BORCHERDS PRODUCTS ON UNITARY GROUP U(2,1) TONGHAIYANGANDDONGXIYE Abstract. Inthisnote,weconstructcanonicalbasesforthespacesofweaklyholomorphicmodular 7 forms with poles supported at the cusp ∞ for Γ0(4) of integral weight k for k≤−1, and we make use of the basis elements for the case k =−1 to construct explicit Borcherds products on unitary 1 group U(2,1). 0 2 n a J 1. Introduction 9 2 In 1998, Borcherds developed a new method to produce memomorphic modular forms on an orthogonal Shimura variety from weakly holomorphic classical modular forms via regularized theta ] T liftings. These memomorphic modular forms have two distinct properties. The first one is the N so-called Boorcherds product expansion at a cusp of the Shimura variety–his original motivation to . prove the Moonshine conjecture. The second is that the divisor of these modular forms are known h t to be a linear combination of special divisors dictated by the principal part of the input weakly a holomorphic forms. The second feature has been extended to produce so-called automorphic green m functions for special divisors using harmonic Maass forms via regularized theta lifting by Bruinier [ ([5]andBruinier-Funke([8]), whichturnedouttobeveryusefultogeneralization of thewell-known 1 Gross-Zagier formula ([13]) and the beautiful Gross-Zagier factorization formula of singular moduli v ([12]) to Shimura varieties of orthogonal type (n,2) and unitary type (n,1) (see for example [10], 6 3 [9], [6], [1], [2], [21], [22]). On the other hand, the Borcherds product expansion and in particular 4 its integral structure is essential to prove modularity of some generating functions of arithmetic 8 divisors on these Shimura varieties ([7], [15]). Borcherds products are also closely related to Mock 0 . theta functions (see for example [18] and references there). 1 We should mention that the analogue of the Borcherds product to unitary Shimura varieties 0 7 of type (n,1) has been worked out by Hofmann ([16]). The Borcherds product expansion in the 1 unitary case is a little more complicated as it is a Fourier-Jacobi expansion rather than Fourier : v expansion. The purpose of this note is to give some explicit examples of these Borcherds product i expansion in concrete term. For this reason, we focus on the Picard modular surface associated to X the Hermitian lattice L =Z[i] Z[i] 1Z[i] with Hermitian form r ⊕ ⊕ 2 a x,y = x y¯ +x y¯ +x y¯ . 1 3 3 1 2 2 h i Our inputs are weakly holomorphic modular forms for Γ (4) of weight 1, character χ := −4 0 −4 − which have poles only at the cusp , which we denote by M!,∞(Γ (4),χk ) with k = 1. Our first ∞ −k 0 −4 (cid:0) (cid:1) result (Theorem 2.1) is to give a canonical basis F (m 1) for the infinitely dimensional vector k,m ≥ space for every k 1. The even k case was given by Haddock and Jenkins in [14] in a slightly ≥ different fashion. Since Γ (4) is normal in SL (Z), a simple conjugation also gives a canonical basis 0 2 for the space of weakly holomorphic forms of Γ (4) with weight k, character χk , and having 0 − −4 poles only at cusp 0 (resp. 1). 2 2010 Mathematics Subject Classification. 11F27, 11F41, 11F55, 11G18, 14G35. Key words and phrases. Borcherds product, unitary modular form, Heegner divisor, unitary modular variety. The research of the first author is supported by a NSFgrant DMS-1500743. 1 Next, we use standard induction procedure to produce vector valued weakly modular forms for SL (Z) using our lattice L which will be used to construct Picard modular forms on U(2,1) 2 (described above). Although the resulting vector valued modular forms for SL (Z) from the three 2 different scalar valued spaces M!,P(Γ (4),χk ), P = ,0, 1 are linearly independent, they don’t −k 0 −4 ∞ 2 generate the whole space. We find it interesting. This concludes Part I of our note, which should be of independent interest. InPartII,wefocusontheunitarygroupU(2,1) associated totheaboveHermitianformandgive explicit Borcherds product expansion of the Picard modular forms constructed from F = F . m 1,m The delicate part is to choose a proper Weyl chamber, which is a dimensional 3 real manifold and described it explicitly and carefully. Our main formula is Theorem 3.4. We remark that the same methodalsoappliestohighdimensionalunitaryShimuravarieties ofunitarytype(n,1)usingforms in M!,P (Γ (4),χk ) where P is a cusp for Γ (4). We restrict to U(2,1) for being as explicit as 1−n 0 −4 0 possible. 2. Part I: Vector Valued Modular Forms In this part, we derive canonical basis for the space M!,∞(Γ (4),χk ) for any integer k 0, and −k 0 −4 ≥ investigate the properties of the vector valued modular forms arising from M!,∞(Γ (4),χk ). For −k 0 −4 completeness, we will also give canonical basis for M!,0(Γ (4),χk ) and M!,12(Γ (4),χk ). −k 0 −4 −k 0 −4 2.1. Canonical Basis for M!,∞(Γ (4),χk ). −k 0 −4 Letχ () := −4 betheKroneckersymbolmodulo4. RecallthatX (4)has3cusps,represented −4 · · 0 by , 0, and 1. For each cusp P, let M!,P(Γ (4),χk ) denote the space of weakly holomorphic ∞ 2(cid:0) (cid:1) −k 0 −4 modular forms, which are holomorphic everywhere except at the cusp P, of weight k on Γ (4) 0 − with character χk . We will focus mainly on the cusp and will remark on other cusps (very −4 ∞ similar) in the end. We will also denote M! (Γ (4),χk ) for the space of weakly holomorphic −k 0 −4 modular forms for Γ (4) of weight k and character χk . 0 − −4 Let τ be a complex number with positive imaginary part, and set q = e(τ) = e2πiτ, and q = r e2πiτ/r. The Dedekind eta function is defined by ∞ η(τ) = q1/24 (1 qn). − n=1 Y Throughout this paper, we write η for η(mτ). The well known Jacobi theta functions are defined m by ∞ ∞ ∞ ϑ00(τ) = qn2, ϑ01(τ) = ( q)n2, ϑ10(τ) = q(n+12)2. − n=−∞ n=−∞ n=−∞ X X X Now we define three functions as follows. 1 η8 (2.1) θ = θ (τ) : = ϑ4 (τ) = 4 = q+O(q2), 1 1 16 10 η4 2 η10 (2.2) θ = θ (τ) : = ϑ2 (τ) = 2 = 1+O(q), 2 2 00 η4η4 1 4 8 η (2.3) ϕ = ϕ (τ) : = 1 = q−1+O(1). ∞ ∞ η (cid:18) 4(cid:19) Here are some basic facts [14] about the functions θ , θ and ϕ . 1 2 ∞ 2 (1) θ (τ) is a holomorphic modular form of weight 2 on Γ (4) with trivial character, has a 1 0 simple zero at the cusp , and vanishes nowhere else. ∞ (2) θ (τ) is a holomorphic modular form of weight 1 on Γ (4) with character χ , has a zero 2 0 −4 of order 1 at the irregular cusp 1, and vanishes nowhere else. 2 2 (3) ϕ (τ) is a modular form of weight 0 on Γ (4) with trivial character, has exactly one simple ∞ 0 pole at the cusp and a simple zero at the cusp 0. ∞ Theorem 2.1. (1) For k 1odd, there isa(canonical) basisF (m 1of M!,∞(Γ (4),χ ) ≥ k,m ≥ −k 0 −4 whose Fourier expansion has the following property: Fk,m = q−k+21−m+1+ c(n)qn. n≥X−k−21 !,∞ (2) For k > 1 even, there is a (canonical) basis F (m 1 of M (Γ (4)) whose Fourier k,m ≥ −k 0 expansion has the following property: Fk,m = q−k2−m+1+ c(n)qn, n≥X−k2+1 Proof. We prove (1) first. Notice that X (4) has no elliptic points [11, Section 3.9]. For F 0 ∈ !,∞ M (Γ (4),χ ), the valence formula for Γ (4) asserts that −k 0 −4 0 k ord (F)+ord (F)+ord (F)+ord (F) = . z ∞ 0 1/2 −2 z∈ΓX0(4)\H This implies ord F 1 (1/2 is the unique irregular cusp), ord (F) k+1. This implies 1/2 ≥ 2 ∞ ≤ − 2 the uniqueness of the basis F if it exists. We prove the existence by inductively construct a k,m { } −k+1 sequence of monic polynomials P (x) of degree m (m 0) such that F = θ θ 2 P (ϕ ) k,m ≥ k,m+1 2 1 k,m ∞ give the basis we seek, i.e., with the following property (2.4) Fk,m+1 = θ2θ1−k+21Pk,m(ϕ∞) = q−k+21−m+ c(n)qn. n≥X−k−21 (The awkward notation F instead of F will be clear in last section.) k,m+1 k,m (1) Notice that θ θ−k+21 M!,∞(Γ (4),χ ) with 2 1 ∈ −k 0 −4 θ2θ1−k+21 = q−k+21 + c(n)qn. n≥X−k−21 So we can and will first define P = 1. k,0 (2) For m 1, assume that P (x) C[x] is constructed with degree m 1, leading k,m−1 ≥ ∈ − coefficient 1, and the property Fk,m = θ2θ1−k+21Pk,m−1(ϕ∞) = q−k+21−m+1+ c(n)qn. n≥X−k−21 Then it is easy to see Fk,mϕ∞ = q−k+21−m+ d(n)qn. n>−Xk+21−m 3 Let −k+1 2 P = xP d(n)P , k,m k,m−1 k,−n − n=−kX+21−m+1 and −k+1 F = θ θ 2 P (ϕ ). k,m+1 2 1 k,m ∞ Then F satisfies (2.4). By induction, we prove the existence of the basis F , and k,m+1 k,m { } (1). The proof of (2) is similar and is left to the reader. The basis F , m 0, has the form k,m+1 { } ≥ ∞ (2.5) Fk,m+1 = θ1−k2Qk,m(ϕ∞)= q−k2−m+ c(n)qn n=X−k2+1 for a unique monic polynomial Q of degree m. k,m (cid:3) Remark 2.2. The canonical basis given in Theorem 2.1(2) was given in a slightly different form first by Haddock and Jenkins [14]. The following corollary follows directly from the proof of Theorem 2.1(1). Corollary 2.3. Every weakly holomorphic modular form f(τ) M!,∞(Γ (4),χk ) with k odd, ∈ −k 0 −4 vanishes at cusp 1/2. 2.2. Vector Valued Modular Form Arising from M!,∞(Γ (4),χk ). −k 0 −4 Let L bean even lattice over Z with symmetricnon-degenerate bilinear form (, ) and associated · · quadratic form Q(x) = 1(x,x). Let L′ be the dual lattice of L. Assume that L has rank 2m+2 2 and signature (2m,2). Then the Weil representation of themetaplectic group Mp (Z) on thegroup 2 algebra C[L′/L] factors through SL (Z). Thus we have a unitary representation ρ of SL (Z) on 2 L 2 C[L′/L], defined by ρ (T)φ = e( Q(µ))φ , L µ µ − 2m−2 √i ρ (S)φ = e((µ,β))φ L µ β L′/L | | β∈XL′/L p 1 1 0 1 where T = 0 1 , S = 1 −0 , φγ for µ ∈ L′/L are the standard basis elements of C[L′/L] (cid:18) (cid:19) (cid:18) (cid:19) and e(z) = e2πiz. We remark that the Weil representation ρ depends only on the finite quadratic L module (L′/L,Q) (called the discriminant group of L), where Q(x+L)= Q(x) (mod 1) Q/Z. ∈ Let k be an integer and F~ be a C[L′/L] valued function on H and let ρ = ρ be a representation L of SL (Z) on C[L′/L]. For γ SL (Z) we define the slash operator by 2 2 ∈ F~ γ (τ) = (cτ +d)−kρ(γ)−1F~(γτ), k,ρ (cid:18) (cid:12) (cid:19) a b (cid:12) where γ = acts on H via(cid:12)γτ = aτ+b. c d cτ+d (cid:18) (cid:19) Definition 2.4. Let k be an integer. A function F~ : H C[L′/L] is called a weakly holomorphic → vector valued modular form of weight k with respect to ρ= ρ if it satisfies L 4 (1) F~ γ = F for all γ SL (Z), 2 k,ρ ∈ (2) F~(cid:12)(cid:12)is holomorphic on H, (3) F~(cid:12)is meromorphic at the cusp . ∞ The space of such forms is denoted by M! . k,ρ The invariance of T-action implies that F~ M! has a Fourier expansion of the form ∈ k,ρ F~ = c(n,φ )qnφ . µ µ µ∈L′/L n∈Q X X n≫−∞ Note that c(n,φ ) = 0 unless n Q(µ) (mod 1). µ ≡ − Fromnowon,wefocusonthespecialcasewiththediscriminantgroupL′/L = Z/2Z Z/2Zwith ∼ × quadratic form Q(x,y) = 1(x2+y2) (mod 1). For our purpose (in last section), it is convenient to 4 identify Z/2Z Z/2Z =Z[i]/2Z[i], where Q(z) = 1zz¯ Q/Z. We write φ , φ , φ and φ for the × ∼ 4 ∈ 0 1 i 1+i basis elements of C[L′/L] corresponding to (0,0), (1,0), (0,1) and (1,1) respectively. !,∞ Let F = F(τ) M (Γ (4),χ ) with k odd and positive. Then using Γ (4)-lifting, we can ∈ −k 0 −4 0 construct a vector valued modular form F~ = F~(τ) arising from F(τ) as follows: 1 (2.6) F~(τ) = (F γ)ρ (γ)−1φ = (F γ)ρ (γ)−1φ . |−k L 0 2 |−k L 0 γ∈Γ0(X4)\SL2(Z) γ∈Γ1(X4)\SL2(Z) Define modular forms F , F and F as follows. Let 0 2 3 ∞ 0 1 F − = a(n)qn. |−k 1 0 4 (cid:18) (cid:19) n=0 X Then for j 0,2,3 , we write ∈ { } ∞ 4n+j F = a(4n+j)q . j 4 n=0 X We also define F to be 1/2 ∞ 1 0 F = F = b(n)qn. 1/2 |−k 2 1 2 (cid:18) (cid:19) n=0 X Then a simple calculation gives (2.7) F~(τ) = ( 2iF +F)φ 2iF φ 2iF φ + 2iF F φ . 0 0 3 1 3 i 2 1/2 1+i − − − − − The following theorem gives some basic facts about F0, F2, F(cid:0)3 and F1/2. (cid:1) Theorem 2.5. With the above definitions, we have (2.8) F M! (Γ (4),χ ), 0 ∈ −k 0 −4 (2.9) F M! (Γ (4),χ ) 3 ∈ −k 0 1 a b where χ (γ) = χ (d)e( ab/4) for γ = Γ (4), 1 −4 − c d ∈ 0 (cid:18) (cid:19) (2.10) (2iF +F ) M! (Γ (4),χ ) 2 1/2 ∈ −k 0 2 a b where χ (γ) = χ (d)e( ab/2) for γ = Γ (4), 2 −4 − c d ∈ 0 (cid:18) (cid:19) 5 and (2.11) F M! (δ−1Γ (4)δ,χ ) 1/2 ∈ −k 0 −4 1 0 where δ = . 2 1 (cid:18) (cid:19) a b Proof. By(2.7),and[19,Section3,p.6]or[20,Proposition4.5],wecanshowthatforγ = c d ∈ (cid:18) (cid:19) Γ (4), 0 (2.12) ( 2iF +F) γ = χ (d)( 2iF +F), − 0 |−k −4 − 0 (2.13) F γ = χ (d)e( ab/4)F , 3|−k −4 − 3 (2.14) ( 2iF F ) γ = χ (d)e( ab/2)( 2iF F ). − 2 − 1/2 −k −4 − − 2 − 1/2 Since F M! (Γ (4),χ ), then (2.(cid:12)12) implies (2.8). Relations (2.9) and (2.10) follow directly ∈ −k 0 −4 (cid:12) from (2.13) and (2.14), respectively. The last relation (2.11) follows from the definition of F , 1/2 1 0 F = F . 1/2 |−k 2 1 (cid:18) (cid:19) (cid:3) Theorem 2.6. Let k be odd. Let F = F(τ) M!,∞(Γ (4),χ ) with ∈ −k 0 −4 ∞ F(τ) = c(n)qn. n=−m X Write ∞ ∞ 0 1 1 0 F − = a(n)qn and F = b(n)qn. |−k 1 0 4 |−k 2 1 2 (cid:18) (cid:19) n=0 (cid:18) (cid:19) n=0 X X And let the Γ (4)-lifting of F be 0 F~(τ) = c(n,φ )qnφ . µ µ µ∈L′/L n∈Q X X n≫−∞ Then we have (i) c(n,φ ) = 2ia(4n)+c(n), 0 − c(n,φ )= c(n,φ ) = 2ia(4n), 1 i − c(n,φ ) = 2ia(4n) b(2n), 1+i − − (ii) the principal part of the vector valued modular form F~(τ) is c( m)q−m+ +c( 1)q−1 φ , 0 − ··· − (iii) the constant term of the φ (cid:0)-component of F~(τ) is (cid:1) 0 m c(0,φ )= (8i)k+1 c( n)P (0)+c(0). 0 − − k,n−k+21 n=Xk+21 6 In particular, when k = 1, the constant term of the φ -component of F~(τ) is 0 m (2.15) c(0,φ ) = c( n) (64χ (n/d)+4χ (d))d2 . 0 −4 −4 −   n=1 d|n X X   Proof. Assertion (i) follows directly from (2.7). For the assertion (ii), since F is holomorphic at 0 and 1, then F for j 0,2,3 and F will not contribute anything to the principal part of F~, 2 j ∈ { } 1/2 and thus by (2.7) the principal part of F~ is c( m)q−m+ +c( 1)q−1 φ . 0 − ··· − For the assertion (iii), we first no(cid:0)te by (i) that (cid:1) c(0,φ ) = 2ia(0)+c(0). 0 − By Theorem 2.1(1), we have −k+1 k+1 −k+1 (2.16) F = c( m)θ θ 2 P (ϕ )+ +c θ θ 2 P (ϕ ) − 2 1 k,m−k+21 ∞ ··· − 2 2 1 k,0 ∞ (cid:18) (cid:19) We can verify that θ2θ1−k+21ϕl∞ 01 −01 = O(q4l), (cid:12)−k(cid:18) (cid:19) (cid:12) and thus θ2θ1−k+21ϕl∞ 01 −01 will not(cid:12)(cid:12)contribute anything to the constant term of F0 when (cid:12)−1(cid:18) (cid:19) l 1. Therefore, (cid:12) ≥ (cid:12) (cid:12) m −k+1 0 1 a(0) =  c(−n)Pk,n−k+21(0)θ2θ1 2 (cid:12)(cid:12) 1 −0  n=Xk+21 (cid:12)(cid:12)−k(cid:18) (cid:19)0  m (cid:12)  (cid:12) = (8i)k+1 c( n)P (0) (cid:12) − − k,n−k+21 n=Xk+21 where (f) denote the constant term of the q-expansion of f. Hence, we have 0 m c(0,φ )= (8i)k+1 c( n)P (0)+c(0). 0 − − k,n−k+21 n=Xk+21 For (2.15), according to (iii), we need to show that m P (0) = χ ((m+1)/d)d2 and c(0) = c( n) 4 χ (d)d2 . 1,m −4 −4 −   d|(m+1) n=1 d|n X X X   For the first formula, we first observe that ℓ θ θ−1ϕℓ = q−ℓ−1+ c ( j)q−j +O(1) 2 1 ∞ ℓ − j=1 X for 0 ℓ m. Thus there are b ,...,b such that 1 m−1 ≤ ≤ h(τ) := θ θ−1ϕm +b θ θ−1ϕm−1+ +b θ θ−1ϕ = q−m−1+a( 1)q−1+O(1) 2 1 ∞ m−1 2 1 ∞ ··· 1 2 1 ∞ − 7 for some constant a( 1). Let g(τ) be defined by − ∞ ∞ g(τ) = χ (n/d)d2 qn = d qn. −4 n   n=1 d|n n=1 X X X   It is known [17] that g(τ) is a weight 3 modular form on Γ (4) with character χ . We note by the 0 −4 basic facts about θ , θ and ϕ that h(τ) vanishes at the cusps 1/2 and 0. Then by [4, Theorem 1 2 ∞ 3.1], we have d +a( 1) = 0, i.e., d = a( 1). m+1 m+1 − − − Therefore P (0) = d = χ ((m+1)/d)d2. 1,m m+1 −4 d|(m+1) X This proves the first formula. For the second one, the proof is similar by noting that h (τ) := θ θ−1P (ϕ ) =q−m−1+C +O(q) 1 2 1 1,m ∞ and ∞ g (τ) = 1+4 χ (d)d2 qn 1 −4   n=1 d|n X X is [17] a weight 3 modular form on Γ (4) with character χ . Then again [4, Theorem 3.1] shows 0 −4 that C = 4 χ (d)d2. −4 d|(m+1) X This together with (2.16) proves the second formula. (cid:3) Example 2.7. Let k = 1 and F(τ) = θ θ−1 = η214 M!,∞(Γ (4),χ ). Then we have 2 1 η14η412 ∈ −1 0 −4 (2.17) F~(τ) = ( 2iF +F)φ 2iF φ 2iF φ + 2iF F φ 0 0 3 1 3 i 2 1/2 1+i − − − − − where F0, F2, F3 and F1/2 are defined as follows; suppose (cid:0) (cid:1) 0 1 η(τ/2)14 F − = 32i |−1 1 0 η(τ/4)4η(τ)12 (cid:18) (cid:19) = 32i 1+12q1/4 +76q2/4 +352q3/4 +1356q +4600q5/4 (cid:0) +14176q6/4 +40512q7/4 + ··· = 32i 1+1356q +O(q2) (cid:1) +32(cid:0)i 12q1/4 +4600q5/(cid:1)4 +O(q9/4) (cid:16) (cid:17) +32i 76q2/4 +14176q6/4 +O(q10/4) (cid:16) (cid:17) +32i 352q3/4 +40512q7/4 +O(q11/4) , (cid:16) (cid:17) then F = 32i 1+1356q +O(q2) , 0 F = 32i(cid:0)76q2/4 +14176q6/4(cid:1)+O(q10/4) 2 (cid:16) (cid:17) F = 32i 352q3/4 +40512q7/4 +O(q11/4) . 3 (cid:16) (cid:17) 8 And 1 0 F = F = 64 q1/2 8q3/2 +42q5/2 +O(q7/2) . 1/2 |−1 2 1 − (cid:18) (cid:19) (cid:16) (cid:17) From (2.17), we note that the principal part of F is e( τ)φ and the constant term of the φ - 0 0 − component is c(0,φ )= 68. 0 2.3. Canonical Basis for M!,0(Γ (4),χk ) and M!,21(Γ (4),χk ). −k 0 −4 −k 0 −4 WecompletethissectionbygivingcanonicalbasisfortheothertwocompanionsofM!,∞(Γ (4),χk ). −k 0 −4 Let θ (τ), ϕ (τ) and ϕ (τ) be defined by 3 0 1/2 η8 (2.18) θ = θ (τ) : =ϑ4 (τ) = 1 = 1+O(q), 3 3 01 η4 2 8 η (2.19) ϕ = ϕ (τ) : = 4 =q+O(q2), 0 0 η (cid:18) 1(cid:19) η8η16 (2.20) ϕ = ϕ (τ) : = 1 4 = q+O(q2). 1/2 1/2 η24 2 Here are some basic facts about θ , ϕ and ϕ : 3 0 1/2 (1) θ (τ) is a weight 2 modular form on Γ (4) with trivial character, has a simple zero at the 3 0 cusp 0, and vanishes nowhere else; (2) ϕ (τ) is a weight 0 modular form on Γ (4) with trivial character, has a simple pole at the 0 0 cusp 0 and a simple zero at the cusp , and vanishes nowhere else; ∞ (3) ϕ (τ) is a weight 0 modular form on Γ (4) with trivial character, has a simple pole at the 1/2 0 cusp 1 and a simple zero at the cusp , and vanishes nowhere else. 2 ∞ Theorem 2.8. Let θ , θ and ϕ be as defined in (2.2), (2.18) and (2.19), respectively. 2 3 0 (1) For k odd, the set θ θ−k+21P (ϕ ) ∞ where P is a monic polynomial of degree m { 2 3 k,m 0 }m=0 k,m such that ∞ θ2θ3−k+21Pk,m(ϕ0) 01 −01 = q4−k+21−m+ c(n)q4n, (cid:12)(cid:12)−k(cid:18) (cid:19) n=X−k−21 (cid:12) !,0 (cid:12) is a canonical basis for M (Γ (4),χ ). −k 0 −4 (2) For k even, the set θ−k2P (ϕ ) ∞ where P is a monic polynomial of degree m such { 3 k,m 0 }m=0 k,m that ∞ θ3−k2Pk,m(ϕ0) 01 −01 = q4−k2−m+ c(n)q4n, (cid:12)(cid:12)−k(cid:18) (cid:19) n=X−k2+1 (cid:12) is a canonical basis for M!,0(cid:12)(Γ (4)). −k 0 Theorem 2.9. Let θ and ϕ be as defined in (2.2) and (2.20), respectively. Then the set 2 1/2 θ−kP (ϕ ) ∞ where P is a monic polynomial of degree m such that { 2 k,m 1/2 }m=0 k,m ∞ θ2−kPk,m(ϕ1/2) 12 01 = q−k2−m+ c(n)qn, −k (cid:12)(cid:12) (cid:18) (cid:19) n=X−k2+1 is a canonical basis for M!,21(Γ (4),χ(cid:12)k ). −k 0 −4 9 Proofs of Theorems 2.8 and 2.9 are similar to that of Theorem 2.1, so we omit the details. Remark 2.10. For a cusp P, denote by M!,P the space of vector valued modular forms induced −k,ρL from M!,P(Γ (4),χk ) via Γ (4)-lifting. We have, by (2.7), −k 0 −4 0 M!,∞ +M!,0 +M!,21 = M!,∞ M!,0 M!,21 . −k,ρL −k,ρL −k,ρL −k,ρL ⊕ −k,ρL ⊕ −k,ρL Clearly, M!,∞ + M!,0 + M!,21 is a subspace of M! . In general, the former space may −k,ρL −k,ρL −k,ρL −k,ρL not be equal to the latter one. We first note that by (2.7) every vector valued modular form in M!,∞ + M!,0 + M!,21 must have the same component functions at φ and φ . We now −k,ρL −k,ρL −k,ρL 1 i give an example of functions in M! that does not have this property. Let F(τ) = θ θ−1 −1,ρL 2 1 ∈ !,∞ M (Γ (4),χ ). Then as above we write the Γ (4)-lifting of F(τ) as −1 0 −4 0 F~(τ) = ( 2iF +F)φ 2iF φ 2iF φ + 2iF F φ 0 0 3 1 3 i 2 1/2 1+i − − − − − where (cid:0) (cid:1) ∞ 4n+j F = a(4n+j)q , j 4 n=0 X ∞ 0 1 F − = a(n)qn |−k 1 0 4 (cid:18) (cid:19) n=0 X and 1 0 F = F . 1/2 |−1 2 1 (cid:18) (cid:19) a b By (2.9), we know that F (τ) M! (Γ (4),χ) where χ = e( b/4). Now we do Γ (4)- 3 ∈ −1 1 c d − 1 (cid:18)(cid:18) (cid:19)(cid:19) lifting on F (τ) against φ and get 3 1 F~ (τ) = 4if φ +(2F +4if )φ +( 4if 2f )φ +4if φ 3 0 0 3 3 1 3 1/2 i 2 1+i − − − where 4n+j f = a˜(4n+j)q , j 4 n∈Z X n≫−∞ 0 1 F3|−1 1 −0 = a˜(n)q4n (cid:18) (cid:19) n∈Z X n≫−∞ and 1 0 f = F . 1/2 3|−1 2 1 (cid:18) (cid:19) Now the component functions at φ and φ are 2F + 4if and 4if 2f , respectively. We 1 i 3 3 3 1/2 − − can compute and verify that they are not the same. Therefore, F~ (τ) is not in the space M!,∞ + 3 −k,ρL M!,0 +M!,21 . −k,ρL −k,ρL 10