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  • 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 group Γ1 := SL2(Z). For z ∈ H, the complex upper half-plane, one defines the Eichler integral attached to f by

    (1.1) Ef (z) := ck ∫ i∞


    f(w)(w − z)k−2dw


    (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 H and


    (2πi)k−1 Ef (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 of both sides of (1.3) are equal and both sides are periodic with period 1, so they differ by a constant which must be zero as seen by letting z tend to i∞.

    Let rf (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 weight 2− k. It is easy to see that

    rf (A)(z) = ck

    ∫ i∞

    −d/c f(w)(w − z)k−2dw (A =


    · · c d




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

    A = S :=


    0 −1 1 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 define a cohomology group of period polynomials (usually called “parabolic cohomology”). The famous Eichler-Shimura theorem states that two copies of the space of cusp forms of weight k for Γ1 are isomorphic to the cohomology group of periods. The theory of Eichler-Shimura plays an important role in the theory of integral weight modular forms, connecting e.g. to elliptic curves, critical values of L-functions and Hecke operators.

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

    All of the above theory generalizes to congruence subgroups Γ ⊂ Γ1. For an interesting further development of Knopp’s results we refer to the monograph [3] by Bruggeman, Choie and 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 is therefore natural to look in some more detail at the case of cusp forms f of weight k ∈ 1 2 + N0. One of the main ideas here (slightly different from the approach in [9]), due to Lawrence and Zagier [11] is to study the non-holomorphic Eichler integral

    Enhf (z) := ck ∫ i∞


    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 choice of 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−ke2πinz

    Γ(k − 1, 4πn|y|) Γ(k − 1) (z ∈ H−, y = ℑ(z))


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

    Γ(σ, 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√ π Γ(


    2 , x) (x > 0)

    where erfc denote the complementary error function, the reader may be advised to also look 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 of Lf (s) (and twists of the latter by additive characters) at half-integral points s = k− 1−n with n ∈ N0. This had an important implication regarding the construction of quantum modular forms attached to f in the sense of Zagier.

    In this paper we would like to start a different approach in developing a cohomology theory in the case of half-integral weight with an attempt to focus again on the connection to 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 λ 0 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(− 1z ) = (−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 the well-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 an additional requirement. The above growth condition will be a crucial hypothesis for our later results. For a further discussion, in particular regarding the case λ = 2, see sect. 2.


  • 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], invariant under the Fricke involution W4N is naturally imbedded into the space of functions f as

    above. Here Γ0(M) = { (

    a b c d


    ∈ Γ1 |M |c}, for any M ∈ N.

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

    1 cos(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), however the 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−)


    (1.7) H(z) := 1√ π


    ezΓ( 1

    2 , z)− 1√



    (ℜ(z) > 0)

    and Γ( 12 , z) denotes the holomorphic continuation of Γ( 1 2 , x) (x > 0) to C. Note that

    H(z) (ℜ(z) > 0) is a holomorphic function. The reader may notice the similarity between (1.4) and (1.6). Using the growth

    condition for the a(n), one easily sees that for k ≥ 32 the series in (1.6) is absolutely locally uniformly convergent on H− and therefore is a holomorphic function on H−. Using inverse Mellin transforms, Theorem 1 and the usual arguments from the proof of Hecke’s converse theorem we will show (Theorem 2) that

    (1.8) E∗f (z)− (iz)k−2E∗f (− 1

    z ) = Pf (z) +



    )−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, respectively in the critical strip.

    We shall actually prove statements analogous to Theorems 1 and 2 under a slightly more 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 can then define a cocycle πf of H(λ) with values in a space Wk of complex valued holomorphic functions, carrying a natural action of H(λ) in weight 2− k and being isomorphic over C to 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 weight k ≥ 32 on H(λ) satisfying the above conditions to the space Z1(H(λ),Wk) of cocycles and


  • to the cohomology groupH1(H(λ),Wk), respectively. We will actually distinguish between the 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 modular forms for Hecke groups. In sect. 3 we will state Theorem 1 in detail and give the proof. Sect. 4 contains a detailed stat