aa r X i v : . [ m a t h . N T ] O c t Logarithmic Vector-Valued Modular Forms
Marvin KnoppGeoffrey Mason ∗ Abstract
We consider logarithmic vector- and matrix-valued modular forms of integral weight k associ-ated with a p -dimensional representation ρ : SL ( Z ) → GL p ( C ) of the modular group, subjectonly to the condition that ρ ( T ) has eigenvalues of absolute value 1. The main result is theconstruction of meromorphic matrix-valued Poincar´e series associated to ρ for all large enoughweights. The component functions are logarithmic q -series, i.e., finite sums of products of q -series and powers of log q . We derive several consequences, in particular we show that thespace H ( ρ ) = ⊕ k H ( k, ρ ) of all holomorphic logarithmic vector-valued modular forms associ-ated to ρ is a free module of rank p over the ring of classical holomorphic modular forms on SL ( Z ). Keywords: Logarithmic vector-valued modular form, Poincar´e series, free module.MSC: 11F11, 11F99, 30F35.
The present work is a natural sequel to our earlier articles on ‘normal’ vector-valued modular forms[KM1], [KM2]. The component functions of a normal vector-valued modular form F are q -series withat worst real exponents. Equivalently, the finite-dimensional representation ρ associated with F hasthe property that ρ ( T ) is (similar to) a matrix that is unitary and diagonal. Here, T = (cid:18) (cid:19) .In the case of a general representation, ρ ( T ) is not necessarily diagonal but may always beassumed to be in Jordan canonical form . This circumstance leads to logarithmic , or polynomial q -expansions for the component functions of a vector-valued modular form associated to ρ (seeSubsection 2.2), which take the form f ( τ ) = t X j =0 (log q ) j h j ( τ ) , (1)where the h j ( τ ) are ordinary q -series. There follow naturally the definition of logarithmic vector-valued modular form and the concomitant notions of logarithmic meromorphic, holomorphic (i.e.,entire in the sense of Hecke) and cuspidal vector-valued modular forms (Subsection 2.3). ∗ Supported by NSA and NSF We actually use a modified Jordan canonical form. See Subsection 2.2 for details. B ρ that must be inserted in thedefinition (cf. Definition 3.1) in order to achieve the desired formal transformation properties withrespect to the representation ρ (Subsection 3.1, following the proof of Lemma 3.2 ).It is useful to compare Definition 3.1 with the corresponding definition in the normal case([KM2], display (18), pp. 1352-1353). Definition 3.1 actually defines a matrix-valued Poincar´eseries, each column of which is a logarithmic vector-valued modular form. In fact, the same istrue of our definition in the normal case, except that in the latter case we define the Poincar´eseries as a single column of the matrix-valued Poincar´e series. Matrix-valued modular forms are avery natural generalization of vector-valued modular forms. In addition to our Poincar´e series, forexample, the modular Wronskian [M1] is the determinant of a matrix-valued modular form. Thepassage from vector-valued modular forms to matrix-valued modular forms is analogous to passingfrom a modular linear differential equation of order p (loc. cit.) to an associated system of p lineardifferential equations of order 1.Subsections 3.2 and 3.3 are devoted, respectively, to the proof of convergence of our matrix-valued Poincar´e series and the determination of the general form of their logarithmic q -series ex-pansions. Our proof of convergence requires the assumption that the eigenvalues of ρ ( T ) haveabsolute value 1, so that the q -series h j ( τ ) in (1) again have at worst real exponents. This conditionwill be implicitly assumed in the remainder of the Introduction.The remainder of the paper is devoted to applications. In Subsection 3.4 we give some conse-quences of an algebraic nature. We show (Theorem 3.13) that if ρ has dimension p , the graded space H ( ρ ) of all holomorphic vector-valued modular forms associated to ρ is a free module of rank p overthe algebra M of (scalar) holomorphic modular forms on Γ. This generalizes the correspondingTheorem proved in [MM] in the normal case. In fact, the proof in [MM] was organized with just sucha generalization in mind. The only additional input that is required is the existence of some nonzeroholomorphic vector-valued modular form associated with ρ , and this is an easy consequence of theexistence of a nonzero meromorphic Poincar´e series. A consequence of the free module Theorem isTheorem 3.14, which implies that if F is a logarithmic vector-valued modular form F then there isa canonical modular linear differential equation whose solution space is spanned by the componentfunctions of F .In Section 4 we derive polynomial growth estimates for the coefficients of entire and cuspidallogarithmic vector-valued modular forms associated to ρ . The method here follows the approachused in [KM1], itself an extension of Hecke’s venerable method for estimating the Fourier coefficientsof scalar modular/automorphic forms, combined with a simple new estimate (Proposition 3.7) thatwe use in Subsection 3.2 to prove convergence of our Poincar´e series.The occurrence of q -expansions of the form (1) is well known in rational and logarithmic con-formal field theory. Indeed, much of the motivation for the present work originates from a need to2evelop a systematic theory of vector-valued modular forms wide enough in scope to cover possibleapplications in such field theories. By results in [DLM] and [M], the eigenvalues of ρ ( T ) for therepresentations that arise in rational and logarithmic conformal field theory are indeed of absolutevalue 1 (in fact, they are roots of unity). Thus this assumption is natural from the perspective ofconformal field theory. Our earlier results [KM1] on polynomial estimates for Fourier coefficients ofentire vector-valued modular forms in the normal case have found a number of applications to thetheory of rational vertex operator algebras, and we expect that the extension to the logarithmiccase that we prove here will be useful in the study of C -cofinite vertex operator algebras, whichconstitute the algebraic underpinning of logarithmic field theory.Other properties of logarithmic vector-valued modular forms are also of interest, from both afoundational and applied perspective. These include a Petersson pairing, generation of the spaceof cusp-forms by Poincar´e series, existence of a natural boundary for the component functions, andexplicit formulas (in terms of Bessel functions and Kloosterman sums) for the Fourier coefficientsof Poincar´e series. This program was carried through in the normal case in [KM2]. We expectthat the more general logarithmic case will yield a similarly rich harvest, but one must expect morecomplications. For example, there are logarithmic vector-valued modular forms with nonconstantcomponent functions that may be extended to the whole of the complex plane, so that the usualnatural boundary result is false per se . Furthermore, our preliminary calculations indicate that theexplicit formulas exhibit genuinely new features. We hope to return to these questions in the future. We start with some notation that will be used throughout. The modular group isΓ = (cid:26)(cid:18) a bc d (cid:19) | a, b, c, d ∈ Z , ad − bc = 1 (cid:27) . It is generated by the matrices S = (cid:18) −
11 0 (cid:19) , T = (cid:18) (cid:19) . (2)The complex upper half-plane is H = { τ ∈ C | ℑ ( τ ) > } . There is a standard left action Γ × H → H given by M¨obius transformations: (cid:18)(cid:18) a bc d (cid:19) , τ (cid:19) aτ + bcτ + d . Let F be the space of holomorphic functions in H . There is a standard 1-cocycle j : Γ → F definedby j ( γ, τ ) = j ( γ )( τ ) = cτ + d, γ = (cid:18) a bc d (cid:19) . : Γ → GL ( p, C ) will always denote a p -dimensional matrix representation of Γ . An unrestrictedvector-valued modular form of weight k with respect to ρ ( k ∈ Z ) is a holomorphic function F : H → C p satisfying ρ ( γ ) F ( τ ) = F | k γ ( τ ) , γ ∈ Γ , where the right-hand-side is the usual stroke operator F | k γ ( τ ) = j ( γ, τ ) − k F ( γτ ) . (3)We could take F ( τ ) to be meromorphic in H , but we will not consider that more general situationhere. Choosing coordinates, we can rewrite (3) in the form ρ ( γ ) f ( τ )... f p ( τ ) = f | k γ ( τ )... f p | k ( γ )( τ ) (4)with each f j ( τ ) ∈ F . We also refer to ( F, ρ ) as an unrestricted vector-valued modular form. q -expansions In this Subsection we consider the q -expansions associated to unrestricted vector-valued modularforms. We make use of the polynomials defined for k ≥ (cid:18) xk (cid:19) = x ( x − . . . ( x − k + 1) k ! , and with (cid:0) x (cid:1) = 1 and (cid:0) xk (cid:1) = 0 for k ≤ − W ⊆ F k that is invariant under T , i.e f ( τ + 1) ∈ W whenever f ( τ ) ∈ W . We introduce the m × m matrix J m,λ = λλ . . .. . . . . . λ λ , (5)i.e. J i,j = λ for i = j or j + 1 and J i,j = 0 otherwise. Lemma 2.1
There is a basis of W with respect to which the matrix ρ ( T ) representing T is in blockdiagonal form ρ ( T ) = J m ,λ . . . J m t ,λ t . (6)4 roof: The existence of such a representation is basically the theory of the Jordan canonical form.The usual
Jordan canonical form is similar to the above, except that the subdiagonal of each blockthen consists of 1’s rather than λ ’s. The λ ’s that appear in (6) are the eigenvalues of ρ ( T ), and inparticular they are nonzero on account of the invertibility of ρ ( T ). Then it is easily checked that(6) is indeed similar to the usual Jordan canonical form, and the Lemma follows. ✷ We refer to (6) as the modified Jordan canonical form of ρ ( T ), and J m i ,λ i as a modified Jordanblock . To a certain extent at least, Lemma 2.1 reduces the study of the functions in W to thoseassociated to one of the Jordan blocks. In this case we have the following basic result. Theorem 2.2
Let W ⊆ F k be a T -invariant subspace of dimension m . Suppose that W has anordered basis ( g ( τ ) , . . . , g m − ( τ )) with respect to which the matrix ρ ( T ) is a single modified Jordanblock J m,λ . Set λ = e πiµ . Then there are m convergent q -expansions h t ( τ ) = P n ∈ Z a t ( n ) q n + µ , ≤ t ≤ m − , such that g j ( τ ) = j X t =0 (cid:18) τt (cid:19) h j − t ( τ ) , ≤ j ≤ m − . (7)The case m = 1 of the Theorem is well known. We will need it for the proof of the general case,so we state it as Lemma 2.3
Let λ = e πiµ , and suppose that f ( τ ) ∈ F satisfies f ( τ + 1) = λf ( τ ) . Then f ( τ ) isrepresented by a convergent q -expansion f ( τ ) = X n ∈ Z a ( n ) q n + µ . (8) ✷ Turning to the proof of the Theorem, we have g j ( τ + 1) = λ ( g j ( τ ) + g j − ( τ )) , ≤ j ≤ m − , (9)where g − ( τ ) = 0. Set h j ( τ ) = j X t =0 ( − t (cid:18) τ + t − t (cid:19) g j − t ( τ ) , ≤ j ≤ m − . These equalities can be displayed as a system of equations. Indeed, B m ( τ ) g ( τ )... g m − ( τ ) = h ( τ )... h m − ( τ ) , (10)where B m ( x ) is the m × m lower triangular matrix with B m ( x ) ij = ( − i − j (cid:18) x + i − j − i − j (cid:19) . (11)5hen B m ( x ) is invertible and B m ( x ) − ij = (cid:18) xi − j (cid:19) . (12)We will show that each h j ( τ ) has a convergent q -expansion. This being the case, (7) holds andthe Theorem will be proved. Using (9), we have h j ( τ + 1) = λ j X t =0 ( − t (cid:18) τ + tt (cid:19) ( g j − t ( τ ) + g j − t − ( τ ))= λ ( j X t =0 ( − t (cid:18) tτ (cid:19) (cid:18) τ + t − t (cid:19) g j − t ( τ ) + j X t =0 ( − t (cid:18) τ + tt (cid:19) g j − t − ( τ ) ) = λ ( h j ( τ ) + j X t =0 ( − t (cid:18) τ + t − t (cid:19) tτ g j − t ( τ ) + j X t =0 ( − t (cid:18) τ + tt (cid:19) g j − t − ( τ ) ) . But the sum of the second and third terms in the braces vanishes, being equal to j X t =1 ( − t (cid:18) τ + t − t (cid:19) tτ g j − t ( τ ) + j X t =1 ( − t − (cid:18) τ + t − t − (cid:19) g j − t ( τ )= j X t =1 ( − t − g j − t ( τ ) (cid:26)(cid:18) τ + t − t − (cid:19) − (cid:18) τ + t − t (cid:19) tτ (cid:27) = 0 . Thus we have established the identity h j ( τ + 1) = λh j ( τ ). By Lemma 2.3, h j ( τ ) is indeedrepresented by a q -expansion of the desired shape, and the proof of Theorem 2.2 is complete. ✷ We call (7) a polynomial q -expansion. The space of polynomials spanned by (cid:0) xt (cid:1) , ≤ t ≤ m − x t , ≤ t ≤ m −
1. Since (2 πiτ ) t = (log q ) t , it follows that in Theorem2.2 we can find a basis { g ′ j ( τ ) } of W such that g ′ j ( τ ) = j X t =0 (log q ) t h ′ j − t ( τ ) (13)with h ′ t ( τ ) = P n ∈ Z a ′ t ( n ) q n + µ . We refer to (13) as a logarithmic q -expansion. We say that a function f ( τ ) with a q -expansion (8) is meromorphic at infinity if f ( τ ) = X n + ℜ ( µ ) ≥ n a ( n ) q n + µ . That is, the Fourier coefficients a ( n ) vanish for exponents n + µ whose real parts are small enough.A polynomial (or logarithmic) q -expansion (7) is holomorphic at infinity if each of the associated6rdinary q -expansions h j − t ( τ ) is holomorphic at infinity. Similarly, f ( τ ) vanishes at ∞ if the Fouriercoefficients a ( n ) vanish for n + ℜ ( µ ) ≤
0; a polynomial q -expansion vanishes at ∞ if the associatedordinary q -expansions vanish at ∞ . These conditions are independent of the chosen representations.Now assume that F ( τ ) = ( f ( τ ) , . . . , f p ( τ )) t is an unrestricted vector-valued modular form ofweight k with respect to ρ . It follows from (4) that the span W of the functions f j ( τ ) is a rightΓ-submodule of F satisfying f j ( τ + 1) ∈ W . Choose a basis of W so that ρ ( T ) is in modified Jordancanonical form. By Theorem 2.2 the basis of W consists of functions g j ( τ ) which have polynomial q -expansions. We call F ( τ ), or ( F, ρ ), a logarithmic meromorphic, holomorphic, or cuspidal vector-valued modular form , respectively, if each of the functions g j ( τ ) is meromorphic, is holomorphic, orvanishes at ∞ , respectively.From now on we generally drop the adjective ‘logarithmic’ from this terminology, and say that F ( τ ) is semisimple if the component functions have ordinary q -expansions, i.e. they are free oflogarithmic terms. This holds if, and only if, ρ ( T ) is a semisimple operator.Let H ( k, ρ ) be the space of holomorphic vector-valued modular forms of weight k with respectto ρ , with H ( ρ ) = ⊕ k ∈ Z H ( k, ρ ) the Z -graded space of all holomorphic vector-valued modular forms. Matrix-valued modular forms are a natural generalization of vector-valued modular forms. Theyarise naturally in several contexts, including (as we shall see) Poincar´e series. Let ρ : Γ → GL p ( C )be a representation, and let M at p × n ( C ) be the space of p × n matrices. Let k = ( k , . . . , k n ) ∈ Z n . Consider a holomorphic map A : H → M at p × n ( C ) satisfying ρ ( γ ) A ( τ ) = A | k γ ( τ ) , γ ∈ Γ , where the right hand side is defined as A | k γ ( τ ) = A ( γτ ) J k ( γ, τ ) − and J is the matrix automorphy factor J k ( γ, τ ) = j ( γ, τ ) k . . . j ( γ, τ ) k n . (14)This defines an unrestricted matrix-valued modular form of weight k with respect to ρ . Let p j : M at p × n ( C ) → M at p × ( C ) be projection onto the j th. column. Then p j ◦ A is an unrestrictedvector-valued modular form of weight k j with respect to ρ , and we say that A ( τ ) is a meromorphic,holomorphic, or cuspidal vector-valued modular form of weight k if each p j ◦ A is meromorphic,holomorphic or cuspidal, respectively. Thus, a matrix-valued modular form associated to ρ consistsof n vector-valued modular forms of weight k , . . . , k n , each associated to ρ with the componentfunctions organized into the columns of a matrix.7 .5 The nontriviality condition Let ρ : Γ → GL p ( C ) be a matrix representation. Because S = − I has order 2, we can choose abasis of the underlying representation space such that ρ ( S ) = (cid:18) I p − I p (cid:19) . Since S is in the center of Γ then ρ (Γ) acts on the two eigenspaces of ρ ( S ) and therefore thematrices ρ ( γ ) = (cid:18) ρ ( γ ) 00 ρ ( γ ) (cid:19) , γ ∈ Γ , (15)are correspondingly in block diagonal form. It follows that if ( F, ρ ) is a vector-valued modular formof weight k , and if we write F ( τ ) = ( F ( τ ) , F ( τ )) with F i ( τ ) having p i components, i = 1 ,
2, then F i ( τ ) is a vector-valued modular form of weight k with respect to the representation ρ ii . More istrue. The equality ρ ( S ) F t ( τ ) = F t | k S ( τ ) says that( F ( τ ) , − F ( τ )) = ( − k ( F ( τ ) , F ( τ )) . Assuming that F = 0, it follows that either F = 0 and k is even , or else F = 0 and k is odd . Itfollows that there are natural identifications H ( k, ρ ) = (cid:26) H ( k, ρ ) k even , H ( k, ρ ) k odd , H ( ρ ) = H ( ρ ) ⊕ H ( ρ ) . (16)The upshot of this discussion is that for most considerations, we may assume that ρ ( S ) is a scalar , i.e. ρ ( S ) = ǫI p , ǫ = ± . (17)In this case, if F ( τ ) ∈ H ( k, ρ ) is nonzero then ǫ = ( − k . (18)This is the nontriviality condition in weight k .In the case of semisimple vector-valued modular forms, it is proved in [KM1] and [M1] thatthere is an integer k such that H ( k, ρ ) = 0 for k < k . The proof in [M1] applies to the general(logarithmic) case. Thus if ρ satisfies (17) then H ( ρ ) = M k ≥ k H ( k + 2 k ) . (19) We develop a theory of Poincar´e series in order to prove existence of nontrivial logarithmic vector-valued modular forms. 8 .1 Definition and formal properties
Fix a representation ρ : Γ → GL ( p, C ). We may, and shall, assume that ρ ( T ) is in modified Jordancanonical form with t blocks, the r th. block being the m r × m r matrix J m r ,λ r (5), (6) and with λ r = e πiµ r the associated eigenvalue of ρ ( T ).We will need several more block diagonal matrices. The matrices in question will all have t blocks, the r th block having the same size as the r th. block of ρ ( T ). Set B ρ ( x ) = diag( B m ( x ) , . . . , B m t ( x )) , (20)where B m ( x ) is given in (11). For ( z , . . . , z t ) ∈ C t letΛ ρ ( z , . . . , z t ) = diag( z I m , . . . , z t I m t ) . (21) Definition 3.1
Let ν = ( ν , . . . , ν t ) ∈ Z t , k = ( k , . . . , k p ) ∈ Z p . The Poincar´e series is defined tobe P k ( ν, τ ) = 1 / X M ρ ( M ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) B ρ ( M τ ) − J k ( M, τ ) − , (22) where M ranges over a set of representatives of the coset space h T i\ Γ and J k ( M, τ ) is the matrixautomorphy factor (14). B ρ ( τ ) − should be considered as an additional matrix automorphy factor. At least formally, P k ( ν, τ )is a p × p matrix-valued function.We interpolate a Lemma. Lemma 3.2
The matrices ρ ( T ) , Λ ρ ( z , . . . , z t ) and B ρ ( τ ) ( τ ∈ H ) commute with each other, andsatisfy ρ ( T ) B ρ ( τ ) − = B ρ ( τ + 1) − Λ ρ ( λ , . . . , λ t ) . Proof:
All of the matrices in question are block diagonal with corresponding blocks of the samesize. So it suffices to show that for a given m and λ , the m × m matrices J m,λ , zI m and B m ( τ )commute and satisfy J m,λ B m ( τ ) − = λB m ( τ + 1) − . (23)The m × m matrices all have the following properties: they are lower triangular and the ( i, j )-entry depends only on i - j . It is easy to check that any two such matrices commute.As for (23), let G ( τ ) and H ( τ ) denote the column vectors of functions that occur in (10), sothat we can write the equation as B m ( τ ) G ( τ ) = H ( τ ) . By definition of G ( τ ) and H ( τ ) (cf. Theorem 2.2) we have J m,λ G ( τ ) = G ( τ + 1) ,H ( τ + 1) = λH ( τ ) . J m,λ B m ( τ ) − H ( τ ) = J m,λ G ( τ ) = G ( τ + 1)= B m ( τ + 1) − H ( τ + 1) = λB m ( τ + 1) − H ( τ ) . Since the components of H ( τ ) are linearly independent, (23) follows. ✷ Now make the replacement M T M in a summand of (22). Using Lemma 3.2 we calculatethat the summand maps to ρ ( T M ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) T Mτ , . . . (cid:1) B ρ ( T M τ ) − J k ( T M, τ ) − = ρ ( M ) − ρ ( T ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) Λ ρ ( λ , . . . , λ t ) B ρ ( M τ + 1) − J k ( M, τ ) − = ρ ( M ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) B ρ ( M τ ) − J k ( M, τ ) − . This calculation confirms that the sum defining P k ( ν, τ ) is independent of the choice of coset rep-resentatives. We also note that P k | k γ ( τ ) = 1 / X M ρ ( M ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mγτ , . . . (cid:1) B ρ ( M γτ ) − J k ( M, γτ ) − J k ( γ, τ ) − = 1 / ρ ( γ ) X M ρ ( M γ ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mγτ , . . . (cid:1) B ρ ( M γτ ) − J k ( M γ, τ ) − = ρ ( γ ) P k ( ν, τ ) , where we used independence of coset representatives for the last equality. This confirms that each P k ( ν, τ ) is, at least formally, a matrix-valued modular form of weight k with respect to ρ . P k ( ν, τ ) From now on we assume that the constants µ r are real , i.e. the eigenvalues λ of ρ ( T ) satisfy | λ | = 1.With this assumption, we show in this Subsection that the Poincar´e series P k ( ν, τ ) is an unrestrictedmatrix-valued modular form for k ≫
0. After the results of the previous Subsection, this amountsto the fact that P k ( ν, τ ) is holomorphic in H as long as the component weights k j of k are largeenough.Define the vertical strip S = { τ ∈ H | |ℜ ( τ ) | ≤ / , ℑ ( τ ) ≥ √ / } . Notice that S contains the closure of the standard fundamental region for Γ. We will prove Theorem 3.3 P k ( ν, τ ) converges absolutely-uniformly in S for k ≫ . It is a consequence of Theorem 3.3 and the formal transformation law for P k ( ν, τ ) (cf. Subsection3.1) that P k ( ν, τ ) is holomorphic throughout H .We split off the two terms of the Poincar´e series corresponding to ± I , so that P k ( ν, τ ) = Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) τ , . . . (cid:1) B ρ ( τ ) − + (24)1 / X M ∈M ∗ ρ ( M ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) B ρ ( M τ ) − J k ( M, τ ) − . M ∗ is a set of representatives of the cosets h T i\ Γ distinct from ±h T i , and ± I are therepresentative of ±h T i . The matrices M ∈ M ∗ have bottom row ( c, d ) with c = 0. The entries of B ρ ( τ ) − are polynomials in τ (cf. (12), (20)), so the first term in (24) is holomorphic. Lemma 3.4
We can choose coset representatives M ∈ M ∗ so that | M τ | is uniformly bounded in S . That is, there is a constant K such that | M τ | ≤ K for all τ ∈ S and all M ∈ M ∗ . Proof:
Suppose that γ = (cid:18) a bc d (cid:19) ∈ Γ with c = 0, and consider the γ -image γ ( S ) of the strip.Apart from two exceptional cases (when c = ± , d = ∓ γ -image of the circle | τ | = 1 is acircle with center b/d and radius at most 1. Moreover γ ( ∞ ) = a/c lies on or inside the boundaryof this circle. From this it is easy to see that γ ( S ) ⊆ { τ ∈ H | |ℜ ( τ ) − a/c | ≤ } . Replacing γ by T l γ for suitable l , the corresponding value of | a/c | can be made less than 1, so that γ ( S ) ⊆ { τ ∈ H | |ℜ ( τ ) | ≤ } . This also holds in the exceptional cases. Therefore we may, and shall, choose a set of coset repre-sentatives M ∗ so that |ℜ ( M τ ) | is uniformly bounded in S for M ∈ M ∗ .On the other hand, it is easy to see that we always have | cτ + d | ≥ c ℑ ( τ ) . Since |ℑ ( τ ) | ≥ √ / τ ∈ S , it follow that ℑ ( γτ ) = ℑ ( τ ) | cτ + d | ≤ ℑ ( τ ) c ℑ ( τ ) = 1 c ℑ ( τ ) ≤ ℑ ( τ ) ≤ √ , so that |ℑ ( γτ ) | is uniformly bounded in S for c = 0. Therefore, with our earlier choice of M ∗ , itfollows that | M τ | is also uniformly bounded in S . This completes the proof of the Lemma. ✷ Henceforth, we assume that M ∗ satisfies the conclusion of Lemma 3.4. Corollary 3.5
The entries of the matrices Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) and B ρ ( M τ ) − are uniformlybounded in S for M ∈ M ∗ . Proof:
For Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) the assertion is an immediate consequence of Lemma 3.4.As for B ρ ( M τ ) − , we have already pointed out that it has polynomial entries, indeed the ( i, j )-entryis (cid:0) Mτi − j (cid:1) . Uniform boundedness in this case is then also a consequence of Lemma 3.4. ✷ Next we state a modification of ([E], p. 169, display (4)) which we call
Eichler’s canonical form for elements of Γ:
Lemma 3.6
Each γ ∈ Γ has a unique representation γ = ( ST l v ) . . . ( ST l )( ST l ) (25) such that ( − j − l j > for ≤ j ≤ v . ✷ l is positive, the l j alternate in sign for j ≥
1, and there is no condition on l .With γ fixed for now, we set P = ST l ,P j +1 = ( ST l j ) P j , ≤ j ≤ v − ,P j = (cid:18) a j b j c j d j (cid:19) , ≤ j ≤ v, (26) γ = (cid:18) a bc d (cid:19) . (27) Proposition 3.7
We have | l l . . . l v | ≤ | d | if l < | l . . . l v | ≤ | d − c | if l = 0; (28) | l l . . . l v | ≤ | c | + | d | if l > . Proof:
Case A: l <
0. We will prove by induction on j ≥ i ) | l l . . . l j | ≤ | d j | , (29)( ii ) ( − j b j d j ≥ . Once this is established, the case j = v of (29)(i) proves (28) in Case A. Now P = (cid:18) − l (cid:19) , and the case j = 0 is clear. For the inductive step, we have P j +1 = (cid:18) − l j +1 (cid:19) (cid:18) a j b j c j d j (cid:19) = (cid:18) − c j − d j a j + l j +1 c j b j + l j +1 d j (cid:19) . (30)Thus ( − j +1 b j +1 d j +1 = ( − j b j d j + ( − j l j +1 d j ≥ − j b j d j and ( − j l j +1 d j are both nonnegative then b j and l j +1 d j have the same sign . Therefore using induction again, we have | l l . . . l j +1 | ≤ | d j l j +1 | ≤| b j | + | l j +1 d j | = | b j + l j +1 d j | = | d j +1 | . This completes the proof of Case A.Case B: l = 0. Notice in this case that γT − = ( ST l v ) . . . ( ST l )( ST − ), which falls into Case Awith l = −
1. Since γT − = (cid:18) a bc d (cid:19) (cid:18) −
10 1 (cid:19) = (cid:18) a b − ac d − c (cid:19) it follows from Case A that | l . . . l v | ≤ | d − c | , as was to be proved.12ase C: l > . We will prove by induction on j that( i ) | l l . . . l j | ≤ | c j | + | d j | , j ≥ ii ) ( − j b j d j , ( − j a j c j ≥ , j ≥ . Once again, the case j = v of (31)(i) proves (28) in Case C, and this will complete the proof of theProposition. Now P = (cid:18) − l (cid:19) , P = (cid:18) − − l l l l − (cid:19) . So when j = 0, (31)(i) is clearly true, and because l , l > − a c = l > , − b d = l ( l l − ≥ . So (31)(ii) holds for j = 1. As for the inductive step, P j +1 is as in (30), and the proof that( − j b j d j ≥ − j +1 a j +1 c j +1 = ( − j c j a j + ( − j l j +1 c j ≥ | l . . . l j +1 | ≤ | c j + d j || l j +1 | < | l j +1 c j | + | l j +1 d j | + | a j | + | b j | = | a j + l j +1 c j | + | b j + l j +1 d j | = | c j +1 | + | d j +1 | . The Proposition is proved. ✷ The
Eichler length of γ is given by L ( γ ) = (cid:26) v + 2 , l = 02 v + 1 , l = 0 (32)By Lam´e’s Theorem we have the estimate L ( γ ) ≤ K (log | c | + 1) (33)with a positive constant K independent of γ .The norm || ρ ( γ ) || , defined to be max i,j | ρ ( γ ) ij | , satisfies || ρ ( γ ) || ≤ || ρ ( S ) || v +1 v Y j =0 || ρ ( T l j ) || . (34) Lemma 3.8
Let s be the maximum of the sizes m j of the Jordan blocks J m j ,λ j of ρ ( T ) (5), (6).There is a constant C s depending only on s such that for l = 0 , || ρ ( T l ) || ≤ C s | l | s − . (35) Proof:
We have J lm,λ = λ l J lm, = λ l ( I m + N ) l = λ l X i ≥ (cid:18) li (cid:19) N i where N is the nilpotent m × m matrix with each ( i, i − i ≥ , and all otherentries zero. Note that N m = 0 and the entries of N i for 1 ≤ i < m are 1 on the i th. subdiagonaland zero elsewhere. Bearing in mind that | λ | = 1, it follows that || J lm,λ || is majorized by themaximum of the binomial coefficients (cid:0) li (cid:1) over the range 0 ≤ i ≤ m −
1. Since (cid:0) li (cid:1) is a polynomialin l of degree i then we certainly have || J lm,λ || ≤ C m | l | m − for a universal constant C m , and sincethis applies to each Jordan block of ρ ( T l ) then the Lemma follows immediately. ✷ orollary 3.9 There are universal constants K , K such that || ρ ( γ ) || ≤ K ( c + d ) K . (36) Moreover the same estimate holds for || ρ ( γ − ) || . Proof:
From Lemma 3.8 and (34) we obtain || ρ ( γ ) || ≤ (cid:26) K v Q vj =0 | l j | s − , l = 0 ,K v Q vj =1 | l j | s − , l = 0 , for a constant K depending only on ρ . Now use (32), (33) and Proposition 3.7 to see that || ρ ( γ ) || ≤ e (log K ) K log( | c | +1) ( | c | + | d | ) ≤ K ( c + d ) K . Concerning the second assertion of the Lemma, since γ − = ( T − l S )( T − l ) . . . ( T − l v S ) then || ρ ( γ − ) || ≤ || ρ ( S ) || v +1 v Y j =0 || ρ ( T ) − l j || , and (35) then holds by Lemma 3.8. The rest of the proof is the same as the previous case, so thatwe indeed obtain the estimate (36) for γ − as well as γ . ✷ We can now prove Theorem 3.3. Let P ∗ k ( ν, τ ) denote the infinite sum in (24). We have byCorollary 3.5 and Corollary 3.9 that || P ∗ k ( ν, τ ) || ≤ X M ∈M ∗ || ρ ( M ) − |||| Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) |||| B ρ ( M τ ) − |||| J k ( M, τ ) − ||≤ K X ( c,d )=1 ( c + d ) K || J k ( M, τ ) − || , with constants K , K that depend only on ρ . We also know ([KM1], display (13)) that c + d ≤ K | cτ + d | (37)for a universal constant K . Because of the nature of the matrix automorphy factor J k (14), itfollows from the previous two displays that if the minimum of the weights k i in k = ( k , . . . , k p ) is large enough , then || P ∗ k ( ν, τ ) || ≤ K X ( c,d )=1 ( cτ + d ) − k , with k >
2. It is well-known that this series converges absolutely-unformly in S , so the same is truefor P ∗ k ( ν, τ ). This completes the proof of Theorem 3.3. ✷ .3 q -expansions of the component functions We now assume (cf. the discussion in Subsection 2.5) that (17) holds. Consider the substitution M
7→ − M in the expression for P k ( ν, τ ). Because the sum is independent of the order of the terms,(17) implies that P k ( ν, τ ) = P k ( ν, τ )Λ ρ ( ǫ ( − k , . . . , ǫ ( − k t ) . If the nontriviality condition (18) holds in weight k j then ǫ ( − k j = 1 and the j th. column of P k ( ν, τ ) is unchanged. If the nontriviality condition does not hold then the j th. column is zero andas such it too is unchanged. We conclude that P k ( ν, τ ) = X M ρ ( M ) − Λ ρ (cid:0) . . . , e πi ( ν r + µ r ) Mτ , . . . (cid:1) B ρ ( M τ ) − J k ( M, τ ) − , (38)where the matrices M now range over an arbitrary set of coset representatives of ±h T i\ Γ.We will show that P k ( ν, τ ) is a meromorphic vector-valued modular form for k ≫
0. Wehave already proved that it is an unrestricted vector-valued modular form, so that the componentfunctions that occur in the matrix representation P k ( ν, τ ) = ( P mn ( τ ))have polynomial q -expansions (7) by Theorem 2.2. It remains to show that these q -expansions aremeromorphic at infinity if the weights are large enough. Note that because of our assumption thatthe constants µ r are real, the q -expansions in question have only real powers of q .To describe P mn ( τ ), let us assume that the nontriviality condition holds in weight k n . Let r besuch that the m th. row of P ( τ ) falls into the r th. Jordan block, and let M r = m + . . . + m r . Thenwe have M r − < m ≤ M r . Now take I as the coset representative of ±h T i\ Γ and set M = M ∗ ∪ { I } . From (38) we have P ( τ ) mn = p X l =1 X M ∈M Λ ρ ( . . . , e πi ( ν s + µ s ) Mτ , . . . ) mm ρ ( M − ) ml B ρ ( M τ ) − ln j ( M, τ ) − k n = e πi ( ν r + µ r ) τ B ρ ( τ ) − mn + p X l =1 ( X M ∈M ∗ e πi ( ν r + µ r ) Mτ ρ ( M − ) ml B ρ ( M τ ) − ln j ( M, τ ) − k n ) . Because of absolute-uniform convergence in the strip S , lim τ → i ∞ may be taken inside the sum-mations. By Lemma 3.4 and Corollary 3.5 we find thatlim τ → i ∞ (cid:8) P ( τ ) mn − e πi ( ν r + µ r ) τ B ρ ( τ ) − mn (cid:9) = 015 k n > q -expansion of P ( τ ) mn − e πi ( ν r + µ r ) B ρ ( τ ) − can have only positive powers of q , so that P ( τ ) mn = d mn (cid:18) τm − n (cid:19) q ν r + µ r + m − M r − − X u =0 (cid:18) τu (cid:19) X l + µ r > ˆ a unr ( l ) q l + µ r , (39) d mn = (cid:26) , M r − < n ≤ m, , otherwise . Notice that the diagonal terms have polynomial q -expansions P ( τ ) mm = q ν r + µ r + regular terms . In particular, if ν r + µ r < i ∞ and if ν r + µ r = 0 the constant term is 1. Soin both cases P mm ( τ ) is nonzero . We have established the following. Theorem 3.10
Suppose that ρ satisfies ρ ( S ) = ǫI p . Then P k ( ν, τ ) is a meromorphic matrix-valuedmodular form of weight k for all k ≫ . If the nontriviality condition holds in all weights k n thenone of the following holds:(a) ν r + µ r > for all r and P k ( ν, τ ) is a cuspidal matrix-valued modular form, possibly zero.(b) ν r + µ r ≥ for all r, ν r + µ r = 0 for some r , and P k ( ν, τ ) is a nonzero , holomorphic matrix-valued modular form of weight k .(c) ν r + µ r < for some r and P k ( ν, τ ) is a nonzero meromorphic matrix-valued modular form ofweight k .If the nontriviality condition is not satisfied in weight k n , then the n th. column of P k ( ν, τ ) vanishesidentically. ✷ We record a consequence of the nature of the q -expansions (39). Theorem 3.11
Suppose that ρ ( S ) = ǫI p . For large enough weight k there is F ( τ ) ∈ H ( k, ρ ) suchthat the component functions of F ( τ ) are linearly independent . Proof:
Let k = ( k, . . . , k ) have constant weight k , and choose k large enough so that P ( τ ) = P k ( ν, τ ) is holomorphic throughout H . This holds for any choice of ν . We may, and shall, alsoassume that the nontriviality condition in weight k is satisfied.Now choose ν so that the exponents ν r + µ r are negative and pairwise distinct for each r inthe range 1 ≤ r ≤ t . Consider the resulting t vector-valued modular forms p M r ◦ P ( τ ) = P ( τ ) M r where p M r is projection onto the M r th column (cf. Subsection 2.4). By (39) we see that thecomponent functions of P ( τ ) M r are holomorphic outside of the r th. block, and in the r th. blockthey have q -expansions q ν r + µ r (cid:0) τu (cid:1) + . . . , ≤ u ≤ m r −
1. Clearly, then, these functions are linearlyindependent.Consider the vector-valued modular form F ( τ ) = ∆( τ ) v t X r =1 P ( τ ) M r , v an integer satisfying v + ν r + µ r ≥ , ≤ r ≤ t . Since the ν r + µ r are pairwise distinct,it follows from the discussion of the preceding paragraph that P r P ( τ ) M r has linearly independentcomponent functions. The choice of v ensures that F ( τ ) is holomorphic at i ∞ and it also haslinearly independent component functions. Since F ( τ ) is a vector-valued modular form associatedwith the same representation ρ , the Theorem follows. ✷ As discussed in Subsection 2.5, ρ is equivalent to the direct sum ρ ⊕ ρ − of a pair of represen-tations ρ ǫ of Γ with the property that ρ ǫ ( S ) = ǫI, ǫ = ±
1. From (16)-(19) it follows that there isa natural identification H ( ρ ) = H ( ρ ) ⊕ H ( ρ − ) , (40)with H ( ρ ) = M k even H ( k, ρ ) , H ( ρ − ) = M k odd H ( k, ρ − ) . In other words, H ( ρ ) and H ( ρ − ) are the even and odd parts respectively of H ( ρ ). Corollary 3.12
For any representation ρ : Γ → GL p ( C ) , there is a nonzero holomorphic vector-valued modular form F ( τ ) ∈ H ( k, ρ ) for large enough weight k . Proof: If ρ ( S ) = ± I p then the Corollary follows immediately from Theorem 3.11. The generalresult is then a consequence of the preceding comments. ✷ .Let M = ⊕ k ≥ M k = C [ Q, R ] be the weighted polynomial algebra of holomorphic modular formsof level 1 on Γ, where Q = E ( τ ) , R = E ( τ ). As in [M1], R = M [ d ]is the ring of differential operators obtained by adjoining to M an element d satisfying df − f d = D ( f ) , f ∈ M , where D is the modular derivative Df = D k f = ( θ + kP ) f ( f ∈ M k ) . (41)Here, θ = qd/dq and P = − /
12 + 2 P n ≥ σ ( n ) q n is the weight 2 quasimodular Eisenstein series,normalized as indicated. R is a 2 Z -graded algebra ( d has degree 2), and H ( ρ ) is a Z -graded R -module in which f ∈ M acts as a multiplication operator and d acts on F ∈ H ( ρ ) by its action on components of F givenby (41). In particular, it follows that R operates on the even and odd parts of H ( ρ ), so that theidentification (40) is one of R -modules. Theorem 3.13 (Free module Theorem) H ( ρ ) is a free M -module of rank p . p weights k , . . . , k p and p vector-valued modular forms F j ( τ ) ∈H ( k j , ρ ) , ≤ j ≤ p , such that every F ( τ ) ∈ H ( k, ρ ) has a unique expression in the form F ( τ ) = p X j =1 f j ( τ ) F j ( τ ) , f j ( τ ) ∈ M k − k j . It is an immediate consequence of this result that the Hilbert-Poincar´e series for H ( ρ ) is a rationalfunction: X k ≥ k dim H ( k, ρ ) t k = P pj =1 t k j (1 − t )(1 − t ) . With Corollary 3.12 available, the remaining details of the proof of Theorem 3.13 are essentiallyidentical to that of the semisimple case given in [MM] and involve mainly arguments from com-mutative algebra. In the few places where the nature of the component functions of vector-valuedmodular forms is relevant, the argument is the same whether the q -expansions are ordinary orlogarithmic. We forgo further discussion.We give an application of the free module Theorem. Let F ( τ ) ∈ H ( k, ρ ). If the elements F, DF, . . . , D p F are linearly independent over M then they span a free M -submodule of H ( ρ ) ofrank p + 1. Since H ( ρ ) has rank p , this is not possible. Therefore, F ( τ ) satisfies an equality of theform (cid:0) g ( τ ) D pk + g ( τ ) D p − k + . . . + g p ( τ ) (cid:1) F = 0 . (42)where g ∈ M l for some weight l and g j ( τ ) ∈ M l +2 j . We may think of (42) as a modular lineardifferential equation (MLDE) [M1] of order at most p , in which case the component functions of F ( τ ) are solutions. Now suppose that the component functions are linearly independent . Since theyare solutions of any MLDE satisfied by F ( τ ), the solution space must have dimension at least p ,and therefore the order of the MLDE must itself be at least p . We have therefore shown that if F ( τ ) ∈ H ( k, ρ ) has linearly independent component functions, it satisfies an MLDE of order p andnone of order less than p .Continuing with the assumption that the component functions of F ( τ ) are linearly independent,let I ⊆ M be the set of all leading coefficients g ( τ ) that occur in order p MLDE’s (42) satisfiedby F . Taking account of the trivial case when all coefficients g j ( τ ) vanish, we see easily that I is agraded ideal. Moreover, our previous comments show that I = 0. We will show that I contains a unique nonzero modular form g ( τ ) of least weight, normalized so that the leading coefficient of its q -expansion is 1, and that I = g ( τ ) M .For nonzero h ( τ ) ∈ I of weight k , we let L h = h ( τ ) D p + h ( τ ) D p − + . . . + h p ( τ )be the unique order p differential operator in R with leading coefficient h ( τ ) and satisfying L h F = 0.Let g ( τ ) be any nonzero element in I of least weight, say m . Then we have L g F = L h F = 0, andtherefore also ( g L h − h L g ) F = 0 . p −
1, and therefore (by our earlierremarks) must vanish identically. It follows that for all indices j we have g h j = h g j . (43)Suppose that the order of vanishing of g ( τ ) at ∞ is greater than that of h ( τ ). By (43) itfollows that all g j ( τ ) vanish to order at least 1 at ∞ , i.e. each g j ( τ ) is divisible by the discriminant∆( τ ) in M . But then L g F = ∆( τ ) L g ′ F = 0, whence L g ′ F = 0 for some nonzero g ′ ( τ ) ∈ M m − .Then g ′ ( τ ) ∈ I , and this contradicts the minimality of the weight of g . Thus we have shown thatthe order of vanishing of g ( τ ) at ∞ is minimal among nonzero elements in I , and that this assertionholds for any nonzero element of least weight in I .If there are two linearly independent elements a ( τ ) , b ( τ ), say, of least weight in I then somelinearly combination of them vanishes at ∞ to an order that exceeds that of at least one of a ( τ )and b ( τ ). By the last paragraph this cannot occur, and we conclude that, up to scalars, g ( τ ) is theunique nonzero element in I of least weight.We use similar arguments to show that g ( τ ) generates I . If not, choose an element h ∈ I ofleast weight n , say, subject to h ( τ ) / ∈ g ( τ ) M . If h ( τ ) has greater order of vanishing at ∞ than g ( τ ), (43) and a previous argument shows that every h j ( τ ) is divisible by ∆( τ ). Then as before, h ( τ ) = ∆( τ ) h ′ ( τ ) with h ′ ( τ ) ∈ I . By minimality of the weight of h ( τ ) we get h ′ ( τ ) ∈ g ( τ ) M , andtherefore also h ( τ ) ∈ g ( τ ) M , contradiction. Therefore, every element of weight n in I \ g ( τ ) M has the same order of vanishing at ∞ as g ( τ ). This again implies the unicity of h ( τ ) up to scalars.If n − m ≥ h ( τ ) + E n − m ( τ ) g ( τ ) has weight n and lies in I \ g ( τ ) M . (Here, E k ( τ )is the usual weight k Eisenstein series.) Thus h ( τ ) is a scalar multiple of h ( τ ) + E n − m ( τ ) g ( τ )and therefore lies in g ( τ ) M , contradiction. Therefore, n − m = 2. In this case we consider h ′ ( τ ) = E ( τ ) h ( τ ) − βE ( τ ) g ( τ ) and h ′′ ( τ ) = E ( τ ) h ( τ ) − γE ( τ ) g ( τ ) for scalars β, γ chosen ineach case so that the order of vanishing at ∞ is greater than that of g ( τ ). A previous argumentshows that L h ′ F = ∆ L h ′ F = 0 for some h ′ ( τ ) of weight n + 4 −
12 = m −
6. Since h ′ ( τ ) ∈ I has weight less than m then h ′ ( τ ) = 0, so that E ( τ ) h ( τ ) = βE ( τ ) g ( τ ). The same reasoningapplied to h ′′ ( τ ) also shows that E ( τ ) h ( τ ) = γE ( τ ) g ( τ ). From these equalities we deduce that g ( τ )( γE ( τ ) − βE ( τ )) = 0. This can only happen if β = γ = 0, whence E ( τ ) h ( τ ) = 0. Thisis impossible since h ( τ ) is nonzero, and we have contradicted the assumed existence of h ( τ ). Tosummarize, we have established Theorem 3.14
Suppose that F ( τ ) ∈ H ( k, ρ ) has linearly independent component functions. Thenthe component functions are a basis of the solution space of a modular linear differential equation (cid:0) g ( τ ) D pk + g ( τ ) D p − k + . . . + g p ( τ ) (cid:1) f = 0 (44) where g j ( τ ) ∈ M l +2 j , ≤ j ≤ p, for some l ≥ . The set of leading coefficients g ( τ ) that can occurin (44) is a (nonzero) principal graded ideal I ⊆ M generated by the unique normalized modularform g ( τ ) of least weight in I . ✷
19f the condition that the component functions of F ( τ ) are linearly independent is not met, onecan replace ρ by the representation ρ ′ of Γ furnished by the span of the component functions. Thenthe Theorem applies to ρ ′ . In this way, we see that to any logarithmic vector-valued modular formwe can associate an MLDE in a canonical way: it is the MLDE of least order and with normalizedleading coefficient of least weight whose solution space is spanned by the component functions of F ( τ ).We can alternatively couch these results in terms of annihilators in the ring of differentialoperators R . For example, we have Corollary 3.15
Let F ∈ H ( k, ρ ) . Then the annihilator Ann R ( F ) is a cyclic R -module. ✷ Let F ( τ ) ∈ H ( k, ρ ) be a logarithmic, holomorphic vector-valued modular form of weight k . Weare going to show that the Fourier coefficients of F ( τ ) satisfy a polynomial growth condition for n → ∞ . Let F ( τ ) = ( f ( τ ) , . . . , f p ( τ )) t . We know by Theorem 2.2 that there are m j q -expansions h l ( τ ) = P n + µ j ≥ a jl ( n ) q n + µ j , ≤ l ≤ m j − F ( τ ) corresponding tothe j th Jordan block are ( f m j − ( τ ) , . . . , f ( τ )) t with f l ( τ ) = l X u =0 (cid:18) τu (cid:19) h l − u ( τ ) , ≤ l ≤ m j − . Here we have relabelled the components in the j th block for notational convenience.The proof is similar to the case treated in [KM1], but with an additional complication due to thefact that we are dealing with polynomial q -expansions rather than ordinary q -expansions. To dealwith this we make use of the estimates that we have obtained in Subsection 3.2. We continue toassume that the eigenvalues of ρ ( T ) are of absolute value 1. We will sometimes drop the subscript j from the notation when it is convenient.We write τ = x + iy for τ ∈ H and let R be the usual fundamental region for Γ. Write z = u + iv for z ∈ R . Choose a real number σ > g l ( τ ) = y σ | f l ( τ ) | . Because F ( τ ) is holomorphic, a l ( n ) = 0 unless n + µ ≥
0. It follows that there is a constant K such that g l ( z ) ≤ K v δ ( σ +1) , ≤ l ≤ p, z ∈ R , (45)where δ = 0 if F ( τ ) is a cusp-form , and is 1 otherwise.Choose γ = (cid:18) a bc d (cid:19) ∈ Γ, set τ = γz , and write γ in Eichler canonical form (25). We now20rgue just as in [KM1] pp. 121-122. Thus g l ( τ ) = g l ( γz ) = ( v | cz + d | − ) σ | f l ( γz ) | = v σ | cz + d | k − σ | f l | k γ ( z ) | = v σ | cz + d | k − σ | ( ρ ( γ ) f ( z )) l | = v σ | cz + d | k − σ | p X m =1 ρ ( γ ) lm f m ( z ) | . Using (45), Lemma 3.9, and (37), we obtain g l ( τ ) ≤ K v δ ( σ +1) | cz + d | k − σ p X m =1 | ρ ( γ ) lm |≤ K v δ ( σ +1) | cz + d | k − σ | c + d | K ≤ K v δ ( σ +1) | cz + d | k − σ + K . Choosing σ = ( k + K ) / g l ( τ ) ≤ K v δ (( k + K ) / . In the cuspidal case we have δ = 0, whence g l ( τ ) is bounded in H . Then | f l ( τ ) | = y − σ g l ( τ ) = O ( y − k − K ) / ) . By a standard argument this implies that the Fourier coefficients of g l ( τ ) satisfy a ( n ) = O ( n ( k + K ) / )for n → ∞ . In the holomorphic case there is a similar argument ([KM1]) wherein the exponent isdoubled. We have proved Theorem 4.1
Suppose that F ( τ ) ∈ H ( k, ρ ) . There is a constant α depending only on ρ such thatthe Fourier coefficients of F ( τ ) satisfy a ( n ) = O ( n k + α ) for n → ∞ . If F ( τ ) is cuspidal then a ( n ) = O ( n k/ α/ ) for n → ∞ . ✷ References [DLM] , C. Dong, H. Li and G. Mason, Modular-invariance of trace functions in orbifold theoryand generalized moonshine, Comm. Math. Phys. (2000), 1-56.[E] M. Eichler, Grenzkreisgruppen und kettenbruchartige Algorithmen, Acta Arith (1965),169-180.[KM1] M. Knopp and G. Mason, On vector-valued modular forms and their Fourier coefficients,Acta Arith. (2003), 117-122.[KM2] M. Knopp and G. Mason, Vector-valued modular forms and Poincar´e series, Ill. J.Math. No. 4 (2004), 1345-1366. 21L] J. Lehner,
Discontinuous groups and automorphic functions , Math. Surveys No. VIII, AMS,Providence, RI, 1964.[M] M. Miyamoto, Modular invariance of vertex operator algebras satisfying C -cofiniteness,Duke J. Math. No. 1 (2004), 51-91.[M1] G. Mason, Vector-valued modular forms and linear differential operators, I.J.N.T. No. 3(2007), 377-390.[M2] G. Mason, 2-dimensional vector-valued modular forms, Ramanujan J.17