The ring of modular forms for the even unimodular lattice of signature (2,18)
aa r X i v : . [ m a t h . AG ] F e b THE RING OF MODULAR FORMS FOR THE EVEN UNIMODULAR LATTICEOF SIGNATURE (2,18)
ATSUHIRA NAGANO AND KAZUSHI UEDA
Abstract.
We show that the ring of modular forms with characters for the even unimodular latticeof signature (2,18) is obtained from the invariant ring of Sym(Sym ( V ) ⊕ Sym ( V )) with respect tothe action of SL( V ) by adding a Borcherds product of weight 132 with one relation of weight 264,where V is a 2-dimensional C -vector space. The proof is based on the study of the moduli space ofelliptic K3 surfaces with a section. Introduction
Let U be the even unimodular hyperbolic lattice of rank 2. A U -polarized K3 surface in the senseof [Nik79] is a pair ( U , j ) of a K3 surface Y and a primitive lattice embedding j : U ֒ → Pic Y. Asexplained, e.g., in [Huy], an elliptic K3 surface with a section corresponds naturally to a pseudo-ample U -polarized K3 surface. Fix a primitive embedding of U to the K3 lattice Λ = U ⊥ U ⊥ U ⊥ E ⊥ E , which is unique up to the left action of O( Λ ), and let T = U ⊥ U ⊥ E ⊥ E be the orthogonal lattice.As explained in [Dol96, Section 3], the global Torelli theorem [PˇSˇS71, BR75] and the surjectivityof the period map [Tod80] show that the period map gives an isomorphism from the coarse modulischeme of pseudo-ample U -polarized K3 surfaces to the quotient M := Γ \D of the bounded Hermitiandomain D := (cid:8) [Ω] ∈ P ( T ⊗ C ) (cid:12)(cid:12) (Ω , Ω) = 0 , (Ω , Ω) > (cid:9) (1.1)of type IV by Γ := O( T ) . The moduli space of elliptic K3 surfaces with a section attracts much attention recently, not onlyfrom the point of view of modular compactification (see e.g. [AB, ABE] and references therein), butalso because of the relation with tropical geometry and mirror symmetry [HU19, OO].A modular form on D with respect to Γ of weight k ∈ Z and character χ ∈ Char(Γ) := Hom(Γ , C × )is a holomorphic function f : e D → C on the total space e D := (cid:8) [Ω] ∈ T ⊗ C (cid:12)(cid:12) (Ω , Ω) = 0 , (Ω , Ω) > (cid:9) (1.2)of a principal C × -bundle on D satisfying(i) f ( αz ) = α − k f ( z ) for any α ∈ C × , and(ii) f ( γz ) = χ ( γ ) f ( z ) for any γ ∈ Γ.The vector spaces A k (Γ , χ ) of modular forms constitute the ring e A (Γ) := ∞ M k =0 M χ ∈ Char(Γ) A k (Γ , χ )(1.3)of modular forms. We also write the subring of modular forms without characters as A (Γ) := ∞ M k =0 A k (Γ) . (1.4)Let V := Spec C [ x, w ] be a 2-dimensional affine space over C . For k ∈ N , we write the k -thsymmetric product of V as Sym k V . The special linear group SL acts naturally on S := Sym V × Sym V considered as an affine variety, whose coordinate ring will be denoted by C [ S ] = C [ u , , u , , . . . , u , , u , , u , , . . . , u , ] . (1.5) A. N. was partially supported by JSPS Kakenhi (18K13383) and MEXT LEADER.K. U. was partially supported by JSPS Kakenhi (16H03930). e let G m act on S in such a way that u i,j has weight ( i + j ) /
2. This G m -action commutes with theSL -action, so that the invariant subring C [ S ] SL has an induced G m -action.Building on [Mir81], it is shown in [OO, Theorem 7.9] that the period map induces an isomorphismfrom Proj C [ S ] SL to the Satake–Baily–Borel compactification of Γ \D , so that one has an isomorphism A (Γ) ∼ = C [ S ] SL (1.6)of graded rings.The main result of this paper is the following: Theorem 1.1.
One has e A (Γ) ∼ = (cid:0) C [ S ] SL (cid:1) [ s ] (cid:14)(cid:0) s − ∆ (cid:1) (1.7) where s is an element of weight and ∆ ∈ C [ S ] SL is an element of weight . The proof is based on the construction of an algebraic stack which is isomorphic to the orbifoldquotient [ D /O( T )] in codimension 1. The same strategy has been used in [HU] and [NU] to determinethe rings of modular forms with characters for the lattices U ⊥ U ⊥ E and U ⊥ U ⊥ A ⊥ A respectively.The modular form s is constructed in [FSM07, Lemma 5.1]. It can also be obtained either as thequasi pull-back [GHS13, Theorem 8.2] of the Borcherds form Φ associated with the even unimodularlattice of signature (2 ,
26) [Bor95, Section 10, Example 2], or by applying [Bor95, Theorem 10.1] tothe nearly holomorphic modular form1728 E E − E = 1 q + 264 + 8244 q + 139520 q + · · · (1.8)where E = 1 + 240 ∞ X n =1 n q n − q n = 1 + 240 q + 2160 q + · · · , (1.9) E = 1 − ∞ X n =1 n q n − q n = 1 − q − q + · · · . (1.10)In particular, it is a cusp form with character det admitting an infinite product expansion. See also[DKW19, Section 5] and references therein for the case of the even unimodular lattice of signature(2,10).Since SL is reductive, the invariant ring C [ S ] SL is finitely generated, and there exists an algorithmfor computing a finite generating set (see e.g. [Stu08] and references therein). The element ∆ canalso be computed algorithmically, and it is an interesting problem to describe them explicitly.2. The coarse moduli space of U -polarized K3 surfaces As is well known (cf. e.g. [SS10, Section 4]), a U -polarized K3 surface admits a Weierstrass modelof the form z = y + g ( x, w ; u ) y + g ( x, w ; u )(2.1)in P (1 , , , g ( x, w ; u ) = X i =0 u − i,i x − i w i (2.2) = u , x + u , x u + · · · + u , w , (2.3) g ( x, w ; u ) = X i =0 u − i,i x − i w i (2.4) = u , x + u , x u + · · · + u , w (2.5)for u = (( u , , . . . , u , ) , ( u , , . . . , u , )) ∈ S. (2.6) he hypersurface in P (1 , , ,
1) defined by (2.1) has a singularity worse than rational double pointson the fiber at a ∈ P if and only if ord a ( g ) ≥ a ( g ) ≥ U ⊂ S be the open subscheme parametrizing hypersurfaces with at worst rational doublepoints.The parameter u describing a given U -polarized K3 surface is unique up to the action of SL × G m ,where G m acts on P (1 , , , × Sym V × Sym V by G m ∋ λ : (( x, y, z, w ) , ( u i,j ) i,j ) ( x, λ y, λ z, w ) , ( λ ( i + j ) / u i,j ) i,j )(2.7)rescaling the holomorphic volume formΩ = Res wdx ∧ dy ∧ dzz − y − g ( x, w ; u ) y − g ( x, w ; u )(2.8)as Ω λu = Res wdx ∧ d ( λ y ) ∧ d ( λ z )( λ z ) − ( λ y ) − g ( x, w ; λ · u )( λ y ) − g ( x, w ; λ · u ) = λ − Ω u . (2.9)The categorical quotient T := U/ SL is the coarse moduli scheme of pairs ( Y, Ω) consisting of a U -polarized K3 surface Y and a holomorphic volume form Ω ∈ H ( ω Y ) on Y . The fact that thecodimension of S \ U is greater than 2 implies an isomorphism C [ S ] SL ∼ = C [ T ](2.10)of graded rings. Since the character of C [ S ] as a SL × G m -module is given by Y i =0 (cid:0) − q i − t (cid:1) − Y i =0 (cid:0) − q i − t (cid:1) − , (2.11)the Hilbert series of the invariant ring is given by ∞ X i =0 dim (cid:0) C [ S ] SL (cid:1) i t i = Res q =0 ( q − − q ) Y i =0 (cid:0) − q i − t (cid:1) − Y i =0 (cid:0) − q i − t (cid:1) − ! (2.12)as explained, e.g., in [Muk03, Section 4.4]. It follows from the global Torelli theorem and thesurjectivity of the period map that the period map induces a ring isomorphism A (Γ) ∼ −→ C [ T ] , (2.13)which preserves the grading by (2.9). The isomorphism (1.6) follows from (2.10) and (2.13).3. Modular forms with characters
The coarse moduli space M := Γ \D of U -polarized K3 surfaces is an open subvariety of its Satake–Baily–Borel compactification Proj A (Γ) ∼ = P (4 , ) // SL . Although M and the orbifold quotient M := [Γ \D ] are closely related, the canonical morphism M → M is not an isomorphism even incodimension 1. In order to obtain an orbifold which is isomorphic to M in codimension 1 (so thatthe total coordinate rings are isomorphic), consider the stack P := (cid:2) P (cid:0) , (cid:1) / SL (cid:3) , (3.1)defined as the quotient of C \ by the action of SL × G m . The morphism M → M lifts to amorphism M → P , which is an isomorphism in codimension 0, since the generic stabilizers are {± id } on both sides.Stabilizers of M along divisors come from reflections. One divisor with a generic stabilizer comesfrom the reflection with respect to a ( − U ⊥ A . In order to describe this locus, first consider thediscriminant h ( x, w ; u ) := 4 g ( x, w ; u ) + 27 g ( x, w ; u ) (3.2)of y + g ( x, w ; u ) y + g ( x, w ; u ) as a polynomial in y , which is homogeneous of degree 24 in ( x, w )and degree 12 in u . Note that the discriminant of a polynomial P ni =0 a i x i w n − i with respect to ( x, w )is homogeneous of degree 2( n −
1) in Z [ a , . . . , a n ] if deg a = · · · = deg a n = 1. It follows that he discriminant k ( u ) of h ( x, w ; u ) with respect to ( x, w ) is a homogeneous polynomial of degree2 · ·
12 = 552 in u . A general point on the divisor D of P defined by k ( u ) corresponds to thelocus where two fibers of Kodaira type I collapse into one fiber. This divisor has two components; ageneral point on one component corresponds to the case when there exists a point p = [ x : w ] on P such that neither g nor g vanishes at p , and a general point on the other component correspondsto the case when both g and g vanishes at p . In the former case, the resulting singular fiber is ofKodaira type I , and the surface acquires an A -singularity. In the latter case, the resulting singularfiber is of Kodaira type II, and the surface does not acquire any new singularity. The defining equationof the latter component is the resultant of g and g . It is given as the determinant r ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u , u , · · · u , u , · · · u , u , . . . . . . . . .. . . . . . . . . u , · · · u , u , u , u , · · · · · · u , u , . . . . . . . . . u , u , · · · · · · u , u , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.3)of the Sylvester matrix, which is homogeneous of degree12 × × . (3.4)As shown in [HU, Lemma 6.1], the polynomial k ( t ) is divisible by r ( t ) , and the quotient∆ ( t ) := k ( t ) /r ( t ) (3.5)defines the reflection hyperplane along a ( − root construction is an operation which adds a stabilizeralong a divisor. Let T be the stack obtained from P by the root construction of order 2 along thedivisor on P defined by ∆ ( t ), which is the quotient of the double cover of P branched along ∆ ( t )by the group G of deck transformations. The Picard group of T (or the G -equivariant Picard groupof P ) is generated by the pull-back O T (1) := p ∗ O P (1) of the generator O P (1) of the Picard group of P by the structure morphism p : T → P and the line bundle O T ( D ) such that the space H ( O T ( D ))is generated by an element s satisfying s = ∆ ∈ H ( O T (264)) ∼ = H ( O P (264)) . Note also that ω P ∼ = O P ( a ) where a = − X i =0 deg u − i,i − X i =0 deg u − i,i = − × − × − . (3.6)The ramification formula for the canonical bundle gives ω T ∼ = p ∗ ω P ⊗ O T ( D )(3.7) ∼ = O T ( − ⊗ O T (132 + ( −
132 + D ))(3.8) ∼ = O T (18) ⊗ O T ( −
132 + D ) . (3.9)Note that O T ( −
132 + D ) is an element of order two in Pic T . By comparing (3.9) with ω M ∼ = O M (dim M ) ⊗ det = O M (18) ⊗ det(3.10)which follows from (the proof of) [HU, Proposition 5.1], one concludes that M has no further stabilizeralong a divisor, so that the lift M → T of M → P is an isomorphism in codimension 1. It follows thatthe injective map Z × Char(Γ) → Pic M , ( i, χ )
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