On the structure of modules of vector valued modular forms
aa r X i v : . [ m a t h . N T ] S e p ON THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS
CAMERON FRANC AND GEOFFREY MASONA
BSTRACT . If ρ denotes a finite dimensional complex representation of SL ( Z ) , thenit is known that the module M ( ρ ) of vector valued modular forms for ρ is free and offinite rank over the ring M of scalar modular forms of level one. This paper initiatesa general study of the structure of M ( ρ ) . Among our results are absolute upper andlower bounds, depending only on the dimension of ρ , on the weights of generatorsfor M ( ρ ) , as well as upper bounds on the multiplicities of weights of generators of M ( ρ ) . We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicitybounds is to show that there exists a free-basis for M ( ρ ) in which the matrix of themodular derivative operator does not contain any copies of the Eisenstein series E ofweight six. C ONTENTS
1. Introduction 12. Weight profiles 33. Exploiting the differential structure 84. Bounds for weight multiplicities 155. Multiplicity tables in low dimensions 17References 201. I
NTRODUCTION If ρ is a finite-dimensional complex representation of SL ( Z ) of dimension d ,then the module M ( ρ ) of vector valued modular forms for ρ is known [8], [2] to befree of rank d over the ring M of classical scalar modular forms of level one. A basicproblem about M ( ρ ) is then to determine the weights of a generating set of modularforms in this module. These are invariants of the isomorphism class of ρ . In [2] itwas observed that this question is tantamount to determining the decomposition of acertain vector bundle V ( ρ ) on the moduli stack of elliptic curves into line bundles. Indimension less than six, some results on this questions have been obtained by Marks[7], but otherwise very little has been proved about the general situation. This paperinitiates a general study of this question.We begin in Section 2 by introducing the weight profile of ρ , which is the tuple ( k , . . . , k d ) of weights of generators for M ( ρ ) , ordered so that k i ≤ k i +1 for all i . In[9] one finds a proof that − d ≤ k for irreducible representations ρ . Using a slightgeneralization of this argument, combined with Serre duality, we show (Lemma 2.4)that there is an upper bound k d ≤ d + 10 as well. In particular, there are only finitelymany weight profiles for irreducible representations in each dimension. Section 2explains how Proposition 2.1 and Westbury’s description of the character variety of SL ( Z ) (recalled as Theorem 2.10 below) can be used to enumerate a finite list con-taining all weight profiles of irreducible representations of SL ( Z ) of fixed dimension(and possibly some weight profiles that do not occur in practice) that is considerablyshorter than the finite list provided by the weight bounds − d ≤ k ≤ k d ≤ d + 10 .Section 5 contains the results of some of these computations.Sections 3 and 4 use the differential structure of M ( ρ ) afforded by the modu-lar derivative to deduce further information about the weight profile of ρ . The mostnotable new results are the no-gap Lemma (Lemma 3.3) and the weight multiplicitybound of Theorem 4.1. The no-gap Lemma states that if ρ is irreducible, then no gaplarger than two occurs in its weight profile. If m , . . . , m r denote the multiplicities ofthe distinct weights in the weight profile of an irreducible representation ρ , then theweight multiplicity bound states that m j ≤ X t ≥ m j +1 − t and m j ≤ X t ≥ m j − t , where m i = 0 if i < or i > r . Since the tuple π ρ = ( m , . . . , m r ) is an orderedpartition of d , this implies in particular that m j ≤ d/ for all j . The proof of Theorem4.1 seems to be new and of considerable interest. It uses the fact, proved in Theorem3.13, that there exists a choice of basis for M ( ρ ) in which the matrix of the modularderivative does not contain any copies of the weight Eisenstein series E .Aside from the intrinsic interest of the weight profiles of representations of SL ( Z ) , there would be practical benefits to understanding them better. For example,if ρ denotes the permutation representation of SL ( Z ) acting on the cosets of some fi-nite index subgroup Γ , then M k ( ρ ) ∼ = M k (Γ) , where M k (Γ) denotes the space of scalarmodular forms for Γ of weight k . If ρ decomposes into irreducible representations ρ ∼ = L i ρ i , then one obtains a corresponding decomposition M k (Γ) ∼ = L i M k ( ρ i ) .Thus, for example,(1) dim M (Γ) = X i dim ρ i · m ( ρ i ) , where m ( ρ i ) denotes the multiplicity of the weight in the weight profile of ρ i (herewe’ve used the fact that the weight profile of a representation of finite image consistsof positive integers).We explored the idea of using the decomposition (1) and the results of [2] tostudy the dimensions of spaces of modular forms of weight one on Γ( p ) for a prime p . We were pleased to observe that when p ≡ , and if ρ and ρ denote theirreducible representations of SL ( F p ) obtained as certain constituents of reducibleprincipal series representations, then the Euler characteristics of the correspondingvector bundles V ( ρ i ) equal (1 ± h ( − p )) , where h ( − p ) denotes the class number of Q ( √− p ) . Using this, it is not too hard to show that dim M ( ρ i ) ≥ (1 + h ( − p )) . Thiselementary argument detects the dihedral theta series of weight one without writingthem down explicitly, and is presumably well-known to experts . Unfortunately, theelementary arguments that [2] enables do not, by themselves, shed any new light onthe question of dimensions of spaces of modular forms of weight one. However, thequestion of the module structure of M ( ρ ) seems to be a richer one, and a methodical Nevertheless, it is worth remarking that the term h ( − p ) in the Euler characteristic arises via theexponents of ρ i ( T ) through Dirichlet’s analytic class number formula. N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 3 study of M ( ρ ) for general ρ could conceivably lead to a better understanding of scalarforms that have so far resisted available techniques. This is part of the impetus thatdrove this work.Let us conclude the introduction by describing our notation and conventions.In this note ρ will always denote a finite-dimensional complex representation of SL ( Z ) , usually irreducible. It will often be convenient to assume that ρ ( S ) is a scalar.Then necessarily ρ ( S ) = ± I . If ρ ( S ) = I then ρ is said to be even , while if ρ ( S ) = − I then ρ is said to be odd . If ρ is even or odd, then the weights of nonzero vectorvalued modular forms for ρ must have the same parity as ρ . Note that all irreduciblerepresentations of SL ( Z ) are either even or odd. The notation ρ ∨ denotes the dualrepresentation of ρ . Let T = (cid:18) (cid:19) , S = (cid:18) −
11 0 (cid:19) , R = ST = (cid:18) −
11 1 (cid:19) . Let χ denote the character of η , so that χ ( T ) = e πi/ . Write ξ = e πi/ . If L is a ma-trix such that ρ ( T ) = e πiL , then we call L a choice of exponents for ρ . Recall that since det e M = e Tr( M ) , the quantity
12 Tr( L ) is an integer for any choice of exponents L for ρ . Let V k,L ( ρ ) denote the vector bundle introduced in [2]. If L has eigenvalues withreal part in [0 , then we write simply V k ( ρ ) for V k,L ( ρ ) . Similarly, if the eigenvalueshave real part in (0 , then we write S k ( ρ ) for V k,L ( ρ ) . The global sections of V k ( ρ ) and S k ( ρ ) are the spaces of weight k vector valued holomorphic modular forms M k ( ρ ) and cusp forms S k ( ρ ) , respectively, for ρ . Note that if L and L denote choices of ex-ponents for ρ ( T ) adapted to [0 , and (1 , , respectively, then Tr( L ) = Tr( L ) + m where m is the multiplicity of one as an eigenvalue of ρ ( T ) .2. W EIGHT PROFILES
In [2], the Euler characterstic of the bundles V k ( ρ ) was computed. When k islarge enough, depending on ρ , the Euler characteristic agrees with the dimension of M k ( ρ ) . The following proposition makes this precise. Proposition 2.1.
Let ρ denote an irreducible representation of dimension d , let L denotea standard choice of exponents for ρ ( T ) , and let L denote a cuspidal choice of exponentsfor ρ ( T ) . Then one has dim M k ( ρ ) = k < d Tr( L ) + 1 − d,χ ( V k ( ρ )) + dim S − k ( ρ ∨ ) d Tr( L ) + 1 − d ≤ k ≤ d Tr( L ) + d − ,χ ( V k ( ρ )) k > d Tr( L ) + d − , and dim S k ( ρ ) = k < d Tr( L ) + 1 − d,χ ( S k ( ρ )) + dim M − k ( ρ ∨ ) d Tr( L ) + 1 − d ≤ k ≤ d Tr( L ) + d − ,χ ( S k ( ρ )) k > d Tr( L ) + d − . Proof.
Recall (Proposition 3.14 of [2]) that if ρ is irreducible of dimension d , L is achoice of exponents for ρ , and k is the minimal integer such that h ( V k,L ( ρ )) = 0 , then k ≥ d Tr( L ) + 1 − d. CAMERON FRANC AND GEOFFREY MASON
Let m be the multiplicity of one as an eigenvalue for ρ ( T ) , let k be the minimal weightfor ρ , and let ℓ be the minimal integer such that S ℓ ( ρ ) = 0 . Then if L is a standardchoice of exponents for ρ , ℓ ≥
12 Tr( L ) + 12 md + 1 − d. However, to apply Serre duality to the computation of dimensions of spaces of mod-ular forms, one wishes to know when S − k ( ρ ∨ ) is nonzero. Note that the multiplicityof one as an eigenvalue of ρ ∨ ( T ) is also m . If L is a standard choice of exponents for ρ , and if L ∨ is a standard choice of exponents for ρ ∨ , then Tr( L ∨ ) = d − Tr( L ) − m .We thus see that if S − k ( ρ ∨ ) = 0 then k ≤ d Tr( L ) + d − This proves the claim about M k ( ρ ) . The proof of the claim for S k ( ρ ) is similar. (cid:3) The middle cases of Proposition 2.1 comprise at most d − weights. Halfof these can be eliminated using parity considerations, but in general the other halfmight be difficult to compute. When d ≤ , however, Proposition 2.1 gives explicitformulae for dim M k ( ρ ) and dim S k ( ρ ) in all weights. For general d one can use Propo-sition 2.1 and positivity to narrow down the possibilities for dim M k ( ρ ) and dim S k ( ρ ) in low weights to a finite number of possibilities – see Theorem 2.9 below.The free module theorem for vector valued modular forms states that the mod-ule M ( ρ ) of vector valued modular forms for ρ is free of rank d = dim ρ over thering M of scalar modular forms of level one. This result follows, for example, fromthe complete decomposability of vector bundles on the moduli stack of elliptic curves[2]. The free-module theorem also holds for the module S ( ρ ) of cusp forms for ρ , andmore generally for the module M L ( ρ ) of modular forms for ρ relative to any givenchoice of exponents L for ρ .Let us write V k ( ρ ) = d M j =1 O ( k − k i ) , S k ( ρ ) = d M j =1 O ( k − ℓ i ) for integers k i , ℓ i with k j ≤ k j +1 and ℓ j ≤ ℓ j +1 for all j . The integers − k i are the roots of ρ . The tuples ( k , . . . , k d ) and ( ℓ , . . . , ℓ d ) are called the weight profile and cuspidalweight profile of ρ , respectively. More generally, if V k,L ( ρ ) = L dj =1 O ( k − k i ) thenwe call ( k , . . . , k d ) the L -adapted weight profile of ρ . If ( k j ) denotes a weight profile,then the type profile , or more simply, the type of ρ is the tuple (0 , k − k , . . . , k d − k ) .Obviously one can recover the weight profile from the knowledge of the type and theminimal weight. Conversely, Lemma 2.2.
Let ρ denote a representation with weight profile ( k , . . . , k d ) and standardchoice of exponents L . Then P dj =1 k j = 12 Tr( L ) . In particular, the minimal weight of ρ is determined by the type of ρ and Tr( L ) for a standard choice of exponents L for ρ .Proof. Proposition 3.6 of [2] observes that det V k ( ρ ) ∼ = O ( dk −
12 Tr( L )) , and the firstclaim follows from this. Thus k = (12 Tr( L ) − P dj =1 ( k j − k )) /d , and this proves thesecond claim. (cid:3) N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 5
Remark . Recall from Proposition 3.13 of [2] that V k ( ρ ) ∨ ∼ = S − k ( ρ ∨ ) . It followsthat if ( k , . . . , k d ) is the weight profile of ρ , then (12 − k d , . . . , − k ) is the cuspidalweight profile of ρ ∨ . Hence if ρ is such that is not an eigenvalue of ρ ( T ) , so that S − k ( ρ ∨ ) = V − k ( ρ ∨ ) , the dual weight profile of ρ is (12 − k d , . . . , − k ) . If moreover ρ is self dual, this implies that k j + k d +1 − j = 12 for all j . Hence Lemma 2.2 implies that Tr( L ) = d/ for such representations ρ . The problem of relating the weight profileof a representation with that of its dual is an interesting and likely tractable openproblem. Lemma 2.4.
Let ρ be an irreducible representation, and let ( k , . . . , k d ) be its weightprofile. Then d Tr( L ) + 1 − d ≤ k ≤ k d ≤ d Tr( L ) + d − . In particular, for all irreducible representations of dimension d , the weight profiles liein the range [1 − d, d + 10] . There are thus finitely many weight profiles for irreduciblerepresentations in each dimension.Proof. The lower bound on k is well-known (see e.g. Proposition 3.14 of [2]). ByRemark 2.3, the cuspidal weight profile of ρ ∨ is (12 − k d , . . . , − k ) . Let L ∨ be achoice of exponents for ρ ∨ adapted to the interval (0 , , so that S k ( ρ ∨ ) = V k,L ∨ ( ρ ∨ ) .Then by the slight generalization of the Wronskian argument given in Proposition3.14 of [2], d Tr( L ∨ ) + 1 − d ≤ − k d , and thus k d ≤
11 + d − d Tr( L ∨ ) . Note that Tr( L ∨ ) = d − Tr( L ) since L ∨ denotes thecuspidal exponents for ρ ∨ . Thus k d ≤ d Tr( L ) + d − . (cid:3) Remark . One can show that the bounds of Lemma 2.4 are sharp.Tuba and Wenzl [10] have described all irreducible representations of SL ( Z ) in dimension less than six. One can use this and Proposition 2.1 to compute alltypes and minimal weights in dimension less than six. The results are below. Thesecomputations are consistent with, and add precision to, the computations in [7]. Example 2.6.
We list the possible types of irreducible representations in dimension ≤ , along with the minimal weight.Dimension Type k L )2 (0 ,
2) 6 Tr( L ) −
13 (0 , ,
4) 4 Tr( L ) −
24 (0 , , ,
6) 3 Tr( L ) −
34 (0 , , ,
4) 3 Tr( L ) − Remark . In [10] is it shown that up to a choice of square root of det( T ) , the eigen-values of ρ ( T ) determine four dimensional irreducible representations of SL ( Z ) . Thetwo possibilities for the type in dimension correspond to the two choices of squareroot. Example 2.8.
The case of five dimensional irreducible representations is more inter-esting. In this case it need not be true that | Tr( L ) . Using [10], one can compute CAMERON FRANC AND GEOFFREY MASON the minimal weights and types. To express the result it is best to write
Tr( L ) = a where ≤ a ≤ . One finds the following possibilities: a (mod 5) Type k , , , ,
8) ( a − /
51 (0 , , , ,
6) ( a − /
52 (0 , , , ,
4) ( a − /
53 (0 , , , ,
4) ( a − /
54 (0 , , , ,
6) ( a − / Our next goal is to describe an algorithm for enumerating a list that contains allpossible types of irreducible representations in a given dimension.
Theorem 2.9.
There exists an algorithm that takes as input an integer d ≥ and theresulting output is a finite list of d -tuples of positive integers that contains all possibletypes of irreducible representations of SL ( Z ) of dimension d .Proof. By the no-gap lemma (Lemma 3.3 below), one could simply enumerate allpossible sequences of integers ( x , . . . , x d ) where x = 0 and such that ≤ x i +1 − x i ≤ for i = 1 , . . . , d − . There are d − such sequences. (cid:3) A large number of the d − possible types given by the no-gap lemma do notoccur in practice. A number of additional restrictions on the types, arising from thedifferential structure on M ( ρ ) , are described in Sections 3 and 4 below. Proposition2.1 and Westbury’s description [11] of the irreducible components of the charactervariety of semistable representations of SL ( Z ) can also be used to cut down thepossibilities dramatically. We describe this next. Theorem 2.10.
The character variety X d classifying d -dimensional semistable repre-sentations of PSL ( Z ) is an affine algebraic variety that decomposes into a disjointunion of irreducible components X d = ∐ α X α indexed by tuples of nonnegative integers α = ( a, b ; x, y, z ) satisfying a + b = x + y + z = d . A given irreducible representation ρ of PSL ( Z ) of dimension d lies on the component X α indexed by α = ( a, b ; x, y, z ) where a and b are the multiplicities of and − , respectively, as eigenvalues of ρ ( S ) , and where x , y and z denote the multiplicities of , ζ = e πi and ζ , respectively, as eigenvalues of ρ ( R ) .Proof. This result was originally proved by Bruce Westbury [11], but it remains un-published. See Section 2 of [1] for more information on the character variety of themodular group. (cid:3)
Remark . If ρ is an odd irreducible representation of SL ( Z ) , then ρ ⊗ χ is anirreducible representation of PSL ( Z ) . Thus, Theorem 2.10 allows one to give asimilar description for the character variety of SL ( Z ) .Fix an irreducible representation ρ of dimension d , let L denote a standardchoice of exponents for ρ ( T ) , so that Tr( L ) ∈ [0 , d ) , and write s = Tr( ρ ( S )) , r =Tr( ρ ( R )) and r = Tr( ρ ( R )) . The quantities s , r and r are constant on each ir-reducible component of the character variety by Theorem 2.10 and Remark 2.11.Similarly, det ρ is constant on components of the character variety, so that Tr( L ) takeson at most d values across each component of the character variety. Thus, if we per-form the following computation for fixed s , r , r and for the d possible values of N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 7
Tr( L ) for representations on the component of the character variety that contains ρ ,then the result is a finite computation that gives all possible types of representationson the irreducible component that contains ρ . Thus, we need only describe how tonarrow down the possibilities for the type of our fixed ρ to a finite list.In order to describe the computation, let ℓ , . . . ℓ r denote the increasing sequenceof integers in the interval between (12 /d ) Tr( L ) + 1 − d and (12 /d ) Tr( L ) + d − withthe same parity as ρ , and set a j = dim S − ℓ j ( ρ ∨ ) for each j . Then by Proposition 2.1,(2) P dj =1 T k j (1 − T )(1 − T ) = X k ≥ ℓ χ ( V k ( ρ )) T k + r X j =1 a j T ℓ j . By Corollary 6.2 of [2], X k ≥ ℓ χ ( V k ( ρ )) T k = T ℓ (cid:18) d −
12 Tr( L )12 11 − T + s i ℓ T + r − ζ ) ξ ℓ − ζ T + r − ζ ) ζ ℓ − ζ T + d ℓ − ( ℓ − T (1 − T ) (cid:19) where ξ = e πi/ and ζ = ξ . It follows that equation (2) yields an explicit andcomputable equation of the form(3) d X j =1 T k j = T ℓ P ( T ) + r X j =1 a j T ℓ j (1 − T )(1 − T ) where P ( T ) is a polynomial of degree at most with integer coefficients. Note that P ( T ) only depends on d , s = Tr( ρ ( S )) , r = Tr( ρ ( R )) , r = Tr( ρ ( R )) and Tr( L ) (since ℓ was defined in terms of Tr( L ) and d via Proposition 2.1). Lemma 2.12.
There are only finitely many solutions to equation (3) in nonnegativeintegers a j .Proof. By comparison with the left side of the equation, the coefficients of the righthand side of equation (3) must be nonnegative integers that are no larger than d .Write P ( T ) = P j =0 b j T j . The coefficient of T ℓ in (3) is of the form p + a . Thus ≤ a ≤ d − p , so that there are finitely many possibilities for a . Similarly, the coefficientof T ℓ j is of the form p k + a j + Q ( a , . . . , a j − ) for some polynomial Q ( a , . . . , a j − ) . Byinduction, this polynomial takes on finitely many values, and so if M is the maximalsuch value, we find that ≤ a j ≤ d − p k − M , for some index k . This proves thelemma. (cid:3) By Lemma 2.12 it is thus possible to enumerate the finitely many solutions toequation (3) in nonnegative integers, and thereby find all types of irreducible repre-sentations ρ of dimension d and with fixed values of Tr( ρ ( S )) , Tr( ρ ( R )) , Tr( ρ ( R )) ,and Tr( L ) . By Theorem 2.10, this allows one to describe a finite list of all types of ir-reducible representations in dimension d . We have implemented these computationsusing Sage , and we were able to run the algorithm in dimensions up to and includingtwelve before the computations began to run into memory limitations. Some of theseresults are listed in Section 5. The number of types that are output by this algorithmtends to be exponential in d , but it is a much smaller number than the d − given bythe no-gap lemma. Nevertheless, there are many types that arise in this way that do CAMERON FRANC AND GEOFFREY MASON not actually occur. Some of these possibilities can be eliminated using the results ofSection 3 below, but in general it seems to be an open problem to determine exactlywhat type profiles do occur in each dimension.We end this section by explaining how to extend this finiteness result to allrepresentations.
Proposition 2.13.
Fix a positive integer d . There are only finitely many possible weightprofiles for representations of SL ( Z ) of dimension d .Proof. We have explained the proof of Proposition 2.13 for irreducible representa-tions. Suppose that → ρ → ρ → ρ → is a short exact sequence of representations of SL ( Z ) . After applying the functor M ,there is an exact sequence → M ( ρ ) → M ( ρ ) → M ( ρ/ρ ) (cf. [8] for more details). This shows that at the level of multisets, the set of weightsfor ρ is contained in the union of the corresponding multisets for ρ and ρ . Taking acomposition series for ρ , this shows that the number of weight profiles in dimension d is no more than d times the maximum of the number of weight profiles for anirreducible of dimension no greater than d . So finiteness in general follows from theirreducible case. (cid:3)
3. E
XPLOITING THE DIFFERENTIAL STRUCTURE
Let R .. = M h D i be the algebra of modular differential operators, where D actson modular forms of weight k via the usual operator D k .. = q ddq − k E . (4)Since we have normalized the Eisenstein series E and E of weights and to haveconstant term equal to , one has D ( E ) = − E and D ( E ) = − E . Formally,elements of R are polynomials P i f i D i in D with coefficients f i ∈ M , however R is noncommutative (although associative). Multiplication is implemented using theidentity Df = f D + D ( f ) ( f ∈ M ) . If we give D degree then R is an N -gradedalgebra.One knows ([9], [8]) that M ( ρ ) is a Z -graded left R -module. Elements of M act by multiplication and D acts via the obvious extension of (4) to vvmfs of weight k . It is the exploitation of this fact that underlies the results in the present Section.Actually, the structure of M ( ρ ) as R -module is an interesting topic in its own right,but we will resist the temptation to axiomatize the situation, and simply record someof the relevant features.The free module theorem ([8], [2]) says that M ( ρ ) is a free M -module of rank dim ρ . On the other hand, M ( ρ ) is a torsion R -module: every element in M ( ρ ) has anonzero annihilator in R . We will use the following more precise version of this factin the case that ρ is irreducible. Lemma 3.1.
Assume that ρ is irreducible of dimension d , and let F ∈ M k ( ρ ) be nonzero.Then the following hold: N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 9 (a)
F, DF, . . . , D d − F are linearly independent over M , (b) F, DF, . . . , D d F are linearly dependent over M (that is, some polynomial ofdegree d annihilates F , but none of degree less than d ), (c) If = N ⊆ M ( ρ ) is a graded R -submodule that is free of rank r as an M -module,then r = d .Proof. To say that a nonzero polynomial of degree n in R annihilates F just means(taking the grading into account) that there is a relation(5) n X i =0 f i D i F = 0 , where each f i ∈ M k ′ − k − i for some fixed k ′ and f n = 0 .Relation (5) tells us that F satisfies a modular linear differential equation, orMLDE (cf. [9], [3]), of order n . If there are no such relations with n = d then F , DF , . . . , D d F are linearly independent over M , whence they span a free M -submoduleof M ( ρ ) of rank d + 1 . Since M ( ρ ) is free of rank d this is not possible, and thiscontradiction establishes part (b).On the other hand, suppose (5) holds with n ≤ d − . As an order n MLDE, thesolution space of (5) is n -dimensional, and therefore the span E of the components of F (a subspace of the solution space) has dimension less than d . However, because ρ isirreducible, the components of F span an SL ( Z ) -module that affords a representationequivalent to ρ . In particular, the span of these components has dimension d . Thiscontradiction proves (a).As for (c), choose a nonzero form F ∈ N ∩ M k ( ρ ) for some k . By part (a), F, . . . , D d − F generate a free M -submodule of N of rank d , so that r ≥ d . On theother hand, r ≤ d because M ( ρ ) is free of rank d . Thus r = d , and the proof of thelemma is complete. (cid:3) Lemma 3.1 implies the no-gap lemma (Lemma 3.3), Lemma 3.8 and Proposition3.12 below.
Definition 3.2.
Let ρ denote an even or odd representation of SL ( Z ) , so that allweights in the weight profile of ρ have the same parity. A gap in the weight profile of ρ is an integer k with the following properties: k has the same parity as the weightsof ρ ; there are weights of ρ which are less than k and weights which are greater than k , but no weights equal to k . Lemma 3.3 (No-gap lemma) . Suppose that ρ is an irreducible representation of SL ( Z ) .Then the weight profile of ρ has no gaps.Proof. Suppose that k is a gap in the weight profile of ρ . Then we can divide a set X of (homogeneous) generators of M ( ρ ) into two nonempty subsets X = X ∪ X suchthat all weights of generators in X are ≤ k − , and all weights of generators in X are ≥ k + 2 . Note that | X | + | X | = dim ρ .Let F ∈ X . Then wt( D ( F )) = wt( F ) + 2 ≤ k , so if we write D ( F ) as an M -linear combination of generators in X , all of those generators have weight ≤ k , andhence they lie in X . This shows that the M -submodule M ⊆ M ( ρ ) spanned by thegenerators in X is in fact an R -submodule. Since X is nonempty, M has M -rank | X | . But | X | < dim ρ , and this contradicts Lemma 3.1(c). (cid:3) Definition 3.4.
Fix a representation ρ of SL ( Z ) of dimension d . We denote by π ρ the ordered partition consisting of the multiplicities of the weights that occur in theweight profile of ρ . Thus π ρ = ( m , . . . , m r ) means that the distinct weights that occurare k ′ < · · · < k ′ r and the weight profile is π ρ = ( k ′ , . . . , k ′ | {z } m , k ′ , . . . , k ′ | {z } m , . . . ) . Similarly, we have the cuspidal analog π Sρ which records the multiplicities of the gen-erating weights in the cuspidal weight profile of ρ . Remark . If ρ is irreducible with weight profile ( k , . . . , k d ) and weight multiplicities ( m , . . . , m r ) , then the following identities hold:(1) if j ≥ and ≤ i ≤ m j then k m + ··· + m j − + i = k + 2 j − ,(2) d = P ri =1 m i ,(3) r = 1 + k d − k . Lemma 3.6. If ρ is an irreducible unitary representation of SL ( Z ) distinct from the1-dimensional trivial representation, then the weights k , . . . , k d in the weight profile of ρ lie in the range [1 , .Proof. This is proved in Section 6 of [2]. We give a second proof: it is proved inSection 3 of [4] (with further details in Section 7 of [5]) that the classical Heckeestimate O ( n k ) continues to hold for the n th Fourier coefficient of any componentof a holomorphic vvmf of weight k associated to a unitary representation ρ . Thena standard argument shows that if ρ is irreducible and nontrivial, the weight of anonzero holomorphic vvmf is necessarily positive . Hence, k ≥ .On the other hand, because ρ is unitary then so is ρ ∨ . By Remark 2.3, the lowestweight in the cuspidal weight profile for ρ ∨ is − k d , and by the argument of theprevious paragraph we have − k d ≥ . (cid:3) Definition 3.7.
Let ρ be a representation of SL ( Z ) . We say that M ( ρ ) is cyclic if it isa cyclic R -module, i.e. there is some weight k vvmf F such that M ( ρ ) = R.F . In thissituation, we also say that ρ itself is cyclic. Lemma 3.8.
Suppose that ρ is irreducible and that π ρ = ( t z }| { , . . . , , . . . ) , i.e. there is t ≥ such that the first t weight multiplicities are . Let F be a nonzero vvmf of minimalweight k . Then either { F, DF, . . . , D t − F } is a complete set of generators, or there is agenerating set that contains { F, DF, ..., D t F } .Proof. By assumption there is a unique generator of weight k (up to scalars), so wecan always include F in a set of free generators. Suppose that F, DF, . . . , D i F are ina free generating set, and that i ≤ t − . By hypothesis, the only free generators withweight between k and k + 2 i − are the D i F ( ≤ i < t ) (up to scalars). If we cannot include F, . . . , D i F, D i +1 F in a generating set, there must be an expression of the form D i +1 F = P j f j G j where f j ∈ M j is a classical modular form of positive weight andeach G j is a free generator. Then wt( G j ) ≤ k + 2 i − , whence each G j is equal tosome D j F ( j < i ) (up to scalars). Now it follows that the M -span of F, . . . , D i F is an R -module, and by Lemma 3.1 it follows that i + 1 = d . Thus t ≥ i + 1 = d ≥ t , whence d = t = i + 1 . N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 11
This shows that if i < t − then we can always adjoin D i +1 F to a set of freegenerators { F, . . . , D i F } to obtain a larger such set of free generators. Similarly, if i = t − then either we can similarly adjoin D t F , or else t = d and { F, . . . , D t − F } isalready a complete set of free generators. The Lemma follows. (cid:3) Lemma 3.9.
Let ρ be an irreducible representation of SL ( Z ) of dimension d . The fol-lowing are equivalent: (a) M ( ρ ) is cyclic, (b) There is F ∈ M k ( ρ ) such that { F, DF, . . . D d − F } freely generates of M ( ρ ) , (c) π ρ = (1 , , . . . , .Proof. If (a) holds, there is a vvmf F of weight k such that M ( ρ ) = R.F . Then M ( ρ ) = P i ≥ M D i F ⊇ P d − i =0 M D i F , and by Lemma 3.8 the last containment is an equality.Now (b) is a consequence of Lemma 3.1(a), and this shows that (a) ⇔ (b). Clearly(b) ⇒ (c), while the converse also follows from Lemma 3.8. So (b) ⇔ (c), and the proofof the Lemma is complete. (cid:3) Remark . For further discussion of the case of cyclic ρ , see Theorem 1.3 andSection 4 of [8]. Example 3.11. (a) For all n ≥ , the n th symmetric power S n ( ρ ) of the defining 2-dimensional representation ρ of SL ( Z ) is irreducible and cyclic. These examples arediscussed at length in [6].(b) Every irreducible ρ of dimension ≤ is cyclic. See [3] for an extensive discussionof these cases.In spite of these examples, it appears that there are not too many classes ofirreducible ρ which are cyclic, and it is an interesting problem to try and classify allexamples. The unitary case seems particularly tractable, because of the next result. Lemma 3.12.
Let ρ be an irreducible, cyclic, unitary representation of SL ( Z ) . Then dim ρ ≤ .Proof. We know from Lemma 3.9 that all weight multiplicities are equal to because ρ is cyclic. On the other hand, by Lemma 3.6 there are no more than distinct weightsthanks to unitarity. The only way to reconcile these statements is if the dimension dim ρ ≤ . (cid:3) This section concludes with a result (Theorem 3.13) that will be used to proveupper bounds on weight multiplicities for irreducible representations (Theorem 4.1).We begin with some preparations.Let F , . . . , F d denote a free basis for M ( ρ ) , chosen so that each F j has integerweight. Let A = ( a ij ) denote the matrix of D in this basis, so that DF j = P di =1 a ij F i .If F denotes the d × d matrix whose columns are the F j , then A is defined by thematrix equation DF = F A . If F is replaced by F P for some invertible matrix P withentries in M = C [ E , E ] , and if A ′ is the matrix of D with respect to this new basis,then F P A ′ = D ( F P ) = D ( F ) P + F D ( P ) = F AP + F D ( P ) and thus A ′ = P − AP + P − D ( P ) . Suppose that P corresponds to replacing a basis vector F j by F j − gF i where i < j and g ∈ M . We call this an elementary replacement operation . The matrix of D changes under such an elementary replacement operation as follows:(1) add g times the j th row of A to the i th row of A and(2) subtract g times the i th column of A from the j th column of A and(3) subtract D ( g ) from the ( i, j ) -entry of the result.We will use elementary replacement operations to find a basis in which the matrixof D has a particularly simple form. The idea will be to methodically winnow awaycopies of E . To this end, if f ∈ M , then let d ( f ) denote the E -degree of f when it isregarded as an element of the polynomial ring C [ E , E ] . For each integer t ≥ , let M tk = { f ∈ M k | d ( f ) ≤ t } . Theorem 3.13.
Let ρ denote an irreducible representation of SL ( Z ) and let L denotea choice of exponents for ρ ( T ) . Then there exists a basis for M L ( ρ ) consisting of integerweight vector valued modular forms such that the matrix of D in this basis contains onlyentries that are multiples of pure monomials of the form E x .Proof. Let ( k , . . . , k d ) be the L -adapted weight profile of ρ and let r = 1 + k d − k .Let m , . . . , m r denote the weight multiplicities. Choose free generators F j for M ( ρ ) ordered by increasing weight. Hence F , . . . , F m are of weight k , F m +1 , . . . , F m + m are of weight k = k + 2 , and so on. Define the matrix A = ( a ij ) of D in this basis bywriting D ( F j ) = P di =1 a ij F i for all j . Then A has the following block shape: m m m m m m m · · · m r − m r m ⋆ r r + 2 m ⋆ r − rm ⋆ ⋆ r − r − m ⋆ ⋆ ⋆ r − r − m ⋆ ⋆ ⋆ ⋆ r − r − m ⋆ ⋆ ⋆ ⋆ ⋆ r −
10 2 r − m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ r −
12 2 r − ... m r − ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ m r ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ The ⋆ entries indicate zeros, the integer entries indicate weights of the entries, andthe row and column labels m j indicate the size of the blocks in the block matrixdecomposition.Notice that the weight of diagonal entries of A is constant. We will slowlyimprove A diagonal by diagonal using elementary replacement operations. Our goalis to use a sequence of elementary replacement operations to find a basis for M ( ρ ) such that the matrix of D in this basis has entries in M k . It will be convenient, tophrase things in a uniform way, to regard D as a matrix with infinitely many columnsmoving to the right and infinitely many rows moving down. Thus, initially the matrix N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 13 of D has the form m m m m m m m m m m m m m ⋆ M M M M M M M M M M M m M ⋆ M M M M M M M M M M m ⋆ M ⋆ M M M M M M M M M m ⋆ ⋆ M ⋆ M M M M M M M M m ⋆ ⋆ ⋆ M ⋆ M M M M M M M m ⋆ ⋆ ⋆ ⋆ M ⋆ M M M M M M · · · m ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M M M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ ...where the m j labels indicate a block of rows or columns of size m j , a block with anentry of the form M tk means that the block matrix contains entries in M tk , and a ⋆ indicates that weight considerations force the entries in those blocks to be zero.Our algorithm proceeds by using elementary replacement operations to changeblock diagonals with entries in M tk to have entries in M t − k , but one must take care inhow one chooses the diagonals. The rule for choosing which block diagonal to adjustis to start looking from the center diagonal of zeros, and move up until you encoutera pair of adjacent diagonals containing entries in M t +12+2 k and M t k . Then, adjust the k th diagonal up from the center, which contains entries in M t +12+2 k . Afterward it willcontain entries in M t − k , and then the algorithm repeats. This alogrithm will involvesome backtracking, and so we must argue that it is possible to do such backtrackingwithout undoing the operations that preceded it.Let us explain the first step of the algorithm very carefully. Consider one ofthe m i × m i +2 block matrices, which contains entries in M = M = h E i . Since D ( E ) = − E , we can replace basis vectors F corresponding with the ( i + 2) thblock column of D with basis vectors of the form F − αE G for α ∈ C and G somebasis vector corresponding with the m block of columns, and appropriate choicesof α will allow us to ensure that all entries on this block diagonal are in M = 0 .Note that these elementary replacement operations will also affect the m i × m i +1 and m i +1 × m i +2 blocks, but it will affect them by adding multiples of E to entries. Thus,the result will still lie in M = M . These operations will also affect block diagonalsabove the weight diagonal, but we don’t care about that at this stage, as we haven’tyet performed any simplifications to that part of the matrix. Thus, after all these replacements we reduce to a matrix for D of the form m m m m m m m m m m m m m ⋆ M M M M M M M M M M M m M ⋆ M M M M M M M M M M m ⋆ M ⋆ M M M M M M M M M m ⋆ ⋆ M ⋆ M M M M M M M M m ⋆ ⋆ ⋆ M ⋆ M M M M M M M m ⋆ ⋆ ⋆ ⋆ M ⋆ M M M M M M · · · m ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M M M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ M m ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ M ⋆ ...Suppose now by induction that we’ve found a basis for M ( ρ ) such that the diag-onals of D have entries in the following spaces: M , M , . . . , M t , M t +2 , M t +4 , . . . , M t − t − , M t t , M t − t +2 , M t t +4 , M t +16 t +6 , M t t +8 , . . . where t ≥ . Note that we have not put any restrictions on the E -degree of entriesin the weight t diagonals and higher. Write entries f ∈ M t t uniquely in the form f = αE t + g for α ∈ C and g ∈ M t − t . Then h = − α E E t − ∈ M t − t − satisfies f − D ( h ) ∈ M t − t . Thus, if we use such forms h to perform elementary replacementoperations, we can force the weight t diagonal to lie in M t − t . This will adjust theentries in the weight t − diagonal by the various h ’s that arise, but since these all liein M t − t − , we will not disrupt this diagonal. Similarly, these elementary replacementoperations will alter diagonals above the weight t diagonal, but since we have notput any restrictions on those diagonals yet, such operations are inconsequential forour goal.Now comes the slightly delicate part: we continue working backwards from theweight t − block diagonal to the weight t + 2 block diagonal, and the issue isthat we’ve adjusted diagonals from the one under consideration up to the weight t diagonal. The saving grace is that there are enough diagonals in low weights that donot contain any copies of E .More precisely, suppose that we’ve reduced to a matrix with diagonals of theform M , . . . , M t , M t +2 , M t +4 , . . . , M t − j +16 t − j − , M t − j t − j , M t − j +16 t − j +2 , . . . , M t − t − , M t − t − , M t − t , . . . When we adjust the weight t − j − diagonal we must be careful not to disruptthe diagonals of weight t − j through weight t , since we have reduced the E degree of each. However, we can ignore diagonals above this, as we have not put anyrestrictions on them yet. The elementary replacement operations that we perform inweight t − j − will involve multiples of h = E x E t − j ∈ M t − j t − j − . Entries in theweight t − j + 2 r diagonals, for r = 0 , . . . , j , will be adjusted by forms in hM r − .The E -degree of zero arises since we have already ensured that the weight through t diagonals have E -degree equal to , and since j ≤ t , these are the diagonals N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 15 that affect the diagonals that we’re worried about when we perform the elementaryreplacement operations. Since hM r − ⊆ M t − j t − j +2 r ⊆ M t − j + r t − j +2 r , we will not undoany of the hard work that we have done between weights t − j − and t . As j increases to t , we wind up with a matrix whose sequence of diagonals looks like M , . . . , M t , M t +2 , M t +4 , M t +6 . . . , M t − t − , M t − t , M t − t +2 , M t t +4 , M t +16 t +6 , M t t +8 , . . . And now we can repeat with the weight t +6 diagonal. This proves the Theorem. (cid:3)
4. B
OUNDS FOR WEIGHT MULTIPLICITIES
Theorem 4.1.
Let ρ be an irreducible representation of SL ( Z ) of dimension dim ρ ≥ .Let π ρ = ( m , . . . , m r ) denote the weight multiplicity tuple of ρ , and define m j = 0 if j < or if j > r . Then for all j ≥ we have m j ≤ X t ≥ m j +1 − t and m j ≤ X t ≥ m j − t . In particular, m j ≤ dim ρ for all j .Proof. We first explain how to establish the first inequality more generally for M L ( ρ ) for any choice of exponents L for ρ ( T ) . Choose a basis for M L ( ρ ) as in Theorem3.13, and assume to the contrary that there exists j such that m j > P t ≥ m j +1 − t .Consider the matrix A obtained from the j th block column of the matrix of D in thechosen basis, but where we ignore the blocks that are known to be zero by weightconsiderations. This is a matrix with P t ≥ m j +1 − t rows and m j columns. Thus, byhypothesis A has a nontrivial kernel consisting of a scalar vector. If b = ( b v ) is such acolumn vector, and if F , . . . , F m j denote the basis vectors of weight corresponding tothe multiplicity m j , then F = P m j v =1 b v F v is nonzero and D ( F ) = 0 , contradicting theirreducibility of ρ since dim ρ ≥ (Lemma 3.1).The second inequality can be deduced from the first by duality, since V k ( ρ ) ∨ ∼ = S − k ( ρ ∨ ) , and since Theorem 3.13 and Lemma 3.1 are valid for any choice of expo-nents. (cid:3) Remark . Computational evidence suggests that the stronger three-term inequality m j ≤ m j +1 + m j − might hold. This would follow if one could find a basis for M ( ρ ) such that the matrix of D contains only constants and constant multiples of E . Wewere unable to prove this stronger result, save for under two different hypotheses:(1) If ρ is irreducible and unitarizable, then it’s known (see Section 6 of [2] orthat the weight profile consists only or Lemma 3.12 above) that there are atmost six multiplicities for ρ . In this case the two inequalities of Theorem 4.1yield the three-term inequality m j ≤ m j +1 + m j − for j = 1 , . . . , .(2) If ρ is an irreducible representation and σ is the standard representation of SL ( Z ) , then it’s easy to relate the weight profiles of ρ and ρ ⊗ σ . Since σ ( T ) has all exponents equal to zero, one has V k ( ρ ⊗ σ ) = V k ( ρ ) ⊗ V ( σ ) . In particular, since V ( σ ) = O (1) ⊕ O ( − , if V ( ρ ) = L dr =1 O ( − k r ) , then V ( ρ ⊗ σ ) = d M r =1 O ( − k r + 1) ⊕ O ( − k r − Let m , . . . m t be the multiplicities for ρ . Then the multiplicities for ρ ⊗ σ are m , m + m , . . . , m t − + m t , m t . Thus the three-term inequality for ρ ⊗ σ boilsdown to ≤ m j +1 + m j − , which is trivially satisfied. This is true regardlessof whether the three-term inequality was satisfied by the weight multiplicitiesof ρ . Remark . It is worth remarking that in the case of the standard representation σ of SL ( Z ) , one has(6) V ( σ ) ∼ = O ( − ⊕ O (1) . This reflects the fact that V ( σ ) can be identified with the relative homology of theuniversal elliptic curve over the moduli stack of generalized elliptic curves. A vectorvalued modular form of minimal weight − for σ is given by F ( τ ) = (cid:18) πiτ πi (cid:19) . The decomposition (6) is the Hodge decomposition for the relative homology of theuniversal elliptic curve, and one might ask to what extent such a relationship holdsfor other representations of SL ( Z ) .We end this Section by looking more closely at the bound m j ≤ d/ for weightmultiplicities given in Theorem 4.1, where d = dim ρ . We will show (Lemma 4.6)that if ℓ is the number of distinct weight multiplicities and e the minimum of the(nonnegative) integers [ d/ − m j ( j ≥ , then ℓ/e ≤ . We can be more precise forsmall e . First we treat the case e = 0 , where we show that ℓ ≤ . Lemma 4.4.
Suppose that ρ is irreducible. There are exactly 2 distinct weight multiplic-ities in the weight profile of ρ if, and only if, dim ρ = 2 .Proof. The result is clear if dim ρ = 2 , so assume that m , m are the two weightmultiplicities. Then we must have m = m = d/ , because neither multiplicity mayexceed d/ . Let F , . . . , F d/ be a basis for the vvmfs of least weight k . Then it iseasy to see that that DF , . . . , DF d/ may be chosen as the free generators of weight k = k + 2 , so that we have relations of the form D F j = E P d/ j =1 a ij F i ( a ij ∈ C ) .Let λ be an eigenvalue of the matrix of coefficients ( a ij ) corresponding to anonzero F in the linear span of the F i s. Then we have D F = λE F , so that F satisfies an order 2 MLDE. Therefore dim ρ = 2 because ρ is irreducible. (cid:3) Lemma 4.5.
Suppose that ρ is irreducible and some weight multiplicity in the weightprofile of ρ is d/ . Then either dim ρ = 2 , or the multiplicity profile has the form ( m , d/ , m ) (and in particular, there are just weights).Proof. Let m j = d/ . By the inequality of Theorem 4.1 we have d/ ≤ X t ≥− m j − − t ≤ d/ . N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 17
Therefore, all nonzero weight multiplicities already appear in the displayed inequali-ties. By the no-gap Lemma, there must be either 2 or 3 nonzero multiplicities, and ifthere are 2 then dim ρ = 2 by Lemma 4.4. If there are 3 then we cannot have m = d/ because m ≤ m , and similarly m = d/ is ruled out. Therefore m = d/ , and theLemma is proved. (cid:3) It is evident that the argument of the last Lemma can be systematized. The gen-eral idea is that the inequality of Theorem 4.1 involves mainly multiplicities m j − − t (the point being that the subscripts have the same parity), whereas the no-gap Lemmasays that there must also be (nonzero) multiplicities for the intermediate multiplici-ties m j ′ with j ′ ≡ j (mod ). In the general case we can argue as follows. For eachweight multiplicity m j , define e j := [ d/ − m j ≥ . By Theorem 4.1 we have m j + X t ≥− m j − − t ≥ m j = 2[ d/ − e j . The number of integers j ′ in the range [1 , j − satisfying j ′ ≡ j (mod ) is [ j − / .By the no-gap Lemma we have m j ′ ≥ for these j ′ , whence we obtain d = X j m j ≥ [ j − /
2] + 2[ d/ − e j . This implies that j ≤ e j + 1) . In a nutshell, if we have a multiplicity m j that is ‘not too far’ from d/ (i.e., e j is small)then j must be small too. For example, if some m j = d/ ⇒ e j = 0 ⇒ j ≤ (because d is even), and we easily recover the results of Lemma 4.5 in this case.Let e .. = min j e j be as before, with e = e j . There may be several such j , butthey all satisfy j ≤ e + 1) . By duality, all of these arguments apply to the cuspidalweight profiles too, and we know that in these cases the weight multiplicities arereversed upon passing from ρ to ρ ∨ (cf. Remark 2.3). Moreover, e is the same for ρ and ρ ∨ . Therefore, not only must the minimum discrepancy e occur by the timewe reach the e + 1) th weight multiplicity, the last time the minimum discrepancyoccurs must be within the same distance of the highest weight. Therefore, as thereare exactly ℓ distinct weight multiplicities, then ℓ ≤ e + 7 . We state this as Lemma 4.6.
Let ρ be irreducible and suppose that π ρ = ( m , . . . , m ℓ ) . Let e be theminimum value of [ d/ − m j (1 ≤ j ≤ ℓ ) . Then ℓ ≤ e + 7 . ✷
5. M
ULTIPLICITY TABLES IN LOW DIMENSIONS
The following lists of multiplicity profiles π ρ for irreducible representations ρ of SL ( Z ) were generated by a computer using the results discussed in Section 2, the no-gap lemma (Lemma 3.3), and Theorem 4.1. They contain all multiplicity profiles thatcan arise from irreducible representations in dimensions six through ten, but our lists may include some examples that do not occur in practice . In each dimension there isa unique multiplicity tuple of length d , all of whose entries are one. This correspondsto the case of cyclic ρ (Lemma 3.9). Similarly, in dimension d ≥ there are d − tuples of length d − , all entries of which are one save for a single two (which cannotoccur in the first or last entries). We omit these from our lists in dimension seven andhigher in order to fit the data within the margins.5.1. d = 6 . Total number of types: ≤ . [ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , m , m ][1 , , , , ,
1] [1 , , , ,
1] [1 , , ,
2] [1 , , , , , ,
1] [1 , , ,
1] [2 , , , , , ,
1] [2 , , ,
1] [2 , , d = 7 . Total number of types: ≤ . [ m , . . . , m ] [ m , . . . , m ] [ m , m , m ][1 , , , ,
2] [1 , , ,
2] [1 , , , , , ,
1] [1 , , ,
2] [2 , , , , , ,
1] [1 , , ,
1] [3 , , , , , ,
1] [1 , , , , , , ,
1] [2 , , , , , , d = 8 . Total number of types: ≤ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , m , m ][1 , , , , ,
2] [1 , , , ,
2] [1 , , ,
3] [1 , , , , , , ,
1] [1 , , , ,
2] [1 , , ,
2] [2 , , , , , , ,
1] [1 , , , ,
1] [1 , , ,
2] [2 , , , , , , ,
1] [1 , , , ,
1] [1 , , ,
1] [3 , , , , , , ,
1] [1 , , , ,
2] [2 , , ,
2] [3 , , , , , , ,
1] [1 , , , ,
1] [2 , , , , , , , ,
1] [1 , , , ,
1] [2 , , , , , , , ,
1] [1 , , , ,
1] [3 , , , , , , , , , , , , , , , Since we do not have explicit equations for the character variety of SL ( Z ) in dimensions six orgreater, we do not know that there in fact exist representations ρ of SL ( Z ) having all of the possibleprescribed values for Tr( ρ ( R )) , Tr( ρ ( S )) and Tr( L ) satisfying the obvious constraints. N THE STRUCTURE OF MODULES OF VECTOR VALUED MODULAR FORMS 19 d = 9 . Total number of types: ≤ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , m , m ][1 , , , , , ,
2] [1 , , , , ,
2] [1 , , , ,
3] [1 , , ,
3] [1 , , , , , , , ,
1] [1 , , , , ,
2] [1 , , , ,
2] [1 , , ,
3] [2 , , , , , , , ,
1] [1 , , , , ,
1] [1 , , , ,
2] [1 , , ,
2] [3 , , , , , , , ,
1] [1 , , , , ,
1] [1 , , , ,
1] [1 , , ,
2] [3 , , , , , , , ,
1] [1 , , , , ,
2] [1 , , , ,
2] [1 , , ,
1] [4 , , , , , , , ,
1] [1 , , , , ,
1] [1 , , , ,
2] [1 , , , , , , , , ,
1] [1 , , , , ,
1] [1 , , , ,
1] [2 , , , , , , , , ,
1] [1 , , , , ,
1] [1 , , , ,
1] [2 , , , , , , , , ,
1] [1 , , , , ,
2] [1 , , , ,
1] [2 , , , , , , , , ,
1] [1 , , , , ,
1] [1 , , , ,
1] [2 , , , , , , , , ,
1] [1 , , , , ,
1] [2 , , , ,
2] [3 , , , , , , , , ,
1] [1 , , , , ,
1] [2 , , , ,
1] [3 , , , , , , , , ,
1] [1 , , , , ,
1] [2 , , , , , , , , , ,
1] [1 , , , , ,
1] [2 , , , , , , , , ,
1] [2 , , , , , , , , ,
1] [3 , , , , , , , , , , , , , , d = 10 . Total number of types: ≤ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , . . . , m ] [ m , m , m ][1 , , , , , , ,
2] [1 , , , , , ,
2] [1 , , , , ,
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Int. J. NumberTheory B and of SL(2 , Z ) . Pacific J.Math. , 197(2):491–510 (2001).[11] Bruce Westbury, On the character varieties of the modular group,