MULTIPLIER SYSTEMSPAUL C. PASLES
Abstract. The multiplier systems of complex weight are considered here, generalizing results of Rademacher and Zuckerman, andof Hassen. These are then connected with an earlier result on modular forms. Along the way we also present an upper bound on thenumber of multipliers possible for a given group and weight.

1. Introduction
A multiplier system is a factor associated with a modular form. Even
more generally, it can be defined without recourse to an underlyingform.
A multiplier system on a subgroup of SL (2, R) is a concept analogous to that of a group character, but with a slightly more complicated consistency condition which includes one or two extra parameters(weights). Because of their importance in modular forms theory, theseobjects were studied by Hecke, Rademacher & Zuckerman, and Petersson, among others [8,28,30,31]. The multiplier systems possible forcomplex weight are considered in the present paper. We also present
an upper bound on the number of multipliers possible for a given group
Date: September 28, 2001, revised September 26, 2002.
1991 Mathematics Subject Classification. Primary 11F11; Secondary 11F37.
Key words and phrases. multiplier system, Hecke group, modular form, Poincaré
series.
1

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PAUL C. PASLES

and weight. Finally, we will apply our approach to extend a convergence result in the theory of modular forms.
Of particular interest to us is the Hecke group Gλ = hSλ , T i , λ > 0,
where Sλ =

1 λ
0 1



and T =



0 −1
.
1 0

Multipliers of a single real weight have been characterized for themodular group G1 [31], for two other Hecke groups [11,33] and fordiscrete (and some nondiscrete) Hecke groups [6,7]. We will obtainlike results for the more general scenario: two “coweights” instead of
one weight; neither coweight necessarily real; no technical assumptions
regarding υ (−I); no bound on the multiplier system υ. The last ofthese is motivated by the fact that the traditional restriction |υ| ≡ 1
has been relaxed in some recent applications [15,24,25,26].
To see that our very general definition is a meaningful one requires ademonstration that such multiplier systems exist for any given coweights.
This was shown for the theta group in [24], but since the focus of thecurrent paper will be triangle groups, we present a construction for oneof the latter.
The important role of multiplier systems in analytic number theory
is well-established, with a close connection — via modular forms —to arithmetic functions such as the partition function p (n) and thesums-of-squares enumerator rs (n) [14,2]. Several recent papers have

MULTIPLIER SYSTEMS

3

also examined modular and automorphic forms and integrals of nonreal weight [24,25,26]; their allied multipliers are among those to beconsidered in this paper. In particular, we will describe a relation between four parameters associated with multipliers: α (often called theweight or first coweight ), β (the coweight or second coweight ), λ (the1width), and κ = 12 + 2πilog [−υ (Sλ )] , all of which will be carefully

defined in due course.
Forms exist on the modular group for any complex weight (or pairof coweights), as exemplified by those we will construct in our first
proof. Nonetheless, Poincaré series on that group diverge absolutely
if the coweight-difference is not real, so the usual method of basisconstruction will not work in that situation. The current author proved
this divergence result [25] except in case |υ (S1)| 6= 1; we will prove thatthe extra assumption is unnecessary, i.e. that |υ (S1 )| = 1 if and only
if the coweight-difference is real.

2. Multiplier systems
We begin with a precise definition of the central concept. We willtake a quite general view, essentially in accordance with [24,25], toobtain a result with broad applicability.

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PAUL C. PASLES

Following [20], take arg (cz + d) ∈ (−π, π] for Im z > 0 and c, d ∈ R(c and d not both zero), and arg w ∈ [−π, π) otherwise. This ensures usefully that arg (cz + d) = − arg (cz + d) and log (cz + d) +log (cz + d) ∈ R, thus simplifying many calculations. As usual define logarithms by log w := log |w| + i arg w and exponentiation by
wr = exp {r log w}.
Definition. If G is a group of real 2 × 2 matrices and α, β ∈ C then
υ : G −→ C \ {0} is called a multiplier system of coweights α, β on G
iff
υ (M3 ) (c3 z + d3 )α (c3 z + d3 )β= υ (M1 ) (c1 M2 z + d1 )α (c1M2 z + d1 )β υ (M2) (c2z + d2 )α (c2 z + d2 )β ,whenever Mj =

aj bj
cj dj


, M3 = M1 M2 , for c, d ∈ R2 \ {(0, 0)} andz ∈ H = the upper half-plane {z : Im z > 0} . (The above equation
is known as the consistency condition.)
When α − β ∈ Z, the consistency condition reduces to multiplicativity: υ is then a homomorphism.

Note. Although a two-coweight approach is not necessary in the multiplier system point-of-view, it makes quite a difference in the immediate
application, namely automorphic forms and integrals. A multiplier ofcoweights 3,0 automatically serves as a multiplier of coweights 5,2, itMULTIPLIER SYSTEMS

5

is true; but the same cannot be said of a form of coweights 3,0. Thus
the two-coweight definition is meaningful.
Several textbooks treat multiplier systems of real weight, for example the now-classic [14,19] and the more recent [9]. As examples, wehave the multiplier systems associated with modular forms such as theclassical eta and theta functions. The reader is referred to various
derivations of the eta multiplier in [30,28,14,2] and the theta multiplierin [14].
Ordinarily one includes further restrictions in the definition, such asthat α ∈ R, β = 0, and |υ| ≡ 1 (e.g. [6,21,13,12]), or simply that|υ (Sλ )| = 1 (as in [25]), or even that υ (Sλ) = 1 (see [24,26]). A few
authors further restrict to α ∈ Z, β = 0, and υ ≡ 1 (e.g. [3,32]). Ourdefinition requires none of these assumptions, and to justify it we will
give a relevant construction for arbitrary α, β.
Thus, our results are quite general in that they allow two coweights,
neither necessarily real, and no restriction on υ (−I) nor on |υ|; υ isnot bounded. In each of these ways our definition is more general thanthose presented elsewhere. On the other hand, our results are specificto the Hecke groups, not to other groups whose multiplier systems havebeen studied [1,5,17,18,29,34].

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PAUL C. PASLES

Our first proposition establishes existence of multiplier systems of
any arbitrary complex coweights, for one important group. Simultaneously, the proof indicates why we do not restrict to |υ| ≡ 1.

Proposition 1 (Existence of multipliers). For any α, β ∈ C, there
exists a multiplier system of coweights α, β on the modular group G1 =


1 1
, 01 −100 1

. (It follows immediately that multiplier systems exist on

the infinitely many Hecke groups Gn , n ∈ Z+ .)Proof. Put α0 = α + 2 |bRe αc| , β0 = β + 2 |bRe βc| and let ∆ de12

note the discriminant function ∆ (z) := (2π) e2πiz

Q∞

2πinz 24) ,n=1 (1 − ewhere Im z > 0. In our terminology, ∆ is a modular form of coweights
12, 0 :∆ (−1/z) = z 12∆ (z) , Im z > 0.
Observe that Re α0, Re β0 > 0. It can be proved that for Re γ > 0,
we have ∆γ (−1/z) = e−6πiγ z 12γ ∆γ (z) and ∆γ (1 + z) = e2πiγ ∆γ (z) ,Im z > 0. (See [24] for a similar calculation using the classical theta
function). Thus, if we define fα,β by
fα,β (z) = ∆α0 /12 (z) ∆β0 /12 (z),then we have
fα0 ,β0 (−1/z) = eπi(β0 −α0 )/2fα0 ,β0 (z)MULTIPLIER SYSTEMS

7

andfα0 ,β0 (z + 1) = eπi(α0 −β0 )/6 fα0 ,β0 (z) .

(This relies on the fact that the operator g → g switches and conjugatesthe coweights of a modular form while conjugating its multiplier system[24]). Since fα0 ,β0 is not identically zero (in fact is never zero), it follows
that υ 10 11 = eπi(α0 −β0 )/6 , υ 01 −10 = eπi(β0 −α0 )/2 generate a multipliersystem on G1 of coweights α0, β0 .
It was previously observed in [25] that the consistency condition incoweights α0, β0 is equivalent to the consistency condition in coweights
α0 +2k+w, β0 +2`+w for any k, ` ∈ Z, w ∈ C. Thus, υ just constructed
is also a multiplier system on G1 of coweights α, β (since α−α0 , β −β0 ∈2Z).
To summarize: one can generate a multiplier system of arbitrary
coweights α, β ∈ C on the modular group G1 by υ 10 11 = ξ, υ 01 −10 =ξ −3 , where ξ = exp {πi [α − β + 2 (|bRe αc| − |bRe βc|)] /6} . 

Note. To our knowledge, the oldest reference to the existence of multiplier systems of absolute value not identically one – aside from a brief
mention by Petersson in 1948 [27] – is by D. James in 1973 [10].

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PAUL C. PASLES

3. Special values of υ
Proposition 2. Let υ be a nonvanishing complex function on any real
2 × 2 matrix group G such that υ satisfies the consistency condition incoweights α, β ∈ C. Then:(i) υ (I) = 1.
(ii) If −I ∈ G, then υ (−I) = ± exp {πi (α − β)} .
(A technical condition, namely that υ (−I)=+ exp {πi (α − β)} , is frequently required so that nontrivial forms with that multiplier exist[14,6,9,22,23]. Since the present paper is concerned with multipliersystems themselves, we impose no such restriction.)(iii) If T =with r =

0 −1
1 0





∈ G, then υ (T ) = ±iβ−α+r ,0 if υ (−I) = eiπ(α−β)
1 if υ (−I) = −eiπ(α−β) .

Thus υ (−I) = υ (T )−2 .
(iv) If Sλ =

1 λ
0 1



∈ G, λ ∈ R, define κ by υ (Sλ ) = exp {2πiκ} ,0 ≤ Re κ < 1. Then υ (Snλ ) = υ (Sλ )n = exp {2πinκ} , for all n ∈ Z.
Furthermore, if Sλ, T ∈ G, then:(v) υ T Sλ−1 T = υ (−I)−1 exp {2πi (α − β − κ)}, and(vi) υ (−T Sλ T ) = exp {2πiκ} = υ (Sλ ) .
Proof. Apply the consistency condition to the equations I = I 2, I =(−I)2 , −I = T 2 to get (i) and to see that (i) ⇒ (ii) ⇒ (iii); for detail,MULTIPLIER SYSTEMS

9see [25], where these values were used to prove a convergence result for
Poincaré series. (iv): Apply the analogous condition for n-fold matrix
products, or use induction. (v),(vi): Use the consistency condition for
3- and 4-fold products; for details of (v) in the case α, κ ∈ R, β = 0,see [6]. The coweight-asymmetry (α − β − κ) in (v) comes from theasymmetry in our argument convention, whence (0z − 1)α (0z − 1)β =(−1)α−β . For the remainder of this section we focus on a more specific type ofgroup. For λ > 0, Gλ is topologically discrete if and only if λ > 2 or λ ∈

 

o
n √
√ √
π
1+ 5
1, 2, 2 , 3, . . . = 2 cos| p = 3, 4, 5, . . . . The discretepHecke groups with λ < 2 are called triangle groups [4,16], so named
because in this case the fundamental region for Gλ is a hyperbolic
triangle (Figure 1).

Figure(Left)
D √1.
E A fundamental region in the upper half-plane
1 3
0 −1
for
, corresponding to the case p = 6. Here R =0 1 , 1 0no√z ∈ C : |z| > 1, Re z < 23 , Im z > 0 . (Right) The same hyperbolic triangle represented in the Poincaré disc, via the conformal map
z − 4iz 7−→ i.
z + 4i

10

PAUL C. PASLES

Our next two results connect the four parameters associated withmultiplier systems on triangle groups. The first is a congruence.
Proposition 3. Let υ be a multipliersystem
 of coweights α, β ∈ C onπ
, p ∈ {3, 4, 5, ...} . Define κa Hecke triangle group Gλ , λ = 2 cospby υ (Sλ ) = exp (2πiκ) , 0 ≤ Re κ < 1. Then(p − 2) (α − β) − 4pκ ∈ Z,
and in particular(p − 2) (α − β) − 4pκ ≡ 2 ± p or p ± p (mod 4)≡ 12 {(p + 2) ± (p + 2) ± 2p} (mod 4) .
Proof. Apply the consistency condition p − 1 times to the groupprelation (Sλ T ) = −I, take z = eπi/p and use Proposition 2(iii). (Thepreceding steps mirror a calculation for real weight and bounded υ in[6].) Modulo 4, then,
2 + p if υ (T ) = ieπi(β−α)/2,
2 − p if υ (T ) = −ieπi(β−α)/2 ,(p − 2) (α − β) − 4pκ ≡2p if υ (T ) = −eπi(β−α)/2,



0
else.
This implies the conclusions of Proposition 3. 
A similar group relation holds for certain of the nondiscrete Hecke
groups as well [7], and so these results can be extended slightly if
desired.
Corollary 1. Let Gλ be a Hecke triangle group, and κ, α, β be defined
as before. Thus υ (Sλ ) = e2πiκ with 0 ≤ Re κ < 1, or equivalently,
1log [−υ (Sλ )]. Then κ ∈ R ⇔ α − β ∈ R.
κ = 12 + 2πi
Proof. By Proposition 3, (p − 2) Im (α − β) = 4p Im κ; the corollaryfollows. 
Corollary 2. For fixed complex coweights, an upper bound on thenumber of multiplier systems on a triangle group is given by n (λ, α, β) =4π/ cos−1 (λ/2) .
Proof. By the proof of Proposition 3,
4j + 2 + p
4j + 2 − p(p − 2) (α − β) − 4pκ =4j + 2p
4jif υ (T ) = ieπi(β−α)/2,
if υ (T ) = −ieπi(β−α)/2,
if υ (T ) = −eπi(β−α)/2,else.

MULTIPLIER SYSTEMS

11

Because 0 ≤ Re κ < 1, j lies in a left-open interval of length p. Sincep ∈ Z+ , in each of the four cases above there are at most p possible
values for j, and thus for κ. This gives us a bound of 4p. (For example,
when p = 3, α = 2 + 3i and β = i, the four cases correspond to 4j lyingin((p − 2) Re (α − β) − 5p − 2, (p − 2) Re (α − β) − p − 2] = (−15, −3] ,((p − 2) Re (α − β) − 3p − 2, (p − 2) Re (α − β) + p − 2] = (−9, 3] ,((p − 2) Re (α − β) − 6p, (p − 2) Re (α − β) − 2p] = (−16, −4] ,((p − 2) Re (α − β) − 4p, (p − 2) Re (α − β)] = (−10, 2] ,respectively, so there are at most 12 multipliers on G1 of coweights 2 +3i, i. This is consistent with the results of Rademacher and Zuckermanon the modular group for real weight [31].)
Therefore a bound is given by n (λ, α, β) = 4p = 4π/ cos−1 (λ/2) .
(Since 0 < πp < π2 , λ = 2 cos πp ⇔ p = π/ cos−1 (λ/2) .) Note that thisbound is independent of the coweights. 
Having calculated the possible values υ can take on the generators T
and Sλ of Gλ (by finding υ(T ) in Proposition 2(iii), Im κ in the proofof Corollary 1, and Re κ in the proof of Corollary 2), we are well on theway to a characterization of the multiplier systems on Hecke trianglegroups. What is still needed is a converse of some sort, establishing
which of these possible multipliers actually do satisfy the consistency
condition for all choices of M1 , M2 .
Now for an application of the foregoing results. In [25] it was provedthat under the restriction |υ (S1 )| = 1, Poincaré series on the modulargroup do not converge nicely when α − β ∈/ R. That restriction can
now be removed, and a more general result proved, with the aid of
Proposition 3. This is the essence of our next corollary.
Corollary 3. If α − β ∈/ R, then Petersson’s Poincaré series on themodular group is not absolutely convergent, for any multiplier system.
Thus for complex weight in general it is not possible to build modular
forms in the usual way, even allowing for a broader definition of multiplier than usual. (While modular forms of nonreal coweight-difference
do exist, they cannot be built from a basis of Poincaré series.)
Proof. [25] gives a sufficient condition for Poincaré series on the modular group to fail to converge absolutely, namely that Im (α − β) +−1(2π) log |υ (S2)| 6= 0. (Omitted parentheses have been restored here.)nThis condition allows one to obtain an expression for |υ ((S2 T ) )| thatcauses the absolute subseries on the cyclic subgroup hS2 T i to diverge.

12PAUL C. PASLES

On the modular group we have λ = 1, p = 3. Also, υ (S2 ) = υ (S12) =υ (S1 )2 = e4πiκ , and so
Im (α − β) + (2π)−1 log |υ (S2)|−1= Im (α − β) + (2π) (−4π Im κ)= Im (α − β) − 2 Im κ
= (p − 2) Im (α − β) − 2 Im κ, because p = 3= 4p Im κ − 2 Im κ, by the proof of Corollary 1= 10 Im κ 6= 0, by Corollary 1, since α − β ∈/ R.
Thus the aforementioned sufficient condition is met, and the Poincaré
series does not converge absolutely. (Therefore the usual method of
forming a basis for the space of modular forms will not work for generalmultipliers.) It bears mention that this divergence problem is not remedied by a
Hecke convergence factor. For, a Poincaré series of coweights α, β withconvergence factor |cz + d|−s inserted can simply be reinterpreted as aform of coweights α + s2 , β + 2s ; the coweight-difference is unchanged,
and Corollary 3 still applies.
4. The theta group
What of the remaining discrete Hecke groups? That is, can anything
be said about κ when λ ≥ 2? Lacking a group relation of the type
(SλT )p = −I, we are without recourse to the arguments used in the
proof of Proposition 3. (For λ ≥ 2, the only group relation is T 2 = −I,
which we have already fully utilized in Proposition 2(iii).)
In particular, consider the limiting case: the theta group, G2 . Since
2 = limp→∞ 2 cos πp , we might hope to glean some information about
at least this one specific case from Proposition3 and/or Corollary 1.
2
Intuitively, since by Proposition 3 it is true that 1 − p (α − β)−4κ ∈R for p = 3, 4, 5..., in the limit one might expect likewise that Im κ =Im (α − β) /4 for multiplier systems on G2 . However, in [25] we findfor any α, β ∈ C there exists a multiplier system on G2 with κ = 0.
Thus no result analogous to Proposition 3 and/or Corollary 1 extends
to λ = 2; the multiple-valued function Υ : λ → κ (λ) is not continuous.
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Department of Mathematical Sciences, Villanova University, 800
Lancaster Avenue, Villanova, PA 19085
E-mail address: paul.pasles@villanova.edu URL : http://www.pasles.org