, j = 3,4, = 0,...,4 and so (14) φ : C (3) 2+1 C(3) 2+2 (1 ω 5 ) 2 ω5 (0,0,1,0) C (4) 2+1 C(4) 2+2 (1 ω 5 ) 2 ω5 (0,0,0,1). A drect verfcaton shows that the map φ defnes an sometry between the module generated by (σp) (C (j) 1 C (j) 2 ), = 0,...,4 and (1 ω 5) 2 Z[ω 5 ], (j = 3,4) f one consders on (1 ω 5 ) 2 Z[ω 5 ] the blnear form assocated to the hermtan form h(α,β) = ταβ where τ = (2 3(ω5 2+ω3 5 )). The real number τ s the square of f = 1 (ω2 5 +ω3 5 ), so the hermtan form above s also h(α,β) = ταβ = fαfβ. So now we consder the Z[ω 5 ]-lattce Z[ω 5 ] 4 wth the hermtan form h gven n (13) and G as a sublattce of {Z[ω 5 ] 4,h}. We show that L 5 = Ω 5. We have to add to the lattce F some classes to obtan the lattce Ω 5, and so we have to add some vectors to the lattce G to obtan the lattce L 5. It s suffcent to

18 18 ALICE GARBAGNATI AND ALESSANDRA SARTI add to F the classes s t 1, C (1) 1 C (2) 1, C(2) 1 C (3) 1, C(3) 1 C (4) 1 and ther mages under σ5. These classes correspond to the followng vectors n Z[ω 5] 4 : s t 1 = (1,1,c,c), C (1) 1 C (2) 1 = (1 ω 5 )(1, 1,0,0), C (2) 1 C (3) 1 = (1 ω 5 )(0,1, (1 + ω 3 5 ),0), C (3) 1 C (4) 1 = (1 ω 5 )(0,0,(1 + ω 3 5 ), (1 + ω3 5 )) where c = ω 5 (2ω 2 5 ω 5 + 2). A bass for the lattce L 5 s then l 1 = (1,1,c,c) l 2 = ω 5 l 1 l 3 = ω5 2l 1 l 4 = ω5 3l 1 l 5 = (1 ω 5 ) 2 (0,0,2 + 4ω 5 + ω ω3 5,0) l 6 = (1 ω 5 ) 2 (1,0,0,0) l 7 = ω 5 l 6 l 8 = ω5 2l 6 l 9 = (1 ω 5 )(1, 1,0,0) l 10 = (1 ω 5 ) 2 (0,1,0,0) l 11 = ω 5 l 10 l 12 = ω5 2l 10 l 13 = (1 ω 5 )(0,1, (1 + ω5 3),0) l 14 = (1 ω 5 ) 2 (0,0,1,0) l 15 = ω 5 l 14 l 16 = ω5 2l 14. The dentfcaton between Ω 5 and L 5 s gven by the map b l. After ths dentfcaton the ntersecton form on Ω 5 s exactly the form b L5 on L 5. Remark. 1) We recall that the densty of a lattce L of rank n s = V n / det L where V n s the volume of the n dmensonal sphere of radus r (called packng radus of the lattce), V n = r n π n/2 /(n/2)!, r = µ/2 and µ s the mnmal norm of a vector of the lattce. The densty of Ω 5 s = π8 1 8! ) The lattce Ω 5 does not admt vectors of norm 2 and can be generated by vectors of norm 4, and a bass s b 1, b 2, b 3, b 4, b 5 b 13 2b 14 3b 15 4b 16, b 6, b 7, b 8, b 9, b 10 + b 11, b 11 + b 12, b 10 + b 11 + b 12, b 13, b 14 + b 15, b 15 + b 16, b 14 + b 15 + b Secton of order seven. Let X be a K3 surface wth an ellptc fbraton whch admts a secton of order seven as descrbed n secton 3. We recall that X has three reducble fbres of type I 7 and three sngular rreducble fbres of type I 1. We have seen that the rank of the Néron-Sever group s 20. We determne now NS(X) and T X. We label the fbers and ther components as descrbed n the secton 3. Let t 1 denote the secton of order seven whch meets the frst fber n C (1) 1. Agan by the formula (4) of secton 3 we have t 1 C (1) 1 = 1, t 1 C (2) 2 = 1, t 1 C (3) 3 = 1, and t 1 C (j) = 0 otherwse. Let σ 7 denote the automorphsm of order seven whch leaves each fber nvarant and s the translaton by t 1, so σ7 (s) = t 1, σ7 (t 1) = t 2, σ7 (t 2) = t 3, σ7 (t 3) = t 4, σ7 (t 4) = t 5, σ7 (t 5) = t 6, σ7 (t 6) = s. The proofs of the next two propostons are very smlar to those of the smlar propostons n the case of the automorphsms of order three and fve, so we omt them. Proposton 4.5. A Z-bass for the lattce NS(X) s gven by s,t 1,t 2,t 3,t 4,t 5,t 6,F,C (1) 1,C(1) 2,C(1) 3,C(1) 4,C(1) 5,C(1) 6,C(2) 1,C(2) 2,C(2) 3,C(2) 4,C(2) 5,C(2) 6. Let U A 3 6 be the lattce generated by the secton, the fber and the rreducble components of the three fbers I 7 whch do not ntersect the zero secton s. It has ndex seven n the

19 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES 19 Néron-Sever group of X, NS(X). The lattce NS(X) has dscrmnant 7 and ts dscrmnant form s Z 7 ( 4 7 ). The transcendental lattce T X s the lattce {Z 2,Υ} where ( ) 4 1 Υ := 1 2 and t has a unque prmtve embeddng n the lattce Λ K The nvarant lattce and ts orthogonal complement. Proposton 4.6. The nvarant sublattce of the Néron-Sever lattce s sometrc to the lattce U(7) and t s generated by the classes F and s + t 1 + t 2 + t 3 + t 4 + t 5 + t 6. The nvarant lattce H 2 (X, Z) σ 7 s sometrc to U(7) TX. Its orthogonal complement Ω 7 := (H 2 (X, Z) σ 7 ) s the negatve defnte eghteen dmensonal lattce {Z 18,M} where M s the blnear form and t s equal to the lattce (NS(X) σ 7 ). The lattce Ω 7 admts a unque prmtve embeddng n the lattce Λ K3. The dscrmnant of Ω 7 s 7 3 and ts dscrmnant form s (Z 7 ( 4 7 )) 3. The sometry σ 7 acts on the dscrmnant group Ω 7 /Ω 7 as the dentty. The bass of (NS(X) σ 7 ) assocated to the matrx M s b 1 = s t 1, b 2 = t 1 t 2, b 3 = t 2 t 3, b 4 = t 3 t 4, b 5 = t 4 t 5, b 6 = t 5 t 6 b 7 = F 7C (2) 6, b = C (1) 7 C(1) 6, = 8,...,12, b 13 = C (1) 1 C (2) 1, b = C (2) 13 C(2) 8, = 14,..., The lattce Ω 7. Let ω 7 be a prmtve seventh root of the unty. In ths secton we prove the followng result Theorem 4.5. The lattce Ω 7 s sometrc to the Z-lattce assocated to the Z[ω 7 ]-lattce {L 7, h L7 } where x 1 x 2 6x 3 mod (1 ω 7 ), L 7 = (x 1,x 2,x 3 ) (Z[ω 7 ]) 3 : (1 + 5ω 7 )x 1 + 3x 2 + 2x 3 0 mod (1 ω 7 ) 2

20 20 ALICE GARBAGNATI AND ALESSANDRA SARTI wth the hermtan form (15) h L7 (α,β) = α 1 β1 + f 1 α 2 f 1 β 2 + f 2 α 3 f 2 β 3, where f 1 = 3 + 2(ω 7 + ω 6 7 ) + (ω2 7 + ω5 7 ), f 2 = 2 + (ω 7 + ω 6 7 ). Proof. As n the prevous cases we defne the lattce F := F7 3. We consder the hermtan form h(α,β) = α 1 β1 + f 1 α 2 f 1 β 2 + f 2 α 3 f 2 β 3 on the lattce Z[ω 7 ] 3, and defne G to be the sublattce G = (1 ω 7 ) 2 Z[ω 7 ] 3 of {Z[ω 7 ] 3,h}. The map φ : F G φ : (σ 7 ) (C (1) 1 C (1) 2 ) (1 ω 7) 2 ω 7 (1,0,0) (σ 7 ) (C (2) 1 C (2) 2 ) (1 ω 7) 2 ω 7 (0,1,0) (σ 7 ) (C (3) 1 C (3) 2 ) (1 ω 7) 2 ω 7 (0,0,1) s an somorphsm between the Z-lattce G Z, wth the blnear form nduced by the hermtan form, and F wth the ntersecton form. We have to add to G some vectors to fnd a lattce L 7 somorphc to Ω 7. These vectors are s t 1 = (1,c,k), C (1) 1 C (2) 1 = (1 ω 7 )(1, (1 + ω7 4),0), C (2) 1 C (3) 1 = (1 ω 7 )(0,(1 + ω7 4), (1 + ω3 7 + ω5 7 )), where c = 1 + 3ω 7 + 3ω7 4 + ω5 7 and k = 5 + ω 7 5ω7 2 3ω4 7 3ω5 7. A bass for the lattce L 7 s l 1 = (1,c,k) l 2 = ω 7 l 1 l 3 = ω7 2l 1 l 4 = ω7 3l 1 l 5 = ω7 4l 1 l 6 = ω7 5l 1 l 7 = (1 ω 7 ) 2 (0,2 + 4ω 7 + 6ω7 2 + ω ω ω5 7,0) l 8 = (1 ω 7 ) 2 (1,0,0) l 9 = ω 7 l 8 l 10 = ω7 2l 8 l 11 = ω7 3l 8 l 12 = ω7 4l 8 l 13 = (1 ω 7 )(1, (1 + ω7 4),0) l 14 = (1 ω 7 ) 2 (0,1,0) l 15 = ω 7 l 14 l 16 = ω7 2l 14 l 17 = ω7 3l 14 l 18 = ω7 4l 14 The dentfcaton between Ω 7 and L 7 s gven by the map b l. After ths dentfcaton the ntersecton form on Ω 7 s exactly the form b L7 on L 7 nduced by the hermtan form (15).. Remark. 1) The densty of Ω 7 s = π9 9! ) As n the prevous cases the lattce Ω 7 does not admt vectors of norm 2 and can be generated by vectors of norm 4, and a bass s b 1, b 2, b 3, b 4, b 5, b 6, b 7 b 13 2b 14 3b 15 4b 16 5b 17 6b 18, b 8 + b 9, b 9 + b 10, b 10 + b 11, b 11 + b 12, b 10 + b 11 + b 12, b 13, b 14 + b 15, b 15 + b 16, b 16 + b 17, b 17 + b 18, b 16 + b 17 + b 18.

21 SYMPLECTIC AUTOMORPHISMS OF PRIME ORDER ON K3 SURFACES Famles of K3 surfaces wth a symplectc automorphsm of order p In the prevous sectons we used ellptc K3 surfaces to descrbe some propertes of the automorphsm σ p. All these K3 surfaces have Pcard number ρ p + 1, where ρ p s the mnmal Pcard number found n the Proposton 1.1. In ths secton we want to descrbe algebrac K3 surfaces wth symplectc automorphsm of order p and wth the mnmal possble Pcard number. Recall that the values of ρ p are p ρ p , and Ω p denote the lattces descrbed n the sectons 4.1, 4.4, 4.6. Proposton 5.1. Let X be a K3 surface wth symplectc automorphsm of order p = 3,5,7 and Pcard number ρ p as above. Let L be a generator of Ω p NS(X), wth L 2 = 2d > 0 and let L p 2d := ZL Ω p. Then we may assume that L s ample and (1) f L 2 2,4,...,2(p 1) mod 2p, then L p 2d = NS(X), (2) f L 2 0 mod 2p, then ether L p 2d = NS(X) or NS(X) = L p 2d wth L p 2d /Lp 2d Z/pZ and n partcular L p 2d s generated by an element (L/p,v/p) wth v2 0 mod 2p and L 2 + v 2 0 mod 2p 2. Proof. Snce L 2 > 0 by Remann Roch theorem we can assume L or L effectve. Hence we assume L effectve. Let N be an effectve ( 2) curve then N = αl + v, wth v Ω p and α > 0 snce Ω p do not contans ( 2)-curves. We have L N = αl 2 > 0, and so L s ample. Moreover recall that L and Ω p are prmtve sublattces of NS(X). Snce the dscrmnant group of L p 2d := ZL Ω p s (Z/2dZ) (Z/pZ) np, wth n 3 = 6, n 5 = 4, n 7 = 3 an element n NS(X) not n L p 2d s of the form (αl/2d,v/p), v Ω p and satsfy the followng condtons: (a) p (αl/2d,v/p) NS(X), (b) (αl/2d,v/p) L Z, (c) (αl/2d,v/p) 2 Z. By usng the condton (a) we obtan p (αl/2d,v/p) (0,v) NS(X) and so pαl 2d NS(X). Hence by the prmtvty of L n NS(X) follows that d 0 mod p, d = pd, d Z >0 and so αl 2d NS(X) whch gves α = 2d and the class (f there s) s (L/p,v/p). Now condton (b) gves (L/p,v/p) L = L 2 /p Z and so L 2 = 2p r, r Z >0, snce the lattce s even. And so f NS(X) = L p 2d, then L2 0 mod 2p. Fnally condton (c) gves (L/p,v/p) 2 = L2 + v 2 and so snce a square s even L 2 + v 2 0 mod 2p 2. p 2