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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# 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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• Other articles by this author:
/ Chang Heon Kim
Published Online: 2018-11-15 | DOI: https://doi.org/10.1515/math-2018-0112

## 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 ${S}_{2k}\left(N\right).$ Throughout this paper let p =1 or a prime number. For $\kappa \in \mathbb{Z}+\frac{1}{2}$ we denote by ${M}_{\kappa }^{!}\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)$ the space of weakly holomorphic modular forms of weight κ on Γ0(4p) . As usual, Mκ0 (4p)) (resp. ${S}_{\kappa }\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)\right)$ 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 ${\mathbb{M}}_{\kappa }^{!}\left(p\right)$ (resp. ℍ2−κ(p) ) denote the subspace of ${M}_{\kappa }^{!}\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)$ (resp. ${H}_{2-\kappa }\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)\right)$, in which each form satisfies Kohnen’s plus space condition, that is, its Fourier expansion is supported only on those n∈ ℤ for which $\left(-1{\right)}^{\kappa -\frac{1}{2}}n\equiv ◻\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}4p\right)$. Let $\kappa =k+\frac{1}{2}$ 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

$dimSk+12(1)=dim⁡S2k(1),$

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 ${\xi }_{2-\kappa }:{H}_{2-\kappa }\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)\to {S}_{\kappa }\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)$ 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 $\kappa \in \mathbb{Z}+\frac{1}{2}$, the Duke-Jenkins basis [5] for ${\mathbb{M}}_{\kappa }^{!}:={\mathbb{M}}_{\kappa }^{!}\left(1\right)$ 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 $f\in {\mathbb{M}}_{\kappa }^{!}$ at i∞ , then by the Shimura correspondence [3] one has

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

A basis for ${\mathbb{M}}_{\kappa }^{!}$ then consists of functions of the form

$fκ,m(τ)=q−m+∑n>Aκaκ(m,n)qn,$(2)

where $m\ge -{A}_{\kappa }$ satisfies $\left(-1{\right)}^{\kappa -3/2}m\equiv 0,1\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}4\right)$. Using (1) and (2) we deduce the following facts.

#### Fact 3

For $\kappa =\frac{3}{2}-k$ with k ∈ {6,8,9,10,11,13}, the maximal order Aκ of a non-zero $f\in {M}_{\kappa }^{!}$ at iis given by Aκ = − 4 . Thus for each m ≥ 4 satisfying (−1)km ≡ 0,1(mod 4) , there exist unique modular forms ${f}_{\frac{3}{2}-k,m}\left(\tau \right)\in {\mathbb{M}}_{\frac{3}{2}-k}^{!}$ with Fourier development

$f32−k,m(τ)=q−m+∑n≥−3a32−k(m,n)qn,$

which form a basis for the space ${\mathbb{M}}_{\frac{3}{2}-k}^{!}$.

#### Fact 4

For $\kappa =k+\frac{1}{2}$ with k ∈ {6,8,9,10,11,13}, the space 𝕊κ is spanned by

$fk:=fκ,−α$

where α is given by

$α=αk=Aκ=Ak+12=1, k even, 3, k odd,$

and fk has the form ${q}^{\alpha }+O\left({q}^{4}\right).$

#### Fact 5

Let $\kappa =k+\frac{1}{2}$ and $f\left(z\right)\in {H}_{\kappa }\left({\mathrm{\Gamma }}_{0}\left(4p\right)\right)$ with Fourier expansion $f\left(\tau \right)=\sum _{n\in \mathbb{Z}}c\left(y;n\right){e}^{2\pi inx}$ where 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∈ ℍ2−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 $f\left(z\right)=\sum _{\genfrac{}{}{0}{}{n\ge 1}{\left(-1{\right)}^{k}n\equiv 0,1\phantom{\rule{thinmathspace}{0ex}}\left(4\right)}}{a}_{f}\left(n\right){q}^{n}\in {\mathbb{S}}_{k+\frac{1}{2}}$, 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 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 $\varphi :\mathfrak{H}\to \mathbb{C}$ is a holomorphic function with ϕ(τ)2 = + d . It is well known that Mp2(ℤ) is generated by $S=\left(\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sqrt{T}\right)$ and $T=\left(\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\right).$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

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

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

$S2=ST3=ZZ=−100−1,i$

from which we note that

$ρL(Z)eγ=ib−−b+e−γ.$(3)

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

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

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

$ργδ(M,ϕ)=<ρL(M,ϕ)eδ,eγ>.$

Following [9], for an integer r we denote by ${H}_{r+1/2,{\rho }_{L}}\left(resp.{M}_{r+1/2,{\rho }_{L}\right)}^{!}$, 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 ${Q}_{r}\left(x\right)=\left(-1{\right)}^{r}{x}^{2}/4p$. Then its dual lattice ${L}_{r}^{\prime }$ is equal to ℤ . For a vector valued modular form $F=\sum _{\gamma }{F}_{\gamma }{\mathfrak{e}}_{\gamma }$, we define a map Φ by

$Φ(F)(τ):=∑γFγ(4pτ).$(4)

It then follows from [8, Theorem 1] that the map Φ defines an isomorphism from ${H}_{r+1/2,{\rho }_{{L}_{r}}}$ to ${\mathbb{H}}_{r+1/2}\left(p\right)$ since ${\overline{\rho }}_{{L}_{r}}={\rho }_{{L}_{r+1}}$.

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 := Lk and Q := Qk . Following [9] we define the vector valued cuspidal Poincaré series ${P}_{\beta ,n}^{L}\left(\tau \right)$ 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 ${P}_{\beta ,n}^{L}\left(\tau \right)$ belongs to the space ${S}_{\kappa ,{\rho }_{L}}$. Let ${\mathfrak{D}}_{k}$ 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 $\kappa =k+\frac{1}{2}$ and L := Lk . For each $D\in {\mathfrak{D}}_{\mathfrak{k}}$, we define

$PD+:=Φ(Pβ,|D|4pL),$

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

1. (i) ${P}_{D}^{+}\in {\mathbb{S}}_{\kappa }\left(p\right)$ and the defintion of ${P}_{D}^{+}$ does not depend on the choice of β .

2. (ii) For each $f=\sum _{n\ge 1}{a}_{f}\left(n\right){q}^{n}\in {\mathbb{S}}_{\kappa }\left(p\right)$ , 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\left(D\right)=\left\{\begin{array}{ll}1,& \text{\hspace{0.17em}if\hspace{0.17em}}p|D,\\ 2,& \text{\hspace{0.17em}otherwise.\hspace{0.17em}}\end{array}\right\$

(iii) The set $\left\{{P}_{D}^{+}\mid D\in {\mathfrak{D}}_{\mathfrak{k}}\right\}$ spans the space 𝕊κ(p) . Moreover, if we let t := dim 𝕊κ( p) and { f1 , f2 , , ft }be a basis for 𝕊κ(p) satisfying ${f}_{i}={q}^{|{D}_{i}|}+O\left({q}^{|{D}_{i}|+1}\right)$ for some ${D}_{i}\in {\mathfrak{D}}_{k}\left(i=1,\dots ,t\right)$ and 0 <| D1 |<| D2 |< <| Dt | , then the set

${PD1+,PD2+,…,PDt+}$

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) $\sum _{i\in I}{\alpha }_{i}{P}_{{D}_{i}}^{+}\left(\tau \right)\equiv 0$ for some ${\alpha }_{i}\in \mathbb{C}$ and DiDk .

2. (b) There exists $g\in {\mathbb{M}}_{2-\kappa }^{!}\left(p\right)$ with the principal part $\sum _{i\in I}\overline{{\alpha }_{i}}|{D}_{i}{|}^{1-\kappa }{q}^{-|{D}_{i}|}$.

#### 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 ${\stackrel{~}{\mathrm{\Gamma }}}_{\mathrm{\infty }}:=$. 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 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 ${F}_{\beta ,m}^{L}$ 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 $\tau =x+iy\in \mathfrak{H}$ 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, ${F}_{\beta ,m}^{L}\left(\tau ,s\right)$ is an eigenfunction of Δ2−κ with an eigenvalue s(1− s) +κ(κ − 2) / 4 . Since ${\mathfrak{e}}_{\beta }\left(\tau \right){|}_{2-\kappa }Z={\mathfrak{e}}_{-\beta }$ by (3), the invariance of ${F}_{\beta ,m}^{L}$ under the action of Z implies ${F}_{\beta ,m}^{L}={F}_{-\beta ,m}^{L}$.

Let $\kappa =k+\frac{1}{2}$ 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 ${F}_{\beta ,m}^{L}\left(\tau ,\frac{\kappa }{2}\right)$ belongs to the space ${H}_{2-\kappa ,{\rho }_{L}}$.

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 ${\lambda }_{k}\left({l}^{2}\right)\in \mathbb{C}$. 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 ${c}_{Q}^{+}\left({\beta }_{k}\right)$ is irrational. Then

$afk(l2αk)=lk−1(−1)k−1βkl−(−1)kαkl if and only if cQ+(l2βk)∈ Q.$

(3) Assume that $\left(\frac{-{\beta }_{k}}{l}\right)=\left(\frac{{\alpha }_{k}}{l}\right)$ and ${c}_{Q}^{+}\left({\beta }_{k}\right)$ 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 $\mathrm{dim}{\mathbb{S}}_{k+\frac{1}{2}}\left(p\right)=1$.

## 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 $\kappa =k+\frac{1}{2}$ for an integer k > 2 and let DDk.

Then the following assertions are true.

(a) For each $G\in {H}_{2-\kappa ,{\rho }_{{L}_{1-k}}}$, we have

$(4p)κ−1Φ∘ξ2−κ(G)=ξ2−κ∘Φ(G).$

(b) For each $f=\sum _{n\ge 1}{c}_{f}\left(n\right){q}^{n}\in {\mathbb{S}}_{\kappa }\left(p\right)$,

$(f,(4p)κ−1Φξ2−κ(Fβ,−|D|4pL1−k(τ,κ2)))=3s(D)⋅cf(|D|), if p = 1 4s(D)⋅cf(|D|), if p = 2 6s(D)⋅cf(|D|), if p > 2.$

#### Lemma 2.2

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

1. (a) For a vector valued function $h={h}_{\beta }\left(\tau \right){\mathfrak{e}}_{\beta }$ one has

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

1. (b) ${\xi }_{2-\kappa }\left(\mathrm{\Gamma }\left(\kappa -1,4\pi ny\right)\right)=-\left(4\pi n{\right)}^{\kappa -1}{e}^{-4\pi ny}.$

2. (c) Let $m=-\frac{|D|}{4p}$. 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 and $\kappa =k+\frac{1}{2}$.

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

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

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

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

where ${c}_{k,{D}_{k}}=\frac{\left(4\pi |{D}_{k}|{\right)}^{\kappa -1}}{3\mathrm{\Gamma }\left(\kappa -1\right)}\in \mathbb{R}$.

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

$PDk+=ckfk$

for some ck ∈ 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

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

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)

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

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

$cQ−(n)=−3||fk||−2⋅(4πn)1−κ⋅afk(n).$(7)

Now we put ${d}_{k}:=-3||{f}_{k}|{|}^{-2}\cdot \left(4\pi {\right)}^{1-\kappa }$. 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

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

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 = βk 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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Accepted: 2018-10-05

Published Online: 2018-11-15

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1335–1343, ISSN (Online) 2391-5455,

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