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formerly Central European Journal of Mathematics

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Volume 16, Issue 1


Volume 13 (2015)

On the algebraicity of coefficients of half-integral weight mock modular forms

SoYoung Choi
  • Department of Mathematics Education and RINS, Gyeongsang National University, 501 Jinjudae-ro, Jinju, 52828, South Korea
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/ Chang Heon Kim
Published Online: 2018-11-15 | DOI: https://doi.org/10.1515/math-2018-0112


Extending works of Ono and Boylan to the half-integral weight case, we relate the algebraicity of Fourier coefficients of half-integral weight mock modular forms to the vanishing of Fourier coefficients of their shadows.

Keywords: Weakly holomorphic modular forms

MSC 2010: 11F11; 11F67; 11F37

1 Introduction and statement of results

Let k be an integer greater than 1 and let N be a positive integer. The space of cusp forms of weight 2k for Γ0(N) is denoted by S2k(N). Throughout this paper let p =1 or a prime number. For κZ+12 we denote by Mκ!(Γ0(4p)) the space of weakly holomorphic modular forms of weight κ on Γ0(4p) . As usual, Mκ0 (4p)) (resp. Sκ(Γ0(4p))) stands for the space of weight κ modular forms (resp. cusp forms) on Γ0(4p) . Let Hκ0(4p)) be the space of weight κ harmonic weak Maass forms on Γ0(4p) . Let Mκ!(p) (resp. ℍ2−κ(p) ) denote the subspace of Mκ!(Γ0(4p)) (resp. H2κ(Γ0(4p))), in which each form satisfies Kohnen’s plus space condition, that is, its Fourier expansion is supported only on those n∈ ℤ for which (1)κ12n(mod4p). Let κ=k+12 and 𝕄κ(p) (resp. 𝕊κ(p) ) denote the subspace of Mκ0 (4p)) (resp. Sκ0 (4 p)) ), in which each form satisfies Kohnen’s plus space condition.

Let Δ(τ) ∈ S 12 (1) be the Ramanujan’s Delta function. The famous Lehmer’s conjecture states that the Fourier coefficients of Δ(τ) never vanish. Concerning this conjecture, Ono [1] related the algebraicity of Fourier coefficients of weight −10 mock modular form whose shadow is Δ(τ) to the vanishing of Fourier coefficients of Δ(τ) . Generalizing Ono’s results, Boylan [2] related the algebraicity of Fourier coefficients of weight 2 − 2k mock modular forms to the vanishing of Fourier coefficients of their shadows when dim S2k (1) =1 . In this paper we will extend their works to the half-integral weight case. In the following we recall some known facts.

Fact 1

By Shimura correspondence [3, Proposition 1] we have


which implies that


Now we assume that k ∈ {6,8,9,10,11,13} . For κ > 2 there is an antilinear differential operator ξ2κ:H2κ(Γ0(4p))Sκ(Γ0(4p)) defined by


Fact 2

For k ∈ {6,8,9,10,11,13}, it follows from [4, Theorem 1.1-(iii) and Lemma 4.2-(c)] that


For any κZ+12, the Duke-Jenkins basis [5] for Mκ!:=Mκ!(1) is constructed as follows. Let 2κ − 1 = 12κ + k′ with uniquely determined κ ∈ ℤ and k′∈ {0, 4,6,8,10,14}. If Aκ denotes the maximal order of a non-zero fMκ! at i∞ , then by the Shimura correspondence [3] one has

Aκ=2κ(1)κ1/2ifκis odd,2κotherwise.(1)

A basis for Mκ! then consists of functions of the form


where mAκ satisfies (1)κ3/2m0,1(mod4). Using (1) and (2) we deduce the following facts.

Fact 3

For κ=32k with k ∈ {6,8,9,10,11,13}, the maximal order Aκ of a non-zero fMκ! at iis given by Aκ = − 4 . Thus for each m ≥ 4 satisfying (−1)km ≡ 0,1(mod 4) , there exist unique modular forms f32k,m(τ)M32k! with Fourier development


which form a basis for the space M32k!.

Fact 4

For κ=k+12 with k ∈ {6,8,9,10,11,13}, the space 𝕊κ is spanned by


where α is given by

α=αk=Aκ=Ak+12=1, k even, 3, k odd, 

and fk has the form qα+O(q4).

Fact 5

Let κ=k+12 and f(z)Hκ(Γ0(4p)) with Fourier expansion f(τ)=nZc(y;n)e2πinx where τ = x + iy . For each prime l with gcd(l, 4 p) =1, the l2 -th Hecke operator is defined by


Then for each ℳ∈ ℍ2−κ (p) , we obtain from [6, (2.6)] or [7, (7.2)] that for κ > 2,


As a corollary of Fact 5, one has that if f(z)=n1(1)kn0,1(4)af(n)qnSk+12, then


(or see [3, Theorem 1-(i)].)

Let (V,Q) be a non-degenerate rational quadratic space of signature (b+ ,b) and L an even lattice with dual L′ . Denote the standard basis elements of the group algebra ℂ[L′ / L] by 𝔢γ for γL′ / L . Let Mp 2 (ℤ) denote the integral metaplectic group, which consists of pairs (γ ,ϕ) , where γ = (abcd) ∈ SL2(ℤ) and ϕ:HC is a holomorphic function with ϕ(τ)2 = + d . It is well known that Mp2(ℤ) is generated by S=0110,T and T=1101,1.Then there is a unitary representation ρL of the group Mp 2(ℤ) on ℂ[L′ / L] , the so-called Weil representation, which is defined by


where e(z) := e2πiz and (X,Y):=Q(X+Y)Q(X)Q(Y) is the associated bilinear form. One has the relations


from which we note that


We write < ⋅,⋅ > for the standard scalar product on ℂ[L′ / L] , i.e.


For γ, δL′ / L and (M,ϕ )∈ Mp 2 (ℤ) the coefficient ργδ(M, ϕ) of the representation ρL is defined by


Following [9], for an integer r we denote by Hr+1/2,ρL(resp.Mr+1/2,ρL)!, the space of ℂ[L′ / L] -valued harmonic weak Maass forms (resp. weakly holomorphic modular forms) of weight r +1/ 2 and type ρL .

Let Lr be the lattice 2 pℤ of signature (1, 0) (resp. (0,1) ) when r is even (resp. odd) equipped with the quadratic form Qr(x)=(1)rx2/4p. Then its dual lattice Lr is equal to ℤ . For a vector valued modular form F=γFγeγ, we define a map Φ by


It then follows from [8, Theorem 1] that the map Φ defines an isomorphism from Hr+1/2,ρLr to Hr+1/2(p) since ρ¯Lr=ρLr+1.

For a ℂ[L′ / L] -valued function f and (M,ϕ) ∈ Mp2 (ℤ) we define the Petersson slash operator by


Let L := Lk and Q := Qk . Following [9] we define the vector valued cuspidal Poincaré series Pβ,nL(τ) as follows: for each β ∈ ℤ / 2 pℤ and n ∈ ℤ + Q(β) with n > 0 ,


Then we know from [9] that Pβ,nL(τ) belongs to the space Sκ,ρL. Let Dk denote the set of all integers D such that (−1)k D > 0 and D is congruent to a square modulo 4 p .

Theorem 1.1

([4, Theorem 1.1]). For an integer k > 2 we let κ=k+12 and L := Lk . For each DDk, we define


where β is an integer such that Dβ2 (mod 4 p) . Then the following assertions are true.

  1. (i) PD+Sκ(p) and the defintion of PD+ does not depend on the choice of β .

  2. (ii) For each f=n1af(n)qnSκ(p) , we have


where (⋅,⋅) denotes the Petersson inner product and

ck,D=(4π|D|)κ1Γ(κ1)s(D)3, if p = 1 (4π|D|)κ1Γ(κ1)s(D)4, if p = 2 (4π|D|)κ1Γ(κ1)s(D)6, if p > 2

with s(D)=1, if p|D,2, otherwise. 

(iii) The set {PD+DDk} spans the space 𝕊κ(p) . Moreover, if we let t := dim 𝕊κ( p) and { f1 , f2 , , ft }be a basis for 𝕊κ(p) satisfying fi=q|Di|+O(q|Di|+1) for some DiDk(i=1,,t) and 0 <| D1 |<| D2 |< <| Dt | , then the set


forms a basis for 𝕊κ(p) .

(iv) Let I be a nonempty finite subset of ℕ . Then the following two conditions are equivalent.

  1. (a) iIαiPDi+(τ)0 for some αiC and DiDk .

  2. (b) There exists gM2κ!(p) with the principal part iIαi¯|Di|1κq|Di|.

Remark 1.2

Let p =1 and take

Dk=1, if k is even3, if k is odd.

Then in Theorem 1.1 one can choose β =1 and


We let Γ~:=<T>. We define for s∈ ℂ and y ∈ ℝ −{0}:


where Mν,μ(z) and Wν,μ(z) denote the usual Whittaker functions. Now we take κ = k +1/ 2 > 2 , L := L1−k , and Q := Q1−k . For each β ∈ ℤ / 2 pℤ and m∈ ℤ +Q(β) with m < 0 , modifying the Poincaré series in [9, $(1.35)$] we define the vector valued Maass Poincaré series Fβ,mL of index (β ,m) by


where τ=x+iyH and s = σ + it∈ ℂ with σ >1. Indeed, since ℳs(4π | m | y) 𝔢 β (mx) is invariant under slash operator |2−κ T , the Maass Poincaré series is well defined. This series has desirable properties as follows. As in Section 1.3 in [9] it converges normally for τ ∈ H and s = σ + itC with σ >1 and hence defines a Mp 2 (ℤ) -invariant function on H under the slash operator |2−κ . Moreover, Fβ,mL(τ,s) is an eigenfunction of Δ2−κ with an eigenvalue s(1− s) +κ(κ − 2) / 4 . Since eβ(τ)|2κZ=eβ by (3), the invariance of Fβ,mL under the action of Z implies Fβ,mL=Fβ,mL.

Let κ=k+12 and L = L1−k with k an integer > 2 . For each β ∈ ℤ / 2 pℤ and m ∈ ℤ + Q(β) with m < 0 , we obtain from [4, Corollary 1.5] that Fβ,mLτ,κ2 belongs to the space H2κ,ρL.



where Q+ = Q+ (k; z) is the holomorphic part of Q(k; z) and Q = Q (k; z) is the nonholomorphic part of Q(k; z) . Let Q(k; z) have the Fourier development as follows:


Now we are ready to state our main results.

Theorem 1.3

With the same notations as above the following assertions are true.

(1) Let


for some λk(l2)C. Then one has


(2) We have


Theorem 1.4

For an odd prime l , the following assertions are true.

(1) We have


(2) Assume that cQ+(βk) is irrational. Then

afk(l2αk)=lk1(1)k1βkl(1)kαkl if and only if cQ+(l2βk)Q.

(3) Assume that βkl=αkl and cQ+(βk) is irrational. Then

afk(l2αk)=0 if and only if cQ+(l2βk)Q.

Remark 1.5

For simplicity, we dealt with the case p =1 in our main results. But we remark that they can be extended to higher level cases whenever dim dimSk+12(p)=1.

2 Proof of Theorem 1.3

First we are in need of two lemmas and one more fact.

Lemma 2.1

([4, Lemma 4.1]). Let κ=k+12 for an integer k > 2 and let DDk.

Then the following assertions are true.

(a) For each GH2κ,ρL1k, we have


(b) For each f=n1cf(n)qnSκ(p),

(f,(4p)κ1Φξ2κ(Fβ,|D|4pL1k(τ,κ2)))=3s(D)cf(|D|), if p = 1 4s(D)cf(|D|), if p = 2 6s(D)cf(|D|), if p > 2.

Lemma 2.2

([4, Lemma 4.2]). With the same notations as in Lemma 2.1, we have the following assertions.

  1. (a) For a vector valued function h=hβ(τ)eβ one has


  1. (b) ξ2κ(Γ(κ1,4πny))=(4πn)κ1e4πny.

  2. (c) Let m=|D|4p. Then one has


Fact 6

Let p =1 and κ=k+12.

  1. (1) It follows from Lemmas 2.1 and 2.2 that


  1. (2) We obtain from Theorem 1.1-(ii) that


where ck,Dk=(4π|Dk|)κ13Γ(κ1)R.

For k ∈ {6,8,9,10,11,13}, it follows from Fact 3, Fact 4, and Theorem 1.1-(iii), (iv) that PDk+ does not vanish and


for some ck ∈ ℂ× . Thus one has from Fact 6 (2) that


which implies


We compute that

ξ2κ(Q(k;z))=ξ2κΦF1,α4L1kτ,κ2=4κ1Φξ2κF1,α4L1kτ,κ2 by Lemma 2.1-(a) =(4π|Dk|)κ1Γ(κ1)PDk+(τ) by Fact 6 (1) =(4π|Dk|)κ1Γ(κ1)ck,Dk1||fk||2fk=3||fk||2fk.(5)



one has


which proves the first assertion. Hence for all n ≥1 with (−1)k n ≡ 0,1(mod 4)


It follows from (5) that


which implies that


Now we put dk:=3||fk||2(4π)1κ. We obtain that for all positive integers n with (−1)k n ≡ 0,1(mod 4) ,

n1κcQ(nl2)(nl2)κ1+l2k1cQ(n/l2)(n/l2)κ1+(1)knllk1cQ(n)nκ1=n1κdkafk(nl2)+afk(n/l2)l2k1+(1)knllk1afk(n) by (7) =n1κdkafk(l2α)+(1)kαllk1afk(n) by (6) =λk(l2)cQ(n).

Thus we have

Q|T32k(l2)=nZcQ(nl2)+(1)knllkcQ(n)+l12kcQ(n/l2)Γ(κ1,4πny)qn=l12knZcQ(nl2)l2k1+(1)knllk1cQ(n)+cQ(n/l2)Γ(κ1,4πny)qn=l12knZdkn1κcQ(nl2)(nl2)κ1dk1+(1)knllk1nκ1cQ(n)dk1+nκ1cQ(n/l2)dk1Γ(κ1,4πny)qn=l12knZdkn1κafk(nl2)+(1)knllk1afk(n)+l2k1afk(n/l2)Γ(κ1,4πny)qn=l12knZλk(l2)cQ(n)Γ(κ1,4πny)qn since fk|Tk+12(l2)=λk(l2)fk=l12kλk(l2)Q.(8)

We obtain that


Indeed, we observe that


Thus we have


which combined with (8) yields the second assertion.

3 Proof of Theorem 1.4

We observe that


So we find that


has integral coefficients and for all positive integers n with (−1)k−1n ≡ 0,1(mod 4) ,


Then for n = βk with

βk=3, k even, 1, k odd, 

we obtain that


As a consequence of the above identity we get the assertions.


We would like to thank KIAS (Korea Institute for Advanced Study) for its hospitality.

Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea goverment (Ministry of Education) (No. 2017R1D1A1A09000691).

Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2018R1D1A1B07045618 and 2016R1A5A1008055).


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About the article

Received: 2018-06-11

Accepted: 2018-10-05

Published Online: 2018-11-15

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1335–1343, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0112.

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© 2018 Choi and Kim, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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