Preliminary remarks

In the section we state dimension formulas for spaces of elliptic modular forms as explicitly as possible. There are various sources for such formulas, and we comment on this in the respective sections. Sometimes the formulas here will be stated somewhat differently from the form which can be found in the literature, but it will always be merely a simple exercise to pass from one form to the other.

The most general dimension formula is the one for vector valued modular forms on SL(2, Z) of integral and half-integral weight. indeed, all of the dimension formulas in this section can be deduced from the one fot vector valued modular forms. An outline of the method to do this is given in the section concerning the general dimension formula for modular forms on half integral weight. In each of the remaining sections we indicate respectively how to obtain the respective dimension formula from the general half-integral weight one. However, the order of the various sections does not follow the indicated logic. Instead we present first the dimension formulas for spaces of ordinary integral weight modular forms (without any character) and come then to the more delicaste ones of half integral weight or with character. Most important, each formula is represented independently from the others (apart from the remarks concerning a proof). In this way the reader can jump immediately to the desired one and apply it without having to care about the others. Also, each formula is followed by a form wher you can ask interactively for numerical tables (as far as this makes sense).

Special cases of the dimension formula for vector valued modular forms can be found (implicitly) in Hecke’s øuvre (keyword: the decomposition of M_{k}(Γ(l)) into irreducible SL(2, Z)-modules.) However, he had not available th epowerful method of trace formulas to succeed completely. The most general dimension formula for vector valued modular forms was first (please tell us if we are wrong) stated and proved in Sko

Finally, we are aware of the fact that there is a myriad of sources for various kinds of dimension formulas. However, we have by far no overview. We are grateful for any interesting hint, in particular to historical issues like for instance priority questions.

The general dimension formula

Theorem. For all subgroups Γ of SL(2, Z), and all integers k, even if −1 belongs to Γ, one has the following dimension formula

dim M_{k}(Γ) − dim S_{2−k}(Γ) = [SL(2, Z) : Γ] · n(Γ) · \frac{k − 1}{24}

− \frac{1}{4} (\frac{−4}{k − 1}) e_{i}(Γ)

− \frac{1}{3} (\frac{k − 1}{3}) e_{ρ}(Γ)

+ \frac{1}{2}# (Γ\P_{1}(Q))_{reg. for k even} .

The notations are as follows:

• n(Γ) = #{±1} ∩ Γ,

• e_{x}(Γ) is the number of orbits Γz in Γ\H such that the stabilizer of z in Γ does contain matrices different from ±1 (“fixed points of Γ”), and such that z is equivalent to x modulo SL(2, Z),

• the last term denotes the number of all cusps of Γ, if k is even, and it denotes the number of regular cusps Γp in Γ\P_{1}(Q), if k is odd. Recall that Γp is called irregular (with respect to Γ) if the stabilizer Γ_{p} of p is cyclic and generated by an element with trace equal to −2.

For specializing this formula to explicitly given Γ the following remark is often useful:

• e_{i}(Γ) and e_{ρ}(Γ) equal the number of fixed points of the permutation of ({±1}·Γ)\SL(2, Z) induced by right multiplication with S respectively ST,

• e_{i}(Γ) = 0 if −1 ∉ Γ,,

• and Γ\P_{1}(Q) equals the number of cycles of the permutation given by right multiplication with T.on the quotient ({±1} · Γ)\SL(2, Z),

• in particular, if Γ is normal in SL(2, Z), one has

#Γ\P_{1}(Q) = \frac{1}{n} [SL(2, Z) : \{±1\} · Γ],

where n is the order of ΓT in the factor group.

• Also, if Γ is normal, either all cusps are irregular or none. There are essentially three methods to derive such a formula:

• by the Riemann Roch theorem, see ?.

• by an explicit formula for the reproducing kernel function with respect to the Petersson scalat product, see ?, which,however,works only for k ≥ 2.

• by counting fixed points on ???, see ?.

The dimension formula for Γ(l)

Theorem. For all l ≥ 3 and all integers k, one has

dim M_{k}(Γ(l)) − dim S_{2−k}(Γ(l)) = ϕ(l)ψ(l) \frac{l(k − 1) + 6}{24}.

For even k ≥ 2,

dim M_{k}(Γ(1)) = \begin{cases}⌊\frac{k}{12}⌋ & if k ≢ 2 mod 12 \\ ⌊\frac{k}{12}⌋ − 1 & if k ≡ 2 mod 12) \end{cases} ,
dim M_{k}(Γ(2)) = \frac{k}{2} + 1.

For Γ(l) one has (with p, q and p′, q′ coprime respectively):

Index in SL(2, Z)  #SL(2, Z/lZ) = lϕ(l)ψ(l)

Number of cusps \frac{1}{2}ϕ(l)ψ(l) for l ≥ 3, and ϕ(l)ψ(l) for l = 1, 2 Equivalence of cusps \frac{p}{q} ≡ \frac{p′}{q′} mod Γ(l) ⇔

(\begin{matrix}p \\ q \end{matrix}) ≡ ± (\begin{matrix}p′ \\ q′ \end{matrix}) mod l

Elliptic fixed points only for l = 1

F ORM

The dimension formula for Γ_{1}(l)

Theorem. For all l ≥ 5 and all integers k one has

dim M_{k}(Γ_{1}(l)) − dim S_{2−k}(Γ_{1}(l)) = ϕ(l)ψ(l)\frac{k − 1}{24} + \frac{1}{4} ∑_{d|l} ϕ(d)ϕ(l/d).

Moreover, for all integers k dim M_{k}(Γ_{1}(3)) − dim S_{2−k}(Γ_{1}(3)) = ⌊\frac{k}{3}⌋ + 1

dim M_{k}(Γ_{1}(4)) − dim S_{2−k}(Γ_{1}(4)) = ⌊\frac{k}{2}⌋ + 1,

and for all even integers

dim M_{k}(Γ_{1}(1)) − dim S_{2−k}(Γ_{1}(1)) = \begin{cases}⌊\frac{k}{12}⌋ & if k ≢ 2 mod 12 \\ ⌊\frac{k}{12}⌋ − 1 & if k ≡ 2 mod 12) \end{cases} ,

dim M_{k}(Γ_{1}(2)) − dim S_{2−k}(Γ_{1}(2)) = ⌊\frac{k}{4}⌋ + 1
Index in SL(2, Z)  ϕ(l)ψ(l)

Number of cusps 1 for l = 1, 2 for l = 2, 3 for l = 4, \frac{1}{2} ∑_{d|l} ϕ(d)ϕ(l/d) otherwise

Equivalence of cusps \frac{p}{q} ≡ \frac{p′}{q′} mod Γ_{1}(l) ⇔

∃ε = ±1 : q ≡ εq′ mod l & p ≡ εp′ mod (q, l)

Irregular cusps 1 for l = 4 (e.g. \frac{1}{2})

Elliptic fixed points i, e^{2πi/3} if l = 1, \frac{1+i}{2} if l = 2

\frac{3+\sqrt{−3}}{6} if l = 3
none for l ≥ 4

F ORM

The dimension formula for Γ_{0}(l)

Theorem. For all l ≥ 1 and all even integers k one has

dim M_{k}(Γ_{0}(l)) − dim S_{2−k}Γ_{0}(l))

= ∑_{d} (d \frac{k − 1}{12} − \frac{1}{4} (\frac{−4}{d}) (\frac{−4}{k − 1}) − \frac{1}{3} (\frac{d}{3}) (\frac{k − 1}{3}) + \frac{1}{2}Q(d)) .

Here d runs through all (positive) divisors of l such that l/d is squarefree.

For the group Γ_{0}(l) one has the following data (with d as in the theorem, and p, q and p′, q′ coprime respectively):

Index in SL(2, Z)  ∑_{d} d
Number of cusps ∑^{d} Q(d)
Equivalence of cusps \frac{p}{q} ≡ \frac{p′}{q′} mod Γ_{0}(l) ⇔
d := (q, l) = (q′, l) & p\frac{q}{d} ≡ p′ \frac{q′}{d} mod (d, \frac{l}{d})

F ORM

Dimension formula for vector valued modular forms

Let DSL(2, Z) be the ususal double cover of SL(2, Z). It is defined as follows: the group elements are the pairs (A, w), where A is a matrix in SL(2, Z) and w is a holomorphic function on H satisfying w^{2}(τ) = cτ + d with c, d the lower row of A. The multiplication of two such pairs is defined by

(A, w(τ)) · (A′, w′(τ)) = (AA′, w(A′τ) · w′(τ)).

For any k ∈ \frac{1}{2}Z we have an action of DSL(2, Z) on functions f : H → ^{n} given by

(f|_{k}(A, w))(τ) = f(Aτ) w(τ)^{−2k}.

Note that for integral k this action factors to an action of SL(2, Z), which is nothing else than the usual “|_{k}”-action of SL(2, Z) given by (f|_{k}A)(τ) = f(Aτ)(cτ + d)^{−k}.

Let ρ : DSL(2, Z) → GL(n,) be a representation with finite image. We let M_{k}(ρ) be the space of all ^{n}-valued holomorphic functions f on H which satisfy

f(Az) = ρ(A, w)w(z)^{2k}f(z)

for all (A, w) in DSL(2, Z), and which are bounded in any region Im(z) ≥ r > 0. The subspace of such f which satisfy even f(z) = o(1) as Im(z) tends to infinity, is denoted by S_{k}(ρ) (and called subspace of “cusp forms”).

Note that M_{k}(ρ) is trivial unless ρ is trivial on the center of DSL(2, Z).

Theorem. Then, for any ρ : DSL(2, Z) → GL(n,) with finite image and trivial on the center of DSL(2, Z), and for any k ∈ \frac{1}{2}Z, one has the formula

dim M_{k}(ρ) − dim S_{2−k}(\overline{ρ}) = \frac{k − 1}{12} · dim ρ + \frac{1}{4}Re(e^{πik/2}trρ((S, \sqrt{τ})))

+ \frac{2}{3\sqrt{3}}Re(e^{πi(2k+1)/6}trρ((ST, \sqrt{τ + 1})))

− ∑_{j=1}^{n} B_{1}(λ_{j}). (1)

Here the λ_{j} (1 ≤ j ≤ n) are (rational) numbers such that e^{2πiλ_{j}} runs through the eigenvalues of ρ((T, 1)), and we use B_{1}(x) = x′ − 1/2 if x ∈ x′ + Z with 0 ≤ x′ < 1. Moreover, for τ ∈ H, we use \sqrt{τ} and \sqrt{τ + 1} for those square roots which have positive real parts.

======REM=== If ρ is a representation of SL(2, Z) and k is integral we use M_{k}(ρ) for M_{k}(ρ ∘ π), where π is the projection of DSL(2, Z) onto the first component. Clearly, in this case the transformation law for the functions F of M_{k}(ρ) is equivalent to F |_{k}A = ρ(A)F for all A ∈ SL(2, Z). In general, if k is integral, the group DSL(2, Z) may be replaced by SL(2, Z) in all of the following considerations.

======Proof sketch

Clearly, if the image of ρ is finite, i.ei̇f the kernel of ρ is of finite index in DSL(2, Z) then the components of an F in M_{k}(ρ) are modular forms of weight k on this kernel. In particular, the space M_{k}(ρ) is then finite dimensional. Formulas for the dimension of these spaces can be obtained as follows: Let V be the complex vector space of row vectors of length n = dim ρ, equipped with the DSL(2, Z)-right action (z, α) ↦ zρ(α). The space M_{k}(ρ) can then be identified with the space Hom_{DSL(2,Z)}(V, M_{k}(Δ)) of DSL(2, Z)- homomorphisms from V to M_{k}(Δ), where Δ = ker ρ, via the correspondence

M_{k}(ρ) ∋ F ↦ the map which associates z ∈ V to z · F ∈ M_{k}(Δ).

By orthogonality of group characters the dimension of Hom_{DSL(2,Z)}(V, M_{k}(Δ)) can be expressed in terms of the traces of the endomorphisms defined by the action of elements of DSL(2, Z) on M_{k}(Δ). These traces in turn can be explicitly computed by using the Eichler-Selberg trace formula. In this way one can derive the following theorem (cf ṡkoruppa, pp˙100 for a complete proof):

For k ≥ 2 the theorem gives an explicit formula for dim M_{k}(ρ) since in this case dim(S_{2−k}(ρ)) = 0 (the components of a vector valued modular form are ordinary modular forms on ker ρ, and there exist no nonzero modular forms of negative weight and no cusp forms of weight 0).

For k = 1/2, 3/2 and ker(ρ) ⊃ Γ(4m)^{♯} it is still possible to give an explicit formula for M_{k}(ρ) skoruppa. However, we do not need those dimension formulas in full generality but need only the following consequence of them:

Supplement to the dimension formula skoruppa: Let ρ : DSL(2, Z) → GL(n,) be an irreducible representation with Γ(4m)^{♯} ⊂ ker(ρ) for some integer m. Then one has dim(M_{1/2}(ρ)) = 0, 1.

A complete list of all those representations ρ for which dim(M_{k}(ρ)) = 1 can be found in skoruppa.

A proof of this supplement can be found in skoruppa. It uses a theorem of Serre-Stark describing explicitly the modular forms of weight 1/2 on congruence subgroups.

Γ_{0}(l) with Nebentypus

Theorem. For all l ≥ 1, all integers k, and for all Dirichlet characters χ modulo l such that χ(−1) = (−1)^{k}, one has

dim M_{k}(Γ_{0}(l), χ) − dim S_{k−2}(Γ_{0}(l), χ)

= ∑_{d} (\frac{k − 1}{12} d − \frac{1}{4} (\frac{−4}{k − 1}) (\frac{−4}{d}) e_{4}(χ) − \frac{1}{3} (\frac{k − 1}{3}) (\frac{d}{3}) e_{3}(χ)

+ \frac{1}{2}ggt(Q(d), \frac{l}{f})).

The sum is over all divisors d of l such that l/d is squarefree, and f is the conductor of f. Let χ = ∏_{p|l} χ_{p} be the decomposition of χ into local components. Then e_{4}(χ) = 0 unless all χ_{p} are even; otherwise e_{4}(χ) = +1 or e_{4}(χ) = −1 acordingly as the restriction of χ to the subgroup of elements of order 4 in the group of all primitive residue classes modul l is trivial or not. Finally, e_{3}(χ) = (−\frac{1}{2})^{h}, where h is the number of prime divisors p of l such that the restriction of χ_{p} to the subgroup of elements of order 3 in the group of primitive residue classes modulo p^{α} is nontrivial, where p^{α} is the exact power of p dividing l.

This formula can be deduced from the dimension formula for vector valued modular forms:

Let ω the character of the SL(2, Z)-representation induced from the character χ on Γ_{0}(l). One has

ω(A) = ∑_{B} χ(BAB^{−1}),

where B runs through a set of representatives for Γ_{0}(l)\SL(2, Z) such that BAB^{−1} belongs to Γ_{0}(l). By a simple calculation

ω(S) = ∑_{xmodl, x^{2}+1=0} χ(x),

with x running through all residue classes modulo l such that x^{2} + 1 = 0. Now the last sum is easily seen to be the product of all the local contributions

ω_{p}(S) = ∑_{xmodp^{α}, x^{2}+1=0} χ_{p}(x),

with p^{α} running through the prime powers dividing l. But then it is easily verified that

ω_{p}(S) = ((\frac{−4}{p^{α}}) + (\frac{−4}{p^{α−1}} )) e_{4}(χ_{p}).

Similarily, the term including e_{3}(χ) comes from the contribution of ST and the formula

ω(ST) = ∑_{xmodl, x^{2}+x+1=0} χ(x).

Finally, we have to analyse the action of T on

V = _{χ} ⊗_{[Γ_{0}(l)]} [SL(2, Z)].

Dimension formulas for half-integral weight

Let c(A, z) be a cocyle of a subgroup Γ of SL(2, Z), i.e. c(A, Bz)c(B, z) = c(AB, z) for all A and B in Γ, and such that c(A, z)| = κ(A)(cz + d)^{k} for suitable constants κ(A) and a suitable k.

Let M_{k}(Γ, c) the space of modular forms on Γ with respect to c. Recall that this is the space of all holomorphic functions f on H such that f(Az) = c(A, z)f(z) for all A ∈ Γ, and such that for all A ∈ SL(2, Z) the function f(Az)J(A, z)^{−2k} remains bounded when z tends to i∞ in a vertical strip.

We can assume that c(−1, z) = 1, if Γ contains −1 (otherwise M_{k}(Γ, c) is obviously trivial). And we assume also that

XXXXXXXXXXXXXXX finite image XXXXXXXXXXXXXX Then

dim M_{k}(Γ, c) = [SL(2, Z) : Γ] \frac{k − 1}{12}

− \frac{1}{4} ∑_{ASA^{−1}∈Γ} Re(e^{πik/2}κ(ASA^{−1}))

− \frac{2}{3\sqrt{3}} ∑_{AST A^{−1}∈Γ} Re(e^{π(2k+1)/6}κ(AST A^{−1}))

− ∑_{p} B_{1}(\frac{1}{2πi} log κ(AT^{n_{p}}A^{−1})).

The first two sums are over a complete set of representatives A for Γ\SL(2, Z) such that the given conditions are respectively fullfilled. The last one runs over a set of representatives p for Γ\P_{1}(Q). For each such p we use n_{p} for the smallest positive integer such that AT^{n_{p}}A^{−1} is in Γ.

Moreover we use

Proof: We can assume −1 ∈ Γ bu extending c(−A, z) = c(A, z) if nexessary.

Set, for (A, w) ∈ DΓ,

χ(A, w) := \frac{c(A, z)}{w(z)^{2k}}.

This defines a character of DΓ. Let ρ be the associated induced representation of SL(2, Z). Since c(±1, z) = 1, we have ρ(±1, w) = w^{−2k} · 1 on the center of DSL(2, Z), and since

Hence we can apply the formula for vector valued modular forms. The S-term equals

∑_{ASA^{−1}∈Γ} χ(A^{∗}S^{∗}A^{∗}^{−1})

(A^{∗} := (A, J(A, z))) But

A^{∗}M^{∗}A^{∗}^{−1} = (AMA^{−1}, sJ(AMA^{−1})),

where

s = ε(M)^{2k}/ε(AMA^{−1})^{2k},

where ε(M) = −1 if c < 0 and tr(A) < 2, and ε(M) = +1 otherwise (see ???). It is easily checked that, for all A, one has s = 1 if M = S or M = ST.

Finally, we have to analyse the action of T^{∗} on

V = _{χ} ⊗_{[DΓ]} [DSL(2, Z)].

For each cuso p write p = A_{p}∞ with a suitable A in SL(2, Z). Then A_{p}T, AT^{2}, ⋯, A_{p}T^{n_{p}} runs through a complete set of representatives for Γ\SL(2, Z). Hence we may take the 1 ⊗ A^{∗}T^{∗}^{2} as basis of V. With respect to this basis the characteristic polynomial f of T^{∗} is easily calculated, namely

f(x) = ∏_{p}(x^{n_{p}} − χ(A^{∗}T^{n_{p}}A^{∗}^{−1}))

From this we obtain as parabolic contribution

∑_{p} ∑_{a=1}^{n_{p}} B_{1}(\frac{1}{2πi} log χ(AT^{n_{p}}A^{−1})^{1/n_{p}} + \frac{a}{n_{p}})) = ∑_{p} B_{1}(\frac{1}{2πi} log χ(AT^{n_{p}}A^{−1})).

But, as before we find χ(AT^{n_{p}}A^{−1})) = κ(AT^{n_{p}}A^{−1}), and hence the desired formula.

Dimension formulas for weight 1