The automorphism group of a generic Jacobian Kummer surface

Shigeyuki Kondō^{1}

Mathematics subject classification: 14J28, 14J50

Abstruct

We shall prove that the automorphism group of the Kummer surface associated with a generic curve of genus two is generated by classical involutions and Keum’s new automorphisms. The proof depends on a description of a fundamental domain of the reflection group of even unimodular lattice of signature (1,25) and a geometry of Leech lattice.

Introduction

Let C be a smooth curve of genus two over the complex number field. The Jacobian variety J(C) of C has a natural involution ι and the quotient variety J(C)/ < ι > has sixteen ordinary nodes. This surface can be embedded as a quartic surface F in P^{3}. The minimal resolution X of J(C)/ < ι > is called the Kummer surface associated with C which is a K3 surface, that is, a simply connected complex surface with trivial canonical line bundle. The X contains sixteen mutually disjoint smooth rational curves {N_{α}} obtained by the resolution of singularities. Also X contains another sixteen mutually disjoint smooth rational curves {T_{α}} which are the images of the theta divisor and its translations by 2-torsion points. The configuration of these thirty-two curves is called Kummer (16)_{6} configuration. We call X generic if the Néron-Severi group of J(C) is generated by the class of C.

At the last century it was known that a generic Kummer surface X has many involutions, that is, sixteen translations induced by those of J(C) by 2-torsion points, sixteen projections of F from nodes, sixteen corellations by means of the tangent plane collinear to a trope, and a switch defined by the dual map of F (Klein [9]). In 1900, Hutchinson [6] found another sixty involutions associated with Göpel tetrads. Since Hutchinson, no other automorphisms have been provided, and it was conjectured that the automorphism group of X is generated by the above involutions.

However, recently, Keum [8] has given new 192 automorphisms of infinite order which are not included in the group generated by classical involutions. The purpose of this paper is to prove:

1 Supported by Max-Planck-Institute in Bonn Theorem. The automorphism group of a generic Kummer surface is generated by classical involutions and Keum’s automorphisms.

The proof depends on a description of the reflection group of even unimodular lattice of signature (1,25) due to Conway [3] and is inspired by a paper of Borcherds [1] in which he calculated a fundamental domain of the reflection group of several lattices of sigunature (1, r) and its automorphism group.

In the following we shall give a sketch of our proof. Let S = S_{X} be the Picard lattice of X which has signature (1, 16) and determinant 64. Then S can be primitively embedded into an even unimodular lattice L of signature (1,25) such that the orthogonal complement T of S contains a root sublattice isomorphic to A_{3} ⊕ A_{1}^{⊕6} (Lemma 4.1). A fundamental domain D of the reflection group of L is described explicitly by Conway [3] in terms of Leech lattice (see Proposition 2.5). After Borcherds, we consider the restriction D′ = D ∩ P (S) where P (S) is a connected component of the set {x ∈ S ⊗ R : ⟨x, x⟩ > 0}. Then we can determine the hyperplanes which bound D′ properly by using a geometry of Leech lattice due to Conway [2]. There are 316 (= 32 + 32 + 60 + 192) such hyperplanes perpendicular to (−2)-, (−4)-, (−4)- or (−12)-vectors in S respectively (Lemmas 4.6 and 4.8). These hyperplanes naturally correspond to thirty-two smooth rational curves on X consisting Kummer (16)_{6} configuration, sixteen projections and sixteen corellations, sixty Cremona transformations and 192 Keum’s automorphisms respectively (§7). Also the projection of the Weyl vector of D corresponds to the ample divisor Σ(N_{α} + T_{α})/4 on X (Lemma 7.1). Here we remark that each Keum’s automorphism κ is not a reflection (it has infinite order), but it works like a reflection. More precisely it sends the half-space defined by the corresponding (−12)-vector to the opposite side of the half-space defined by the (−12)-vector which defines the inverse of κ (Lemma 6.5). We can also see that Aut(D′) is an extension of (Z/2Z)^{5} by Sp(4, F_{2}) where (Z/2Z)^{5} is generated by sixteen translations and a switch, and Sp(4, F_{2}) ≃ S_{6} is naturally isomorphic to the group of permutations of six Weierstrass points of the curve C (Lemma 7.2). Moreover D′ is like a fundamental domain of the group generated by reflections corresponding to (−2)-vectors in S, projections, corellations, Cremona transformations and Keum’s automorphisms (Lemma 7.3). We remark that the reflection group of S_{X} (corresponding to (−2)-, (−4)-vectors) is not of finite index in O(S_{X}).

Acknowledgements.

This paper was written during the author’s stay at the Max-Planck-Insitut für Mathematik in Bonn 1996-1997. He express his heartly thanks to the Insitut for giving him the opportunity of this work in the stimulating working atmosphere. He also thanks Professor Jong Keum for sending him an unpublished manuscript and for valuable conversations.

§1. Lattices

A lattice L is a free Z-module of finite rank endowed with an integral symmetric bilinear form ⟨ , ⟩. If L_{1} and L_{2} are lattices, then L_{1}⊕L_{2} denotes the orthogonal direct sum of L_{1} and L_{2}. Also we denote by L^{⊕m} the orthogonal direct sum of m-copies of L. An isomorphism of lattices preserving the bilinear forms is called an isometry. For a lattice L, we denote by O(L) the group of self-isometries of L. A sublattice S of L is called primitive if L/S is torsion free.

A lattice L is even if ⟨x, x⟩ is even for each x ∈ L. A lattice L is non-degenerate if the discriminant d(L) of its bilinear form is non zero, and unimodular if d(L) = ±1. If L is a non-degenerate lattice, the signature of L is a pair (t_{+}, t_{−}) where t_{±} denotes the multiplicity of the eigenvalues ±1 for the quadratic form on L⊗R.

Let L be a non-degenerate even lattice. The bilinear form of L determines a canonical embedding L ⊂ L^{∗} = Hom(L, Z). The factor group L^{∗}/L, which is denoted by A_{L}, is an abelian group of order |d(L)|. We denote by l(L) the number of minimal generators of A_{L}. We extend the bilinear form on L to the one on L^{∗}, taking value in Q, and define

q_{L} : A_{L} → Q/2Z, q_{L}(x + L) = ⟨x, x⟩ + 2Z (x ∈ L^{∗}).

We call q_{L} the discriminant quadratic form of L.

Let A be a finite abelian group. A non-degenerate finite quadratic form is a map q : A ⟶ Q/2Z satisfying the following conditions:

(1) q(na) = n^{2}q(a) for all n ∈ Z and a ∈ A.
(2) q(a + a′) − q(a) − q(a′) ≡ 2b(a, a′) mod 2Z,

where b : A × A ⟶ Q/2Z is a non-degenerate symmetric bilinear form. We denote by u_{2} (resp. v_{2}) a finite quadratic form on (Z/2Z)^{⊕2} defined by q(x) = q(y) = 0, b(x, y) = 1/2 (resp. q(x) = q(y) = 1, b(x, y) = 1/2). We also denote by q_{a}(2^{n}) a finite quadratic form on (Z/2^{n}Z) defined by q_{a}(2^{n})(x) = a/2^{n}. It is easy to see that u_{2}^{⊕2} ≃ v_{2}^{⊕2}.

We denote by U the hyperbolic lattice defined by ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) which is an even unimodular

lattice of signature (1,1), and by A_{m}, D_{n} or E_{l} an even negative definite lattice associated with the Cartan matrix of type A_{m}, D_{n} or E_{l} (m ≥ 1, n ≥ 4, l = 6,7,8). A root of an even lattice L is a (−2)-element in L. A root lattice of a negative definite lattice L is the sublattice of L generated by all roots in L, which is isometric to a direct sum of A_{m}, D_{n}, E_{l}. For a lattice L and an integer m, L(m) is the lattice whose bilinear from is the one on L multiplied by m. For example q_{A_{1}} ≃ q_{−1}(2), q_{A_{3}} ≃ q_{−3}(4), q_{D_{4}} ≃ v_{2} and q_{U(2)} ≃ u_{2}.

Next we recall a structure of the orthogonal group O(L) of an even lattice L of signature (1, r−1). Let P (L) be a connected component of the set {x ∈ L⊗R : ⟨x, x⟩ > 0}. Let Δ be the set of all roots in L. For each δ, an isometry s_{δ} defined by

s_{δ} : x → x + ⟨x, δ⟩δ

is called a reflection associated with δ. Let W (L)^{(2)} be the group generated by all reflections associated with δ ∈ Δ. Let H_{δ} be the hyperplane orthogonal to δ ∈ Δ. Then { H_{δ} : δ ∈ Δ } divide P (L) into regions. The connected components of P (L) \ (⋃ H_{δ}) are called the chambers each of which is a fundamental domain with respect to the action of W (L)^{(2)} on P (L). Fix a chamber D(L). Then we have a partition Δ = Δ^{+} ∪ Δ^{−} such that Δ^{+} consists of roots δ with ⟨δ, ρ⟩ > 0 where ρ is an interior point of D(L). Let G(L) be the group of symmetries of D(L). Then O(L) is a split extension of {±1} · W (L)^{(2)} by G(L). For more details we refer the reader to Vinberg [15].

§2. Leech lattice

Let Ω be the projective line P L(23) over the field F_{23} : Ω = {∞, 0, 1, ..., 23}. We consider the set P (Ω) of all subsets of Ω with symmetric difference as a 24-dimensional vector space over F_{2}. Let C be the binary Goray code which is a 12-dimensional subspace of P (Ω) defined by the 24 sets N_{i} where N = Ω \ {x^{2} : x ∈ F_{23}}, N_{∞} = Ω and N_{i} = {n − i : n ∈ N} for i ≠ ∞ (see Conway [2]). We call a set in C a C-set. A C-set consists of 0, 8, 12 or 16 elements. A 8-elements C-set (resp. 12-elements C-set) is called an octad (resp. dodecad), and a system of 6 tetrads is called a sextet if it satisfies that the union of any two tetrads is an octad. We denote by C(8) (resp. C(12)) the set of all octads (resp. dodecads).

Proposition 2.1 (Conway [2], §3, Tables 1, 2). (1) The number of octads through m-points is 759, 253, 77, 21, 5 or 1 for m = 0, 1, 2, 3, 4 or 5 respectively.

(2) The number of dodecads through m-points is 2576, 1288, 616, 280, 120, 48 or 16 for m = 0, 1, 2, 3, 4, 5 or 6 respectively.

(3) For any set of 4 elements, there exists a unique sextet.

The stabilizer subgroup of the binary Goray code in the permutation group of Ω is called the Mathieu group (denoted by M_{24}) which is a sporadic simple group of order 2^{10}·3^{3}·5·7·11·23.

Next we recall the Leech lattice. Let R^{24} be spanned by the orthogonal basis ν_{i} (i ∈ Ω), and define ν_{S} to be Σν_{i}(i ∈ Ω) for S ⊂ Ω. Let Λ be the lattice spanned by the vectors 2ν_{K} for K ∈ C(8) and ν_{Ω} − 4ν_{∞}.

Proposition 2.2 (Conway [2], §4, Theorem 2). The vector (ξ_{∞}, ξ_{0}, ..., ξ_{23}) is in Λ if and only if

(i) the coordinates ξ_{i} are all congruent modulo 2, to some m;

(ii) the set of i for which ξ_{i} takes any given value modulo 4 is a C-set;

(iii) the coordinate-sum is congruent to 4m modulo 8.

For x, y ∈ Λ the scalar product x·y is a multiple of 8 and x·x is a multiple of 16.

For x, y ∈ Λ, define ⟨x, y⟩ = −x·y/8. Then (Λ, ⟨, ⟩) is an even negative-definite unimodular lattice which is called the Leech lattice. We denote by Λ_{n} the set of all vectors x in Λ with ⟨x, x⟩ = −n. We remark that Λ_{2} is empty.

Proposition 2.3 (Conway-Sloane [4], p.133, Table 4.13). The complete lists of Λ_{4}, Λ_{6} are as follows :

Λ_{4} = {(±2^{8}, 0^{16}), (±3, ±1^{23}), (±4^{2}, 0^{22})},
Λ_{6} = {(±2^{12}, 0^{12}), (±3^{3}, ±1^{21}), (±4, ±2^{8}, 0^{15}), (±5, ±1^{23})}

where the signs are taken to satisfy the conditions in Proposition 2.2.

Finally we recall a structure of the orthogonal group of even unimodular lattice of signature (1, 25). In the following in this paper we write L for an even unimodular lattice of signature (1, 25), which is unique up to isomorphisms and isomorphic to Λ ⊕ U. Fix a decomposition L = Λ ⊕ U. We write (λ, m, n) for a vector in L where λ in Λ, m, n are integers, the norm is given by 2mn + ⟨λ, λ⟩. If w = (0, 0, 1), then a root r in L with ⟨r, w⟩ = 1 is called a Leech root and w is called a Weyl vector. The set of all Leech roots bijectively corresponds to the set Λ as follows (Conway-Sloane [4], Chap. 26, Theorem 3):

L ∋ r = (λ, 1, −1 − ⟨λ, λ⟩/2) ⟷ λ ∈ Λ.

Remark 2.4. For Leech roots r, r′ ∈ L and the corresponding vectors λ, λ′ in Λ,

⟨r, r′⟩ = \{ \begin{matrix}0, & if λ − λ′ ∈ Λ_{4}; \\ 1, & if λ − λ′ ∈ Λ_{6}. \end{matrix}

Put

D = {x ∈ P (L) | ⟨x, r⟩ > 0 for any Leech root r}.

Proposition 2.5 (Conway [3]). D is a fundamental domain of the reflection group W (L)^{(2)} and O(L) is a split extension of W (L)^{(2)} by Aut(D) so that Aut(D) is isomorphic to the group ·∞ of affine automorphisms of Λ, which is a split extension of Z^{24} by ·0 = O(Λ)

§3. Kummer surfaces

Let C be a compact Riemann surface of genus two. There is a double covering π : C ⟶ P^{1} ramified at six points p_{0}, ..., p_{5}. We denote by ι the covering transformation of π. Let J(C) be the Jacobian variety of C. The map

φ : C ⟶ J(C)

defined by φ(p) = [p − p_{0}] gives an embedding of C into J(C), and the points

μ_{i} = [p_{i} − p_{0}], i = 0, ..., 5, μ_{ij} = [p_{i} + p_{j} − 2p_{0}], 1 ≤ i < j ≤ 5

are 2-torsion points of J(C). We identify the group of 2-torsion points of J(C) with the set of their indices, that is, J(C)_{2} = {i, jk : 0 ≤ i ≤ 5, 1 ≤ j < k ≤ 5}. Let

Θ = Θ_{0}

be the theta divisor and put

Θ_{i} = Θ + μ_{i}, Θ_{ij} = Θ + μ_{ij}.

Then each of these divisors contains six 2-torsion points and, conversely, each 2-torsion point is contained in six of these divisors. The J(C) has a natural involution z ⟶ −z with φ(ι(p)) = −φ(p).

Let X = Km(J(C)) be the minimal resolution of the quotient surface J(C)/ < ι >, which is called the Kummer surface associated with C. The X has mutually disjoint sixteen smooth rational curves {N_{α}} obtained by resolution of singularities corresponding to 2-torsion points μ_{α}. Also X contains mutually disjoint sixteen smooth rational curves {T_{β}} which are the images of Θ_{β}, β = i, ij. Each member of {N_{α}} meets exactly six members of {T_{β}} and vice versa. More precisely,

⟨N_{α}, T_{β}⟩ = 1 if and only if α − β ∈ {0, 1, 2, 3, 4, 5}

This configuration is called Kummer (16)_{6} configuration.

The linear system | 2Θ | gives a 2 to 1 morphism from J(C) to a quartic surface F in P^{3} with sixteen ordinary nodes. The minimal resolution of F is isomorphic to X = Km(J(C)) and {N_{α}} are exceptional divisors. There are sixteen planes in P^{3} each of which touches F along a conic. These conics (called tropes) are the images of T_{β}. Let H be a divisor on X which is the pull back of the hyperplane section of F. Then the linear system

| 2H − (1/2)ΣN_{α} |=| Σ(N_{α} + T_{α})/4 |

gives an embedding of X onto a complete intersection of three quadrics in P^{5}. For more details, we refer the reader to Hudson [7], Griffiths-Harris [5].

It is well known that X = Km(J(C)) is a K3 surface, i.e. it is simply connected and its canonical line bundle is trivial. There is a canonical structure of lattice on H^{2}(X, Z) with the cup product which is even unimodular lattice of signature (3, 19), and hence isometric to U^{⊕3} ⊕ E_{8}^{⊕2}. Let S_{X} be the Picard lattice of X:

S_{X} = {x ∈ H^{2}(X, Z) : ⟨x, ω_{X}⟩ = 0}

where ω_{X} is a nowhere vanishing holomorphic 2-form on X. Let T_{X} be the orthogonal complement of S_{X}, which is called the transcendental lattice of X. In our case, X is algebraic and contains linearly independent sixteen smooth rational curves. Hence rank(S_{X}) ≥ 17. We call X generic if the Néron-Severi group of J(C) is generated by the class of C. In this case T_{X} ≃ U(2)^{⊕2} ⊕ ⟨−4⟩ and S_{X} is an even lattice of signature (1, 16) with q_{S_{X}} ≃ u_{2}^{⊕2} ⊕ q_{1}(4).

Lemma 3.1 (Keum [8], Lemma 3.1). Assume that X is generic. Then

(i) {H, N_{α}} is a Q-base of S_{X}, and {N_{α}, T_{β}} generate S_{X} over Z.

(ii) A_{S_{X}} is generated by (N_{2} + N_{12} + N_{3} + N_{13})/2, (N_{1} + N_{12} + N_{4} + N_{24})/2, (N_{2} + N_{12} + N_{4} + N_{14})/2, (N_{1} + N_{12} + N_{3} + N_{23})/2, H/4 + (N_{0} + N_{1} + N_{2} + N_{12})/2.

Lemma 3.2 (Keum [8], (1.8)). T_{α} can be written as a linear combination of {H, N_{α}} as follows :

T_{0} = (H − N_{0} − N_{1} − N_{2} − N_{3} − N_{4} − N_{5})/2

T_{1} = (H − N_{0} − N_{1} − N_{12} − N_{13} − N_{14} − N_{15})/2

T_{2} = (H − N_{0} − N_{2} − N_{12} − N_{23} − N_{24} − N_{25})/2

T_{12} = (H − N_{1} − N_{2} − N_{12} − N_{45} − N_{35} − N_{34})/2

T_{3} = (H − N_{0} − N_{3} − N_{13} − N_{23} − N_{35} − N_{34})/2

T_{13} = (H − N_{1} − N_{3} − N_{13} − N_{45} − N_{24} − N_{25})/2

T_{23} = (H − N_{2} − N_{3} − N_{23} − N_{45} − N_{14} − N_{15})/2

T_{45} = (H − N_{12} − N_{13} − N_{23} − N_{45} − N_{4} − N_{5})/2

T_{4} = (H − N_{0} − N_{45} − N_{4} − N_{14} − N_{24} − N_{34})/2

T_{14} = (H − N_{1} − N_{23} − N_{4} − N_{14} − N_{35} − N_{25})/2

T_{24} = (H − N_{2} − N_{13} − N_{4} − N_{24} − N_{35} − N_{15})/2

T_{35} = (H − N_{12} − N_{3} − N_{14} − N_{24} − N_{35} − N_{5})/2

T_{34} = (H − N_{12} − N_{3} − N_{4} − N_{34} − N_{25} − N_{15})/2

T_{25} = (H − N_{2} − N_{13} − N_{14} − N_{34} − N_{25} − N_{5})/2

T_{15} = (H − N_{1} − N_{23} − N_{24} − N_{34} − N_{15} − N_{5})/2

T_{5} = (H − N_{0} − N_{45} − N_{35} − N_{25} − N_{15} − N_{5})/2.

Definition 3.3. A standard graph Γ is a graph which is isomorphic to the dual graph of thirty-two nodal curves {N_{α}, T_{β}}. We denote by B(Γ) (resp. R(Γ)) the set of vertices (resp. edges) of Γ.

Let S be a lattice which has a generator system {ξ_{a}}_{a∈B(Γ)} satisfying

⟨ξ_{a}, ξ_{b}⟩ = \{ \begin{matrix}−2, & if a = b; \\ 1, & if \{a, b\} ∈ R(Γ); \\ 0, & if a ≠ b, \{a, b\} ∉ R(Γ). \end{matrix}

Obviously S is isometric to S_{X} for a generic Kummer surface X. We remark here that the automorphism group Aut(Γ) of Γ is an extension of (Z/2Z)^{5} by S_{6} (Nikulin [10], Corollary 2.7). By Lemma 3.1, Aut(Γ) ⊂ O(S). It is known (Nikulin [10], Theorem 4.5) that

ξ = Σ_{a∈B(Γ)}ξ_{a}/4 ∈ S.

Proposition 3.4 (Nikulin [10]). Let X be a K3 surface whose Picard lattice S_{X} contains S as a sublattice. Assume that

⟨ξ, r⟩ ≠ 0 for any r ∈ S_{X} with ⟨r, r⟩ = −2.

Then X is the Kummer surface associated with a curve of genus two. Moreover there exists an isometry ϕ in W (S_{X})^{(2)} such that {ϕ(ξ_{a})} is the set of the classes of thirty-two nodal curves {N_{α}, T_{β}} on X.

Let P (S_{X}) be the connected component of the cone {x ∈ S_{X} ⊗ R : ⟨x, x⟩ > 0} which contains an ample divisor. Put

D(S_{X}) = {x ∈ P (S_{X}) : ⟨x, δ⟩ > 0 for any classes δ of smooth rational curves on X}

which is called the Kähler cone of X and a fundamental domain with respect to the natural action of W (S_{X})^{(2)} on P (S_{X}). It follows from Torelli theorem for K3 surfaces due to Piatetsukii-Shapiro, Shafarevich [13] that Aut(X) is isomorphic to O(S_{X})/W (S_{X})^{(2)} up to finite groups. For any g ∈ Aut(X), denote by m(g) the order of g^{∗} | T_{X}. Then it follows from Nikulin [11], Theorem 3.1 that φ(m(g)) divides the rank of T_{X}, where φ is the Euler function. If X is a generic Kummer surface, then rank(T_{X}) = 5. Hence g^{∗} | T_{X} = ±1 as φ(m) is even if m > 2. Hence we have an injection

Aut(X) ⟶ Γ(S_{X})/W (S_{X})^{(2)}

where Γ(S_{X}) = {g ∈ O(S_{X}) : g^{∗} | A_{S_{X}} = ±1}. Moreover there exists an automorphism g of X with g^{∗} | A_{S_{X}} = −1 (e.g. a switch σ in §5). It now follows from Nikulin [12], Proposition 1.6.1 and Torelli theorem that

Proposition 3.5 (Nikulin [10], Corollary 6.4, Keum [8], Theorem 4.1).

Aut(X) ≃ Γ(S_{X})/W (S_{X})^{(2)}.

We remark that O(q_{S_{X}}) is isomorphic to an extension of Z/2Z by Sp(4, F_{2}) where Sp(4, F_{2}) is isomorphic to the orthogonal group of the form obtained from q_{S_{X}} by restricting it on the 2-elementary subgroup (Z/2Z)^{5} of A_{S_{X}}.

§4. A relation between the Picard lattice and Leech roots

In the following we fix a decomposition L = Λ ⊕ U and take w = (0, 0, 1) as a Weyl vector. Consider the following vectors in Λ:

X = 4ν_{∞} + ν_{Ω}, Y = 4ν_{0} + ν_{Ω}, Z = 0, P = 4ν_{∞} + 4ν_{0}, Q_{i} = 2ν_{K_{i}} (i = 1,..., 5)

where K_{i} ∈ C(8), K_{i} ∋ 0, ∞ and | K_{i} ∩ K_{j} |= 4 for i ≠ j, i.e. K_{1}, ..., K_{5} are the five octads containing common four points. Obviously

⟨X, X⟩ = ⟨Y, Y ⟩ = −6, ⟨P, P ⟩ = ⟨Q_{i}, Q_{i}⟩ = ⟨X, Y ⟩ = −4 and ⟨P, Q_{i}⟩ = ⟨Q_{i}, Q_{j}⟩ = −2 for i ≠ j.

The corresponding Leech roots

x = (X, 1, 2), y = (Y, 1, 2), z = (0, 1, −1), x_{0} = (P, 1, 1) and x_{i} = (Q_{i}, 1, 1)

genetare a root lattice R isomorphic to A_{3} ⊕ A_{1}^{⊕6}. Put

t = (x + 2z + 3y)/4, t_{i} = x_{i}/2 (i = 0,...,5).

Then {t, t_{0}, ..., t_{5}} is a generator of A_{R}. Note that q_{R} ≃ q_{−3}(4) ⊕ (q_{−1}(2))^{⊕6}. Let

θ = 2t + t_{0} + ⋯ + t_{5}

which is isotropic with respect to q_{R}. Hence we have an over lattice T of R corresponding to θ : T is generated by R and θ. Note that q_{T} ≃ q | θ^{⊥}/θ ≃< t_{0} + t_{1}, t_{0} + t_{2}, t_{3} + t_{4}, t_{3} + t_{5}, t + t_{0} + t_{1} + t_{2} > mod T ≃ v_{2}^{⊕2} ⊕ q_{−1}(4) ≃ u_{2}^{⊕2} ⊕ q_{−1}(4) ≃ q_{T_{X}}.

Lemma 4.1. T is a primitive sublattice of L whose root lattice is R.

Proof. First we shall see that T = T^{∗} ∩ L mod T. Let

θ′ = at + Σa_{i}t_{i} (0 ≤ a < 4, 0 ≤ a_{i} < 2) ∈ T^{∗} ∩ L.

Since θ′ ∈ T^{∗}, ⟨θ, θ′⟩ = −3a/2 − Σa_{i}/2 ∈ Z. By definition,

θ′ = ((a + 2a_{0})ν_{∞} + (3a + 2a_{0})ν_{0} + aν_{Ω} + Σa_{j}ν_{K_{j}}, (3a + Σa_{i})/2, (3a + Σa_{i})/2) ∈ Λ ⊕ U.

It is easy to see that if a is odd, then θ′ does not satisfy the condition (i) in Proposition 2.2. Hence a = 0 or 2. Again by Proposition 2.2, (i), and the fact that Σa_{i} ≡ 0 mod 2, we can see that all a_{i} are even or odd. If a_{i} = 0 for all i, then θ′ ∈ R or θ′ ≡ (2ν_{∞} + 6ν_{0}, 0, 0) mod L. In the last case, θ′ does not satisfy the condition (ii) in Proposition 2.2. If a_{i} = 1 for all i, then

θ′ = ((a + 2)ν_{∞} + (3a + 2)ν_{0} + aν_{Ω} + Σν_{K_{j}}, (3a + 6)/2, (3a + 6)/2).

If a = 0, then θ′ does not satisfy the condition (ii) in Proposition 2.2. If a = 2, then θ′ = θ ∈ T.

Next we shall see that the root sublattice of T is R. For r ∈ T, write r = ax + by + cz + Σa_{i}x_{i} (a, b, c, a_{i} ∈ Q). If r ∉ R, then no a_{i} vanish, and hence r^{2} < −2. This implies the assersion. q.e.d

Let S be the orthogonal complement of T in L. Since q_{T} ≃ q_{T_{X}}, S ≃ S_{X} (Nikulin [12], Theorem 1.14.2). It is easy to see that the symmetry group Z/2Z × S_{6} of the Dynkin diagram R can be extended to isometries of T because it preserves θ. Moreover they generate O(q_{T}) ≃ O(q_{S}). Hence the natural map O(T) ⟶ O(q_{T}) is surjective. It follows from Nikulin [12], Proposition 1.6.1 that any g ∈ O(S) can be extended to an isometry in O(L).

Now put

D′ = D ∩ P (S).

Then it is known that D′ is non-empty, has only a finite number of faces and of finite volume. Moreover D′ contains the projection w′ of the Weyl vector w = (0, 0, 1) ∈ Λ ⊕ U as an interior point (Borcherds [1], Lemmas 4.1, 4.2, 4.3). First we determine the Leech roots orthogonal to R.

Lemma 4.2. There are exactly thirty-two Leech roots which are orthogonal to R.

Proof. By Proposition 2.3 and Remark 2.4, we can see that such Leech root r corresponds to a vector

2ν_{K} in Λ where K ∈ C(8), K ∋ 0, ∞, | K ∩ K_{i} |= 4, i = 1, ..., 5.

The number of these octads is counted as follows: Take a point A in K_{1}∩K_{2} with A ≠ 0, ∞ and a point B in K_{1} \ K_{2}. Then the set {0, ∞, A, B} determines a unique sextet, in other words, there exists exactly five octads containing 0, ∞, A, B. Note that any two distinct octads meet 0, 2 or 4 points because C-set consists of 0, 8, 12, 16 or 24 elements. It follows easily from this fact that the above five octads satisfy the condition | K ∩ K_{i} |= 4 for all i. We should exclude the octad K_{1} from the set of these five octads. Hence the number of desired octads is \binom{2}{1} × \binom{4}{1} × 4 = 32. q.e.d.

Remark 4.3, (i). In his paper [14], Todd gave the 759 octads of the Steiner system S(5, 8, 24) (p.219, Table I). The octads in the proof of Lemma 4.2 correspond to the following {E_{α}, L_{β}} in his table. Here we assume K_{1} = {∞, 0, 1, 2, 3, 5, 14, 17}, K_{2} = {∞, 0, 1, 2, 4, 13, 16, 22}, K_{3} = {∞, 0, 1, 2, 6, 7, 19, 21}, K_{4} = {∞, 0, 1, 2, 8, 11, 12, 18}, K_{5} = {∞, 0, 1, 2, 9, 10, 15, 20}. Then E_{45} = {∞, 0, 1, 3, 4, 11, 19, 20}, E_{5} = {∞, 0, 1, 3, 6, 8, 10, 13}, E_{24} = {∞, 0, 1, 3, 7, 9, 16, 18}, E_{2} = {∞, 0, 1, 3, 12, 15, 21, 22}, E_{3} = {∞, 0, 1, 4, 5, 7, 8, 15}, E_{35} = {∞, 0, 1, 4, 6, 9, 12, 17}, E_{4} = {∞, 0, 1, 4, 10, 14, 18, 21}, E_{1} = {∞, 0, 1, 5, 6, 18, 20, 22}, E_{34} = {∞, 0, 1, 5, 9, 11, 13, 21}, E_{14} = {∞, 0, 1, 5, 10, 12, 16, 19}, E_{13} = {∞, 0, 1, 6, 11, 14, 15, 16}, E_{15} = {∞, 0, 1, 7, 10, 11, 17, 22}, E_{0} = {∞, 0, 1, 7, 12, 13, 14, 20}, E_{25} = {∞, 0, 1, 8, 9, 14, 19, 22}, E_{23} = {∞, 0, 1, 8, 16, 17, 20, 21}, E_{12} = {∞, 0, 1, 13, 15, 17, 18, 19}, L_{1} = {∞, 0, 2, 3, 4, 8, 9, 21}, L_{34} = {∞, 0, 2, 3, 6, 12, 16, 20}, L_{14} = {∞, 0, 2, 3, 7, 11, 13, 15}, L_{3} = {∞, 0, 2, 3, 10, 18, 19, 22}, L_{2} = {∞, 0, 2, 4, 5, 6, 10, 11}, L_{25} = {∞, 0, 2, 4, 7, 17, 18, 20}, L_{15} = {∞, 0, 2, 4, 12, 14, 15, 19}, L_{45} = {∞, 0, 2, 5, 7, 9, 12, 22}, L_{24} = {∞, 0, 2, 5, 8, 13, 19, 20}, L_{5} = {∞, 0, 2, 5, 15, 16, 18, 21}, L_{4} = {∞, 0, 2, 6, 8, 15, 17, 22}, L_{23} = {∞, 0, 2, 6, 9, 13, 14, 18}, L_{12} = {∞, 0, 2, 7, 8, 10, 14, 16}, L_{0} = {∞, 0, 2, 9, 11, 16, 17, 19}, L_{13} = {∞, 0, 2, 10, 12, 13, 17, 21}, L_{35} = {∞, 0, 2, 11, 14, 20, 21, 22}

(ii) We use the same notation as in Remark 4.3, (i). For simplicity we denote by the same symbols {E_{α}, L_{β}} for the corresponding Leech roots. Then {E_{α}} and {L_{β}} are both mutually orthogonal sixteen roots and they form a standard graph Γ (see Definition 3.3). Moreover {E_{α}} and {L_{β}} are switched by the isometry σ induced from the permutation

(0, ∞)(1, 2)(3, 5)(4, 22)(6, 21)(7, 19)(8, 18)(9, 20)(10, 15)(11, 12)(13, 16)(14, 17) on Ω (Todd [14], §3, (xvii)) : σ(E_{α}) = L_{α} for any α. Consider the following involutions in M_{24} which commute with σ, fixes ∞, 0, 1, 2 and preserves five octads K_{1}, ..., K_{5} :

τ_{1} = (3, 17)(4, 16)(5, 14)(6, 7)(8, 11)(12, 18)(13, 22)(19, 21),
τ_{2} = (3, 14)(4, 16)(5, 17)(6, 19)(7, 21)(9, 10)(13, 22)(15, 20),
τ_{3} = (3, 17)(4, 13)(5, 14)(8, 12)(9, 10)(11, 18)(15, 20)(16, 22),
τ_{4} = (4, 13)(6, 19)(7, 21)(8, 11)(9, 15)(10, 20)(12, 18)(16, 22),
τ_{5} = (3, 14)(5, 17)(6, 7)(8, 12)(9, 15)(10, 20)(11, 18)(19, 21).

It is easy to see that {τ_{1}, ..., τ_{5}, σ} generate a 2-elementary abelian group of order 32 with τ_{1} ∘ τ_{2} ∘ τ_{3} ∘ τ_{4} ∘ τ_{5} = 1. The set of indices appeared in {E_{α}} has a group structure such that the translation by α is equal to τ_{α}, α = 1, ..., 5, that is, it coincides with J(C)_{2} = {i, jk : 0 ≤ i ≤ 5, 1 ≤ j < k ≤ 5}.

Lemma 4.4. Let w′ be the projection of the Weyl vector w = (0, 0, 1) ∈ Λ ⊕ U. Then

w′ = (ΣE_{α} + ΣL_{β})/4 ∈ S.

Proof. First note that w = w′ + x^{∗} + y^{∗} + z^{∗} + Σx_{i}^{∗} and w^{2} = 0 where x^{∗} = −(3x + y + 2z)/4, y^{∗} = −(x + 3y + 2z)/4, z^{∗} = −(x + y + 2z)/2 and x_{i}^{∗} = −x_{i}/2. Hence (w′)^{2} = 8. On the other hand, ⟨w′, E_{α}⟩ = ⟨w′, L_{β}⟩ = 1. The right hand side of the equation in Lemma 4.4 also has this property. Since {E_{α}, L_{α}} generate S (Lemma 3.1), this implies the assersion. q.e.d

We denote by Aut(D′) the group of isometries of S which preserves D′. As mentioned before, each isometry φ ∈ Aut(D′) can be extended to an isometry of L which preserves D. Hence Aut(D′) is isomorphic to the subgroup G of Aut(D) which preserves R.

Lemma 4.5. Aut(D′) is an extension of 2-elementary abelian group of order 32 by S_{6} in which a 2-elementary abelian subgroup of order 16 acts on R trivially and the quotient by it is the symmetry group of the Dynkin diagram of R.

Proof. Obviously the restriction of G on R is a subgroup of the symmetry group Z/2Z×S_{6} of the Dynkin diagram of R. Let G_{0} be the subgroup of G which acts trivially on R. Then G_{0} is a subgroup of M_{22} because G_{0} fixes X, Y, Z and P (see Conway-Sloane [4], Chap.10, §3.5). Moreover G_{0} preserves five octads K_{i}. By the table of maximal subgroups of M_{22} (Conway [2], Table 3), G_{0} is isomorphic to a 2-elementary abelian group of order 16. Thus G is a subgroup of a group 2^{5} · S_{6}. On the other hand, O(S) contains a subgroup (Z/2Z)^{5} · S_{6} induced from the automorphism group Aut(Γ) of the standard graph Γ whose vertices are {E_{α}, L_{β}} (see §3). By Lemma 4.4, they preserves w′ and hence D′. Thus Aut(D′) ≃ G ≃ (Z/2Z)^{5} · S_{6}. q.e.d.

Next we shall classify the hyperplanes bounding D′. By Lemma 4.1, the root sublattice of T is R. Since T is primitive in L, there are no sublattice of rank 9 in L generated by some roots which contains R properly. Hence it suffices to consider Leech roots r such that r and R generate a root lattice of rank 10 because otherwise the hyperplane H_{r} in P (L) does not meet P (S).

Lemma 4.6. Let r be a Leech root. Assume r and R generate a root lattice R′ of rank 10. Then the following cases occur :

(0) R′ = A_{3} ⊕ A_{1}^{⊕7}, (1) R′ = D_{4} ⊕ A_{1}^{⊕6}, (2) R′ = A_{3} ⊕ A_{3} ⊕ A_{1}^{⊕4}, (3) R′ = A_{5} ⊕ A_{1}^{⊕5}.

Moreover the number of Leech roots of type (0), (1), (2) or (3) is 32, 32, 60 or 192 respectively.

Proof. It is obvious that R′ is one of the following nine cases:

(0)A_{3} ⊕ A_{1}^{⊕7}, (1)D_{4} ⊕ A_{1}^{⊕6}, (2)A_{3} ⊕ A_{3} ⊕ A_{1}^{⊕4}, (3)A_{5} ⊕ A_{1}^{⊕5}, (4)A_{4} ⊕ A_{1}^{⊕6}, (5)A_{3} ⊕ A_{2} ⊕ A_{1}^{⊕5},

(6)D_{5} ⊕ A_{1}^{⊕5}, (7)D_{6} ⊕ A_{1}^{⊕4}, (8)A_{3} ⊕ D_{4} ⊕ A_{1}^{⊕3}.

We have seen the Case (0) in Lemma 4.2.

Case (1): By Proposition 2.3 and Remark 2.4, in this case, the root r corresponds to a vector

(ξ_{0}, ξ_{∞}, ξ_{j_{1}}, ..., ξ_{j_{6}}, ξ_{j_{7}}, ..., ξ_{j_{22}}) = (3, 3, 3, −1, ..., −1, 1, ..., 1)

in Λ where K = {0, ∞, j_{1}, ..., j_{6}} is an octad satisfying | K ∩ K_{i} |= 4 and K_{i} ∋ j_{1} for any i.

By the same reason as in Case (0), the number of desired Leech roots is equal to 32.

Case (2): If r meets x_{0} and x_{i}, then r corresponds to a vector

ν_{Ω} − 4ν_{k}(k ≠ 0, ∞), k ∈ K_{i}, k ∉ K_{j} for j ≠ i,

and if r meets x_{i} and x_{j}, then r corresponds to a Leech vector

2ν_{K}, K ∈ C(8), K ∋ 0, ∞, | K ∩ K_{i} |=| K ∩ K_{j} |= 2, | K ∩ K_{l} |= 4 for l ≠ i, j.

Obviously the number of the above roots r meeting x_{0} and x_{i} is equal to 4. It follows from Lemma 4.5 that the number of desired Leech roots is \binom{6}{2} × 4 = 60.

Case (3): If r meets y and x_{0}, then r corresponds to a vector

2ν_{K}, ∞ ∈ K, 0 ∉ K, | K ∩ K_{i} |= 4 for all i,

and if r meets y and x_{i}, then r corresponds to

(ξ_{∞}, ξ_{j_{1}}, ..., ξ_{j_{7}}, ξ_{j_{8}}, ..., ξ_{j_{22}}, ξ_{0}) = (3, −1, ..., −1, 1, ..., 1)

where K = {∞, j_{1}, ..., j_{7}} is an octad with | K ∩ K_{i} |= 4 and | K ∩ K_{j} |= 2 for j ≠ i. Now we count the number of Leech roots. We may assume that r meets y and x_{0}. Take a point A ∈ K_{1}\K_{2} and consider the five octads containing the four points ∞, A, K_{1}∩K_{2}\{0, ∞}. Each of them satisfies the above condition, however, in which we should exclude the octad K_{1}. The number of such octads is equal to \binom{4}{1} × 4 = 16. Hence the total number of the desired Leech roots is \binom{2}{1} × \binom{6}{1} × 16 = 192.

In other cases (4)-(8), by Proposition 2.3 and case-by-case argument, we can see that there are no Leech roots r satisfying < r, R >≃ R′. We leave the proof to the reader. Thus we have now finished the proof. q.e.d.

Remark 4.7. We use the same notation as in Remark 4.3. Then the octads appeared in the proof of Lemma 4.6 correspond to the followings in Todd’s list [14]:

Case (1) : The same octads as in Remark 4.3.

Case (2) : 40 octads from {∞, 0, 3, 4, 5, 12, 13, 18} to {∞, 0, 12, 14, 16, 17, 18, 22} in p. 219.

Case (3) : Sixteen octads from {∞, 1, 2, 3, 4, 6, 15, 18} to {∞, 1, 2, 14, 16, 18, 19, 20} in p.219, 96 octads from {∞, 3, 4, 5, 6, 8, 17, 20} to {0, 1, 2, 12, 17, 19, 20, 22} in p.220 and 80 octads from {0, 3, 4, 5, 7, 9, 11, 17} to {0, 11, 12, 13, 14, 15, 18, 21} in p. 221.

Lemma 4.8. Let r be a Leech root as in Lemma 4.6. Let r = r′ + r′′ where r′ ∈ S^{∗} and r′′ ∈ T^{∗}. Then

Case (0) : r = r′; Cases (1), (2) : ⟨r′, r′⟩ = −1; Case (3) : ⟨r′, r′⟩ = −3/4.

Proof. We shall see the assersion in each case:

Case (0): obvious.

Case (1): Let {x, y, z} be a base of A_{3} as above. Then r = z^{∗} + r′ where z^{∗} = −(x + 2z + y)/2 is the dual of z. Since ⟨r, r⟩ = −2, ⟨r′, r′⟩ = −1.

Case (2): We may assume that r meets x_{1} and x_{2}. Then r = −x_{1}/2 − x_{2}/2 + r′. Hence ⟨r′, r′⟩ = −1.

Case (3): We may assume that r meets y and x_{1}. Then r = y^{∗} − x_{1}/2 + r′ where y^{∗} = −(x + 2z + 3y)/4 is the dual of y. Hence ⟨r′, r′⟩ = −3/4. q.e.d

§5. Classical involutions

In this section we recall classical involutions of X. Let F be a quartic surface in P^{3} with sixteen nodes {n_{α}} which is the image of J(C) by the map given by | 2Θ |.

(1) Sixteen translations (Klein [9]):

A 2-torsion point μ ∈ J(C)_{2} acts on J(C) as translations. These involutions descend to involutions t_{μ} on X which are induced from projective automorphisms of P^{3}. There are no other projective automorphisms if rank(S_{X}) = 17 (Keum [8]).

(2) Sixteen switches (Klein [9]):

The dual surface F^{∗} is also a quartic surface with sixteen ordinary nodes (corresponding to sixteen tropes) and is projectively isomorphic to F (e.g. Griffiths-Harris [5], Chap.6). Hence we have an involution σ which sends {N_{α}} to {T_{α}}. We call this involution a switch. Let σ be a switch which sends N_{α} to T_{α} for any α. Then sixteen switches are of the form σ ∘ t_{μ}, μ ∈ J(C)_{2}. We use the same symbol σ for a switch and for the isometry stated in Remark 4.3, (ii) because they coincide under the identification of {N_{α}, T_{β}} and {E_{α}, L_{β}} (see §6).

(3) Sixteen projections (Klein [9]):

The projection from a node n_{α} on F gives an involution p_{α} of X.

(4) Sixteen correlations (Klein [9]):

A projection of F^{∗} induces an involution of X : σ ∘ p_{α} ∘ σ.

(5) Sixty Cremona transformations referred to Göpel tetrads (Hutchinson [6]):

A Göpel tetrad is a set of four nodes {n_{α}, n_{β}, n_{γ}, n_{δ}} such that no tropes contain three of four nodes. It is known that there are sixty Göpel tetrads of F. For each Göpel tetrad, F can be defined by the equation:

A(x^{2}t^{2} + y^{2}z^{2}) + B(y^{2}t^{2} + z^{2}x^{2}) + C(z^{2}t^{2} + x^{2}y^{2}) + Dxyzt + F (yt + zx)(zt + xy)+

G(zt + xy)(xt + yz) + H(xt + yz)(yt + zx) = 0.

The Cremona transformation

(x, y, z, t) ⟶ (yzt, ztx, txy, xyz)

preserves F and so we have sixty involutions c_{α,β,γ,δ} of X.

By definitions and Lemma 3.2, we have:

Proposition 5.1 (Keum [8], §4, Table). Assume that X is generic. Then the actions of classical involutions on S_{X} are as follows :

(1) t_{α}(N_{β}) = N_{α+β}, t_{α}(H) = H.

(2) σ(N_{α}) = T_{α}, σ(H) = 3H − ΣN_{α}.

(3) p_{α} acts on the Picard lattice as reflection with respect to (−4)-root H − 2N_{α}.

(4) c_{α,β,γ,δ} = r_{α,β,γ,δ} ∘ t_{α,β,γ,δ} where r_{α,β,γ,δ} is a reflection with respect to (−4)-root H −N_{α}−N_{β} −N_{γ} −N_{δ} and t = t_{α,β,γ,δ} fixes H, N_{α}, N_{β}, N_{γ}, N_{δ} and permutes the remaining 12 nodal curves as follows : t interchanges the two tropes containing any two of the four nodes n_{α}, n_{β}, n_{γ}, n_{δ}. This determines uniquely the action of t on sixteen nodal curves N_{α}.

Remark 5.2(Keum [8], §6, Warning). Neither r_{α,β,γ,δ} nor t_{α,β,γ,δ} can be realized as an automorphism of X.

§6. Keum’s automorphisms

Recently Keum [8] has found 192 automorphisms of a generic Kummer surface X which are not contained in the group generated by the classical automorphisms. In this section we shall give the Definition and properties of Keum’s automorphisms. Consider thirtytwo roots {E_{α}, L_{β}} stated in Remark 4.3. Then {E_{α}, L_{β}} forms a standard graph Γ (see Definition 3.3). By Lemma 4.4 and Proposition 3.4, we can identify {E_{α}, L_{β}} to thirty-two nodal curves {N_{α}, T_{β}} on a generic Kummer surface X associated with a curve of genus two. Under this identification, S_{X} is a primitive sublattice of L. Note that D′ = D ∩ P (S_{X}) ⊂ D(S_{X}) because D′ contains an ample divisor w′ = Σ(N_{α} + T_{α})/4 (Lemma 4.4).

Lemma 6.1. Let r be a Leech root as in Lemma 4.6, Case (3). Then there exists exactly two hexads Ψ_{r}, Φ_{r} ⊂ J(C)_{2} which satisfy the following :

(1) ⟨r, N_{α}⟩ = ⟨r, T_{β}⟩ = \{ \begin{matrix}1 & for α ∈ Ψ_{r}, β ∈ Φ_{r}; \\ 0 & for α ∉ Ψ_{r}, β ∉ Φ_{r}. \end{matrix}

(2) | Ψ_{r} ∩ Φ_{r} |= 1.

(3) For each α ∈ Φ_{r}, there exists a unique β_{α} ∈ Ψ_{r} with ⟨N_{α}, T_{β_{α}}⟩ = 1, and vice versa.

(4) For each α ∈ Ψ_{r} \ Φ_{r}, N_{α} meets exactly three members T_{α}, T_{α_{1}}, T_{α_{2}} in {T_{β} : β ∈ Ψ_{r}} and I(T_{α_{1}}) ∩ I(T_{α_{2}}) = {N_{α}, N_{α′}} where α′ ∈ J(C)_{2} \ (Ψ_{r} ∪ Φ_{r}) and I(T_{β}) = {N_{α} : ⟨N_{α}, T_{β}⟩ = 1}.

(5) For each α ∈ J(C)_{2} \ (Ψ_{r} ∪ Φ_{r}), N_{α} meets exactly three members T_{Ψ_{r}∩Φ_{r}}, T_{β_{1}}, T_{β_{2}} in {T_{β} : β ∈ Φ_{r}} and I(T_{β_{1}}) ∩ I(T_{β_{2}}) = {N_{α}, N_{α′′}} where α′′ ∈ Φ_{r} \ Ψ_{r}.

(6) Let σ be the isometry defined in Remark 4.5. Then

Ψ_{σ(r)} = Φ_{r}, Φ_{σ(r)} = Ψ_{r}.

Proof. Consider the action of Aut(D′) on the set of Leech roots as in Case (3). First we shall show that the stabilizer of a such Leech root is isomorphic to S_{5}. Let r be a Leech root corresponding to a vector 2ν_{K}, ∞ ∈ K, 0 ∉ K, | K ∩ K_{i} |= 4 for all i. The stabilizer of 2ν_{K} is the subgroup of M_{22} which preserves K , fixes two points 0, ∞ and has three orbits on K of sizes 1, 2, 5. It is known (Conway [2], §3, Theorem 6) that this group is isomorphic to S_{5}. Since [Aut(D′) : S_{5}] = 192, Aut(D′) acts on this set transitively. Therefore it suffices to prove the assersion for one such Leech root.

Now we use the same notation as in Remarks 4.3, 4.7. Let K = {∞, 1, 2, 3, 4, 6, 15, 18} as in Remark 4.7, Case (3). Let r be the Leech root corresponding to 2ν_{K}. Then

Ψ_{r} = {0, 14, 15, 23, 25, 34}, Φ_{r} = {0, 12, 13, 24, 35, 45}.

The assersions (1)-(2) are now obvious. The assersion (3) follows from Lemma 3.2. The assersion (4) follows from the facts that Ψ_{r} \ Φ_{r} = {14, 15, 23, 25, 34} and

I(T_{23})∩I(T_{25}) = {N_{2}, N_{14}}, I(T_{23})∩I(T_{34}) = {N_{3}, N_{15}}, I(T_{14})∩I(T_{15}) = {N_{1}, N_{23}},

I(T_{14}) ∩ I(T_{34}) = {N_{4}, N_{25}}, I(T_{15}) ∩ I(T_{25}) = {N_{5}, N_{34}}.

Also the assersion (5) follows from the facts that J(C)_{2} \ (Ψ_{r} ∪ Φ_{r}) = {1, 2, 3, 4, 5} and I(T_{12})∩I(T_{13}) = {N_{1}, N_{45}}, I(T_{12})∩I(T_{24}) = {N_{2}, N_{35}}, I(T_{13})∩I(T_{35}) = {N_{3}, N_{24}},

I(T_{24}) ∩ I(T_{45}) = {N_{4}, N_{13}}, I(T_{35}) ∩ I(T_{45}) = {N_{5}, N_{12}}.

Finally the assersion (6) follows from the fact that σ(N_{α}) = T_{α} for any α. q.e.d

Definition 6.2. Let r be a Leech root as in Lemma 4.6, Case (3), and Ψ_{r}, Φ_{r} as in Lemma 6.1. Put

K_{r} = Σ_{α∈Ψ_{r}}N_{α}.

We define an isometry ψ_{r} of S_{X} ⊗ Q by the following equations:

ψ_{r}(H) = 7H − 4K_{r},

ψ_{r}(N_{α}) = \{ \begin{matrix}2H − K_{r} − N_{β_{α}} & for α ∈ Φ_{r}, \\ N_{α′} & for α ∈ Ψ_{r} \ Φ_{r}, \\ N_{α′′} & for α ∈ J(C)_{2} \ (Ψ_{r} ∪ Φ_{r}), \end{matrix}

where β_{α}, α′, α′′ are as in Lemma 6.1.

Remark 6.3. (1) Let r′ be the projection of r into S_{X}^{∗}. By Lemma 4.8, (r′)^{2} = −3/4. In particular, 4r′ = 3H − 2K_{r}.

(2) By using the relations in Lemma 3.2, we can see that ψ_{r}(T_{β}) = σ(ψ_{σ(r)}(N_{β})). In particular ψ_{r} preserves S_{X}.

(3) ψ_{σ(r)} = ψ_{r}^{−1}.

Lemma 6.4. ψ_{r} is an isometry of S_{X} which acts on A_{S_{X}} as identity.

Proof. By using Lemma 3.1, (ii) and Lemma 3.2, we can see that ψ_{r} acts on A_{S_{X}} as identity. q.e.d.

Lemma 6.5 (Keum). Let D be a divisor on X. Then ⟨D, 3H − 2K_{σ(r)}⟩ ≥ 0 if and only if ⟨ψ_{r}(D), 3H − 2K_{r}⟩ ≤ 0.

Proof. Write D as D = aH − Σb_{α}N_{α}. Then ⟨D, 3H − 2K_{σ(r)}⟩ = 12a − 4Σ_{α∈Φ_{r}}b_{α} and ⟨ψ_{r}(D), 3H − 2K_{r}⟩ = −12a + 4Σ_{α∈Φ_{r}}b_{α}. q.e.d

Proposition 6.6 (Keum [8]). ψ_{r} preserves the Kähler cone D(S_{X}) of X.

Proof. See Keum [8], the proof of Proposition 6.16. q.e.d.

Definition 6.7. By Lemma 6.4 and Proposition 6.6, ψ_{r} can be extended to a Hodge isometry of H^{2}(X, Z) which preserves the Kähler cone. It follows from the Torelli theorem for K3 surfaces (Piatetsukii-Shapiro, Shafarevich [13]) that ψ_{r} is induced from an automorphism κ_{r} : ψ_{r} = κ_{r}^{∗}. We call κ_{r} Keum’s automorphism associated with a Leech root r.

Remark 6.8. Our definition of ψ_{r} is slightly different from that of Keum. In [8], he defines an isometry f associated with a good hexad {α, β, γ, δ, ε, λ} ⊂ J(C)_{2}, that is, no four of six elements are contained in a trope or a Göpel tetrad. The above hexads Ψ_{r}, Φ_{r} are good hexads in this sense. For example, if Ψ_{r} = {0, 14, 15, 23, 25, 34}, then ψ_{r} = f ∘ t_{34} where t_{34} is the translation by 2-torsion point μ_{34}. He also showed that f^{−1} is induced by the linear system | 7H − 4(N_{α} + N_{β} + N_{γ} + N_{δ} + N_{ε} + N_{λ}) | by using Torelli theorem for K3 surfaces.

§7. The automorphism group of a generic Kummer surface

In this section we shall complete a proof of the main theorem. We identify thirty-two Leech roots {E_{α}, L_{β}} to thirty-two nodal curves {N_{α}, T_{β}} on X .

Lemma 7.1. Let w′ be the projection of the Weyl vector w in S_{X}. Then

w′ = (ΣN_{α} + ΣT_{β})/4 = 2H − ΣN_{α}/2.

Proof. We have shown that the first equality and (w′)^{2} = 8 (Lemma 4.4). The second assersion follows from the fact ⟨w′, N_{α}⟩ = 1. q.e.d.

Lemma 7.2. Aut(D′) is generated by sixteen translations t_{μ}, a swiche σ and the permutations of six Weierstrass points of the curve C.

Proof. Recall that O(S_{X}) contains isometries induced from sixteen translations t_{α}, a switch σ, and the permutations of six Weierstrass points of the curve C. It follows from Proposition 7.1 that these isometries preserve w′ and hence D′. Hence these isometies generate the group Aut(D′) ≃ (Z/2Z)^{5} · S_{6} (see Lemma 4.5). q.e.d

Next we consider automorphisms corresponding to Leech roots in Lemma 4.6, Cases (1) and (2).

Case (1): Let r′ be the projection of a Leech root r into S_{X}^{∗}. Let λ in Λ corresponding to r (see Remark 2.4). Recall that r is defined by an octad. This octad also defines a (−2)-vector in S (see Remark 4.7). First we assume that this octad defines N_{α}. Then 2ν_{N_{β}} − λ is a vector of the following form:

(3^{5}, −1^{19}) if α = β and (3, ±1^{23}) if α ≠ β.

Hence ⟨r, N_{β}⟩ = 2δ_{α,β}. If we write

2r′ = aH − Σb_{α}N_{α},

then the above implies

2r′ = H − 2N_{α}

because (2r′)^{2} = −4. Hence the action of the reflection s_{r′} on S_{X} coincides with the action of the projection p_{α} (see Proposition 5.1).

Next we assume that the above octad defines T_{α}. Then 2ν_{N_{β}} − λ is:

(±3^{3}, ±1^{21}) if ⟨T_{α}, N_{β}⟩ = 0, (−3, ±1^{23}) if ⟨T_{α}, N_{β}⟩ = 1.

Hence

⟨N_{β}, r⟩ = \{ \begin{matrix}1 & if ⟨T_{α}, N_{β}⟩ = 0; \\ 0 & if ⟨T_{α}, N_{β}⟩ = 1. \end{matrix}

This implies

2r′ = 3H − ΣN_{ν} − 2T_{α} = σ(H − 2N_{α}).

Thus s_{r′} = σ ∘ p_{α}^{∗} ∘ σ.

Case (2): First we shall see that for each r there are exactly four nodal curves N_{α}, N_{β}, N_{γ}, N_{δ} in {N_{ν}} which meet r. We use the same notation as in the proof of Lemma 4.6. It suffices to see the claim in the case that r corresponding to a vector ν_{Ω}−4ν_{k} because there exists an isometry in Aut(D′) which sends ν_{Ω} − 4ν_{k} to 2ν_{K}. Then ⟨r, N_{α}⟩ = 1 if and only if the corresponding octad to N_{α} contains k. The number of such octads is obviously 4 (see the proof of Lemma 4.2). Next we shall see that {N_{α}, ..., N_{δ}} is a Göpel tetrad. We may assume that k = 3. Then {N_{α}, ..., N_{δ}} = {N_{2}, N_{5}, N_{24}, N_{45}} (see Remark 4.3, (i)) and

I(N_{2}) = {T_{0}, T_{2}, T_{12}, T_{23}, T_{24}, T_{25}}, I(N_{5}) = {T_{0}, T_{5}, T_{15}, T_{25}, T_{35}, T_{45}},
I(N_{24}) = {T_{2}, T_{4}, T_{13}, T_{15}, T_{24}, T_{35}}, I(N_{45}) = {T_{4}, T_{5}, T_{12}, T_{13}, T_{23}, T_{45}},

where I(N_{α}) = {T_{β} : ⟨N_{α}, T_{β}⟩ = 1}. Now the assersion is obvious.

Since (r′)^{2} = −1 (Lemma 4.8), we have

2r′ = H − (N_{α} + N_{β} + N_{γ} + N_{δ}).

Hence s_{r′} = r_{α,β,γ,δ} (see Proposition 5.1).

Let r be a Leech root in Lemma 4.6, Case (1), (2) or (3). Denote by ι_{r} the isometry induced from the corresponding automorphism (i.e. classical involution or Keum’s automorphism). Let N be the group generated by all ι_{r}. Obviously N acts on the Kähler cone D(S_{X}).

Lemma 7.3. Let ϕ be an isometry of S_{X} which preserves D(S_{X}). Then there exists a g ∈ N such that g ∘ ϕ ∈ Aut(D′).

Proof. Let w′ be the projection of the Weyl vector w : w′ = 2H − ΣN_{ν}/2. Put v = ϕ(w′). Take an element g ∈ N which attains the minimum min{⟨g(v), w′⟩ : g ∈ N}. Then for any r,

⟨g(v), w′⟩ ≤ ⟨ι_{r}(g(v)), w′⟩ = ⟨g(v), ι_{r}^{−1}(w′)⟩.

If ι_{r} corresponds to a classical involution, then by Proposition 5.1,

ι_{r}^{−1}(w′) = ι_{r}(w′) = w′ + 2⟨w′, r′⟩r′.

Hence

⟨g(v), w′⟩ ≤ ⟨g(v), w′⟩ + 2⟨w′, r′⟩⟨r′, g(v)⟩.

Since ⟨w′, r′⟩ > 0, we have ⟨r′, g(v)⟩ > 0.

If ι_{r} corresponds to a Keum’s automorphism, then by Definition 6.2 and Remark 6.3, (1),

ι_{r}^{−1}(w′) = w′ + 8σ(r′).

Hence

⟨g(v), w′⟩ ≤ ⟨g(v), w′⟩ + 8⟨g(v), σ(r′)⟩.

Thus ⟨g(v), σ(r′)⟩ > 0, and hence g(v) ∈ D′. q.e.d.

Combining Proposition 3.5, Lemmas 7.2, 7.3, we have:

Theorem 7.4. The automorphism group of a generic Kummer surface is a split extension of 2-elementary abelian group 2^{5} of oreder 32 by a group N where 2^{5} is generated by sixteen translations t_{μ} and a switch σ, and N is generated by sixteen projections p_{α}, sixteen corellations σ ∘ p_{α} ∘ σ, sixty Cremona transformations c_{α,β,γ,δ} and 192 Keum’s automorphisms κ_{r}.

Remark 7.5. The method is applicable to other K3 surfaces. For example, by taking E_{6}, D_{6} as a root lattice R (= T), we have the two singular K3 surfaces with

T_{X} ≃ ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) , ( \begin{matrix}2 & 0 \\ 0 & 2 \end{matrix} )

respectively. In these cases, the reflection group is of finite index in O(S_{X}) and the automorphism groups of these K3 surfaces were calculated by Vinberg [16] and also by Borcherds [1] by our method.

References

[1] R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133-153.

[2] J.H. Conway, Three lectures on exceptional groups, in Finite simple group, pp. 215-247, Academic Press, New York 1971.

[3] J.H. Conway, The automorphism group of the 26 dimensional even Lorentzian lattice, J. Algebra 80 (1983), 159-163.

[4] J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss. Bd 290, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York 1988.

[5] P. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley and Suns, New York 1978.

[6] J.I. Hutchinson, On some birational transformations of the Kummer surface into itself, Bull. Amer. Math. Soc. (2) 7 (1901), 211-217.

[7] R.W.H.T. Hudson, Kummer’s quartic surface, Cambridge Univ. Press. 1905.

[8] J.H. Keum, Automorphisms of Jacobian Kummer surfaces, to appear in Compositio Math.

[9] F. Klein, Ueber Configurationen, Welche der Kummer’schen Fläche zugleich

eingeschrieben und umgeschrieben sind, Math. Ann. 27 (1886), 106-142.

[10] V.V. Nikulin, An analogue of the Torelli theorem for Kummer surfaces of Jacobians, Math. USSR Izv., 8 (1974), 21-41.

[11] V.V. Nikulin, Finite groups of automorphisms of Kählerian surfaces of type K3, Moscow Math. Soc., 38 (1980), 71-137.

[12] V.V. Nikulin, Integral symmetric bilinear forms and its applications, Math. USSR Izv., 14 (1980), 103-167.

[13] I. Piatetsukii-Shapiro, I.R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5 (1971), 547-587.

[14] J.A. Todd, A representation of the Mathieu group M_{24} as a collineation group, Ann. Mat. Pure Appl. 71 (1966), 199-238.

[15] E.B. Vinberg, Some arithmetic discrete groups in Lobachevskii spaces, in Proc. Int. Coll. on Discrete Subgroups of Lie Groups and appl. to Moduli (Bombay 1973), pp. 323-348. Oxford Univ. Press 1975.

[16] E.B. Vinberg, The two most algebraic K3 surfaces, Math. Ann. 265 (1983), 1-21.

Graduate school of Polymathematics Nagoya University Nagoya, 464-01 Japan e-mail: kondo@math.nagoya-u.ac.jp

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