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

SoYoung Choi; Chang Heon Kim

doi:10.1515/math-2018-0112

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

Published/Copyright: November 15, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

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 κ∈ Z+12 we denote by Mκ!(Γ0(4p)) the space of weakly holomorphic modular forms of weight κ on Γ₀(4p) . As usual, M_κ(Γ₀ (4p)) (resp. Sκ(Γ0(4p))) stands for the space of weight κ modular forms (resp. cusp forms) on Γ₀(4p) . Let H_κ(Γ₀(4p)) be the space of weight κ harmonic weak Maass forms on Γ₀(4p) . Let Mκ!(p) (resp. ℍ_2−κ(p) ) denote the subspace of Mκ!(Γ0(4p)) (resp. H2−κ(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)κ−12n≡◻(mod4p). Let κ=k+12 and 𝕄_κ(p) (resp. 𝕊_κ(p) ) denote the subspace of M_κ(Γ₀ (4p)) (resp. S_κ(Γ₀ (4 p)) ), in which each form satisfies Kohnen’s plus space condition.

Let Δ(τ) ∈ S₁₂ (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 S_2k (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

dimSk+12(1)=dim⁡S2k(1),

which implies that

dimSk+12(1)=1⟺k=6,8,9,10,11,13⟺1−k=−5,−7,−8,−9,−10,−12.

Now we assume that k ∈ {6,8,9,10,11,13} . For κ > 2 there is an antilinear differential operator ξ2−κ:H2−κ(Γ0(4p))→Sκ(Γ0(4p)) defined by

ξ2−κ(f)(τ):=2iy2−κ⋅∂f∂τ¯¯.

Fact 2

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

ξ32−k:H32−k(1)→Sk+12(1)issurjective.

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

(1)Aκ=2ℓκ−(−1)κ−1/2ifℓκis odd,2ℓκotherwise.

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

(2)fκ,m(τ)=q−m+∑n>Aκaκ(m,n)qn,

where m≥−Aκ satisfies (−1)κ−3/2m≡0,1(mod4). Using (1) and (2) we deduce the following facts.

Fact 3

Forκ=32−kwith k ∈ {6,8,9,10,11,13}, the maximal order A_κ of a non-zerof∈ Mκ!at i∞ is given by A_κ = − 4 . Thus for each m ≥ 4 satisfying (−1)^km ≡ 0,1(mod 4) , there exist unique modular formsf32−k,m(τ)∈ M32−k!with Fourier development

f32−k,m(τ)=q−m+∑n≥−3a32−k(m,n)qn,

which form a basis for the space M32−k!.

Fact 4

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

fk:=fκ,−α

where α is given by

α=αk=Aκ=Ak+12=1, k even, 3, k odd,

and f_k has the form qα+O(q4).

Fact 5

Letκ=k+12andf(z)∈ Hκ(Γ0(4p))with Fourier expansionf(τ)=∑n∈ Zc(y;n)e2πinxwhere τ = x + iy . For each prime l with gcd(l, 4 p) =1, the l²-th Hecke operator is defined by

f|κT(l2)(τ)=∑n∈ Zc(y/l2;nl2)+(−1)knllk−1c(y;n)+l2k−1c(l2y;n/l2))e2πinx.

Then for each ℳ∈ ℍ_2−κ (p) , we obtain from [6, (2.6)] or [7, (7.2)] that for κ > 2,

ξ2−κ(M)|κT(l2)=l2κ−2ξ2−κ(M|2−κT(l2)).

As a corollary of Fact 5, one has that if f(z)=∑n≥1(−1)kn≡0,1(4)af(n)qn∈ Sk+12, then

f|k+12T(l2)=∑n≥1(−1)kn≡0,1(4)af(l2n)+(−1)knllk−1af(n)+l2k−1af(n/l2)qn∈ Sk+12.

(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 𝔢_γ for γ ∈ L′ / L . Let Mp₂ (ℤ) denote the integral metaplectic group, which consists of pairs (γ ,ϕ) , where γ = (abcd) ∈ SL₂(ℤ) and ϕ:H→C is a holomorphic function with ϕ(τ)² = cτ + d . It is well known that Mp₂(ℤ) is generated by S=0−110,T and T=1101,1.Then there is a unitary representation ρ_L of the group Mp₂(ℤ) on ℂ[L′ / L] , the so-called Weil representation, which is defined by

ρL(T)(eγ):=e(Q(γ))eγ,ρL(S)(eγ):=e((b−−b+)/8)|L′/L|∑δ∈ L′/Le(−(γ,δ))eδ,

where e(z) := e^2πiz and (X,Y):=Q(X+Y)−Q(X)−Q(Y) is the associated bilinear form. One has the relations

S2=ST3=ZZ=−100−1,i

from which we note that

(3)ρL(Z)eγ=ib−−b+e−γ.

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

<∑γ∈ L′/Lλγeγ,∑γ∈ L′/Lμγeγ>=∑γ∈ L′/Lλγμγ¯.

For γ, δ ∈ L′ / L and (M,ϕ )∈ Mp₂ (ℤ) the coefficient ρ_γδ(M, ϕ) of the representation ρ_L is defined by

ργδ(M,ϕ)=<ρL(M,ϕ)eδ,eγ>.

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 ρ_L .

Let L_r 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

(4)Φ(F)(τ):=∑γFγ(4pτ).

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

(f|r+1/2L(M,ϕ))(τ)=ϕ(τ)−2r−1ρL(M,ϕ)−1f(Mτ).

Let L := L_k and Q := Q_k . 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 ,

Pβ,nL(τ):=12∑(M,ϕ)∈ Γ~∞∖Mp2(Z)eβ(nτ)|κ(M,ϕ).

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 letκ=k+12and L := L_k . For eachD∈ Dk, we define

PD+:=Φ(Pβ,|D|4pL),

where β is an integer such that D ≡ β² (mod 4 p) . Then the following assertions are true.

(i) PD+∈Sκ(p) and the defintion of PD+ does not depend on the choice of β .
(ii) For each f=∑n≥1af(n)qn∈ Sκ(p) , we have

(f,ck,DPD+)=af(|D|)

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

ck,D=(4π|D|)κ−1Γ(κ−1)⋅s(D)3, if p = 1 (4π|D|)κ−1Γ(κ−1)⋅s(D)4, if p = 2 (4π|D|)κ−1Γ(κ−1)⋅s(D)6, if p > 2

with s(D)=1, if p|D,2, otherwise.

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

{PD1+,PD2+,…,PDt+}

forms a basis for 𝕊_κ(p) .

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

(a) ∑i∈ IαiPDi+(τ)≡0 for some αi∈ C and D_i ∈ D_k .
(b) There exists g∈ M2−κ!(p) with the principal part ∑i∈ Iαi¯|Di|1−κq−|Di|.

Remark 1.2

Let p =1 and take

Dk=1, if k is even−3, if k is odd.

Then in Theorem 1.1 one can choose β =1 and

PDk+:=Φ(P1,|Dk|4pLk).

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

Ms(y)=y−(2−κ)/2M−(2−κ)/2,s−1/2(y)(y>0),Ws(y)=|y|−(2−κ)/2W2−κ2sgn(y),s−1/2(|y|)

where M_ν,μ(z) and W_ν,μ(z) denote the usual Whittaker functions. Now we take κ = k +1/ 2 > 2 , L := L_1−k , and Q := Q_1−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

Fβ,mL(τ,s):=12Γ(2s)∑(M,ϕ)∈ Γ~∞∖Mp2(Z)[Ms(4π|m|y)eβ(mx)]|2−κL(M,ϕ)

where τ=x+iy∈ H and s = σ + it∈ ℂ with σ >1. Indeed, since ℳ_s(4π | m | y) 𝔢_β (mx) is invariant under slash operator |_2−κT , 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 + it∈ C with σ >1 and hence defines a Mp₂ (ℤ) -invariant function on H under the slash operator |_2−κ . Moreover, Fβ,mL(τ,s) is an eigenfunction of Δ_2−κ with an eigenvalue s(1− s) +κ(κ − 2) / 4 . Since eβ(τ)|2−κZ=e−β by (3), the invariance of Fβ,mL under the action of Z implies Fβ,mL=F−β,mL.

Let κ=k+12 and L = L_1−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 H2−κ,ρL.

Let

Q=Q(k;z):=ΦF1,−α4L1−kτ,κ2=Q++Q−

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:

Q(k;z)=2q−α+cQ+(0)+∑n≥1(−1)k−1n≡0,1(4)cQ+(n)qn+∑n≥1(−1)kn≡0,1(4)cQ−(n)Γ(κ−1,4πny)q−n.

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

fk|k+12T(l2)=λk(l2)fk

for some λk(l2)∈ C. Then one has

λk(l2)=afk(l2α)+(−1)kαllk−1.

(2) We have

Q|32−kT(l2)−l1−2kλk(l2)Q=Q+|32−kT(l2)−l1−2kλk(l2)Q+∈ M32−k!.

Theorem 1.4

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

(1) We have

cQ+(l2βk)∈ Z[cQ+(βk)]l2k−1⊆Q(cQ+(βk)).

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

afk(l2αk)=lk−1(−1)k−1βkl−(−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]). Letκ=k+12for an integer k > 2 and let D ∈ Dk.

Then the following assertions are true.

(a) For each G∈ H2−κ,ρL1−k, we have

(4p)κ−1Φ∘ξ2−κ(G)=ξ2−κ∘Φ(G).

(b) For each f=∑n≥1cf(n)qn∈ Sκ(p),

(f,(4p)κ−1Φξ2−κ(Fβ,−|D|4pL1−k(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.

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

ξ2−κ(h|2−κL1−k(M,ϕ))=(ξ2−κ(h))|κLk(M,ϕ).

(b) ξ2−κ(Γ(κ−1,4πny))=−(4πn)κ−1e−4πny.
(c) Let m=−|D|4p. Then one has

ξ2−κ(Fβ,mL1−k(τ,κ2))=(4π|m|)κ−1Γ(κ−1)Pβ,|m|Lk(τ).

Fact 6

Let p =1 andκ=k+12.

(1) It follows from Lemmas 2.1 and 2.2 that

Φξ2−κF1,−|Dk|4L1−kτ,κ2=Φ(π|Dk|)κ−1Γ(κ−1)P1,|Dk|4Lk(τ)=(π|Dk|)κ−1Γ(κ−1)PDk+(τ)∈ Sk+12.

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

(fk,ck,DkPDk+)=afk(|Dk|)=1

where ck,Dk=(4π|Dk|)κ−13Γ(κ−1)∈ 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

PDk+=ckfk

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

1=(fk,ck,Dkckfk)=ck¯ck,Dk||fk||2,

which implies

ck=ck,Dk−1||fk||−2.

We compute that

(5)ξ2−κ(Q(k;z))=ξ2−κΦF1,−α4L1−kτ,κ2=4κ−1Φξ2−κF1,−α4L1−kτ,κ2 by Lemma 2.1-(a) =(4π|Dk|)κ−1Γ(κ−1)PDk+(τ) by Fact 6 (1) =(4π|Dk|)κ−1Γ(κ−1)ck,Dk−1||fk||−2fk=3||fk||−2fk.

Since

fk|k+12T(l2)=∑n≥1(−1)kn≡0,1(4)afk(l2n)+(−1)knllk−1afk(n)+l2k−1afk(n/l2)qn=λk(l2)fk,

one has

λk(l2)=afk(l2α)+(−1)kαllk−1,

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

(6)afk(l2α)+(−1)kαllk−1afk(n)=afk(l2n)+(−1)knllk−1afk(n)+l2k−1afk(n/l2).

It follows from (5) that

3||fk||−2fk(z)=ξ2−κ(Q(k;z))=−∑n≥α(−1)kn≡0,1(4)(4πn)κ−1cQ−(n)¯qn,

which implies that

(7)cQ−(n)=−3||fk||−2⋅(4πn)1−κ⋅afk(n).

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

n1−κcQ−(nl2)(nl2)κ−1+l2k−1cQ−(n/l2)(n/l2)κ−1+(−1)knllk−1cQ−(n)nκ−1=n1−κdkafk(nl2)+afk(n/l2)l2k−1+(−1)knllk−1afk(n) by (7) =n1−κdkafk(l2α)+(−1)kαllk−1afk(n) by (6) =λk(l2)cQ−(n).

Thus we have

(8)Q−|T32−k(l2)=∑n∈ZcQ−(nl2)+(−1)knll−kcQ−(n)+l1−2kcQ−(n/l2)Γ(κ−1,4πny)q−n=l1−2k∑n∈ZcQ−(nl2)l2k−1+(−1)knllk−1cQ−(n)+cQ−(n/l2)Γ(κ−1,4πny)q−n=l1−2k∑n∈Zdkn1−κcQ−(nl2)(nl2)κ−1dk−1+(−1)knllk−1nκ−1cQ−(n)dk−1+nκ−1cQ−(n/l2)dk−1Γ(κ−1,4πny)q−n=l1−2k∑n∈Zdkn1−κafk(nl2)+(−1)knllk−1afk(n)+l2k−1afk(n/l2)Γ(κ−1,4πny)q−n=l1−2k∑n∈Zλk(l2)cQ−(n)Γ(κ−1,4πny)q−n since fk|Tk+12(l2)=λk(l2)fk=l1−2kλk(l2)Q−.

We obtain that

l2κ−2ξ2−κQ|2−κTl2=ξ2−κQ|κTl2byFact5=2fk−2fk|κTl2by5=3fk−2λkl2fk=ξ2−κλkl2Qsinceλkl2∈R.

Indeed, we observe that

λk(l2)=afk(l2α)+(−1)kαllk−1∈ Z.

Thus we have

l2κ−2Q|2−κT(l2)−λk(l2)Q∈ M2−κ!,

which combined with (8) yields the second assertion.

3 Proof of Theorem 1.4

We observe that

Q|32−kT(l2)−l1−2kλk(l2)Q=Q+|32−kT(l2)−l1−2kλk(l2)Q+=2q−α+cQ+(0)+∑n≥1(−1)k−1n≡0,1(4)cQ+(n)qn|32−kT(l2)−l1−2kλk(l2)2q−α+cQ+(0)+∑n≥1(−1)k−1n≡0,1(4)cQ+(n)qn=2l1−2kq−αl2+2(−1)kαll−k−l1−2kλk(l2)q−α+cQ+(0)(1+l1−2k−l1−2kλk(l2))+∑n≥1(−1)k−1n≡0,1(4)cQ+(l2n)+(−1)k−1nll−kcQ+(n)+l1−2kcQ+(n/l2)−l1−2kλk(l2)cQ+(n)qn=2l1−2kf32−k,αl2since−α≥−3.

So we find that

2f32−k,αl2=l2k−1Q+|32−kT(l2)−λk(l2)Q+

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

l2k−1cQ+(l2n)+(−1)k−1nllk−1cQ+(n)+cQ+(n/l2)−λk(l2)cQ+(n)=cQ+(n)(−1)k−1nllk−1−(−1)kαllk−1−afk(l2α)+l2k−1cQ+(l2n)+cQ+(n/l2)∈ 2Z.

Then for n = β_k with

βk=3, k even, 1, k odd,

we obtain that

cQ+(βk)(−1)k−1βkl−(−1)kαllk−1−afk(l2α)+l2k−1cQ+(l2βk)∈ 2Z.

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

Acknowledgement

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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Received: 2018-06-11

Accepted: 2018-10-05

Published Online: 2018-11-15

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