Special values of Hecke L-functions of modular forms

of half-integral weight and cohomology

Winfried Kohnen and Wissam Raji

1. Introduction

Let f be a cusp form of integral weight k, say for simplicity for the full modular groupΓ1 := SL2(Z). For z ∈ H, the complex upper half-plane, one defines the Eichler integralattached to f by

(1.1) Ef (z) := ck

∫ i∞

z

f(w)(w − z)k−2dw

where

(1.2) ck :=(−2πi)k−1

Γ(k − 1)

is a normalization constant.

The value of the integral is independent of the choice of the path from z to i∞ in Hand

1

(2πi)k−1Ef (z)

is a (k − 1)-fold integral of f(z).

We note that

(1.3) Ef (z) =∑

n≥1

a(n)n1−ke2πinz (z ∈ H),

where a(n) (n ≥ 1) are the Fourier coefficients of f . Indeed, the (k − 1)-th derivatives ofboth sides of (1.3) are equal and both sides are periodic with period 1, so they differ by aconstant which must be zero as seen by letting z tend to i∞.

Letrf (A)(z) := (Ef − Ef |2−kA)(z) (A ∈ Γ1, z ∈ H)

where |2−k denotes the usual action of Γ1 on complex valued functions on H in weight2− k. It is easy to see that

rf (A)(z) = ck

∫ i∞

−d/c

f(w)(w − z)k−2dw (A =

(

· ·c d

)

)

1

and hence rf (A) is a polynomial of degree at most k− 2, called a period polynomial. Notethat in the special case

A = S :=

(

0 −11 0

)

the coefficients of rf (A) essentially are the special values of the Hecke L-function Lf (s)attached to f at integral points in the critical strip 0 < σ := ℜ(s) < k.

The period polynomials satisfy certain cocycle conditions and thus one can definea cohomology group of period polynomials (usually called “parabolic cohomology”). Thefamous Eichler-Shimura theorem states that two copies of the space of cusp forms of weightk for Γ1 are isomorphic to the cohomology group of periods. The theory of Eichler-Shimuraplays an important role in the theory of integral weight modular forms, connecting e.g. toelliptic curves, critical values of L-functions and Hecke operators.

In the case where the weight k no longer is an integer, one cannot define Eichlerintegrals as before due to e.g. branching problems. Also the notion of a “(k − 1)-foldintegral” no longer makes sense. To overcome this problem, Knopp [9] replaced w by w inthe expression (w − z)k−2 in (1.1) and restores holomorphicity by complex conjugation ofthe whole integral. In this way he obtains an anti-linear map from the space of cusp formsto the first cohomology group of Γ1 with values in a very large module (of over-countablerank over C), consisting of holomorphic functions on H with polynomial growth whenapproaching zero or i∞. He showed that for real weights k < 0 or k > 2 this map isa bijection and conjectured this to be true for all k. Together with Mawi he proved theconjecture in [10].

All of the above theory generalizes to congruence subgroups Γ ⊂ Γ1. For an interestingfurther development of Knopp’s results we refer to the monograph [3] by Bruggeman, Choieand Diamantis.

It has been known for quite some time, starting with Shimura’s lifting theorem [16]that half-integral weight modular forms also encode deep arithmetic information. It istherefore natural to look in some more detail at the case of cusp forms f of weight k ∈12 + N0. One of the main ideas here (slightly different from the approach in [9]), due toLawrence and Zagier [11] is to study the non-holomorphic Eichler integral

Enhf (z) := ck

∫ i∞

z

f(w)(w − z)k−2dw

where z now is in the lower half-plane H− and ck again is defined by (1.2). (For the choiceof the square roots see “Notations” at the end of this section.)

It is not difficult to see that one has a Fourier expansion

(1.4) Enhf (z) =

∑

n≥1

a(n)n1−ke2πinzΓ(k − 1, 4πn|y|)

Γ(k − 1)(z ∈ H−, y = ℑ(z))

2

where again a(n) (n ≥ 1) are the Fourier coefficients of f and we denote by

Γ(σ, x) :=

∫ ∞

x

tσ−1e−tdt (σ > 0, x > 0)

the incomplete Γ-function.

For properties of the incomplete Γ-function, in particular at the argument σ = 12

which will be frequently used later, we refer the reader to [1, sect. 7]. Since

(1.5) erfc√x =

1√πΓ(

1

2, x) (x > 0)

where erfc denote the complementary error function, the reader may be advised to alsolook at [1, sect. 6] in this connection.

Note that equation (1.4) can be considered as a non-holomorphic analogue of identity(1.3).

In [2] Bringmann and Rolen used the function Enhf (z) to study the asymptotic expan-

sion for t → 0 of certain power series in t whose coefficients are given by special values ofLf (s) (and twists of the latter by additive characters) at half-integral points s = k− 1−nwith n ∈ N0. This had an important implication regarding the construction of quantummodular forms attached to f in the sense of Zagier.

In this paper we would like to start a different approach in developing a cohomologytheory in the case of half-integral weight with an attempt to focus again on the connectionto special values of Lf (s) at half-integral and integral points inside the “critical strip”0 < σ < k, similar as in the case of integral weight. We will work with the Hecke groups

H(λ) ⊂ SL2(R) generated by the translation Tλ :=

(

1 λ0 1

)

acting on H by z 7→ z + λ

and the inversion S (see above) acting by z 7→ − 1z , where λ is a positive real number

satisfying some conditions to be specified later.

We are concerned with holomorphic functions f : H → C such that f(z + λ) = f(z)and f(− 1

z ) = (−iz)kf(z), and such that f has a Fourier expansion

f(z) =∑

n≥1

a(n)e2πinz/λ.

If λ < 2 such an f is a cusp form of weight k on H(λ) in the usual sense, and by thewell-known Hecke argument the coefficients can easily be shown to satisfy the bound

a(n) ≪f nk/2 (n ≥ 1).

If λ ≥ 2, this is not necessarily the case, and we will add the latter growth condition as anadditional requirement. The above growth condition will be a crucial hypothesis for ourlater results. For a further discussion, in particular regarding the case λ = 2, see sect. 2.

3

In the range λ ≥ 2, the case λ = 2√N with N ∈ N is of special importance, since the

space of cusp forms of weight k on the group Γ0(4N) in the sense of Shimura [16], invariantunder the Fricke involution W4N is naturally imbedded into the space of functions f as

above. Here Γ0(M) = {(

a bc d

)

∈ Γ1 |M |c}, for any M ∈ N.

Now let f be as above. Our first result (Theorem 1) gives a functional equation for theshifted L-function φf (s) := Lf (s+k−1) when completed with a Γ-factor and the function

1cos(πs) , under s 7→ 2 − k − s. The proof follows an observation due to Razar [14] in the

integral weight case and is easily deduced from the functional equation of Lf (s), howeverthe appearance of trigonometric functions seems to be special for half-integral weight.

Now define a formal series

(1.6) E∗f (z) :=

∑

n≥1

a(n)n1−kH(2πinz/λ) (z ∈ H−)

where

(1.7) H(z) :=1√π

(

ezΓ(1

2, z)− 1√

z

)

(ℜ(z) > 0)

and Γ( 12 , z) denotes the holomorphic continuation of Γ( 12 , x) (x > 0) to C. Note thatH(z) (ℜ(z) > 0) is a holomorphic function.

The reader may notice the similarity between (1.4) and (1.6). Using the growthcondition for the a(n), one easily sees that for k ≥ 3

2 the series in (1.6) is absolutely locallyuniformly convergent on H− and therefore is a holomorphic function on H−. Using inverseMellin transforms, Theorem 1 and the usual arguments from the proof of Hecke’s conversetheorem we will show (Theorem 2) that

(1.8) E∗f (z)− (iz)k−2E∗

f (−1

z) = Pf (z) +

(2πiz

λ

)−1/2

Qf (z)

where Pf (z) and Qf (z) are polynomials of degree at most k− 32 whose coefficients are essen-

tially given by the special values of Lf (s) at half-integral and integral points, respectivelyin the critical strip.

We shall actually prove statements analogous to Theorems 1 and 2 under a slightlymore general condition, admitting f to have a multiplier C = ±1.

Using a well-known description of H(λ) in terms of generators and relations, we canthen define a cocycle πf of H(λ) with values in a space Wk of complex valued holomorphicfunctions, carrying a natural action of H(λ) in weight 2− k and being isomorphic over Cto countably infinitely many copies of the space Vk−3/2 of complex polynomials of degree

at most k − 32 (Theorems 3 and 4). In particular πf (S) will be given by (1.8).

In this way we will obtain a linear map f 7→ πf from the space of functions f of weightk ≥ 3

2 on H(λ) satisfying the above conditions to the space Z1(H(λ),Wk) of cocycles and

4

to the cohomology groupH1(H(λ),Wk), respectively. We will actually distinguish betweenthe case λ < 2 (Theorem 3) and the case λ ≥ 2 (Theorem 4).

The paper is organized as follows. In sect. 2 we will recall basic facts on modularforms for Hecke groups. In sect. 3 we will state Theorem 1 in detail and give the proof.Sect. 4 contains a detailed statement and the proof of Theorem 2, while in sect. 5 we willdo some technical computations needed later. Finally, sect. 6 and 7 are concerned withthe statements and proofs of Theorems 3 and 4. We will also discuss some open questionswhich naturally appear. In addition, in sect. 8 we will address the case of modular formsof half-integral weight in the sense of Shimura, as briefly mentioned above.

Notations. For z ∈ C we write z = x+ iy. If z 6= 0 we define√z = z1/2 := e

12log z where

log z (z 6= 0) denotes the principal branch of the complex logarithm.

We let H and H− be the upper and lower complex half-plane, respectively. The actionof GL+

2 (R) on H and H−, respectively by fractional linear transformations will be denotedby

z 7→ Az =az + b

cz + d(A =

(

a bc d

)

).

Regarding the incomplete Γ-function we refer the reader to the remarks given above.

For ℓ ∈ N we let Vℓ−1 be the space of complex polynomials of degree not larger thanℓ− 1.

In connection with L-functions depending on the variable s, we write as usual s =σ + it.

The letter λ denotes a positive real number. The letter k always denotes an half-integer, k ∈ 1

2 + Z. We often write k = 12 + ℓ.

2. Basic facts on modular forms on Hecke groups

We let G∧ be the group of all pairs γ = (A, φ(z)) where A =

(

a bc d

)

∈ GL+2 (R) and

φ(z) is a holomorphic function on H such that

φ(z)2 = t(detA)−1/2(cz + d)

with t ∈ C, |t| = 1, and with group law given by

(A, φ(z))(B,ψ(z)) = (AB, φ(Bz)ψ(z)).

Let k ∈ 12 + Z. Then G∧ operates on functions g : H → C “in weight k” by

(g|kγ)(z) = φ(z)−2kg(Az).

5

In a similar way, for purely formal purposes we can define a group G∨ by replacing Hwith H− in the above definition. It operates on functions g : H− → C in the same wayas above, mutatis mutandis. We note that G∧ and G∨ are naturally isomorphic under themap (A, φ(z)) 7→ (A, φ(−z)).

For a detailed discussion of modular forms on Hecke groups we refer e.g. to [13]. Herewe recall only some basic facts.

For λ > 0 we define the Hecke group H(λ) to be the subgroup of SL2(R) generated

by Tλ :=

(

1 λ0 1

)

and S =

(

0 −11 0

)

. It was proved by Hecke that H(λ) is discrete if

and only if λ ≥ 2 or λ = 2 cos(πq ) with q ∈ N, q ≥ 3. It is arithmetic if λ = 1,√2,√3, 2.

Clearly H(1) = Γ1. If λ = 2 cos(πq ) as above, then H(λ) has exactly one cusp, namely i∞and the hyperbolic volume of H(λ)\H is finite. If λ = 2, then H(λ) ⊂ Γ1 is Γ1 -conjugateto Γ0(2) and so there are two cusps (represented by i∞ and say −1), and the volume isfinite again. Finally, if λ > 2 then the hyperbolic volume is infinite.

Fix C = ±1 and define elements in G∧ by

T∧λ := (Tλ, 1), S

∧ := (S,C(−iz)1/2) (z ∈ H).

(We suppress the dependence of S∧ on C in the notation since there will be no danger ofconfusion.) We let H∧(λ,C) be the subgroup of G∧ generated by T∧

λ and S∧. If C = 1 wesimply write H∧(λ). Note that

(S∧)2 = (−E, 1)is in H∧(λ,C).

In a similar way (again for formal reasons), we define

T∨λ := (Tλ, 1), S

∨ := (S,C(iz)1/2) (z ∈ H−)

and let H∨(λ,C) be the subgroup of G∨ generated by T∨λ and S∨. If C = 1 we simply

write H∨(λ). Again(S∨)2 = (−E, 1)

is in H∨(λ,C).

Let k ∈ 12 +Z. We denote byMk(λ,C) the space of holomorphic functions f : H → C

such that

(2.1) f |kγ = f (∀γ ∈ H∧(λ,C))

and f has a Fourier expansion

f(z) =∑

n≥0

a(n)e2πinz/λ.

If C = 1 we write Mk(λ). Note that one has to check (2.1) only for generators.

6

Now suppose that λ < 2. Then the above functions are modular forms of weight k onH(λ) with multiplier C in the usual sense. A fundamental domain for the action of H(λ)on H is given by

Fλ = {z ∈ H | |x| ≤ λ

2, |z| ≥ 1}

and the intersection of Fλ with {z ∈ H | y ≥ c} (c > 0) is compact. If f is in Mk(λ,C) andis cuspidal (so a(0) = 0), then yk/2f(z) is bounded on H and tends to zero for y → ∞,hence by the usual argument one sees that

(2.2) a(n) ≪f nk/2 (n ≥ 1).

If λ ≥ 2, then (2.2) in general is not automatically satisfied. We therefore –for arbitraryλ > 0– make the definition

(2.3) S0k(λ,C) := {f ∈Mk(λ,C) | a(0) = 0, a(n) ≪f n

k/2 (∀n ≥ 1)}.

This is the space we will be working with in the next sections. We will simply write S0k(λ)

if C = 1.

We recall that Mk(λ,C) (resp. S0k(λ,C)) is the space of modular forms (resp. cusp

forms) of weight k for H(λ) with multiplier C if λ < 2. If λ = 2 this in fact is also true.Indeed, the growth condition for the a(n) (even with the exponent k/2 replaced by anyc > 0) implies that f is holomorphic at −1 (see [13, sect. I, pp. 33 ff.]). However, the cusp−1 is k-irregular (recall that k is an half-integer), so f then automatically vanishes at −1.

3. A functional equation for L-functions

Letf(z) =

∑

n≥1

a(n)e2πinz/λ (z ∈ H)

be in S0k(λ,C) (see (2.3)). Then (2.2) holds, hence the Hecke L-function

Lf (s) =∑

n≥1

a(n)n−s

is absolutely convergent for σ > 1 + k2 . Recall that the completed function

Λf (s) :=(2π

λ

)−s

Γ(s)Lf (s)

has holomorphic continuation to the entire complex plane, is bounded in every verticalstrip and satisfies the functional equation

(3.1) Λf (k − s) = CΛf (s).

7

In fact, as follows from the Phragmen-Lindelof principle, for any µ > 0 one has

|Λf (s)| ≪f,σ1,σ2|t|−µ (|t| → ∞)

uniformly in any strip σ1 ≤ σ ≤ σ2 where the constant implied in ≪ depends on f, σ1 andσ2 (cf. e.g. [12, pp. 118-121]).

We defineφf (s) := Lf (s+ k − 1) (s ∈ C).

Note that the general coefficient of the Dirichlet series φf (s) (σ ≫ 1) equals a(n)n1−k. Weset

Φf (s) :=(2π

λ

)−s Γ(s)

cos(πs)φf (s)

which is a meromorphic function on C. Note that by definition

(3.2) Φf (s) :=(2π

λ

)k−1 Γ(s)

Γ(s+ k − 1) cos(πs)Λf (s+ k − 1).

Theorem 1. i) The functional equation

Φf (2− k − s) = CΦf (s)

holds.

ii) The function Φf (s) is holomorphic on C except for possible simple poles at thepoints s = −n with n ∈ N0 and s = 1

2 +n with n ∈ Z, n ≥ 1− ℓ, respectively. The residuesat these points equal

ress=−nΦf (s) =(2π

λ

)n 1

n!Lf (ℓ−

1

2− n)

and

ress= 12+nΦf (s) =

(2π

λ

)−n−1/2

· (−1)n

π· Γ(1

2+ n)Lf (ℓ+ n),

respectively.

Proof. i) Replacing s by 2− k − s in (3.2) we find that

(3.3) Φf (2− k − s) =(2π

λ

)k−1 Γ(2− k − s)

Γ(1− s) cos(π(2− k − s))Λf (1− s).

From the functional equation (3.1) with s replaced by 1− s we see that

Λf (1− s) = CΛf (s+ k − 1).

8

Also by Euler’s reflection principle

Γ(1− s) =π

Γ(s) sin(πs)

andΓ(2− k − s) =

π

Γ(s+ k − 1) sin(π(s+ k − 1)).

Therefore (3.3) says that

(3.4) Φf (2− k − s) = C(2π

λ

)k−1 sin(πs)

cos(π(2− k − s)) sin(π(s+ k − 1))

· Γ(s)

Γ(s+ k − 1)Λf (s+ k − 1).

Bearing in mind that k = 12 + ℓ with ℓ ∈ Z and using standard identities for the functions

sin and cos we see that

cos(π(2− k − s)) = (−1)ℓ+1 sin(πs), sin(π(s+ k − 1)) = (−1)ℓ+1 cos(πs).

Hence from (3.4) using (3.2) again we finally obtain that

Φf (2− k − s) = CΦf (s)

as claimed.

ii) The function φf (s) is entire. Hence possible poles of Φf (s) can arise only from thepoles of Γ(s), i.e. the points s = 0,−1,−2, . . . and the zeros of cos(πs), i.e. the pointss = 1

2 + n with n ∈ Z. Note that these poles and zeros, respectively, are simple.

Since Λf (s) is entire and Γ(s) has poles as given above, we must have Lf (s) = 0 fors = 0,−1,−2, . . ., hence φf (s) = 0 for s = 1

2 − ℓ,− 12 − ℓ, . . . and so the zeros of φf (s) and

cos(πs) at the latter points cancel.

We have

ress=−nΓ(s) =(−1)n

n!(n = 0, 1, . . .)

and

ress= 12+n

1

cos(πs)=

(−1)n

π(n ∈ Z).

Alsocos(−πn) = (−1)n (n = 0, 1, . . .).

From this our assertions follow immediately.

4. A functional equation for period functions

9

Letf(z) =

∑

n≥1

a(n)e2πinz/λ

be in S0k(λ,C) and define as before E∗

f (z) (z ∈ H−) by (1.6) and H(z) (x > 0) by (1.7).

Lemma 1. Suppose that k ≥ 32 . Then the series E∗

f (z) is absolutely locally uniformlyconvergent on H− and therefore is a holomorphic function on H−.

Proof. We recall that [loc. cit.]

√πezerfc

√z =

1√z+O(|z|−3/2) (|z| → ∞, x > 0)

and hence (note (1.5))

(4.1) H(z) =1√π

(√πezerfc

√z − 1√

z

)

= O(|z|−3/2) (|z| → ∞, x > 0).

For z ranging in a compact set K ⊂ H−, using (4.1) we therefore find that

∑

n≥1

|a(n)|n1−k|H(2πinz/λ)| ≪K

∑

n≥1

n1−k/2 · n−3/2

≪K

∑

n≥1

n−k+1

2

<∞,

since by hypothesis k ≥ 32 . This proves our claim.

We now come to the main result of the paper. Recall that we defined (sect. 2)

S∨ = (S,C(iz)1/2) ∈ H∨(λ,C).

Theorem 2. Suppose that k ≥ 32 . Then the identity

(4.2)(

E∗f − CE∗

f |2−kS∨)

(z) = Pf (z) +(2πiz

λ

)−1/2

Qf (z) (z ∈ H−)

holds, where Pf and Qf are polynomials over C of degree at most ℓ− 1 given by

Pf (z) =ℓ−1∑

n=0

Lf (k − (n+ 1))

Γ(n+ 1)

(2πiz

λ

)n

10

and

Qf (z) =ℓ−1∑

n=0

Lf (k − (n+ 12 ))

Γ(n+ 12 )

(2πiz

λ

)n

,

respectively.

Remark. Observe that the coefficients of Pf and Qf essentially are the values of Lf (s) atthe half-integral and integral points inside the critical strip 0 < σ < k.

Proof. The proof follows from Theorem 1 and the well-known arguments in the proof ofthe Hecke converse theorem. However, one carefully has to check the main details whichare bit more involved here.

Because of holomorphic continuation it suffices to show (4.2) for z = iy with y < 0.Changing y to −y we therefore have to show that

(4.3) E∗f (−iy)− Cyk−2E∗

f (−i

y) = Pf (−iy) +

(2πy

λ

)−1/2

Qf (−iy)

for y > 0.

By definition

E∗f (−iy) =

∑

n≥1

a(n)n1−k · 1√π

(

e2πny/λΓ(1

2, 2πny/λ)− 1

√

2πny/λ

)

(y > 0).

We now use inverse Mellin transforms and apply formula (14) on p. 348 in [7] with n = 1in the notation there. We then find that

(4.4)1√π

(

eyΓ(1

2, y)− 1√

y

)

=1

2πi

∫ c+i∞

c−i∞y−s Γ(s)

cos(πs)ds (y > 0)

for any c with 12 < c < 3

2 (note (1.5)). We then deduce from (4.3) and (4.4) in the usualway that

E∗f (−iy) =

∑

n≥1

a(n)n1−k · 1

2πi

∫ c+i∞

c−i∞

(2πny

λ

)−s Γ(s)

cos(πs)ds

=1

2πi

∫ c+i∞

c−i∞y−s ·

(2π

λ

)−s Γ(s)

cos(πs)φf (s)ds,

i.e.

(4.5) E∗f (−iy) =

1

2πi

∫ c+i∞

c−i∞y−sΦf (s)ds.

Because of (2.2), the series φf (s) is absolutely convergent for σ > 2− k2 . Therefore, taking

into account that k ≥ 32 by hypothesis, one sees that the interchange of summation and

integration above is justified if we choose c close to the left of 32 .

11

We now want to shift the line of integration in (4.5) to the left to a line σ = c′,appropriately chosen. We want to do this in such a way that when reflecting back s 7→2− k − s and using the functional equation (Theorem 1, i)), we end up on a vertical lineclose to the left of the line σ = 3

2 , so that we can evaluate the integral again as before. Itis therefore natural to choose c′ close to the right of 1

2 − k = −ℓ.We recall that

Φf (s) :=(2π

λ

)k−1 Γ(s)

Γ(s+ k − 1) cos(πs)Λf (s+ k − 1).

For any µ > 0, the function Λf (s+k− 1) is O(|t|−µ) for |t| → ∞ uniformly in any verticalstrip, cf. sect. 3. Also, by Stirling’s formula

Γ(σ + it) ≍√2π|t|σ−1/2e−

π2|t| (|t| → ∞),

henceΓ(σ + it)

Γ(σ + k − 1 + it)≍ 1

|t|k−1(|t| → ∞).

Furthermore, the identity

4 cos(πs)2 = e2πt + e−2πt + 2 cos(πσ)

≥ e2πt + e−2πt − 2

shows that 1cos(πs) is universally bounded from above when s stays away from the zeros of

cos(πs).

The above arguments show that now one can continue in the usual way, and for thereader’s convenience we shortly give the details.

The values of the integrals of y−sΦf (s) over horizontal lines of bounded lenght of“height” |T | go to zero for |T | → ∞, and hence we find that

(4.6)1

2πi

∫ c′+i∞

c′−i∞y−sΦf (s)ds+

1

2πi

∫ c+i∞

c−i∞y−sΦf (s)ds = R(y)

where R(y) is the sum of the residues of the poles of y−sΦf (s) on the line segment 12 −k <

σ < 32 , i.e. −ℓ < σ < 3

2 .

The second term of the left-hand side of (4.6) is equal to E∗f (−iy). Using the functional

equation and the substitution s 7→ 2 − k − s, we see that the first term of the left-handside of (4.6) equals

C

2πi

∫ c′+i∞

c′−i∞y−sΦf (2− k − s)ds

=−C2πi

∫ c+i∞

c−i∞y−(2−k−s)Φf (s)ds

12

= −Cyk−2 · 1

2πi

∫ c+i∞

c−i∞ysΦf (s)ds.

We therefore conclude that

E∗f (−iy)− Cyk−2E∗

f (−i

y) = R(y).

By Theorem 1, ii) the possible poles of y−sΦf (s) on the line segment −ℓ < σ < 32

occur exactly at the points s = 0,−1, . . . ,−ℓ + 1 and s = 12 ,− 1

2 , . . . ,−ℓ + 12 . We have

corresponding residues

ress=−n y−sΦf (s) = yn

(2π

λ

)n 1

n!Lf (ℓ−

1

2− n) (n = 0, 1, . . . , ℓ− 1)

and

ress= 12+n y

−sΦf (s) = y−n−1/2(2π

λ

)−n− 12 (−1)n

πΓ(

1

2+n)Lf (ℓ+n) (n = 0,−1, . . . , 1−ℓ).

Therefore we find that

(4.6) R(y) =ℓ−1∑

n=0

Lf (ℓ− 12 − n)

n!

(2πy

λ

)n

+(2πy

λ

)−1/2

· 1π

ℓ−1∑

n=0

(−1)nΓ(1

2− n)Lf (ℓ− n)

(2πy

λ

)n

.

Using Euler’s formula

Γ(s)Γ(1− s) =π

sin(πs)

with s = 12 + n once more and observing that sin(π( 12 + n)) = (−1)n we get

Γ(1

2− n) =

(−1)nπ

Γ( 12 + n).

Inserting this into (4.6) we finally find that

R(y) = Pf (−iy) +(2πy

λ

)−1/2

Qf (−iy)

as claimed.

Remark. In the proof of Theorem 2, ii) we may very well choose c′ far left of −ℓ. Inevaluating the integral then we pick up more residues on the left and when reflectings 7→ 2− k− s we are pushed back to lines σ = c1 with c1 much larger than c. To evaluate

13

the corresponding integral we can then use an inverse Mellin formula analogous to (4.4),with c replaced by c1, where on the left there appears a polynomial in 1√

y as a “correcting

term”, rather than just 1√y , cf. formula (14) on p. 348 in [7].

5. Some technical results

In this section we shall prove some technical results needed later. We will alwayssuppose that C = 1.

We remark that if λ < 2, λ = 2 cos(πq ) with q ∈ N, q ≥ 3, then there exists a non-zero

modular form of weight k (half-integral) on H(λ) for some k if and only if q ≥ 10, q ≡ 2(mod 8). Indeed, this follows from a well-known dimension formula, see [13, I-23, Theorem3].

Lemma 2. Let P : G∨ → GL+2 (R) be the natural projection onto the first factor. If

λ < 2, suppose that λ = 2 cos(πq ) with q ≥ 10, q ≡ 2 (mod 8). Then the restriction of P

to H∨(λ) gives an isomorphism of H∨(λ) onto H(λ).

Proof. Let γ =(

(

a bc d

)

, φ(z))

∈ H∨(λ). If φ(z)2 = t|cz + d| with t ∈ T , the circle

group, then t must be a fourth root of unity. Indeed, H∨(λ) is generated by T∨λ = (Tλ, 1)

and S∨ = (S, (iz)1/2), and the assertion is true for the generators by definition, hence musthold for any element in the group.

The kernel of the map P is equal to T , hence we only have to show that H∨(λ) hasno element, other than the identity, of the form (E, t) with t4 = 1. Suppose the contraryis true. By taking a square if necessary we may suppose that (E,−1) ∈ H∨(λ). Recallthat H∨(λ) and H∧(λ) are isomorphic under the map that replaces z by −z in the secondcomponent. It then follows that (E,−1) ∈ H∧(λ). Note that the element (E,−1) acts onfunctions f : H → C by

f 7→ (−1)2kf = −f.Hence it follows that Mk(λ) = {0} (cf. sect. 2) for any k. This is a contradiction. Indeed,if λ ≥ 2 it is well-known that dimMk(λ) > 0 for k > 0 [13, I-13, Theorem 2 and I-40], andif λ < 2 this follows from our hypothesis as explained above. This proves our assertion.

We will denote the inverse of the map of Lemma 2 by A 7→ A∨. This is consistent withearlier notation. Note that we have an induced isomorphism from PH(λ) := H(λ)/{±E}onto PH∨(λ) := H∨(λ)/{(±E, 1)}.

To state the next result, we let

H∞(λ) := {(

a bc d

)

∈ H(λ)) | c = 0}.

Note that H∞(λ) contains the subgroup generated by Tλ. However, in general it seems tobe unknown if both agree. If C∨ = (C, φC(z)) ∈ H∨

∞(λ), then

φC(z) = t ∈ C∗

14

is constant, hence for any A∨ = (A, φA(z)) ∈ H∨(λ) we have

φCA(z) = φC(Az)φA(z)

= tφA(z).

We let Wk be the subspace of the space of complex valued holomorphic functions onH− generated by all the subspaces φ−1

A Vℓ−1 where A∨ = (A, φA(z)) runs over all classesin H∨

∞(λ)\H∨(λ). Note that Wk is well-defined.

Lemma 3. i) The action of H∨(λ) on functions f : H− → C in weight 2 − k leaves Wk

stable. In fact, one has

(5.1)(

(φ−1A P )|2−kB

∨)

(z) = φ−1AB(z)Q(z)

for any P ∈ Vℓ−1 and B ∈ H(λ), where

Q(z) =(

φ2B(z))ℓ−1

P (Bz) ∈ Vℓ−1.

ii) One has a direct sum decomposition

(5.2) Wk =⊕

A∨∈H∨

∞(λ)\H∨(λ)

φ−1A Vℓ−1.

iii) Suppose that F ∈Wk and F |2−kTλ = F , i.e. F is periodic with period λ. Then Fis constant.

Proof. i) It suffices to show (5.1) which directly follows from the definitions. Indeed,

(

(φ−1A P )|2−kB

∨)

(z) = φB(z)2ℓ−3φ−1

A (Bz)P (Bz)

= φ−1B (z)φ−1

A (Bz)φB(z)2ℓ−2P (Bz)

= φ−1AB(z)Q(z).

ii) Suppose we have a relation

(5.3) P (z) +∑

A

φ−1A (z)PA(z) = 0

where the sum runs over a finite number of representatives A of classes of H∞(λ)\H(λ),with A 6∈ H∞(λ) and PA ∈ Vℓ−1, and P ∈ Vℓ−1 corresponds to the trivial class.

15

Fix A0 =

(

· ·c0 d0

)

and write A∨0 = (A0, φ0(z)) where φ0(z)

2 = t0(c0z + d0) and

|t0| = 1. We abbreviate P0 := PA0.

Since A0 6∈ H∞(λ) we have c0 6= 0. Choose a sequence (zν)ν≥1 in H− with lim zν =−d0

c0. Then limφ0(zν) = 0.

On the other hand, if A =

(

· ·c d

)

occurs on the right of (5.3), A is not equivalent

to A0 and A∨ = (A, φA(z)), then

limφA(zν)2 = t

(

c(−d0c0

) + d)2

= t(c0d− d0c)

2

c20

for some t with |t| = 1. If the latter expression would be zero, then

AA−10 =

(

· ·c d

)(

d0 ·−c0 ·

)

=

(

· ·0 ·

)

would be in H∞(λ), contradicting our assumption. Therefore limφA(zν)2 is not zero

and so by an obvious elementary argument there exists a subsequence (zνµ)µ≥1 such that

(φA(zνµ))µ≥1 converges to a non-zero limit.

To treat all the different A as above, at each step we successively choose a subsequenceof the previous sequence. After a finite number of steps, we then arrive at a sequence(wν)ν≥1 in H− with limwν = −d0

c0, and such that limφ0(wν) = 0 and limφA(wν) 6= 0, for

all A as above.

Since P (−d0

c0) and PA(−d0

c0) are finite, we now conclude from (5.3) that P0(−d0

c0) = 0.

Thus we can write

P0(z) = (z +d0c0

)Q0(z)

with Q0(z) a polynomial of degree one unit less than that of P0(z).

We now compute the derivative of φ−10 (z)P0(z). We have

φ−10 (z)P0(z) = φ−1

0 (z)(z +d0c0

)Q0(z)

=1

t0c0φ0(z)Q0(z).

16

Using that

φ′0(z) =t0c0

2φ0(z)

we findd

dz

(

φ−10 (z)P0(z)

)

=1

t0c0

(

φ′0(z)Q0(z) + φ0(z)Q′0(z)

)

=1

t0c0

( t0c02φ0(z)

Q0(z) +t0(cz0 + d0)

φ0(z)Q′

0(z))

=1

φ0(z)

[1

2Q0(z) + (z +

d0c0

)Q′0(z)

]

.

If degQ0 = N and aN is the highest non-vanishing coefficient of Q0, then the highest term

in the polynomial P(1)0 in the square bracket above is aN ( 12 + N) 6= 0. Thus degP

(1)0 =

degQ0 = degP0 − 1.

If we proceed in the same way as above with A0 replaced by any A occurring on theright of (5.3) and take derivatives on both sides of (5.3), then we arrive at an equation

P ′(z) +∑

A

φ−1A (z)P

(1)A (z) = 0,

where P(1)A ∈ Vℓ−1 and degP

(1)A = degPA − 1.

We thus see that we can use an induction argument to conclude that all the terms in(5.3) must be zero. Indeed, assume the contrary and suppose that PA1

, . . . , PAr(r ≥ 1)

are the polynomials on the right of (5.3) of highest degree, say N ≥ 1. Differentiating (5.3)N times we arrive at an equation

(5.4) P (N)(z) +

r∑

ν=1

λAνφ−1Aν

(z) = 0

where λAνare non-zero constants. Arguing exactly as in the beginning of the proof, we

obtain a contradiction from (5.4).

iii) Suppose that F ∈Wk and F |Tλ = F (where we have written | instead of |2−k). LetprB be the projection map onto the B-th summand φ−1

B Vℓ−1 in (5.2). By ii) prB commuteswith the action of Tλ. Hence it suffices to show that if F ∈ φ−1

A Vℓ−1 and F |Tλ = F , thenF is constant if A ∈ H∞(λ) and is zero otherwise. Since F ∈ Vℓ−1 for A ∈ H∞(λ) the first

assertion is obvious. So suppose that A =

(

· ·c d

)

6∈ H∞(λ) and there exists Q ∈ Vℓ−1

such that φ−1A Q has period λ. Then

Q2(z + λ)

c(z + λ) + d=Q2(z)

cz + d,

17

i.e. Q2(z)cz+d is a rational function with period λ, hence must be constant. Since c 6= 0,

counting degrees we deduce that Q = 0. This proves our assertion.

Remark. The proof of Lemma 3, ii) is ad hoc and rather computational. It certainly wouldbe desirable to give a more conceptual proof.

We finally recall the structure of the group H(λ) resp. PH(λ). Suppose first thatλ < 2, λ = 2 cos(πq ) with q ≥ 3. Then PH(λ) ∼= C2 ∗ Cq is isomorphic to the free productof a cyclic group C2 of order 2 and a cyclic group Cq of order q, with C2 generated by Sand Cq generated by STλ. (Here we identify A with −A in SL2(R).) It has a presentation

PH(λ) = 〈S, STλ |S2 = (STλ)q = E〉.

Recall that this means that PH(λ) is isomorphic to the quotient of the free group generatedby S and STλ by the normal closure of the subgroup generated by S2 and (STλ)

q. Forthese facts see e.g. [4,5,15].

Now suppose that λ ≥ 2. Then according to [17] , PH(λ) ∼= C2 ∗C∞ is isomorphic tothe free product of a cyclic group C2 of order 2 generated by S and an infinite cyclic group

C∞ generated by STλ. Note that STλ =

(

0 −11 λ

)

is parabolic if and only if λ = 2. One

has a presentationPH(λ) = 〈S, Tλ |S2 = E〉.

6. The case λ < 2

We will now first suppose that λ < 2, λ = 2 cos(πq ) and q ≥ 10, q ≡ 2 (mod 8), as

explained in the beginning of sect. 5. We will also assume that k ≥ 32 and that C = 1, as

in sect. 5.

Furthermore, here and in the next section, for convenience we define an action ofPH(λ) on functions g : H− → C by setting g|2−kA := g|2−kA

∨, cf. Lemma 2. In additionwe often simply write | instead of |2−k.

Let f ∈ S0k(λ) (see (2.3)). We put

(6.1) πf (S) := E∗f − E∗

f |S

(see (4.2)) and

(6.2) πf (STλ) := 0.

Theorem 3. Under the given hypotheses, the following assertions hold:

i) The rules (6.1) and (6.2) extend uniquely to a cocycle πf of the group PH(λ) withvalues in the space Wk.

18

ii) The mapS0k(λ) → Z1(PH(λ),Wk), f 7→ πf

is a C-linear homomorphism.

Proof. i) Using the presentation of PH(λ) given in sect. 5, we can define πf on generatorsas above and then extend πf to a map from the whole group by induction, requiring thecocycle relations, i.e.

(6.3) πf (AB) = πf (A)|B + πf (B),

provided that (6.1) and (6.2) are consistent with the group relations among the generators.The latter assertion, however is obvious. Indeed, the equation

πf (S2) = πf (S)|S + πf (S)

= 0

immediately follows from the definition. Also one has

πf ((STλ)q) = πf (STλ)|

(

E + STλ + . . .+ (STλ)q−1

)

= 0

by (6.2).

Then πf is a cocycle and the image of πf is contained in Wk. Indeed, by Theorem 2,πf (S) is in Wk, hence the assertion is true for generators and then follows by inductionfor πf (A) with A any element in the group, using (6.3) and Lemma 3, i).

ii) The assertion is clear. Indeed, to check say that πf+g(A) = πf (A) + πg(A) (wheref and g are in S0

k(λ) and A ∈ H(λ)), it is sufficient to do this for a generator A, since bothπf+g and πf + πg are cocycles, and for the latter the claim is obvious.

Remarks. The following questions naturally arise: i) What are the coboundaries? Notethat if πf is a coboundary, so πf (A) = F |(E−A) for all A ∈ H(λ) with some fixed F ∈Wk,then taking A = STλ we necessarily have F |STλ = F . By Lemma 3, i) the projection mapfrom Wk onto its B-th component commutes with the action of STλ, so to find the Fsatisfying the latter condition it is sufficient to look at the single components of Wk.

ii) Is the map f 7→ πf from S0k(λ) to Z1(PH(λ),Wk) and H1(PH(λ),Wk), respec-

tively injective? Note that by Theorem 2 and the definitions (6.1) and (6.2) the firstassertion is equivalent to saying that if all the special values Lf (n) with n integral or half-integral within the critical strip 0 < σ < k are zero, then f must be zero. Observe thatthe number of n as above is equal to 2k − 1 which is not less than dimS0

k(λ) as followsfrom well-known dimension formulas [13, loc. cit.].

iii) Can one give a “nice” description of the image of S0k(λ)?

19

7. The case λ ≥ 2

Now suppose that λ ≥ 2. We will also suppose again that k ≥ 32 .

As described at the end of sect. 5, in this case there is only one relation, namelyS2 = E. For f ∈ S0

k(λ) we can therefore define a cocycle πf with values in Wk by setting

(7.1) πf (S) := E∗f − E∗

f |S

and

(7.2) πf (Tλ) := 0,

and requiring

πf (AB) = πf (A)|B + πf (B),

for A,B ∈ H(λ) as before.

Theorem 4. Under the given hypotheses, the following assertions hold:

i) The rules (7.1) and (7.2) in a unique way define a cocycle πf of the group PH(λ)with values in Wk.

ii) The map

S0k(λ) → Z1(PH(λ),Wk), f 7→ πf

is a C-linear homomorphism.

iii) The only coboundaries in Z1(PH(λ),Wk) are the constant functions.

Proof. The proofs of i) and ii) are as before in the case λ < 2. Regarding iii), supposethat πf (A) = F |(E − A) for all A, with a fixed F in Wk. Taking A = Tλ, we find from(7.2) that F |Tλ = F , and so our assertion follows from Lemma 3, iii).

Of course, one can ask questions ii) and iii) in the Remarks in sect. 6 also in the caseλ = 2. See also sect. 8 for a discussion in this case. (We recall that S0

k(λ) has infinitedimension for λ > 2, cf. [13, I-13, Thm. 2].)

8. Modular forms of half-integral weight in the sense of Shimura

We now look at the special case λ = 2√N (N ∈ N). We denote byMk(Γ0(4N)) (resp.

Sk(Γ0(4N))) the space of modular forms (resp. cusp forms) of weight k on Γ0(4N)) in thesense of Shimura [16]. Recall that Mk(Γ0(4N)) consists of complex valued holomorphicfunctions f on H which satisfy the transformation law

f(az + b

cz + d) = (

c

d)(−4

d)−k(cz + d)kf(z)

20

for all

(

a bc d

)

∈ Γ0(4N) and which are holomorphic at the cusps. Here ( cd ) is a generalized

Legendre symbol. As usual, f is called a cusp form if it vanishes at all cusps.

The Fricke involution W4N given by

f(z) 7→ (−2i√Nz)−kf(− 1

4Nz)

acts on Mk(Γ0(4N)) and leaves cusp forms stable. We denote by M(+)k (Γ0(4N)) (resp.

S(+)k (Γ0(4N))) the (+1)-eigenspace. The following Proposition essentially seems to be

well-known, and we will only give a sketch of the proof. For notation, see sect. 2.

Proposition. Suppose that k ≥ 32 . The following assertions hold:

i) The map

(8.1) f(z) 7→ g(z) := f(z

2√N

) (z ∈ H)

gives a linear embedding of S(+)k (Γ0(4N)) into S0

k(2√N), preserving Hecke L-functions.

ii) If N = 1, the map (8.1) gives an isomorphism between S(+)k (Γ0(4)) and S0

k(2),preserving Hecke L-functions.

Proof. The statement about L-functions is true, by definition.

i) Checking the action of the translation and of W4N , one finds that

g(z + 2√N) = g(z), g(−1

z) = (−iz)kg(z).

Also, since f is a cusp form, f has a Fourier expansion at i∞ with positive terms only,hence the same is true for g. Moreover, the Fourier coefficients of f satisfy the Heckebound ≪f n

k/2 as is well-known. Hence g ∈ S0k(2

√N).

ii) Suppose that λ = 2. The group Γ0(4) has three cusps, represented by i∞, 0 and12 . The cusps i∞ and 0 are permuted by W4, and

12 is k-irregular. One has

dimM(+)k (Γ0(4)) = 1 + [

k

4],

cf. e.g. [8, Propos. 3.5]. (Note that the notation used in [8] is different from ours.) Henceby using explicit Eisenstein series (cf. [6]) one finds that

dimS(+)k (Γ0(4)) = [

k

4].

21

On the other hand, the space Mk(2) (resp. Sk(2)) is equal to the space of modularforms (resp. cusp forms) of weight k for H(2), as explained at the end of sect. 2. On has

dimMk(2) = 1 + [k

4],

cf. [13, I-40, ll. 4-6]. Since (up to multiplication by non-zero scalars) there is exactly onenon-zero Eisenstein series in Mk(2), it follows that

dimS0k(2) = [

k

4].

This proves our assertion.

Remark. We note –as is well-known– that the Proposition gives some interesting finite-dimensional subspaces of Mk(λ) (resp. S

0k(λ)).

Using the Proposition, one can reformulate the assertions of Theorems 1, 2 and 4 formodular forms of half-integral weight of level 4N in the sense of [16], invariant under theFricke involution. The role of the matrix S in Theorems 1 and 4 will then be taken overby the matrix

(

0 −1/2√N

2√N 0

)

.

The details are rather obvious, and we will leave it to the reader to formulate the precisestatements

Remark. Can one give a simple and nice description of the kernel and of the image of the

cocycle map restricted to S(+)k (Γ0(4N)), respectively?

References

[1] M. Abramowitz and I. Stegun: Handbook of mathematical functions. New York: Dover1965

[2] K. Bringmann and L. Rolen: Half-integral weight Eichler integrals and quantum mod-ular forms. Preprint 2017

[3] R. Bruggeman, Y.J. Choie and N. Diamantis: Holomorphic automorphic forms andcohomology, arXiv: 1404.6718 (2017)

[4] B. Chen and P.Z. Yuan: Congruence subgroups of Hecke groups, Acta MathematicaSinica, English series, vol. 25, no. 6. pp. 931-944 (2009)

[5] Y.J. Choie: Hecke operators on rational period functions on the Hecke groups, Resultsin Mathematics vol. 25, 40-49 (1994)

[6] H. Cohen: Sums involving the values at negative integers of L-functions of quadraticcharacters, Math. Ann. 217, 271-285 (1976)

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[7] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi: Table of integral trans-forms, vol. 1. McGraw-Hill, New York: 1954

[8] S. Gun, M. Manickam and B. Ramakrishnan: A canonical subspace of modular formsof half-integral weight, Math. Ann. 347, no. 4, 899-916 (2010)

[9] M. Knopp: Some new results on the Eichler cohomology of automorphic forms, Bull.AMS 80.4, 607-632 (1974)

[10] M. Knopp and H. Mawi: Eichler cohomology theorem for automorphic forms of smallweights, Proc. AMS 138.2, 395-404 (2010)

[11] R. Lawrence and D. Zagier: Modular forms and quantum invariants of 3-manifolds,Sir Michael Atiyah: a great mathematician of the twentieth century, Asian J. Math. 3,no. 1, 93-107 (1999)

[12] T. Miyake: Modular forms. Springer: Berlin Heidelberg 1989

[13] A. Ogg: Modular forms and Dirichlet series. Benjamin: New York 1969

[14] M. Razar: Values of Dirichlet series at integers in the critical strip. In: Modularfunctions of one variable VI (eds: J.-P. Serre and D. Zagier), Lecture Notes in Math. 627,Springer: Berlin 1977

[15] Z. Sarigedik, S. Ikikardes and R. Sahin: Power subgroups of the extended Heckegroups, Miskolc Math. Notes, vol. 16, no. 1, 483-490 (2015)

[16] G. Shimura: On modular forms of half-integral weight, Ann. Math. 97, 440-481 (1973)

[17] O. Yilmaz and N. Cangul: On the groups structure and parabolic points of the Heckegroup H(λ), Proc. Estonian Acad. Sci. Phys. Math., vol. 55 (1), 35-46 (2002)

Winfried Kohnen, Universitat Heidelberg, Mathematisches Institut, INF 205,D-69120 Heidelberg, GermanyE-mail: winfried@mathi.uni-heidelberg.de

Wissam Raji, Dept. of Mathematics, 317 A Bliss Hall, American University of Beirut,Beirut, LebanonE-mail: wissam.raji@gmail.com

23