Ramanujan J (2008) 17: 405–427 DOI 10.1007/s11139-007-9054-4
2-Dimensional vector-valued modular forms Geoffrey Mason
Received: 22 August 2006 / Accepted: 20 August 2007 / Published online: 12 October 2007 © Springer Science+Business Media, LLC 2007
Abstract We study the graded space H(ρ) of holomorphic vector-valued modular forms of integral weight associated to a 2-dimensional irreducible representation ρ of SL(2, Z). When ρ(T ) is unitary, a complete description of H(ρ) is given: the Poincaré series is calculated and it is shown that H(ρ) is a free module of rank 2 over the ring of (classical) holomorphic modular forms on SL(2, Z). Keywords Vector-valued modular form · Representation of the modular group · Linear differential equation Mathematics Subject Classification (2000) 11F12 · 11F99
1 Introduction Let = SL(2, Z) and ρ : → GL(2, C) a 2-dimensional representation. A holomorphic vector-valued modular form of weight k with respect to ρ is a pair of functions F (τ ) = (f1 (τ ), f2 (τ ))t , holomorphic in the complex upper half-plane H, and which satisfy a b ρ(γ )F (τ ) = (cτ + d)−k F (γ τ ), γ = ∈ , c d where the superscript t denotes the transpose. Moreover, the component functions fi (τ ) are required to have q-expansions which are holomorphic at infinity in the usual Supported by the NSA, the NSF and the Committee on Research at the University of California, Santa Cruz. G. Mason () Department of Mathematics, University of California, Santa Cruz, CA 95064, USA e-mail:
[email protected]
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sense of modular forms. (See Sect. 2 for more details.) We consider only the case of integral weights in the present paper. The set H(k, ρ) of holomorphic vector-valued modular forms of weight k is a complex linear space of finite dimension [6]. A theory of vector-valued modular forms (of any finite dimension) was pursued in [4–6], where a number of general properties were established. In the present paper we will make a precise study of 2-dimensional holomorphic vector-valued modular forms. As in [5]and[6], we are mainly interested in those ρ which are irreducible and such that ρ 10 11 is equivalent to a unitary matrix. For such ρ, we will give a precise description of all holomorphic vector-valued modular forms of any integral weight k. To describe our main results, let H(ρ) = H(k, ρ) (1) k
be the Z-graded space of holomorphic vector-valued modular forms. H(ρ) admits an action of both the ring M of (classical, or scalar) holomorphic modular forms on and the vector-valued analog of the weight-changing operator D : H(k, ρ) → H(k + 2, ρ). M and D together generate a ring R of skew polynomials (a.k.a. Ore extension, or algebra of differential operators) acting on H(ρ). Elements of R may be regarded as polynomials in D with coefficients in M, though the ring is noncommutative. Among other things, we will establish the following results about the structure of H(ρ): 1. Let the eigenvalues of ρ 10 11 be e2πim1 , e2πim2 , 1 > m1 > m2 ≥ 0. Set k0 = 6(m1 + m2 ) − 1 (∈ Z). Then dim H(k, ρ)t k = t k0 /(1 − t 2 )(1 − t 6 ), k
i.e. dim H(k0 + 2k, ρ) = 1 + [k/3], k ≥ 0. 2. Let F0 span H(k0 , ρ). Then F0 generates H(ρ) considered as R-module. 3. There is an isomorphism of Z-graded linear spaces H(ρ) ∼ = R/RQ where Q ∈ R is a certain quadratic skew polynomial. 4. H(ρ) is a free M-module of rank 2 with basis F0 , F1 = DF0 . Thus, every F ∈ H(k, ρ) has a unique representation F = αF0 + βF1 with α, β ∈ M. We make some further comments about the nature of the proofs, and the contents of the paper. In Sect. 2 we provide further background concerning vector-valued modular forms and discuss the ring R in greater detail. In Sect. 3 we give (Theorem 3.1) the classification of the irreducible 2-dimensional representations ρ of for which 11 ρ 0 1 is semisimple. This is probably well-known, but in any case it is not difficult and it plays an important rôle at several subsequent points in the paper. We also discuss those ρ which are unitarizable and those whose image ρ() is finite. The
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characterization of the latter class of representations requires a bit of effort; it turns out (Theorem 3.7) that there are exactly 54 equivalence classes of such irreducible representations, and in each case the kernel of ρ is a congruence subgroup of level dividing 120. It is just this set of 54 classes of representations for which the component functions fi (τ ) of any (f1 (τ ), f2 (τ ))t ∈ H(k, ρ) are classical modular forms of weight k. In Sect. 4 we make a start on the assertions (1)–(4) listed above. We show (Theorem 4.2) that H(k, ρ) = 0 if k < k0 and is 1-dimensional if k = k0 . The methods here depend heavily on our earlier paper [8]. There is a general problem of how to construct holomorphic vector-valued modular forms when standard analytic approaches such as those used in [5, 6] are inapplicable. The idea of [8] is to make use of the theory of linear differential equations, and we implement this idea here. The modular Wronskian associated to a vector-valued modular form (loc. cit.) is employed to show nonexistence for k < k0 , and we show how to write down an order 2 modular linear differential equation (loc. cit.) whose solution space is spanned by the components of a (nonzero) F0 ∈ H(k0 , ρ). This approach is the conceptual heart of the present paper, and is related to D-module theory. The LDE satisfied by the components of F0 defines a quadratic polynomial Q(D) ∈ R, and it is just this Q which intervenes in assertion (3). In Sect. 5 we complete the proofs of assertions (1)–(4). After the results of Sect. 4 these are mainly formal in nature, although a second application of the method of LDEs is needed at one point. In Sect. 6 we consider 2-dimensional vector-valued Eisenstein series. These were introduced in [6] (at least for large enough weight) and involve vector-valued modular forms (f1 (τ ), f2 (τ ))t for which the Fourier expansion of f2 (τ ) (say) lies in C[[q]] (though this is not the definition). It turns out that there are just 3 equivalence classes of ρ for which nonzero Eisenstein series exist, and in each case the image of ρ is finite. So the component functions are scalar forms on a congruence subgroup of , and we indicate how to compute the q-expansion for f2 (τ ) using results about vector-valued Poincaré series established in [6]. This involves evaluation of the generalized Kloosterman sums (loc. cit.) determined by ρ. In the present situation they turn out to be essentially Gauss sums. Differential equations similar to the ones we consider have appeared several times in the recent literature, though except for [8] the perspective of vector-valued modular forms is not utilized. We mention explicitly the paper of Kaneko and Koike [3], which has a more arithmetic focus than the present paper. We hope to consider this aspect of our 2-dimensional vector-valued modular forms in a later work.
2 Background We introduce standard matrices −1 1 R= , −1 0
S=
0 −1 , 1 0
T=
1 1 0 1
in . Any two of them generate , indeed = R, S, T | RS = T , R 3 = S 4 = I .
(2)
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The commutator quotient / is cyclic of order 12, and is generated by the coset T . The group of characters (i.e. 1-dimensional representations) of is also cyclic of order 12, with generator χ which satisfies χ(T ) = e2πi/12 .
(3)
acts in the standard way on H, i.e. aτ + b γ :τ → , cτ + d
γ=
a c
b d
∈ .
Let j (γ , τ ) = cτ + d denote the usual 1-cocycle on , with F the space of holomorphic functions on H. For k ∈ Z, f |k γ (τ ) = f (γ τ )j (γ , τ )−k
(4)
defines a right action of on F . The action of on F extends in a componentwise fashion to F 2 . Now suppose that ρ : → GL(2, C) is a representation of . There is then another right action of on F 2 defined by γ : F → ρ(γ )−1 F |k γ . (Note that (ρ(γ )−1 F )|k γ = ρ(γ )−1 (F |k γ ).) A weak vector-valued modular form of weight k (with respect to ρ) is a -invariant with respect to this action. In other words, we have F |k γ = ρ(γ )F,
γ ∈ .
(5)
A meromorphic vector-valued modular form of weight k is a weak vector-valued modular form F = (f1 , f2 )t ∈ F 2 of weight k such that each component function fi has a q-expansion, convergent in a neighborhood of infinity, of the shape fi (τ ) = q λi an (i)q n , λi ∈ R. (6) n≥0
(Here and below, q = e2πiτ .) If λj is nonnegative we say that fj is holomorphic at infinity, and if each fj satisfies this condition then F is called a (holomorphic) vectorvalued modular form. F is a cusp-form if each λj is positive. One may consider more general kinds of vector-valued modular forms, but these are the ones of interest in the present paper. The space H(k, ρ) of holomorphic vector-valued modular forms of weight k and the space H(ρ) of all holomorphic vector-valued modular forms associated to ρ have already been introduced in Sect. 1. They contain respectively the space S(k, ρ) of cuspidal vector-valued modular forms of weight k and the S(ρ) = k∈Z S(k, ρ) of all vector-valued cusp-forms. By [6], Lemma 2.4 and Theorem 2.5, H(k, ρ) has finite dimension and is zero for k 0. If ρ = 1 is the trivial 1-dimensional representation
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of then H(1) is the ring of classical holomorphic modular forms on , which we denote by M. As usual, a pair of representations ρ, ρ of are called equivalent if there is an invertible matrix A such that ρ (γ ) = Aρ(γ )A−1 ,
γ ∈ .
It is easy to see that in this situation, there is a linear isomorphism H(k, ρ) → H(k, ρ ), F → AF. Because of this, when proving statements about vector-valued modular forms (e.g. assertions (1)–(4) in the Introduction), one may usually replace ρ by any representation to which it is equivalent. In our case, because ρ(T ) is assumed to be unitarizable, we may (and usually shall) assume that 2πim 1 e 0 ρ(T ) = , 1 > m1 > m2 ≥ 0. (7) 0 e2πim2 We discuss the weight-changing operator D in the context of vector-valued modular forms [8]. The ‘unmodular’ Eisenstein series of weight 2 is dq n . (8) E2 (q) = −1/12 + 2 n≥1 d|n
For an integer k we have the operator Dk : F → F , f →
1 df + kE2 f. 2πi dτ
(9)
It is well-known [7] that the following is satisfied: Dk (f |k γ ) = (Dk f )|k+2 γ ,
γ ∈ .
(10)
In particular, restricting the action of Dk to modular forms, we find that Dk : H(k, 1) → H(k + 2, 1). Define D : M → M so that the restriction of D to H(k, 1) coincides with the action of Dk . D is a graded derivation of the algebra M, that is D(fg) = D(f )g + f D(g).
(11)
Extend Dk and D to linear operators on H(ρ) in a componentwise fashion, so that Dk : H(k, ρ) → H(k + 2, ρ), D : H(ρ) → H(ρ). Note that H(ρ) is a Z-graded module over M, more precisely multiplication induces natural bilinear maps H(k, 1) × H(l, ρ) → H(k + l, ρ).
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We combine the actions of M and D. Introduce the ring of skew polynomials R = M[d] = {f0 + f1 d + · · · fn d n | f0 , . . . , fn ∈ M, n ≥ 0}. The elements are (skew) polynomials in an indeterminate d with coefficients in M. Addition is defined in the usual way and multiplication is (uniquely) determined by the identity df − f d = D(f ),
f ∈ M.
(12)
Note that in general d does not commute with the elements of M, and one should be careful not to be misled by the word ‘polynomial’. In any case, this defines an associative algebra structure on R, a well-known fact which follows from the derivation property (11). R is also known as the Ore extension or algebra of differential operators associated to M and D. If we give d degree 2 then R is a Z-graded algebra with M a graded subalgebra. The obvious vector-valued analog of (11) holds, i.e. D(f F ) = D(f )F + f D(F ),
f ∈ M, F ∈ H(ρ).
(13)
It follows that H(ρ) carries the structure of a Z-graded (left) R-module.
3 2-Dimensional representations of In this section we will consider the equivalence classes of 2-dimensional irreducible representations ρ : → GL2 (C) such that ρ(T ) is semisimple. This includes the representations we are mainly interested in, when ρ(T ) is unitarizable. It is easy enough to enumerate the remaining 2-dimensional representations of , which are either not irreducible or for which ρ(T ) is not semisimple, though we will not do that here. For later purposes, it will be useful to first assume only that ρ is indecomposable. This has the following consequences: No one of ρ(R), ρ(S), ρ(T ) is a scalar matrix; ρ(−I ) = ±I. Because det ρ(−I ) = 1, det ρ is a character of order dividing 6. Thus σ = det(T ) is a sixth root of unity. Since ρ(T ) is diagonalizable we may, and shall, assume that λ 0 ρ(T ) = (14) 0 λ−1 σ for some λ ∈ C∗ . Since ρ(T ) is not a scalar then σ = λ2 . Now the eigenvalues of ρ(S) are either ±1 or ±i, and in either case we have trρ(S) = 0. Since det ρ(S) = det ρ(T 3 ) = σ 3 , it follows that there are scalars a, b, c
2-Dimensional vector-valued modular forms
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such that ρ(S) =
a c
b , −a
a 2 + bc = −σ 3 .
Observe that if bc = 0 then ρ(S), ρ(T ) are both upper or both lower triangular, in which case ρ is not irreducible. Now assume that bc = 0. The matrices which commute with ρ(T ) are the diagonal matrices D = u0 v0 , and if uv = 0 then a bu/v Dρ(S)D −1 = cv/u −a . Conjugating ρ(S) by suitable D we may choose b = 1. So we may, and shall, assume that a 1 ρ(S) = (15) −a 2 − σ 3 −a with a 2 = −σ 3 . Now ρ(R) = ρ(T )ρ(S
−1
)=σ
3
−aλ σ λ−1 (a 2 + σ 3 )
−λ aσ λ−1
.
(16)
Since det ρ(R) = det ρ(T 4 ) = σ 4 and det ρ(R) + trρ(R) = 0, then σ 2 = −aσ 3 (σ λ−1 − λ). That is, a=
1 . σ (λ − σ λ−1 )
(17)
One checks that the condition a 2 = −σ 3 is then equivalent to σ λ−2 is not a primitive 6th root of unity.
(18)
This proves that if ρ(T ) is as in (14) then, up to equivalence, there is at most one irreducible representation ρ of . Conversely, suppose that λ ∈ C∗ , σ is a 6th. root of unity unequal to λ2 , and we take a as in (17) and ρ(R), ρ(S) as above. Then ρ(R)3 = I, ρ(S)2 = ±I , so that the two matrices ρ(R), ρ(S) generate a quotient group of and thereby define a representation of . The representation is irreducible if, and only if, (18) holds. Otherwise, it is indecomposable but not irreducible. We have established Theorem 3.1 Let μ1 , μ2 be a pair of complex numbers satisfying (μ1 μ2 )6 = 1 and μ2 /μ1 ∈ / {1, ±e2πi/6 }. Up to equivalence, there is exactly one 2-dimensional indecomposable representation ρ of such that ρ(T ) = diag(μ1 , μ2 ), and ρ is irreducible. Every irreducible representation of such that ρ(T ) is semisimple is equivalent to such a ρ. There is an alternate approach which is useful. Namely, after having observed that bc = 0 in the previous√ proof, it is clear that by choosing the conjugating diagonal matrix D to be diag(1, b/c), the conjugated S-matrix Dρ(S)D −1 is symmetric. So
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up to equivalence, we may choose ρ(S) = κ −1
√1 r
√ r , −1
(19)
where κ = σ (λ − σ λ−1 ), r = −σ 3 κ 2 − 1.
(20)
Thus we have Theorem 3.2 Each 2-dimensional irreducible representation of for which ρ(T ) is semisimple is equivalent to a representation ρ with ρ(T ) the diagonal matrix (14) and ρ(S) the symmetric matrix (19). Theorem 3.3 The representation ρ defined by (14), (19) and (20) is unitary if, and only if, |λ| = 1, r is a positive real number, and κ = −σ 3 κ. Any 2-dimensional irreducible representation of which is unitarizable is equivalent to such a ρ. Proof ρ is unitary if, and only if, ρ(T ) and ρ(S) are unitary, and we easily find that this is equivalent to the stated numerical restrictions on λ, r and κ. To complete the proof of the theorem, we must show that if ρ(S) is not unitary, then ρ is not unitarizable. Otherwise, there is an invertible diagonal matrix D = diag(1, b) such that Dρ(S)D −1 is unitary. Now √ 1 r/b −1 −1 √ Dρ(S)D = κ −1 b r and the unitarity of this matrix entails the identity bb¯ = 1. Then D is unitary, whence so too is ρ(S). We now consider the question of which 2-dimensional irreducible representations ρ have finite image. Let us say that ρ is a basic representation if it is irreducible and satisfies the following additional conditions: (a) ρ : → SU(2, C) (b) ρ() is finite (c) ρ(T ) does not have twice odd order. Lemma 3.4 Let ρ be a 2-dimensional irreducible representation of . Then ρ() is finite if, and only if, there is an integer s such that χ s ρ is equivalent to a basic representation. Proof Since ker χ has index 12 in , sufficiency is clear. Now assume that ρ() is finite. Then ρ is certainly unitarizable, so we may, and shall, assume that ρ is unitary.
2-Dimensional vector-valued modular forms
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For any s, χ s ρ is also unitary with finite image, and we have to show that we can pick s so that (a) and (c) both hold. We have already observed that det ρ is a character of of order dividing 6. It therefore lies in the group of characters generated by χ 2 . Thus there is an integer s such that det ρ = χ −2s . Then we have det(χ s ρ) = χ 2s det ρ = 1. So χ s ρ maps into SU(2, C) and (a) is satisfied. Finally, observe that χ s+6 ρ also has determinant 1, and if χ(T )s ρ(T ) has twice odd order then χ(T )s+6 ρ(T ) has odd order. So at least one of the two representations χ s ρ, χ s+6 ρ satisfies (a)–(c) and hence is a basic representation. For any character λ of , we call the representation λρ a twist of ρ. Lemma 3.4 reduces the classification of the irreducible ρ with finite image to the classification of the basic representations and their twists. We will see that there are just six basic representations, up to equivalence. The next result contains the main calculation. Theorem 3.5 Let ρ be a basic representation of . Set K = ker ρ, G = /K, and let N be the order of ρ(T ). There are just four possibilities for K, and one of the following holds: (a) (b) (c) (d)
G is the binary dihedral group of order 12, N = 4 G is the binary tetradral group of order 24, N = 3 G is the binary octahedral group of order 48, N = 8 G is the binary icosahedral group of order 120, N = 5.
In each case (N) ⊆ K. Proof It is readily checked that the groups G in (a)–(d) are quotients of . It is the uniqueness of K in each case that is the main issue. We set ¯ = /K and use the bar ¯ notation, whereby a bar over an element or subgroup of denotes its image in . ¯ Since ρ is a basic representation, is a finite irreducible subgroup of SU(2, C). By a well-known result of F. Klein, ¯ is a binary dihedral, tetrahedral, octahedral or icosahedral group. Recall the normal closure of T N , that is
(N ) = γ T N γ −1 | γ ∈ . The following result is well-known (e.g. [10]).
(N ) = (N )
if (and only if) N ≤ 5.
(21)
By definition, we have (N ) ⊆ K, so if N ≤ 5 then (N ) ⊆ K. We use this fact several times below. ¯ , ¯ and so ¯ = Case 1: ¯ is binary dihedral. Since R¯ has odd order then R ¯ ¯ ¯ ¯ ¯ R, S = RS. Hence N = 4 and has order 12. As for uniqueness of K, we have (4) ⊆ K. Moreover (2)/ (4) ∼ = Z32 , and as GF(2)R-module it is the direct sum
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of ±I (4) and an irreducible R-module V / (4). Then we must have V ⊆ K, and since V with /V the binary dihedral group of order 12, then in fact K = V . So K is uniquely determined in this case. Case 2: ¯ is binary tetrahedral. Because T¯ cannot have order 6 by hypothesis then we must have N = 3. So (3) ⊆ K, and since / (3) is the binary tetrahedral group, we have K = (3). Case 3: ¯ is the icosahedral. This is similar to the previous case: we must have N = 5 and K = (5). Case 4: ¯ is binary octahedral. This final case is more intricate than the others. At any rate, we have N ≥ 4, and if this is equality then (4) ⊆ K. But / (4) and the binary octahedral group both have order 48 and are not isomorphic groups. So in fact N = 4. The only remaining possibility is N = 8. The unique involution of ¯ is −I , which coincides with T 4 . This shows that (4) ⊆ ±I K. Having already established that K = (4), it follows that K0 = (4) ∩ K has index 2 in both (4) and K. Let L be the unique normal subgroup of (4) minimal with respect to the property that (4)/L has exponent 2. Since (4) is a free group of rank 5 then (4)/L ∼ = Z52 . Note that L ⊆ K0 . On the other hand, ∼ one readily computes that (4)/ (8) = Z32 , so we also have L ⊆ (8). We are going to show that (8) ⊆ K0 . We work in the quotient group ˜ = /L. Introduce the elements 1 2 1 0 α= L, β= L. 0 1 2 1 The matrices (without the adorning L) are free generators of (2), and their images in ˜ (2)/ (4) generate the group V / (4) which intervened in Case 1. Set P = (4) = 5 2 ∼ ∼ ˜ ˜ (4)/L = Z2 , Q = (8) = (8)/L = Z2 . We have to show that Q ⊆ K0 . Direct calculation shows that [P , α], [P , β] ⊆ Q. Note that [P , α], [P , β] and [P , αβ] are permuted transitively among themselves via conjugation by R, in particular all three groups have the same order. Suppose that [P , α] = [P , β] has order at most 2. Then ˜ ˜ W = [α, β], [P , α] has order at most 4, W (2), and (2)/W is abelian. But this implies that (2) has an abelian quotient which cannot be generated by two elements, which is impossible since (2) is free of rank 2. So either [P , α] has order 4, or else it has order 2 and is distinct from [P , β]. In either case it follows that [P , α][P , β] = Q. Now if [P , α] has order 2 then the three subgroups [P , α], [P , β], [P , αβ] must be the three distinct subgroups of Q of order 2. As they are permuted transitively by R then Q = [Q, R], in which case the containment Q ⊆ K˜0 follows. The remaining case is when Q = [P , α]. In this case CQ (α) has order 8 (look at the Jordan normal form for α considered as a linear automorphism of the GF(2)-space Q). Now consider K˜0 as GF(2)α-module: |K˜0 | = 16, so either |[K˜0 , α]| = 4 or |CK˜0 (α)| = 8 (Jordan decomposition again). In the first case we have Q = [P , α] = [K0 , α] ⊆ K˜0 . In the second case Q ⊆ CP (α) = CK˜0 (α) ⊆ K˜0 . Having established that K0 , and hence K, contains (8), it remains to show that K is uniquely determined. Indeed, we must have K0 = (8)[(4), R]. Moreover the quotient /K0 has exactly 3 subgroups of order 2, two of which are (4)/K0 and ±I K0 . Since K contains neither (4) nor ±I then K/K0 must be the unique
2-Dimensional vector-valued modular forms
415
remaining subgroup of order 2 in /K0 . This establishes the uniqueness of K, and completes the proof of the theorem. Corollary 3.6 has exactly 6 inequivalent basic representations. They may be labelled ρi , 1 ≤ i ≤ 6, so that (a) (b) (c) (d)
ρ1 () is the binary dihedral group of order 12 ρ2 () is the binary tetradral group ρ3 () ∼ = ρ4 () is the binary octahedral group ρ5 () ∼ = ρ6 () is the binary icosahedral group.
Proof We have to enumerate the isomorphisms of /K into SU (2, C) for each of the cases (a)–(d) in Theorem 3.5. This is straightforward, and one finds that up to equivalence, there is a unique such representation in cases (a) and (b) and a pair of dual representations in cases (c) and (d). Concerning the twists of ρi , one can readily check (e.g. using Theorem 3.1) that the following holds: those for ρ1 are equivalent in pairs; no two twists of ρ2 , ρ5 and ρ6 are equivalent; all 12 twists of ρ3 are inequivalent, and each is equivalent to a twist of ρ4 . Using Lemma 3.4, we arrive at Theorem 3.7 has exactly 54 equivalence classes of irreducible representations ρ for which the image ρ() is finite. They consist of 6 twists of ρ1 and 12 twists of each of ρ2 , ρ3 , ρ5 and ρ6 . The image of under a twist ρ of a basic representation ρi is generally not a subgroup of SU(2, C), and the level of ker ρ is generally not equal to that of ker ρi . However, because K in Theorem 3.5 is a congruence subgroup, ker ρ is itself a congruence subgroup. It turns out that the level can be any divisor of 120 except for 1, 40 and 120. We enumerate each of the 54 possibilities in Tables 1–5 below. We list the values m1 , m2 where ρ(T ) is as in (7) (this determines ρ up to equivalence by Theorem 3.1), and the level N . For later convenience we also record the integer k0 = 6(m1 + m2 ) − 1. Tables 1–5 correspond to ρ1 , ρ2 , ρ3 , ρ5 and ρ6 respectively. The ith row of each table is the data associated to the representation χ i−1 ρj . Table 1 Twists of ρ1 m1
m2
N
k0
3/4
1/4
4
5/6
1/3
6
6
11/12
5/12
12
7
5
1/2
0
2
2
7/12
1/12
12
3
2/3
1/6
6
4
416 Table 2 Twists of ρ2
Table 3 Twists of ρ3
Table 4 Twists of ρ5
G. Mason m1
m2
N
k0
2/3
1/3
3
5
3/4
5/12
12
6
5/6
1/2
6
7
11/12
7/12
12
8
2/3
0
3/4
1/12
3
3
12
4
5/6
1/6
6
5
11/12
1/4
12
6
1/3
0
5/12
1/12
3
1
12
2
1/2
1/6
6
3
7/12
1/4
12
4
m2
N
k0
m1 7/8
1/8
8
5
23/24
5/24
24
6
7/24
1/24
24
1
3/8
1/8
8
2
11/24
5/24
24
3
13/24
7/24
24
4
5/8
3/8
8
5
17/24
11/24
24
6
19/24
13/24
24
7
7/8
5/8
8
8
23/24
17/24
24
9
19/24
1/24
24
4
N
k0
m1 4/5
m2 1/5
5
5
60
6
53/60
17/60
29/30
11/30
30
7
9/20
1/20
20
2
8/15
2/15
15
3
37/60
13/60
60
4 5
7/10
3/10
10
47/60
23/60
60
6
13/15
7/15
15
7
19/20
11/20
20
8
19/30
1/30
30
3
43/60
7/60
60
4
2-Dimensional vector-valued modular forms Table 5 Twists of ρ6
m1
417 m2
N
k0
3/5
2/5
5
5
41/60
29/60
60
6
23/30
17/30
30
7
17/20
13/20
20
8
14/15
11/15
15
9
49/60
1/60
60
4 5
9/10
1/10
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1/15
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1
7/20
3/20
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4
4 Vector-valued modular forms of low weight We consider a 2-dimensional irreducible representation ρ of such that ρ(T ) is as in (7). Set m = m1 + m2 . Thanks to the irreducibility of ρ, we have ρ(S 2 ) = ±I . Therefore det ρ(S 2 ) = det ρ(T 6 ) = e12πim = 1. It follows that 6m ∈ Z, 1 ≤ 6m < 12, and det ρ = χ 12m where χ is as in (3). Lemma 4.1 Suppose that 0 = F ∈ H(k, ρ). Then ρ(S 2 ) = (−1)k I. Proof We have (−1)k F = F |k S 2 = ρ(S 2 )F. Since ρ(S 2 ) is a scalar matrix, the result follows.
Now suppose that F = (f1 , f2 )t ∈ H(k, ρ) is nonzero. Replacing ρ by a suitable equivalent representation if necessary we may, and shall, assume that the q-expansions of the component functions are f1 (τ ) = q λ1 + · · · , f2 (τ ) = q λ2 + · · · , where λ1 , λ2 ≥ 0 satisfy λj − mj ∈ Z,
j = 1, 2.
(22)
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We also have λj ≥ mj ,
j = 1, 2.
(23)
Moreover λ1 − λ2 ∈ / Z, and in particular f1 (τ ), f2 (τ ) are linearly independent. We now prove Theorem 4.2 We have dim H(k, ρ) =
0, k < 6m − 1, 1, k = 6m − 1.
Proof We use results of [8]. Suppose that 0 = F = (f1 , f2 )t ∈ H(k, ρ), and let the notation be as above. First we show that k ≥ 6m − 1. As introduced in [8], the modular Wronskian associated to F is the determinant
f1 Df1
. W (F ) =
f2 Df2
It is a certain classical modular form. Indeed, taking into account the fact that ρ has dimension 2 in the present context, Theorem 3.6 (loc. cit.) says that there are a nonzero constant υ and a nonzero modular form G ∈ M such that W (F ) = η24m G ∈ H(2k + 2, χ 12m ).
(24)
Since G has nonnegative weight, we obtain 2k + 2 ≥ 12m,
(25)
as desired. Now suppose that k = 6m − 1. By (23–25), W (F ) is necessarily a nonzero multiple of η24m . From [8], Sect. 4, especially Theorems 4.1 and 4.3, this in turn holds if, and only if, f1 and f2 form a fundamental system of solutions of an order 2 differential equation of the shape Dk2 f + κ1 E4 f = 0. Here, κ1 is a constant, E4 is the weight 4 Eisenstein series E4 = 1 + 240 σ3 (n)q n ,
(26)
(27)
n≥0
Dk is the operator (9), and Dk2 = Dk+2 ◦ Dk . The LDE (26) has q = 0 as a regular singular point, and the roots of its indicial equation must be λ1 = m1 and λ2 = m2 . This requirement determines κ1 uniquely. Indeed, we have κ1 = m1 m2 − k(k + 2)/144.
(28)
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We have therefore shown that if there is a nonzero vector-valued modular form F ∈ H(6m − 1, ρ) then its component functions span the solution space of a uniquely determined linear differential equation. As a result, F is unique up to scalars, so that dim H(6m − 1, ρ) ≤ 1. To show the existence of a nonzero element in H(6m − 1, ρ), we argue along similar lines. With κ1 as in (28), the LDE (26) has indicial equation with roots m1 , m2 . Since m1 and m2 do not differ by an integer, the Fuchsian theory [1, 2] shows that there is a fundamental system of solutions fj , j = 1, 2 with q-expansions fj = q mj + · · ·. By [8], Sect. 4, the solution space V of (26) is a -module with respect to the kth. slash action |k . Hence, (f1 , f2 )t is a nonzero element of H(k, ρ ) where ρ is the representation of afforded by V . It remains to show that ρ is equivalent to ρ. It is clear that we may take ρ (T ) to coincide with ρ(T ). By Theorem 3.1, the isomorphism class of ρ is uniquely determined among 2-dimensional indecomposable representations of by the matrix ρ(T ). So if ρ is indecomposable it is necessarily equivalent to ρ, and we are done. We will derive a contradiction if ρ is assumed to be not indecomposable. For in this case, ρ is (equivalent to) the direct sum of a pair of 1-dimensional representations ρ1 , ρ2 , say, with e2πimj = ρj (T ), j = 1, 2. Moreover fj is a classical modular form in H(k, χ 12mj ) with q-expansion fj = q λj + · · ·. The q-expansion shows that fj /η24λj is a classical holomorphic modular form, and in particular fj has weight at least 12λj . We then have 12m − 2 = wtf1 + wtf2 ≥ 12λ1 + 12λ2 = 12m, which is the desired contradiction. This completes the proof of the theorem.
5 Structure of H(ρ) We continue with the assumptions and notation of the previous section. Set k0 = 6m − 1. According to Theorem 4.2, H(k0 , ρ) is 1-dimensional and H(k, ρ) = 0 for k < k0 . Using Lemma 4.1 we then find that H(ρ) = H(k0 + 2k, ρ). (29) k≥0
We are going to compute the Poincaré series of the Z-graded space (29). As before we may, and shall, choose F0 = (f1 , f2 )t ∈ H(k0 , ρ) so that fj (τ ) = q mj + · · · ,
j = 1, 2.
Consider the action of the skew polynomial ring R on H(ρ) (cf. Sect. 2). We will show that F0 generates H(ρ) considered as an R-module: H(ρ) = RF0 .
(30)
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First we compute the annihilator of F0 in R. We know that f1 and f2 form a fundamental system of solutions of the differential equation (26) (with k = k0 ). In terms of the R-action, this says that (d 2 + κ1 E4 )F0 = 0, where κ1 is as in (28). Let Q = Q(d) = d 2 + κ1 E4 be the quadratic polynomial that annihilates F0 . Lemma 5.1 The annihilator AnnR (F0 ) of F0 in R is the left ideal RQ generated by Q. Proof It is clear from what we have already said that RQ ⊆ AnnR (F0 ). Choose p(d) ∈ AnnR (F0 ). Notwithstanding the fact that the elements of R are skew polynomials, it is not hard to see that there are m(d), r(d) ∈ R such that r(d) has degree at most 1 and p(d) = m(d)Q + r(d).
(31)
Thus r(d) annihilates F0 . If r(d) is nonzero then it follows from r(d)F0 = 0 that the components of F0 satisfy a LDE of order 1, which is impossible. Therefore, r(d) = 0 and p(d) ∈ RQ. This completes the proof of the lemma. From the last lemma it follows that there is an injective morphism of Z-graded R-modules R/RQ → H(ρ), p(d) + RQ → p(d)F0 .
(32)
We will show that (32) is an isomorphism. Note that (31) is a general fact which applies for any p(d) ∈ R, with r(d) of degree at most one. As a result, there is a direct sum decomposition into M-modules R = RQ ⊕ M ⊕ Md. Then the claimed isomorphism (32) is equivalent to the following assertion: H(ρ) = MF0 ⊕ MdF0 .
(33)
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In words, H(ρ) is a free M-module of rank 2 with free generators F0 and dF0 . Assuming the truth of (33), it follows that dim H(k0 + 2k, ρ) = dim H(2k, 1) + dim H(2k − 2, 1). Using the well-known formula for dim H(k, 1), one checks that for all k ≥ 0, dim H(k0 + 2k, ρ) = 1 + [k/3].
(34)
On the other hand, if (34) holds then so too does (33). The upshot is that (30), (33) and (34) are equivalent statements. We will thus have proved all of them once we have proved (34), to which we now turn our attention. Lemma 5.2 If 0 ≤ k ≤ 2 then H(k0 + 2k, ρ) is spanned by d k F0 , and in particular is 1-dimensional. Proof The case k = 0 is a restatement of Theorem 4.2, so take k = 1 or 2. It is clear that d k F0 is a nonzero element of H(k0 + 2k, ρ), so the dimension is at least one. If the lemma is false then we can choose G ∈ H(k0 + 2k, ρ) such that aq m1 + n≥1 an q n+m1 g1 = (n2 ≥ 1, a = 0). G= (35) g2 q m2 +n2 + n≥1 bn q n+m2 +n2 Consider the modular Wronskian
g W (G) =
1 g2
Dg1
. Dg2
Using [8], the irreducibility of ρ guarantees that g1 and g2 are linearly independent and that therefore W (G) is a nonzero classical modular form of weight 2(k0 + 2k + 2) with q-expansion beginning υq m+n2 + · · · ,
υ = 0.
Thus the weight of W (G) is bounded below by 12(m + n2 ) and we have the inequalities 12(m + 1) ≤ 12(m + n2 ) ≤ 2(k0 + 2k + 2) = 2(6m + 2k + 1). But then 4k ≥ 10, i.e. k ≥ 3, a contradiction which completes the proof of the lemma. Lemma 5.3 H(k0 + 6, ρ) contains a vector-valued modular form G3 of shape m1 a q + n≥1 an q n+m1 , a = 0. G3 = q m2 +1 + n≥1 bn q n+m2 +1 Proof Consider the LDE Dk2 f + κ1 E4 f = 0,
(36)
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G. Mason
where k = k0 + 6, κ1 = m1 (m2 + 1) − k (k + 2)/144. This is nothing more than (26) and (28) with the previous k, κ1 replaced by k , κ1 respectively. We find, as in the proof of Theorem 4.2, that a fundamental system of solutions of (36) provides the vector-valued modular form G that we seek. In the following, E2k = 1 + · · · ∈ M is the usual Eisenstein series of weight 2k, and = η24 = q + · · · the cusp-form in M of weight 12. Lemma 5.4 Suppose that k ≥ 3. There is a decomposition H(k0 + 2k, ρ) = U ⊕ H(k0 + 2k − 12, ρ), where U has dimension 2. Proof First observe that given three linearly independent vector-valued modular forms in H(k0 + 2k, ρ), a nonzero linear combination of them can be found of the shape F =
a q m1 +n1 + n≥1 an q n+m1 +n1 q m2 +n2 + n≥1 bn q n+m2 +n2
with n1 , n2 ≥ 1. Then −1 F is a nonzero, holomorphic, vector-valued modular form of weight k0 + 2k − 12, that is F ∈ H(k0 + 2k − 12, ρ). This shows that
H(k0 + 2k − 12, ρ) has codimension at most 2 in H(k0 + 2k, ρ). If k ≥ 3 we have to exhibit a 2-dimensional subspace U whose intersection with H(k0 + 2k − 12, ρ) is trivial. If k = 4 we may take U = E2k F0 , E2k−6 G3 , as we see from Lemma 5.3. Now consider the residual case k = 4. It follows from Lemma 5.3 that H(k0 + 8, ρ) contains the 2-dimensional subspace E8 F0 , dG3 , and if this subspace intersects
H(k0 + 2k − 12, ρ) trivially we are done. Otherwise, we can find 0 = F ∈ H(k0 + 8, ρ)∩ H(ρ). But then 0 = −1 F ∈ H(k0 −4, ρ), and this contradicts Theorem 4.2. The proof of the lemma is complete. We now establish (34). Define δ(k) = dim H(k0 +2k, ρ) for k ≥ 0 and δ(k) = 0 for k < 0. We have to prove that δ(k) = 1 + [k/3] for k ≥ 0. If 0 ≤ k ≤ 2 then Lemma 5.2 yields the desired formula. By Lemma 5.4 it follows that if k ≥ 3 then δ(k) = 2 + δ(k − 6).
(37)
Then we find inductively that δ(k) = 2 + (1 + [(k − 6)/3]) = 1 + [k/3], as required. We have proved
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Theorem 5.5 Suppose that ρ is a 2-dimensional irreducible representation of with ρ(T ) as in (7). Then the following hold: (a) H(ρ) is generated (as R-module) by a vector-valued modular form F0 of weight k0 = 6(m1 + m2 ) − 1; (b) H(ρ) is a free M-module of rank 2 with free generators F0 , dF0 ; (c) dim H(k0 + 2k, ρ) = 1 + [k/3], k ≥ 0. From part (c) of the theorem it follows that the Poincaré series of H(ρ) is given by
dim H(k0 + 2k, ρ)t k0 +2k =
k≥0
t k0 . (1 − t 2 )(1 − t 6 )
Suppose that F ∈ H(k0 + 2k, ρ) is a nonzero vector-valued modular form whose component functions are classical modular forms on . In this case the representation ρ necessarily factors through a (finite) quotient of by a congruence subgroup, and hence is one of the 54 representations described in Theorem 3.7. Conversely, because the kernel of each of these representations is a congruence subgroup (cf. remarks following the proof of Theorem 3.7), all vector-valued modular forms associated to such a ρ have component functions which are classical forms. Hence, Theorem 5.6 Let ρ be a 2-dimensional irreducible representation of . A nonzero vector-valued modular form in H(ρ) has component functions which are classical modular forms if, and only if, ρ is equivalent to one of the 54 classes of irreducible representations which have finite image. In these cases, a nonzero vector-valued modular form F0 of least weight itself has classical modular forms as component functions. The weight k0 and level N are given for each case in Tables 1–5. With this information one is able to identify the classical forms in question.
6 Vector-valued Eisenstein series In this final section we make some comments about vector-valued Eisenstein series. These were introduced in [6] in the context of vector-valued Poincaré series. If the underlying representation ρ is unitary then the theory runs parallel to the classical theory, and applies to vector-valued modular forms of weight greater than 2. Lemma 6.1 There are exactly three inequivalent 2-dimensional irreducible representations ρ of such that ρ(T ) is semisimple and has an eigenvalue equal to 1. In each case, ρ() is finite. Proof Since ρ is irreducible and ρ(T ) is semisimple, Theorem 3.1 implies that the eigenvalue of ρ(T ) not equal to 1 is a primitive second or third root of unity. Thus ρ(T ) has finite order N = 2 or 3 respectively, and (N ) ⊆ ker ρ. By (21) we get
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G. Mason
(N) ⊆ ker ρ, so ρ() is finite, as asserted. From Tables 1–5 in Sect. 3 we can see that there are just three possible choices of ρ, with levels N = 2, 3, 3. For the rest of this section, we refer to the three irreducible representations occurring in the last lemma as the exceptional representations. If ρ is not exceptional then the component functions of a nonzero vector-valued modular form necessarily have q-expansions which involve no integral powers of q. In particular, there is no constant term. So we have Corollary 6.2 Suppose that ρ is a 2-dimensional irreducible representation of which is not exceptional and such that ρ(T ) is as in (7). Then H(ρ) = S(ρ). For all choices of ρ which are not exceptional, we naturally define the space of Eisenstein series to be 0. In the three exceptional cases ρ() is finite, so that we may, and shall, assume that ρ is unitary. For k > 2 we may then define the space E(k, ρ) of weight k vector-valued Eisenstein series as in [6], Theorem 5.5, i.e. the orthogonal complement to the space of weight k cusp-forms with respect to the Petersson pairing defined in (loc. cit). If k ≤ 2 then H(k, ρ) has dimension at most 1 (Theorem 5.6), and we define E(k, ρ) = H(k, ρ). We have now defined E(k, ρ) for all 2-dimensional irreducible representations ρ. Then E(ρ) =
∞
E(k0 + 2k, ρ)
k=0
is the full space of Eisenstein series. Lemma 6.3 Suppose that ρ is one of the three exceptional representations. Then for all k ≥ 0, dim E(k0 + 2k, ρ) = 1. Proof It is clear from Theorem 5.5 that F0 ∈ E(k0 , ρ) and dF0 ∈ E(k0 + 2, ρ). Moreover if k ≥ 2, E2k F0 is not a cusp-form. Since dim E(k0 + 2k, ρ) ≤ 1, the lemma follows. Consider first a nonzero minimal weight form F0 associated to one of the exceptional ρ. F0 has weight k0 = 1, 2 or 3. Let V be the space spanned by the component functions of F0 . We will identify V with a specific space of scalar forms in each case. Suppose first that N = k0 = 2. It is well-known ([9], Chap. VIII) that the space of holomorphic scalar forms of weight 2 on (2) has dimension 2 and is spanned by the second division values of the Weierstrass ℘-function. So V is this space. It contains, for example, the weight 2 form on 0 (2) with q-expansion E2∗ (τ ) = 1 − 24 n≥1 d|n,d odd dq n . Next consider the case N = k0 = 3. One knows (loc. cit.) that the space J of scalar forms of weight 3 on (3) is 4-dimensional. Now J is spanned by the primitive Eisenstein series of level 2 and weight 3, and these are permuted among themselves (up to a sign ±1) by the action |3 of . That is, as a module for / (3) (the binary tetrahedral group), J is induced from an order 2 linear character of a cyclic subgroup of order 6.
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The binary tetrahedral group has exactly 3 inequivalent, irreducible representations of dimension 2, and we denote them by Vt where t ranges over the cube roots of unity. We can choose notation so that an element of order 3 has trace −t on Vt . From our description of J, we see that there is an isomorphism of modules J ∼ = Vω ⊕ Vω¯ where ω = e2πi/3 . Note that ρ is (equivalent to) the representation afforded by Vω , so that the space V is Vω . We leave it to the reader to confirm that in the case N = 3, k0 = 1, V is the space of level 3, weight 1 scalar Eisenstein series described in [9], Theorem 10∗ , P. 175. Whenever the weight k is greater than 2, Theorem 3.2 of [6] provides formulas for the component functions of a nonzero element in E(k, ρ). By way of example, we sketch how this can be used to calculate explicit q-expansions in the case when N = k0 = 3. In the notation of Sect. 3, the exceptional representation we are now considering satisfies ρ = χ 4 ρ2 . Note that ker ρ = (3). Take the second eigenvalue of ρ(T ) equal to 1 as in Table 2. Let Pk be the second component function of the vector-valued Eisenstein series of (odd) weight k defined in [6]. Slightly changing notation compared to (loc. cit.), we have: ∞
(2πi)k Pk = 1 − cn q n , (k − 1)!
(38)
n=1
cn = nk−1
∞ (2,2) Ac (n) c=1
Ac(2,2) (n) =
ck
1−c≤d ≤0, (c,d )=1
(39)
,
−1 2πind /c ρ(Mc,d , )(2,2) e
(40)
Here, M ranges over coset representatives of T in , and more precisely Mc,d is that M which has bottom row (c, d ). (2,2) Ac (n) is the generalized Kloosterman sum of [6]. We will show how to calculate it. In the present case, ρ() is the binary tetrahedral group and ρ(T ) has order 3. Thus ρ() is a semidirect product of the group generated by ρ(T ) and a (normal) quaternion group of order 8. Consequently, we may choose the matrices M in such a way that ρ(M) ranges over the elements of the quaternion subgroup. The explicit matrices ρ(M) can be calculated from Theorem 3.3 and (19), (20). Thus we may take ρ such that 1 0 ±(3) → ± , 0 1 √ 1 0 −1 1 2 √ ± (3) → ± , 1 0 2 −1 ω − ω¯ (41) √ 1 1 −2 1 2ω¯ √ ± (3) → ± , 1 −1 2ω −1 ω − ω¯ √ 1 2ω 2 1 √1 ± (3) → ± 2 ω ¯ −1 1 1 ω − ω¯ with ω = e2πi/3 .
426 Table 6
G. Mason d
−1 ρ(Mc,d )2,2
0
1
1
0
−1
−1
1
0
(ω − ω) ¯ −1
−1
0
(ω¯ − ω)−1
1
−1
(ω − ω) ¯ −1
−1
1
(ω¯ − ω)−1
1
1
(ω − ω) ¯ −1
−1
−1
(ω¯ − ω)−1
c
Lemma 6.4 Let χ3 be the nontrivial Dirichlet character (mod 3). Then if 3|c, χ3 (d ) −1 ρ(Mc,d )2,2 = χ3 (c)(ω − ω) ¯ −1 if (c, 3) = 1. Proof From display (41) we distill the Table 6 of values. The lemma follows from this.
For 3|c let χc be the Dirichlet character (mod c) which satisfies χc (f ) = χ3 (f ) if (f, c) = 1. From the last lemma and (40) we obtain (for 3|c) Ac(2,2) (n) = −
c−1
χc (f )e−2πinf/c .
f =0
If (c, 3) = 1 then we similarly obtain Ac(2,2) (n)
c−1 χ3 (c) = χ0 (f )e2πinf/c , ω − ω¯ f =0
where χ0 is the trivial Dirichlet character (mod c). In either case the expression for (2,2) Ac (n) involves a Gauss sum whose evaluation is standard. When fed into (38) and (39) we obtain the q-expansion of Pk .
References 1. Hille, E.: Ordinary Differential Equations in the Complex Domain. Dover, New York (1976) 2. Ince, E.: Ordinary Differential Equations. Dover, New York (1956) 3. Kaneko, M., Koike, M.: On modular forms arising from a differential equation of hypergeometric type. Ramanujan J. 7(1–3), 145–164 (2003) 4. Knopp, M., Mason, G.: Generalized modular forms. J. Number Theory 99, 1–28 (2003) 5. Knopp, M., Mason, G.: On vector-valued modular forms and their Fourier coefficients. Acta Arith. 110(2), 117–124 (2003) 6. Knopp, M., Mason, G.: Vector-valued modular forms and Poincaré series. Ill. J. Math. 48(4), 1345– 1366 (2004)
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7. Lang, S.: Introduction to Modular Forms. Grundlehren Math. Wiss., vol. 222. Springer, New York (1976) 8. Mason, G.: Vector-valued modular forms and linear differential equations. Int. J. Number Theory 3(3), 1–14 (2007) 9. Schoeneberg, B.: Elliptic Modular Functions. Grundlehren Math. Wiss. Springer, New York (1974) 10. Wohlfahrt, K.: An extension of F. Klein’s level concept. Int. J. Math. 13, 529–535 (1964)
