2 L-series of vector-valued weakly holomorphic modular forms

In this section, we introduce the L-series of a vector-valued weakly holomorphic modular form and prove some of its properties. In particular, we prove a converse theorem. We start by introducing some notation.

Let $\Gamma =\mathrm {SL}_2(\mathbb {Z})$ . Let $k\in \frac 12\mathbb {Z}$ and $\chi $ a unitary multiplier system of weight k on $\Gamma $ , i.e., $\chi :\Gamma \to \mathbb {C}$ satisfies the following conditions:

1. (1) $|\chi (\gamma )| = 1$ for all $\gamma \in \Gamma $ .

2. (2) $\chi $ satisfies the consistency condition

   $$\begin{align*}\chi(\gamma_3) (c_3\tau + d_3)^k =\chi(\gamma_1)\chi(\gamma_2) (c_1\gamma_2\tau + d_1)^k (c_2\tau+d_2)^k, \end{align*}$$
   where $\gamma _3 =\gamma _1\gamma _2$ and $\gamma _i =\left (\begin {smallmatrix} a_i&b_i\\c_i&d_i\end {smallmatrix}\right )\in \Gamma $ for $i=1,2$ , and $3$ .

Let m be a positive integer and $\rho :\Gamma \to \mathrm {GL}_m(\mathbb {C})$ an m-dimensional unitary complex representation such that $\rho (T)$ is diagonal, where $T:=\left (\begin {smallmatrix} 1&1\\0&1\end {smallmatrix}\right )$ . We denote the diagonal entries of $\chi (T)\rho (T)$ by $e^{2\pi i\kappa _1},\ldots , e^{2\pi i\kappa _m}$ , where $\kappa _1,\ldots ,\kappa _m$ are real numbers with $0\leq \kappa _i <1$ for all $1\leq i\leq m$ . Let $\{\mathbf {e}_1,\ldots ,\mathbf {e}_m\}$ denote the standard basis of $\mathbb {C}^m$ . For a vector-valued function $f =\sum _{j=1}^m f_j\mathbf {e}_j$ on $\mathbb {H}$ and $\gamma =\left (\begin {smallmatrix} a&b\\c&d\end {smallmatrix}\right )\in \Gamma $ , define a slash operator by

$$\begin{align*}(f|_{k,\chi,\rho}\gamma)(\tau):= (c\tau+d)^{-k}\chi^{-1}(\gamma)\rho(\gamma)^{-1} f(\gamma\tau). \end{align*}$$

The definition of the vector-valued modular forms is given as follows.

The space of all vector-valued weakly holomorphic modular forms of weight k, multiplier system $\chi $ , and type $\rho $ on $\Gamma $ is denoted by $M^!_{k,\chi ,\rho }$ . There is a subspace $S_{k,\chi ,\rho }$ of vector-valued cusp forms for which we require each $a_j(n) = 0$ when $n+\kappa _j$ is non-positive.

For a vector-valued cusp form $f(\tau ) =\sum _{j=1}^m\sum _{n+\kappa _j>0} a_{f,j}(n)e^{2\pi i(n+\kappa _j)\tau }\mathbf {e}_j\in S_{k,\chi ,\rho }$ , then $a_{f,j}(n) = O(n^{k/2})$ for every $1\leq j\leq m$ as $n\to \infty $ by the same argument as for classical modular forms (for more details, see [Reference Knopp and Mason16]). Then, the vector-valued L-series defined by

(2.2) $$ \begin{align} L(f,s) :=\sum_{j=1}^m\sum_{n+\kappa_j>0}\frac{a_j(n)}{(n+\kappa_j)^s}\mathbf{e}_j \end{align} $$

converges absolutely for $\mathrm {Re}(s)\gg 0$ . This has a (vector-valued) integral representation

$$\begin{align*}\frac{\Gamma(s)}{(2\pi)^s} L(f,s) =\int_0^\infty f(iv) v^s\frac{dv}{v}. \end{align*}$$

From this, we see that it has an analytic continuation to $\mathbb {C}$ and a functional equation

$$\begin{align*}L^*(f,s) = i^k\chi(S)\rho(S) L^*(f,k-s), \end{align*}$$

where $L^*(f,s) =\frac {\Gamma (s)}{(2\pi )^s} L(f,s)$ and $S :=\left (\begin {smallmatrix} 0&-1\\1&0\end {smallmatrix}\right )$ (for example, see [Reference Jin and Lim13, Reference Lim and Raji17, Reference Lim and Raji18]).

Let f be a vector-valued weakly holomorphic modular form in $M^!_{k,\chi ,\rho }$ with Fourier expansion of the form (2.1). Let $n_0\in \mathbb {N}$ be such that $f_j(\tau )$ are $O(e^{2\pi n_0 v})$ as $v=\mathrm {Im}(\tau )\to \infty $ for each $1\leq j\leq m$ . Let $\mathcal {F}_f$ be the space of test functions $\varphi :\mathbb {R}_+\to \mathbb {C}$ such that

1. (1) $(\mathcal {L}\varphi )(s)$ converges for all s with $\mathrm {Re}(s)\geq -2\pi n_0$ ,

2. (2) the series

   (2.3) $$ \begin{align} \sum_{n\gg-\infty} | a_{f,j}(n)| (\mathcal{L}|\varphi|)(2\pi (n+\kappa_j)) \end{align} $$
   converges for each $1\leq j\leq m$ .

The space $\mathcal {F}_f$ contains the compactly supported smooth functions on $\mathbb {R}_+$ because of the growth of $a_{f,j}(n)$ (for more details of the growth of $a_{f,j}(n)$ , see [Reference Bruinier and Funke5]). We define the vector-valued L-series map $L_f :\mathcal {F}_f\to \mathbb {C}^m$ by

$$\begin{align*}L_f(\varphi) :=\sum_{j=1}^m\sum_{n\gg-\infty} a_{f,j}(n) (\mathcal{L}\varphi)(2\pi (n+\kappa_j))\mathbf{e}_j. \end{align*}$$

Then, we prove that this L-series has an integral representation and satisfies a functional equation.

Proof (1) For $u>0$ and $\mathrm {Re}(s)>0$ , we have

$$\begin{align*}(\mathcal{L} I_s)(u) =\frac{(2\pi)^s}{\Gamma(s)}\int_0^\infty e^{-ut} t^{s-1}dt =\left(\frac{2\pi}{u}\right)^s. \end{align*}$$

Therefore, if f is a vector-valued cusp form, then $L_f(I_s)$ is the usual L-series of f defined in (2.2).

(2) Suppose that f has a Fourier expansion as in (2.1). By the definition of $L_f(\varphi )$ , we have

$$ \begin{align*} L_f(\varphi) &=\sum_{j=1}^m\sum_{n\gg-\infty} a_{f,j}(n) (\mathcal{L}\varphi)(2\pi (n+\kappa_j))\mathbf{e}_j\\ &=\sum_{j=1}^m\sum_{n\gg-\infty} a_{f,j}(n)\int_0^\infty e^{-2\pi (n+\kappa_j)t}\varphi(t)dt\mathbf{e}_j\\ &=\sum_{j=1}^m\left(\int_0^\infty f_j(it)\varphi(t) dt\right)\mathbf{e}_j. \end{align*} $$

The last equality follows from the fact that we can interchange the order of summation and integration since $\varphi \in \mathcal {F}_f$ .

(3) Since $f\in M^!_{k,\chi ,\rho }$ , we see that

$$\begin{align*}f(iy) = (iy)^{-k}\chi^{-1}(S)\rho(S)^{-1} f\left(i\frac1y\right) \end{align*}$$

for $y>0$ . Therefore, we have

$$ \begin{align*} L_f(\varphi) &=\int_0^\infty f(iy)\varphi(y)dy =\int_0^\infty f\left( i\frac 1y\right)\varphi\left(\frac 1y\right) y^{-2}dy\\ &=\int_0^\infty (iy)^k\chi(S)\rho(S) f(iy)\varphi\left(\frac 1y\right) y^{-2}dy\\ &= i^k\rho(S)\int_0^\infty f(iy) y^{k-2}\chi(S)\varphi\left(\frac 1y\right) dy. \end{align*} $$

Let $\mathcal {C}(\mathbb {R},\mathbb {C})$ be the space of piece-wise smooth complex-valued functions on $\mathbb {R}$ . For $s\in \mathbb {C}$ and $\varphi \in \mathcal {C}(\mathbb {R},\mathbb {C})$ , we define $\varphi _s(x) :=\varphi (x) x^{s-1}$ . We also define the series

$$\begin{align*}L(s,f,\varphi) := L_f(\varphi_s). \end{align*}$$

Then, we prove that this series has an analytic continuation to all $s\in \mathbb {C}$ and satisfies a functional equation.

Theorem 2.3 Let $f\in M^!_{k,\chi ,\rho }$ , and $n_0\in \mathbb {N}$ be such that $f_j(\tau )$ are $O(e^{2\pi n_0 v})$ as $v=\mathrm {Im}(\tau )\to \infty $ for each $1\leq j\leq m$ . Suppose that $\varphi \in \mathcal {C}(\mathbb {R},\mathbb {C})$ is a non-zero function such that, for some $\epsilon>0$ , $\varphi (x)$ and $\varphi (x^{-1})$ are $o(e^{-2\pi (n_0+\epsilon )x})$ as $x\to \infty $ . We further assume that series (2.3) converges. Then the series $L(s,f,\varphi )$ converges absolutely for $\mathrm {Re}(s)>\frac 12$ , has an analytic continuation to all $s\in \mathbb {C}$ and satisfies the functional equation

$$\begin{align*}L(s,f,\varphi) = i^k\rho(S) L(1-s, f,\varphi|_{1-k,\chi^{-1}} S). \end{align*}$$

Proof By the growth of $\varphi $ , we see that $\mathcal {L}(|\varphi |^2)(y)$ converges absolutely for $y\geq -2\pi n_0$ . For $y>0$ and $s\in \mathbb {C}$ with $\mathrm {Re}(s)>\frac 12$ , Cauchy–Schwarz inequality implies that

$$\begin{align*}(\mathcal{L}|\varphi_s|)(y)\leq (\mathcal{L}(|\varphi|^2)(y))^{\frac12} y^{-\mathrm{Re}(s)+\frac12} (\Gamma(2\mathrm{Re}(s)-1))^{\frac12}. \end{align*}$$

Therefore, $\varphi _s\in \mathcal {F}_f$ for $\mathrm {Re}(s)>\frac 12$ .

Recall that

$$\begin{align*}f(iy) = (iy)^{-k}\chi^{-1}(S)\rho(S)^{-1} f\left(i\frac1y\right) \end{align*}$$

for $y>0$ . Therefore, we have

$$ \begin{align*} L(s, f,\varphi) &= L_f(\varphi_s) =\int_0^\infty f(iy)\varphi_s(y)dy\\ &=\int_0^\infty f(iy)\varphi(s) y^{s-1} dy\\ &=\int_1^\infty f(iy)\varphi(y)y^{s-1} dy +\int_0^1 f(iy)\varphi(y)y^{s-1}dy\\ &=\int_1^\infty f(iy)\varphi(y)y^{s-1} dy +\int_1^\infty f\left( i\frac1y\right)\varphi\left(\frac 1y\right) y^{1-s}y^{-2}dy\\ &=\int_1^\infty f(iy)\varphi(y)y^{s-1} dy + i^k\rho(S)\int_1^\infty f(iy)\left(\varphi|_{1-k,\chi^{-1}}S\right)(y) y^{-s}dy. \end{align*} $$

By the growth of $\varphi $ at $0$ and $\infty $ , we see that the integrals

$$\begin{align*}\int_1^\infty f(iy)\varphi(y)y^{s-1} dy \quad \text{and} \int_1^\infty f(iy)\left(\varphi|_{1-k,\chi^{-1}}S\right)(y) y^{-s}dy \end{align*}$$

are well-defined for all $s\in \mathbb {C}$ , and give holomorphic functions.

Since $f\in M^!_{k,\chi ,\rho }$ , we obtain that $\rho (-I_2)\chi (-I_2) = (-1)^{-k} I_m$ , where $I_m$ denotes the identity matrix of size m. Therefore, we get the desired functional equation.

Let $S_c(\mathbb {R}_+)$ be a set of complex-valued, compactly supported, and piecewise smooth functions on $\mathbb {R}_+$ , which satisfies the following condition: for any $y\in \mathbb {R}_+$ , there exists $\varphi \in S_c(\mathbb {R}_+)$ such that $\varphi (y)\neq 0$ . We write $\langle \cdot ,\cdot \rangle $ for the standard scalar product on $\mathbb {C}^m$ , i.e.,

$$\begin{align*}\left\langle\sum_{j=1}^m\lambda_j\mathbf{e}_j,\sum_{j=1}^m\mu_j\mathbf{e}_j\right\rangle =\sum_{j=1}^m\lambda_j\overline{\mu_j}. \end{align*}$$

We now state the converse theorem in the case of vector-valued weakly holomorphic modular forms.

Theorem 2.4 For each $1\leq j\leq m$ , let $(a_{f,j}(n))_{n\geq -n_0}$ be’ a sequence of complex numbers such that $a_{f,j}(n) = O(e^{C\sqrt {n}})$ as $n\to \infty $ , for some $C>0$ . For each $\tau \in \mathbb {H}$ , set

$$\begin{align*}f(\tau) :=\sum_{j=1}^m\sum_{n=-n_0}^\infty a_{f,j}(n)e^{2\pi i(n+\kappa_j)\tau}\mathbf{e}_j. \end{align*}$$

Suppose that for each $\varphi \in S_c(\mathbb {R}_+)$ , the function $L_f(\varphi )$ satisfies

$$\begin{align*}L_f(\varphi) = i^k\rho(S) L_f(\varphi|_{2-k,\chi^{-1}} S). \end{align*}$$

Then, f is a vector-valued weakly holomorphic modular form in $M^!_{k,\chi ,\rho }$ .

Proof By the bound for $a_{f,j}(n)$ , we see that $f_j(\tau )$ converges absolutely to a smooth function on $\mathbb {H}$ for $1\leq j\leq m$ . Note that $\varphi _s\in S_c(\mathbb {R}_+)$ for any $s\in \mathbb {C}$ and $\varphi \in S_c(\mathbb {R}_+)$ . Since $\varphi \in S_c(\mathbb {R}_+)$ , there exist $0<c_1<c_2$ and $c_3>0$ such that $\mathrm {Supp}(\varphi )\subset [c_1, c_2]$ and $|\varphi (y)|\leq c_3$ for any $y>0$ . Then, for each $1\leq j\leq m$ and $n+\kappa _j>0$ , we have

$$ \begin{align*} & |a_{f,j}(n)| (\mathcal{L}|\varphi_s|) (2\pi (n+\kappa_j)) \\ &\quad \leq c_3 |a_{f,j}(n)| e^{-2\pi (n+\kappa_j)c_1} (c_2-c_1)\max\{c_1^{\mathrm{Re}(s)-1}, c_2^{\mathrm{Re}(s)-1}\}. \end{align*} $$

This implies that for each $1\leq j\leq m$ , we have

$$ \begin{align*} & \sum_{n\in\mathbb{Z}\atop n+\kappa_j>0} |a_{f,j}(n)| (\mathcal{L}|\varphi_s|)(2\pi (n+\kappa_j)) \\ &\quad \leq c_3(c_2-c_1)\max\{c_1^{\mathrm{Re}(s)-1}, c_2^{\mathrm{Re}(s)-1}\}\sum_{n\in\mathbb{Z}\atop n+\kappa_j>0} |a_{f,j}(n)| e^{-2\pi (n+\kappa_j) c_1}. \end{align*} $$

Therefore, $\varphi _s\in \mathcal {F}_f$ , and $L_f(\varphi _s)$ is an analytic function on $s\in \mathbb {C}$ by Weierstrass theorem.

Recall that by Theorem 2.2, we have

$$\begin{align*}L_f(\varphi_s) =\int_0^\infty f(iy)\varphi_s(y) dy. \end{align*}$$

By the Mellin inversion formula, we have

$$\begin{align*}f(iy)\varphi(y) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_s) y^{-s} ds \end{align*}$$

for all $c\in \mathbb {R}$ and $y>0$ . By integration by parts, we see that for each $1\leq j\leq m$ ,

$$\begin{align*}|\langle L_f(\varphi_s),\mathbf{e}_j\rangle |\leq\frac{1}{|s|}\int_0^\infty\left|\frac{d}{dy} (f_j(iy)\varphi(y))\right| y^{\mathrm{Re}(s)} dy. \end{align*}$$

Therefore, $L_f(\varphi _s)\to 0$ as $\mathrm {Im}(s)\to \infty $ . From this, we have

$$ \begin{align*} f(iy)\varphi(y) &=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_s) y^{-s}ds=\frac{1}{2\pi i}\int_{k-c-i\infty}^{k-c+i\infty} L_f(\varphi_s) y^{-s}ds\\ &=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}) y^{-k+s} ds, \end{align*} $$

and by Theorem 2.2, we see that

(2.4) $$ \begin{align} f(iy)\varphi(y) =\frac{i^k}{2\pi i}\rho(S)\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}|_{2-k,\chi^{-1}} S)y^{-k+s}ds \end{align} $$

for any $y>0$ .

Since we have

$$\begin{align*}L_f(\varphi_{k-s}|_{2-k} S) =\int_0^\infty f(iy)(\varphi_{k-s}|_{2-k,\chi^{-1}} S)(y)dy =\int_0^\infty f(iy)\chi(S)\varphi\left(\frac 1y\right) y^{s-1} dy, \end{align*}$$

the Mellin inversion formula implies that

$$\begin{align*}f(iy)\chi(S)\varphi\left(\frac1y\right) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}|_{2-k,\chi^{-1}}S)y^{-s}dy \end{align*}$$

for all $c\in \mathbb {R}$ and $y>0$ . Therefore, we have

(2.5) $$ \begin{align} f\left(\frac{i}{y}\right)\chi(S)\varphi(y) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}|_{2-k,\chi^{-1}}S) y^{s} dy, \end{align} $$

and from (2.4) and (2.5), we see that

$$\begin{align*}f(iy)\varphi(y) = i^k y^{-k}\rho(S)\chi(S) f\left(\frac iy\right)\varphi(y) \end{align*}$$

for any $y>0$ . Since for each $y>0$ , there exists $\varphi \in S_c(\mathbb {R}_+)$ such $\varphi (y)\neq 0$ , we have

(2.6) $$ \begin{align} f(iy)= i^k y^{-k}\rho(S)\chi(S) f\left(\frac iy\right) \end{align} $$

for any $y>0$ . Since f is a holomorphic function, this implies that f satisfies

$$\begin{align*}f\left( -\frac{1}{\tau}\right) =\tau^k\chi(S)\rho(S) f(\tau) \end{align*}$$

for $\tau \in \mathbb {H}$ . Hence, f satisfies

$$\begin{align*}f|_{k,\chi,\rho}T = f \text{and}\ f|_{k,\chi,\rho}S = f. \end{align*}$$

Since T and S generates $\mathrm {SL}_2(\mathbb {Z})$ , we see that f is a weakly holomorphic modular form in $M^!_{k,\chi ,\rho }$ .

3 L-series of vector-valued harmonic weak Maass forms

In this section, we review basic definitions of vector-valued harmonic weak Maass forms and their L-series. We prove that the properties of L-series in Section 2 also hold in the case of vector-valued harmonic weak Maass forms. Moreover, we prove a converse theorem and a summation formula for vector-valued harmonic weak Maass forms.

Following [Reference Bruinier and Funke5, Reference Jin and Lim14], we introduce vector-valued harmonic weak Maass forms.

We use $H_{k,\chi ,\rho }$ to denote the space of such forms. We write $f^{+}$ (resp. $f^-$ ) for the first (resp. second) summation of (3.1) and call it the holomorphic (resp. non-holomorphic) part of f. By [Reference Bruinier and Funke5, Lemma 3.4], if $f\in H_{k,\chi ,\rho }$ with its Fourier expansion as in (3.1), then there is a constant $C>0$ such that the Fourier coefficients satisfy

$$ \begin{align*} c^+_{f,j}(n) &= O(e^{C\sqrt{n}}),\ n\to +\infty,\\ c^-_{f,j}(n) &= O(|n|^{k/2}),\ n\to-\infty. \end{align*} $$

Let $\mathcal {C}(\mathbb {R},\mathbb {C})$ be the space of piecewise smooth complex-valued functions on $\mathbb {R}$ . Suppose that $k\in \frac 12\mathbb {Z}$ and $f:\mathbb {H}\to \mathbb {C}^m$ is a vector-valued function on $\mathbb {H}$ given by the absolutely convergent series as in (3.1). Let $n_0\in \mathbb {N}$ be such that $f_j(\tau )$ are $O(e^{2\pi n_0 v})$ as $v=\mathrm {Im}(\tau )\to \infty $ for each $1\leq j\leq m$ . Let $\mathcal {F}_f$ be the space of functions $\varphi \in C(\mathbb {R},\mathbb {C})$ such that

1. (1) $(\mathcal {L}\varphi )(s)$ converges absolutely for all s with $\mathrm {Re}(s)\geq -2\pi n_0$ ,

2. (2) $(\mathcal \varphi _{2-k})(s)$ converges absolutely for all s with $\mathrm {Re}(s)>0$ ,

3. (3) for each $1\leq j\leq m$ , the series

   (3.2) $$ \begin{align} &\sum_{n\gg-\infty} |c^+_{f,j}(n)| (\mathcal{L} |\varphi|)(2\pi (n+\kappa_j))\\ \nonumber & +\sum_{n+\kappa_j<0} |c^-_{f,j}(n)| (4\pi |n+\kappa_j|)^{1-k}\int_0^\infty\frac{(\mathcal{L}|\varphi_{2-k}|)(-2\pi (n+\kappa_j)(2t+1))}{(1+t)^k} dt \end{align} $$
   converges.

For $\varphi \in \mathcal {F}_f$ , we define the L-series of f by

(3.3) $$ \begin{align} & L_f(\varphi) :=\sum_{j=1}^m\sum_{n\gg-\infty} c^+_{f,j}(n) (\mathcal{L}\varphi) (2\pi (n+\kappa_j))\quad \mathbf{e}_j\\ &\quad +\sum_{j=1}^m\sum_{n+\kappa_j<0} c^-_{f,j}(n)(-4\pi (n+\kappa_j))^{1-k}\int_0^\infty\frac{(\mathcal{L}\varphi_{2-k})(-2\pi (n+\kappa_j)(2t+1))}{(1+t)^k} dt\quad \mathbf{e}_j.\nonumber \end{align} $$

We also define the following derivative

$$\begin{align*}(\delta_k f)(\tau) :=\tau\frac{\partial f}{\partial u}(\tau) +\frac{k}{2} f(\tau). \end{align*}$$

Then, we have

$$ \begin{align*} (\delta_k f)(\tau) &=\frac{k}{2} f(\tau) +\sum_{j=1}^m\sum_{n\gg-\infty} c^+_{f,j}(n) (2\pi i(n+\kappa_j)\tau)e^{2\pi i(n+\kappa_j)\tau}\quad\mathbf{e}_j\\ &\quad +\sum_{j=1}^m\sum_{n+\kappa_j<0} c^-_{f,j}(n) (2\pi i(n+\kappa_j))\Gamma(1-k, -4\pi (n+\kappa_j)v) e^{2\pi i(n+\kappa_j)\tau}\quad\mathbf{e}_j. \end{align*} $$

Let $\mathcal {F}_{\delta _k f}$ be the space of functions $\varphi \in C(\mathbb {R},\mathbb {C})$ such that

1. (1) $(\mathcal {L}\varphi )(s)$ converges absolutely for all s with $\mathrm {Re}(s)\geq -2\pi n_0$ ,

2. (2) $(\mathcal \varphi _{3-k})(s)$ converges absolutely for all s with $\mathrm {Re}(s)>0$ ,

3. (3) for each $1\leq j\leq m$ the series

   (3.4) $$ \begin{align} \hspace{-1pc}&\sum_{n\gg-\infty} |c^+_{f,j}(n)(n+\kappa_j)| (\mathcal{L} |\varphi_2|)(2\pi (n+\kappa_j))\\ \nonumber & +\sum_{n+\kappa_j<0} |c^-_{f,j}(n)(n+\kappa_j)| (4\pi |n+\kappa_j|)^{1-k}\int_0^\infty\frac{(\mathcal{L}|\varphi_{3-k}|)(-2\pi (n+\kappa_j)(2t+1))}{(1+t)^k} dt \end{align} $$
   converges.

For $\varphi \in \mathcal {F}_{\delta _k f}$ , we define $L_{\delta _k f}(\varphi )$ by

(3.5) $$ \begin{align} \nonumber L_{\delta_k f}(\varphi) &:=\frac{k}{2}L_f(\varphi) -2\pi\sum_{j=1}^m\sum_{n\gg-\infty} c^+_{f,j}(n) (n+\kappa_j) (\mathcal{L}\varphi_2) (2\pi (n+\kappa_j))\quad \mathbf{e}_j\\ &\qquad -2\pi\sum_{j=1}^m\sum_{n+\kappa_j<0} c^-_{f,j}(n)(n+\kappa_j)(-4\pi (n+\kappa_j))^{1-k}\\ \nonumber &\qquad\qquad\times\int_0^\infty\frac{(\mathcal{L}\varphi_{3-k})(-2\pi (n+\kappa_j)(2t+1))}{(1+t)^k} dt\quad \mathbf{e}_j. \end{align} $$

Then, the series $L_f(\varphi )$ and $L_{\delta _k f}(\varphi )$ have integral representations.

Proof We only prove (1) since the same proof works for (2). For the holomorphic part of f, we have

$$ \begin{align*} &\sum_{j=1}^m\sum_{n\gg-\infty} c^+_{f,j}(n) (\mathcal{L}\varphi_j)(2\pi (n+\kappa_j))\mathbf{e}_j\\ &=\sum_{j=1}^m\sum_{n\gg-\infty} c^+_{f,j}(n)\int_0^\infty e^{-2\pi (n+\kappa_j)t}\varphi_j(t)dt\mathbf{e}_j\\ &=\sum_{j=1}^m\left(\int_0^\infty f^+_j(it)\varphi_j(t) dt\right)\mathbf{e}_j. \end{align*} $$

The last equality follows from the fact that we can interchange the order of summation and integration since $\varphi \in \mathcal {F}_f$ .

For the non-holomorphic part of f, note that

$$\begin{align*}\Gamma(a, z) = z^a e^{-z}\int_0^\infty\frac{e^{-zt}}{(1+t)^{1-a}}dt, \end{align*}$$

is valid for $\mathrm {Re}(z)>0$ [Reference Olver, Lozier, Boisvert and Clark21, (8.6.5)]. Therefore, for $n+\kappa _j<0$ , we have

$$ \begin{align*} &\int_0^\infty\Gamma(1-k, -4\pi(n+\kappa_j) y) e^{-2\pi (n+\kappa_j) u}\varphi(y) dy\\ &=\int_0^\infty (-4\pi(n+\kappa_j) y)^{1-k} e^{4\pi (n+\kappa_j)y} e^{-2\pi (n+\kappa_j) y}\varphi(y)\int_0^\infty\frac{e^{4\pi(n+\kappa_j)yt}}{(1+t)^k} dtdy\\ &= (-4\pi(n+\kappa_j))^{1-k} \int_0^\infty\int_0^\infty y^{1-k}e^{2\pi(n+\kappa_j)y(2t+1)}\varphi(y)dy\frac{1}{(1+t)^k}dt\\ & = (-4\pi(n+\kappa_j))^{1-k}\int_0^\infty\frac{(\mathcal{L}\varphi_{2-k})(-2\pi (n+\kappa_j)(2t+1))}{(1+t)^k} dt. \end{align*} $$

Therefore, we obtain

$$ \begin{align*} &\sum_{j=1}^m\sum_{n+\kappa_j<0} c^-_{f,j}(n)(-4\pi (n+\kappa_j))^{1-k}\int_0^\infty\frac{(\mathcal{L}\varphi_{2-k})(-2\pi (n+\kappa_j)(2t+1))}{(1+t)^k} dt\quad \mathbf{e}_j\\ &=\sum_{j=1}^m\sum_{n+\kappa_j<0} c^-_{f,j}(n)\int_0^\infty\Gamma(1-k, -4\pi(n+\kappa_j)y) e^{-2\pi (n+\kappa_j)y}\varphi(y)dy\mathbf{e}_j\\ &=\int_0^\infty f^-(iy)\varphi(y)dy. \end{align*} $$

We now prove the functional equations of $L_f(\varphi )$ and $L_{\delta _k f}(\varphi )$ .

Theorem 3.3 Let f be a vector-valued harmonic weak Maass form in $H_{k,\chi ,\rho }$ . We assume that the series (3.2) and (3.4) converge. Then, we have the following functional equations:

$$\begin{align*}L_f(\varphi) = i^k\rho(S) L_f(\varphi|_{2-k,\chi^{-1}} S) \end{align*}$$

and

$$\begin{align*}L_{\delta_k f}(\varphi) = -i^k\rho(S) L_{\delta_k f}(\varphi|_{2-k,\chi^{-1}} S). \end{align*}$$

Proof For the first equality, note that since $f\in H_{k,\chi ,\rho }$ , we see that

$$\begin{align*}f(iy) = (iy)^{-k}\chi^{-1}(S)\rho(S)^{-1} f\left(i\frac1y\right) \end{align*}$$

for $y>0$ . Then, by Theorem 3.2, we have

$$ \begin{align*} L_f(\varphi) &=\int_0^\infty f(iy)\varphi(y)dy\\ &=\int_0^\infty f\left( i\frac 1y\right)\varphi\left(\frac 1y\right) y^{-2}dy\\ &=\int_0^\infty (iy)^k\chi(S)\rho(S) f(iy)\varphi\left(\frac 1y\right) y^{-2}dy\\ &= i^k\rho(S)\int_0^\infty f(iy) y^{k-2}\chi(S)\varphi\left(\frac 1y\right) dy\\ & = i^k\rho(S) L_f(\varphi|_{2-k,\chi^{-1}}S). \end{align*} $$

For the second equality, note that

$$ \begin{align*} (\delta_k(f|_k S))(\tau) &=\tau (-k)\tau^{-k-1}\chi^{-1}(S)\rho(S)^{-1} f\left(-\frac1\tau\right) +\tau\tau^{-k}\chi^{-1}(S)\rho(S)^{-1}\frac{\partial f}{\partial u}\left(-\frac 1\tau\right)\frac{1}{\tau^2}\\ &\quad+\frac k2\tau^{-k}\chi^{-1}(S)\rho(S)^{-1} f\left( -\frac 1\tau\right)\\ &= -\frac k2\tau^{-k}\chi^{-1}(S)\rho(S)^{-1} f\left( -\frac 1\tau\right) -\tau^{-k}\chi^{-1}(S)\rho(S)^{-1}\frac{\partial f}{\partial u}\left( -\frac 1\tau\right)\frac 1\tau\\ &= - (\delta_k(f)|_k S)(\tau). \end{align*} $$

Therefore, we have

$$\begin{align*}((\delta_k f)|S)(\tau) = -(\delta_k(f|_k S))(\tau) = -(\delta_k f)(\tau). \end{align*}$$

From this and Theorem 3.2, we see that

$$ \begin{align*} L_{\delta_k f}(\varphi) &=\int_0^\infty (\delta_k f)(iy)\varphi(iy) dy\\ &=\int_0^\infty (\delta_k f)\left( i\frac 1y\right)\varphi\left(\frac1y\right) y^{-2} dy\\ &= i^k\rho(S)\int_0^\infty (\delta_k f|_k S)(iy) (\varphi|_{2-k,\chi^{-1}}S)(y)dy\\ &= -i^k\rho(S)\int_0^\infty (\delta_k f)(iy) (\varphi_{2-k,\chi^{-1}}S)(y)dy\\ &=-i^k\rho(S) L_{\delta_k f}(\varphi|_{2-k,\chi^{-1}}S). \end{align*} $$

Recall that $\varphi _s(x) =\varphi (x) x^{s-1}$ . We define the series $L(s,f,\varphi )$ by

$$\begin{align*}L(s,f,\varphi) := L_f(\varphi_s). \end{align*}$$

Then, we prove that the series $L(s,f,\varphi )$ has an analytic continuation and satisfies a functional equation.

Theorem 3.4 Let $f\in H_{k,\chi ,\rho }$ , and $n_0\in \mathbb {N}$ be such that $f_j(\tau )$ are $O(e^{2\pi n_0 v})$ as $v=\mathrm {Im}(\tau )\to \infty $ for each $1\leq j\leq m$ . Suppose that $\varphi \in \mathcal {C}(\mathbb {R},\mathbb {C})$ is a non-zero function such that, for some $\epsilon>0$ , $\varphi (x)$ and $\varphi (x^{-1})$ are $o(e^{-2\pi (n_0+\epsilon )x})$ as $x\to \infty $ . Then the series $ L(s,f,\varphi ) $ converges absolutely for $\mathrm {Re}(s)>\frac 12$ , has an analytic continuation to all $s\in \mathbb {C}$ and satisfies the functional equation

$$\begin{align*}L(s,f,\varphi) = i^k\rho(S) L(1-s, f,\varphi|_{1-k,\chi^{-1}} S). \end{align*}$$

Proof By the growth of $\varphi $ , we see that $\mathcal {L}(|\varphi |^2)(y)$ converges absolutely for $y\geq -2\pi n_0$ . For $y>0$ and $s\in \mathbb {C}$ with $\mathrm {Re}(s)>\frac 12$ , Cauchy–Schwarz inequality implies that

$$\begin{align*}(\mathcal{L}|\varphi_s|)(y)\leq (\mathcal{L}(|\varphi|^2)(y))^{\frac12} y^{-\mathrm{Re}(s)+\frac12} (\Gamma(2\mathrm{Re}(s)-1))^{\frac12}. \end{align*}$$

Therefore, $\varphi _s\in \mathcal {F}_f$ for $\mathrm {Re}(s)>\frac 12$ .

Recall that

$$\begin{align*}f(iy) = (iy)^{-k}\chi^{-1}(S)\rho(S)^{-1} f\left(i\frac1y\right) \end{align*}$$

for $y>0$ . Therefore, we have

$$ \begin{align*} L(s, f,\varphi) &= L_f(\varphi_s) =\int_0^\infty f(iy)\varphi_s(y)dy\\ &=\int_0^\infty f(iy)\varphi(s) y^{s-1} dy\\ &=\int_1^\infty f(iy)\varphi(y)y^{s-1} dy +\int_0^1 f(iy)\varphi(y)y^{s-1}dy\\ &=\int_1^\infty f(iy)\varphi(y)y^{s-1} dy +\int_1^\infty f\left( i\frac1y\right)\varphi\left(\frac 1y\right) y^{1-s}y^{-2}dy\\ &=\int_1^\infty f(iy)\varphi(y)y^{s-1} dy + i^k\rho(S)\int_1^\infty f(iy)\left(\varphi|_{1-k,\chi^{-1}}S\right)(y) y^{-s}dy. \end{align*} $$

By the growth of $\varphi $ at $0$ and $\infty $ , we see that the integrals

$$\begin{align*}\int_1^\infty f(iy)\varphi(y)y^{s-1} dy \quad \text{and} \int_1^\infty f(iy)\left(\varphi|_{1-k,\chi^{-1}}S\right)(y) y^{-s}dy \end{align*}$$

are well-defined for all $s\in \mathbb {C}$ , and give holomorphic functions. Since $f\in H_{k,\chi ,\rho }$ , we obtain that $\rho (-I_2)\chi (-I_2) = (-1)^{-k} I_m$ . Therefore, we get the desired functional equation.

We now state the converse theorem in the case of vector-valued harmonic weak Maass forms.

Theorem 3.5 For each $1\leq j\leq m$ , let $(c^+_{f,j}(n))_{n\geq -n_0}$ and $(c^-_{f,j}(n))_{n+\kappa _j<0}$ be sequences of complex numbers such that $c^+_{f,j}(n), c^-_{f,j}(n) = O(e^{C\sqrt {|n|}})$ as $|n|\to \infty $ , for some $C>0$ . For each $\tau \in \mathbb {H}$ , set

$$\begin{align*}f(\tau) & :=\sum_{j=1}^m\sum_{n=-n_0}^\infty c^+_{f,j}(n)e^{2\pi i(n+\kappa_j)\tau}\mathbf{e}_j \\ &\quad +\sum_{j=1}^m\sum_{n+\kappa_j<0}c^-_{f,j}(n)\Gamma(1-k, -4\pi (n+\kappa_j)v)e^{2\pi i(n+\kappa_j)\tau}\mathbf{e}_j. \end{align*}$$

Suppose that for each $\varphi \in S_c(\mathbb {R}_+)$ , the functions $L_f(\varphi )$ and $L_{\delta _k f}(\varphi )$ satisfy

$$\begin{align*}L_f(\varphi) = i^k\rho(S) L_f(\varphi|_{2-k,\chi^{-1}} S) \end{align*}$$

and

$$\begin{align*}L_{\delta_k f}(\varphi) = -i^k\rho(S) L_{\delta_k f}(\varphi|_{2-k,\chi^{-1}}S). \end{align*}$$

Then, f is a vector-valued harmonic weak Maass form in $H_{k,\chi ,\rho }$ .

Proof By the bounds for $c^+_{f,j}(n)$ and $c^-_{f,j}(n)$ , we see that $f_j(\tau )$ and $(\delta _k(f))_j$ converge absolutely to smooth functions on $\mathbb {H}$ for $1\leq j\leq m$ . Note that $\varphi _s\in S_c(\mathbb {R}_+)$ for any $s\in \mathbb {C}$ and $\varphi \in S_c(\mathbb {R}_+)$ . Since $\varphi \in S_c(\mathbb {R}_+)$ , there exist $0<c_1<c_2$ and $c_3>0$ such that $\mathrm {Supp}(\varphi )\subset [c_1, c_2]$ and $|\varphi (y)|\leq c_3$ for any $y>0$ . Then, for each $1\leq j\leq m$ and $n+\kappa _j>0$ , we have

$$ \begin{align*} & |c^+_{f,j}(n)| (\mathcal{L}|\varphi_s|) (2\pi (n+\kappa_j)) \\ &\quad \leq c_3 |c^+_{f,j}(n)| e^{-2\pi (n+\kappa_j)c_1} (c_2-c_1)\max\{c_1^{\mathrm{Re}(s)-1}, c_2^{\mathrm{Re}(s)-1}\}. \end{align*} $$

This implies that for each $1\leq j\leq m$ , we have

$$ \begin{align*} & \sum_{n\in\mathbb{Z}\atop n+\kappa_j>0} |c^+_{f,j}(n)| (\mathcal{L}|\varphi_s|)(2\pi (n+\kappa_j)) \\ &\quad\leq c_3(c_2-c_1)\max\{c_1^{\mathrm{Re}(s)-1}, c_2^{\mathrm{Re}(s)-1}\}\sum_{n\in\mathbb{Z}\atop n+\kappa_j>0} |c^+_{f,j}(n)| e^{-2\pi (n+\kappa_j) c_1}. \end{align*} $$

For each $1\leq j\leq m$ and $n+\kappa _j<0$ , we have

$$ \begin{align*} (\mathcal{L}|\varphi_{s+1-k}|)(-2\pi (n+\kappa_j)(2t+1)) &\ll\int_{c_1}^{c_2} e^{2\pi(n+\kappa_j)(2t+1)} y^{\mathrm{Re}(s)-k} dy\\ &\ll e^{2\pi(n+\kappa_j)c_1(2t+1)}\max\{c_1^{\mathrm{Re}(s)-k}, c_2^{\mathrm{Re}(s)-k}\}. \end{align*} $$

This implies that

$$ \begin{align*} &\sum_{n+\kappa_j<0} |c_{f,j}^-(n)| |4\pi (n+\kappa_j)|^{1-k}\int_0^\infty\frac{(\mathcal{L}|\varphi_{s+1-k})(-2\pi (n+\kappa_j)(2t+1)}{(1+t)^k} dt\\ & \quad \ll\max\{c_1^{\mathrm{Re}(s)-k}, c_2^{\mathrm{Re}(s)-k}\}|4\pi (n+\kappa_j)|^{1-k} \\ &\quad \times \int_0^\infty\frac{e^{-4\pi t c_1 n_1}}{(1+t)^k}dt\sum_{n+\kappa_j<0} e^{2\pi (n+\kappa_j)c_1} |c_{f,j}^-(n)|, \end{align*} $$

where $n_1$ is defined by

$$\begin{align*}n_1 := \begin{cases} 1 &\text{if } \kappa_j = 0,\\ \kappa_j &\text{if } \kappa_j\neq0. \end{cases} \end{align*}$$

Therefore, $\varphi _s\in \mathcal {F}_f$ , and $L_f(\varphi _s)$ is an analytic function on $s\in \mathbb {C}$ by Weierstrass theorem. In the same way, we see that $L_{\delta _k f}(\varphi _s)$ is an analytic function for $s\in \mathbb {C}$ .

Recall that by Theorem 3.2, we have

$$\begin{align*}L_f(\varphi_s) =\int_0^\infty f(iy)\varphi_s(y) dy \end{align*}$$

and

$$\begin{align*}L_{\delta_k f}(\varphi) =\int_0^\infty (\delta_k f)(iy)\varphi(y) dy. \end{align*}$$

By the Mellin inversion formula, we have

$$\begin{align*}f(iy)\varphi(y) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_s) y^{-s} ds \end{align*}$$

and

$$\begin{align*}(\delta_k f)(iy)\varphi(y) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_{\delta_k f}(\varphi_s) y^{-s}ds \end{align*}$$

for all $c\in \mathbb {R}$ and $y>0$ .

By integration by parts, we see that for each $1\leq j\leq m$ ,

$$\begin{align*}|\langle L_f(\varphi_s),\mathbf{e}_j\rangle |\leq\frac{1}{|s|}\int_0^\infty\left|\frac{d}{dy} (f_j(iy)\varphi(y))\right| y^{\mathrm{Re}(s)} dy. \end{align*}$$

Therefore, $L_f(\varphi _s)\to 0$ as $\mathrm {Im}(s)\to \infty $ . The corresponding fact for $L_{\delta _k f}(\varphi _s)$ can be proved in the same way. From this, we have

$$ \begin{align*} f(iy)\varphi(y) &=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_s) y^{-s}ds=\frac{1}{2\pi i}\int_{k-c-i\infty}^{k-c+i\infty} L_f(\varphi_s) y^{-s}ds\\ &=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}) y^{-k+s} ds, \end{align*} $$

and by Theorem 3.3, we see that

(3.6) $$ \begin{align} f(iy)\varphi(y) =\frac{i^k}{2\pi i}\rho(S)\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}|_{2-k,\chi^{-1}} S)y^{-k+s}ds \end{align} $$

for any $y>0$ . In the same way, we have

$$ \begin{align*} (\delta_k f)(iy)\varphi(y) &=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_{\delta_k(f)}(\varphi_{k-s}) y^{-k+s}ds\\ &= -\frac{i^k}{2\pi i}\rho(S)\int_{c-i\infty}^{c+i\infty} L_{\delta_k(f)}(\varphi_{k-s}|_{2-k,\chi^{-1}}S)y^{-k+s}ds \end{align*} $$

for any $y>0$ .

Since we have

$$\begin{align*}L_f(\varphi_{k-s}|_{2-k} S) =\int_0^\infty f(iy)(\varphi_{k-s}|_{2-k,\chi^{-1}} S)(y)dy =\int_0^\infty f(iy)\chi(S)\varphi\left(\frac 1y\right) y^{s-1} dy, \end{align*}$$

the Mellin inversion formula implies that

$$\begin{align*}f(iy)\chi(S)\varphi\left(\frac1y\right) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}|_{2-k,\chi^{-1}}S)y^{-s}dy \end{align*}$$

for all $c\in \mathbb {R}$ and $y>0$ . Therefore, we have

(3.7) $$ \begin{align} f\left(\frac{i}{y}\right)\chi(S)\varphi(y) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_f(\varphi_{k-s}|_{2-k,\chi^{-1}}S) y^{s} dy, \end{align} $$

and from (3.6) and (3.7), we see that

$$\begin{align*}f(iy)\varphi(y) = i^k y^{-k}\rho(S)\chi(S) f\left(\frac iy\right)\varphi(y) \end{align*}$$

for any $y>0$ . Since for each $y>0$ , there exists $\varphi \in S_c(\mathbb {R}_+)$ such $\varphi (y)\neq 0$ , we have

(3.8) $$ \begin{align} f(iy)= i^k y^{-k}\rho(S)\chi(S) f\left(\frac iy\right) \end{align} $$

for any $y>0$ . In the same way, we see that

$$ \begin{align*} L_{\delta_k f}(\varphi_{k-2}|_{2-k,\chi^{-1}}S) =\int_0^\infty (\delta_k f)(iy)\chi(S)\varphi\left(\frac 1y\right) y^{s-1}dy, \end{align*} $$

and hence we have

$$ \begin{align*} (\delta_k f)(iy)\left(i\frac 1y\right)\chi(S)\varphi(y) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} L_{\delta_k f}(\varphi_{k-s}|_{2-k,\chi^{-1}}S)y^s dy. \end{align*} $$

Therefore, we obtain

(3.9) $$ \begin{align} (\delta_k f)(iy) = -i^k y^{-k}\rho(S)\chi(S) (\delta_k f)\left( i\frac 1y\right). \end{align} $$

We now define

$$\begin{align*}F(\tau) = f(\tau) - (f|_{k,\chi,\rho}S)(\tau) \end{align*}$$

for $\tau \in \mathbb {H}$ . By (3.8) and (3.9), we see that $F(iv) = 0$ and $\frac {\partial }{\partial u} F(iv) = 0$ . This implies that $F\equiv 0$ since F is an eigenfunction of the Laplace operator. Therefore,

$$\begin{align*}f = f|_{k,\chi,\rho} S. \end{align*}$$

Since T and S generates $\mathrm {SL}_2(\mathbb {Z})$ , we see that f is a harmonic weak Maass form in $H_{k,\chi ,\rho }$ .

We define the operator $\xi _{2-k}$ by

$$\begin{align*}\xi_{2-k}:=2iv^{2-k}\overline{\frac{\partial}{\partial\overline{\tau}}}. \end{align*}$$

Then, it defines a surjective map [Reference Bruinier and Funke5, Proposition 3.2]

$$ \begin{align*} \xi_{2-k}: H_{2-k,\chi,\rho}\to S_{k,\overline{\chi},\overline{\rho}}. \end{align*} $$

Note that if $f\in S_{k,\overline {\chi },\overline {\rho }}$ , then f has a Fourier expansion of the form

(3.10) $$ \begin{align} f(\tau) =\sum_{j=1}^m\sum_{n-\kappa_j>0} a_{f,j}(n)e^{2\pi i(n-\kappa_j)\tau}\mathbf{e}_j. \end{align} $$

Let $\mathcal {C}_c^\infty (\mathbb {R},\mathbb {R})$ be the space of piecewise smooth, compactly supported functions on $\mathbb {R}$ with values in $\mathbb {R}$ . Then, we have the following summation formula for harmonic weak Maass forms.

Theorem 3.6 Let $k\in 2\mathbb {N}$ and let $f\in S_{k,\bar {\chi },\bar {\rho }}$ with Fourier expansion as in (3.10). Suppose that g is an element of $H_{2-k,\chi ,\rho }$ such that $\xi _{2-k}(g) = f$ with Fourier expansion

$$\begin{align*}g(\tau) &=\sum_{j=1}^m\sum_{n\gg-\infty} c_{g,j}^+(n) e^{2\pi i(n+\kappa_j)\tau}\mathbf{e}_j \\&\quad +\sum_{j=1}^m\sum_{n+\kappa_j<0} c_{g,j}^-(n)\Gamma(k-1, -4\pi (n+\kappa_j) v) e^{2\pi i(n+\kappa_j)\tau}\mathbf{e}_j. \end{align*}$$

For every $\varphi \in \mathcal {C}_c^\infty $ and $1\leq j\leq m$ , we have

$$ \begin{align*} &\sum_{n\gg-\infty} c_{g,j}^+(n)\int_0^\infty\varphi(y)\left( e^{-2\pi (n+\kappa_j)y} -(-iy)^{k-2}e^{-2\pi(n+\kappa_j)/y}\right)dy\\ &=\sum_{l=0}^{k-2}\sum_{n-\kappa_j>0}\overline{a_{f,j}(n)}\bigg(\frac{(k-2)!}{l!} (4\pi (n-\kappa_j))^{1-k+l}\int_0^\infty e^{-2\pi(n-\kappa_j)y}y^l\varphi(y) dy\\ &+\frac{2^{l+1}}{(k-1)} (8\pi (n-\kappa_j))^{-\frac{k}{2}}\int_0^\infty e^{-\pi (n-\kappa_j)y}y^{\frac k2 - 1}\varphi(y) M_{1-\frac k2 + l,\frac{k-1}2}(2\pi (n-\kappa_j)y)dy\bigg), \end{align*} $$

where $M_{\kappa ,\mu }(z)$ denotes the Whittaker hypergeometric function (see [Reference Olver, Lozier, Boisvert and Clark21, Section 13.14]).

Proof Note that $\mathcal {C}_c^\infty (\mathbb {R},\mathbb {R})\subset \mathcal {F}_f\cap \mathcal {F}_g$ . Since $a_{f,j}(n) = -\overline {c_{g,j}^-(n)}(4\pi (n-\kappa _j))^{k-1}$ for $n-\kappa _j>0$ , by Theorem 3.2, we have

$$ \begin{align*} L_g(\varphi) &=\int_0^\infty g^+(iy)\varphi(y)dy -\overline{L_f(\Phi(\varphi))}, \end{align*} $$

where $\Phi (\varphi )$ is defined by

$$\begin{align*}\Phi(\varphi):=\mathcal{L}^{-1}\left( (2u)^{1-k}\int_0^\infty\Gamma(k-1, 2uy)e^{uy}\varphi(y)dy\right). \end{align*}$$

Recall that $L_g(\varphi )$ satisfies a functional equation

$$\begin{align*}L_g(\varphi) = i^{2-k}\rho(S) L_g(\varphi|_{k,\chi^{-1}}S). \end{align*}$$

Therefore, we have

$$\begin{align*} & \int_0^\infty g^+(iy)\varphi(y)dy - i^{2-k}\rho(S)\int_0^\infty g^+(iy) (\varphi|_{k,\chi^{-1}}S)(y)dy \\ &\quad =\overline{L_f(\Phi(\varphi))} -i^{2-k}\rho(S)\overline{L_f(\Phi(\varphi|_{k,\chi^{-1}}S))}. \end{align*}$$

Note that by the functional equation of $L_f(\varphi )$ , we have

$$ \begin{align*} i^{2-k}\rho(S)\overline{L_f(\Phi(\varphi|_{k,\chi^{-1}}S))} &= i^{2-k}\rho(S)\overline{i^k\overline{\rho(S)} L_f(\Phi(\varphi|_{k,\chi^{-1}}S)|_{2-k,\chi}S)}\\ &= -\rho(-I)\overline{ L_f(\Phi(\varphi|_{k,\chi^{-1}}S)|_{2-k,\chi}S)}. \end{align*} $$

By [Reference Erdélyi, Magnus, Oberhettinger and Tricomi12, 4.1 (25)], we have

$$ \begin{align*} \mathcal{L}\left(u^{v-1} f\left(\frac 1u\right)\right)(x) = x^{-\frac12 v}\int_0^\infty u^{\frac12 v}J_v\left(2u^{\frac12}x^{\frac12}\right)\mathcal{L}(f)(u)du \end{align*} $$

for $\mathrm {Re}(v)>-1$ , where $J_v(z)$ denotes the Bessel function defined in [Reference Olver, Lozier, Boisvert and Clark21, Section 10.2.2]. Therefore, we see that

(3.11) $$ \begin{align} &\mathcal{L}(\Phi(\varphi|_{k,\chi^{-1}}S)|_{2-k,\chi^{-1}} S)(2\pi (n-\kappa_j)) \nonumber \\ &=\mathcal{L}\left(\chi^{-1}(S) x^{k-2}\mathcal{L}^{-1}\left((2u)^{1-k}\int_0^\infty\Gamma(k-1, 2uy) e^{uy}\varphi\left(\frac 1y\right)\chi(S) y^{-k} dy\right)\left(\frac 1x\right)\right) \nonumber \\ &\quad\times (2\pi (n-\kappa_j)\nonumber \\ &= (2\pi (n-\kappa_j))^{-\frac12(k-1)}\chi^{-1}(S)\int_0^\infty u^{\frac12 (k-1)} J_{k-1}\left(2u^{\frac12}(2\pi (n-\kappa_j))^{\frac12}\right)\nonumber \\ &\quad\times (2u)^{1-k}\int_0^\infty\Gamma(k-1, 2uy) e^{uy}\varphi\left(\frac 1y\right)\chi(S) y^{-k}dydu\nonumber \\ &= (2\pi (n-\kappa_j))^{-\frac12(k-1)}\chi^{-1}(S)\int_0^\infty u^{\frac12 (k-1)} J_{k-1}\left(2u^{\frac12}(2\pi (n-\kappa_j))^{\frac12}\right)\nonumber \\ &\quad\times (2u)^{1-k}\int_0^\infty\Gamma(k-1, 2u/y) e^{u/y}\varphi\left( y\right)\chi(S) y^{k-2}dydu \nonumber \\ &= (8\pi (n-\kappa_j))^{\frac 12 (1-k)}\int_0^\infty\varphi(y)y^{k-2} \nonumber \\ & \quad \times \int_0^\infty u^{\frac12 (1-k)} J_{k-1}\left(\sqrt{8\pi (n-\kappa_j) u}\right)\Gamma(k-1, 2u/y)e^{u/y} dudy. \nonumber\\ \end{align} $$

Note that by [Reference Olver, Lozier, Boisvert and Clark21, (8.4.8)] we have

$$\begin{align*}\Gamma(n+1, z) = n! e^{-z}\sum_{l=0}^n\frac{z^l}{l!}. \end{align*}$$

Therefore, (3.11) is equal to

$$ \begin{align*} &(8\pi (n-\kappa_j))^{\frac12 (1-k)} (k-2)! \\ &\qquad \times\sum_{l=0}^{k-2}\frac{2^l}{l!}\int_0^\infty\varphi(y)y^{k-2-l}\int_0^\infty u^{\frac12 (1-k)+l} J_{k-1}\left(\sqrt{8\pi (n-\kappa_j) u}\right) e^{-u/y} dudy\\ &\quad = (8\pi (n-\kappa_j))^{\frac12 (1-k)} (k-2)!\\ &\qquad\times\sum_{l=0}^{k-2}\frac{2^{l+1}}{l!}\int_0^\infty\varphi(y)y^{k-2-l}\int_0^\infty u^{2-k+2l} J_{k-1}\left(\sqrt{8\pi (n-\kappa_j) } u\right) e^{-u^2/y} dudy. \end{align*} $$

By [Reference Erdélyi, Magnus, Oberhettinger and Tricomi12, (8.4.8)], we have

$$\begin{align*}\int_0^\infty e^{-\beta^2 x^2} J_v(ax) x^{s-1}dx =\frac{\Gamma\left(\frac 12v +\frac12 s\right)}{a\beta^{s-1}\Gamma(v+1)} e^{-\frac{a^2}{8\beta^2}} M_{\frac12 s-\frac12,\frac12 v}\left(\frac{a^2}{4\beta^2}\right) \end{align*}$$

for $|\mathrm {arg}\beta |<\frac {\pi }{4}$ and $\mathrm {Re}(s)> -\mathrm {Re}(v)$ . Therefore, (3.11) is equal to

$$ \begin{align*} \frac{(8\pi (n-\kappa_j))^{\frac {-k}{2}}}{k-1}\sum_{l=0}^{k-2} 2^{l+1}\int_0^\infty\varphi(y)y^{\frac k2-1} e^{-\pi (n-\kappa_j)y} M_{1+l-\frac k2,\frac12 (k-1)}(2\pi (n-\kappa_j)y) dy. \end{align*} $$

We can also see that

$$ \begin{align*} &\mathcal{L}(\Phi(\varphi))(2\pi(n-\kappa_j))\\ &\quad = (4\pi(n-\kappa_j))^{1-k}\int_0^\infty\Gamma(k-1, 4\pi(n-\kappa_j)y)e^{2\pi(n-\kappa_j)y}\varphi(y)dy\\ &\quad=\sum_{l=0}^{k-2}\frac{(k-2)!}{l!} (4\pi (n-\kappa_j))^{1-k+l}\int_0^\infty e^{-2\pi(n+\kappa_j)y} y^l\varphi(y) dy.\\[-34pt] \end{align*} $$

4 L-series of harmonic Maass Jacobi forms

In this section, we consider the case of Jacobi forms and prove a converse theorem as well. We review basic notions of Jacobi forms (for more details, see [Reference Choi and Lim7, Section 3.1] and [Reference Eichler and Zagier11, Section 5]).

Let k be a positive even integer and m be a positive integer. From now on, we use the notation $\tau = u+iv\in \mathbb {H}$ and $z = x+iy\in \mathbb {C}$ . Let F be a complex-valued function on $\mathbb {H}\times \mathbb {C}$ . For $\gamma =\left (\begin {smallmatrix} a&b\\c&d\end {smallmatrix}\right )\in \mathrm {SL}_2(\mathbb {Z}) , X = (\lambda ,\mu )\in \mathbb {Z}^2$ , we define

$$\begin{align*}(F|_{k,m}\gamma)(\tau,z) := (c\tau+d)^{-k}e^{-2\pi im\frac{cz^2}{c\tau+d}}F(\gamma(\tau,z))\end{align*}$$

and

$$\begin{align*}(F|_m X)(\tau,z) :=e^{2\pi i m (\lambda^2\tau + 2\lambda z)}F(\tau,z+\lambda\tau+\mu),\end{align*}$$

where $\gamma (\tau ,z) = (\frac {a\tau +b}{c\tau +d},\frac {z}{c\tau +d})$ .

We now define a Jacobi form.

We denote by $J^!_{k,m}$ the space of all weakly holomorphic Jacobi forms of weight k and index m on $\mathrm {SL}_2(\mathbb {Z})$ . If a Jacobi form satisfies the condition $a(l,r)\neq 0$ only if $4ml - r^2\geq 0$ (resp. $4ml-r^2>0$ ), then it is called a Jacobi form (resp. Jacobi cusp form). We denote by $J_{k,m}$ (resp. $S_{k,m}$ ) the space of all Jacobi forms (resp., Jacobi cusp forms) of weight k and index m on $\mathrm {SL}_2(\mathbb {Z})$ .

We now recall some definitions and facts about harmonic Maass–Jacobi forms, which were introduced in [Reference Bringmann, Raum and Richter3] and [Reference Bringmann and Richter4]. The Casimir operators $\mathcal {C}_{k,m}$ are defined by

$$ \begin{align*} \mathcal{C}_{k,m} &:=-2(\tau-\overline{\tau})^2\partial_{\tau\overline{\tau}}-(2k-1)(\tau-\overline{\tau})\partial_{\overline{\tau}}+\frac{(\tau-\overline{\tau})^2}{4\pi i m}\partial_{\tau z z}+\frac{k(\tau-\overline{\tau})}{4\pi i m}\partial_{z\overline{z}}\\ &\quad +\frac{(\tau-\overline{\tau})(z-\overline{z})}{4\pi i m}\partial_{zz\overline{z}}-2(\tau-\overline{\tau})(z-\overline{z})\partial_{\tau\overline{z}}+(1-k)(z-\overline{z})\partial_{\overline{z}}\\ &\quad +\frac{(\tau-\overline{\tau})^2}{4\pi i m}\partial_{\tau\overline{z}\overline{z}}+\left(\frac{(z-\overline{z})^2}{2}+\frac{k(\tau-\overline{\tau})}{4\pi i m}\right)\partial_{\overline{z}\overline{z}}+\frac{(\tau-\overline{\tau})(z-\overline{z}) }{4\pi i m}\partial_{z\overline{z}\overline{z}}. \end{align*} $$

We use $\hat {J}_{k,m}$ to denote the space of such forms. Any $F\in \hat {J}_{k,m}$ has a Fourier expansion of the form

$$\begin{align*} F(\tau,z) &=\sum_{l,r\in\mathbb{Z}\atop 4ml-r^2\gg -\infty} c^+_{F}(l,r)e^{2\pi il\tau} e^{2\pi irz} \nonumber \\ &\quad +\sum_{l,r\in\mathbb{Z}\atop 4ml-r^2>0} c^-_{F}(l,r)\Gamma\left(1-k,\frac{(4ml-r^2)\pi v}{m}\right)e^{2\pi il\tau} e^{2\pi irz}. \end{align*}$$

A harmonic Maass–Jacobi form has the theta decomposition (see [Reference Bringmann, Raum and Richter3, Section 5], [Reference Bringmann and Richter4, Section 6], and [Reference Eichler and Zagier11, Section 5])

(4.2) $$ \begin{align} F(\tau,z)=\sum_{j=1}^{2m}F_j(\tau)\theta_{m,j}(\tau,z), \end{align} $$

where $\theta _{m,j}$ is defined by

$$\begin{align*}\theta_{m,j}(\tau,z):=\sum_{r\equiv j\ \pmod{2m}} e^{\pi ir^2/(2m)} e^{2\pi irz}. \end{align*}$$

Moreover, $\sum _{j=1}^{2m} F_j\mathbf {e}_j$ is a vector-valued harmonic weak Maass form. To explain this, we recall the definition of metaplectic groups. The metaplectic group $\mathrm {Mp}_2(\mathbb {R})$ consists of pairs $(g,\omega (\tau ))$ , where $g=\left (\begin {smallmatrix} a&b\\c&d\end {smallmatrix}\right )\in \mathrm {SL}_2(\mathbb {R})$ and $\omega :\mathbb {H}\to \mathbb {C}$ is a holomorphic function satisfying $\omega (\tau )^2= c\tau +d$ , with group law

$$\begin{align*}(g,\omega(\tau))(g',\omega'(\tau)):=(gg', (\omega\circ g')(\tau)\omega'(\tau)). \end{align*}$$

We use $\mathrm {Mp}_2(\mathbb {Z})$ to denote the inverse image of $\mathrm {SL}_2(\mathbb {Z})$ in $\mathrm {Mp}_2(\mathbb {R})$ . Let m be a positive integer. We also recall the Weil representation $\rho _m:\mathrm {Mp}_2(\mathbb {Z})\to \mathrm {GL}_{2m}(\mathbb {C})$ given by

$$ \begin{align*} &\rho_m(\tilde{T})\mathbf{e}_l:=e_{4m}(l^2)\mathbf{e}_l,\\ &\rho_m(\tilde{S})\mathbf{e}_l:=\frac{1}{\sqrt{2im}}\sum_{l'=1}^{2m}e_{2m}(-ll')\mathbf{e}_{l'}, \end{align*} $$

where we use the notation $e_{m}(w):=e^{\frac {2\pi i w}{m}}$ . Here, $\tilde {T}:=\big (\left (\begin {smallmatrix} 1&1\\0&1\end {smallmatrix}\right ) ,1\big )$ and $\tilde {S}:=\big (\left (\begin {smallmatrix} 0&{-1}\\1&0\end {smallmatrix}\right ),\sqrt {\tau }\big )$ are two generators of $\mathrm {Mp}_2(\mathbb {Z})$ . Throughout this article, we use the convention that $\sqrt {\tau }$ is chosen so that $\arg (\sqrt {\tau })\in (-\pi /2,\pi /2]$ . The map

$$\begin{align*}\left(\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\right)\mapsto\widetilde{\left(\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\right)} = (\left(\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\right),\sqrt{c\tau+d})\end{align*}$$

defines a locally isomorphic embedding of $\mathrm {SL}_2(\mathbb {R})$ into $\mathrm {Mp}_2(\mathbb {R})$ . We then define a representation $\rho ^{\prime }_{m} :\mathrm {SL}_2(\mathbb {Z})\to \mathrm {GL}_{2m}(\mathbb {C})$ by

$$\begin{align*}\rho_{m}'(\gamma) :=\rho_m(\widetilde{\gamma})\chi_\eta(\gamma)\end{align*}$$

for $\gamma \in \mathrm {SL}_2(\mathbb {Z})$ . It is known ([Reference Choi and Lim7], p.281) that $\rho _m'$ is unitary representation of $\mathrm {SL}_2(\mathbb {Z})$ . The theta decomposition induces an isomorphism $\phi _{k,m}$ between $H_{k-\frac 12,\overline {\rho _m},\overline {\chi _\eta }}$ and $\hat {J}_{k,m}$ (see [Reference Bringmann, Raum and Richter3, Section 5] and [Reference Cho and Choie6]):

$$\begin{align*}F\mapsto\sum_{j=1}^{2m} F_j(\tau)\theta_{m,j}(\tau,z), \end{align*}$$

where $\chi _\eta $ is the eta-multiplier system defined by

$$\begin{align*}\chi_\eta(\gamma) :=\frac{\eta(\gamma\tau)}{\sqrt{c\tau+d}\quad \eta(\tau)} \end{align*}$$

for $\gamma =\left (\begin {smallmatrix} a&b\\c&d\end {smallmatrix}\right )\in \mathrm {SL}_2(\mathbb {Z})$ .

Let F be a Jacobi cusp form $F\in S_{k,m}$ with its Fourier expansion

$$\begin{align*}F(\tau,z) = \sum_{\substack{l, r\in\mathbb{Z}\\ 4ml - r^2>0}}a_F(l,r)e^{2\pi il\tau}e^{2\pi irz}. \end{align*}$$

Then, $F_j$ has the Fourier expansion

$$\begin{align*}F_j(\tau) =\sum_{\substack{n>0\\ n+j^2\equiv 0\pmod{4m}}} a_F\left(\frac{n+j^2}{4m}, j\right)e^{2\pi in\tau/(4m)}. \end{align*}$$

We define the partial L-series of F by

$$\begin{align*}L(F,j,s) :=\sum_{\substack{n>0\\ n+j^2\equiv 0\pmod{4m}}}\frac{a_F\left(\frac{n+j^2}{4m}, j\right)} {\left(\frac{n}{4m}\right)^{s}} \end{align*}$$

for $1\leq j\leq 2m$ . This L-series was studied in [Reference Berndt1, Reference Choi and Lim7, Reference Lim and Raji17, Reference Lim and Raji18].

We now consider L-series of harmonic Maass Jacobi forms. Let $F\in \hat {J}_{k,m}$ . Then, it has the theta decomposition as in (4.3), and $F_j$ has the Fourier expansion

$$ \begin{align*} F_j(\tau) &=\sum_{n\gg-\infty\atop n+j^2\equiv 0\pmod{4m}} c^+_{F}\left(\frac{n+j^2}{4m}, j\right) e^{2\pi in\tau/(4m)}\\ &\quad +\sum_{n<0\atop n+j^2\equiv 0\pmod{4m}} c^-_F\left(\frac{n+j^2}{4m}, j\right)\Gamma\left(\frac32-k, -\frac{-\pi nv}{m}\right) e^{2\pi in\tau/(4m)}. \end{align*} $$

Let $n_0\in \mathbb {N}$ be such that $F_j(\tau )$ are $O(e^{2\pi n_0 v})$ as $v=\mathrm {Im}(\tau )\to \infty $ for each $1\leq j\leq m$ . Let $\mathcal {F}_F$ be the space of functions $\varphi \in \mathcal {C}(\mathbb {R},\mathbb {C})$ such that

1. (1) $(\mathcal {L}\varphi )(s)$ converges absolutely for all s with $\mathrm {Re}(s)\geq -2\pi n_0$ ,

2. (2) $(\mathcal \varphi _{\frac 52-k})(s)$ converges absolutely for all s with $\mathrm {Re}(s)>0$ ,

3. (3) for each $1\leq j\leq m$ , the series

   $$ \begin{align*} &\sum_{n\gg-\infty\atop n+j^2\equiv 0\pmod{4m}}\left|c^+_{F}\left(\frac{n+j^2}{4m}, j\right)\right| (\mathcal{L} |\varphi|)(\pi n/(2m))\\ &\quad +\sum_{n<0\atop n+j^2\equiv 0\pmod{4m}}\left|c^-_{F}\left(\frac{n+j^2}{4m}, j\right)\right| (\pi |n+\kappa_j|/m)^{\frac32-k} \\ &\quad \times\int_0^\infty\frac{(\mathcal{L}|\varphi_{\frac52-k}|)(-\pi n(2t+1)/(2m))}{(1+t)^{k-\frac12}} dt \end{align*} $$
   converges.

For $\varphi \in \mathcal {F}_F$ , we define the L-series of F by

$$ \begin{align*} L_F(\varphi) &:=\sum_{j=1}^{2m}\sum_{n\gg-\infty\atop n+j^2\equiv 0\pmod{4m}} c^+_{F}\left(\frac{n+j^2}{4m}, j\right) (\mathcal{L}\varphi) (\pi (n+\kappa_j)/(2m))\quad \mathbf{e}_j\\ &\quad +\sum_{j=1}^{2m}\sum_{n<0\atop n+j^2\equiv 0\pmod{4m}} c^-_{F}\left(\frac{n+j^2}{4m}, j\right) (-\pi (n+\kappa_j)/m)^{\frac32-k}\\ &\quad\times\int_0^\infty\frac{(\mathcal{L}\varphi_{\frac52-k})(-\pi n(2t+1)/(2m))}{(1+t)^{k-\frac12}} dt \quad \mathbf{e}_j. \end{align*} $$

Let $L_m$ be the heat operator defined by

$$\begin{align*}L_{m}:=\frac{2m}{\pi i}\frac{\partial}{\partial\tau}-\frac{1}{(2\pi i)^2}\frac{\partial^2}{\partial z^2}. \end{align*}$$

We define a differential operator $\alpha _k$ by

$$\begin{align*}(\alpha_k F)(\tau,z) L=\tau\left(\frac{\partial F}{\partial\bar{\tau}} +\frac{\pi i}{2m}L_m(F)\right)(\tau,z) +\frac{2k-1}{4}F(\tau,z). \end{align*}$$

Then, we prove that the corresponding vector-valued function for $\alpha _k F$ is the image of the corresponding vector-valued harmonic weak Maass form for F under the operator $\delta _{k-\frac 12}$ .

Lemma 4.3 If $F\in \hat {J}_{k,m}$ , then we have

$$\begin{align*}\phi_{k,m}\left(\alpha_k F\right) =\delta_{k-\frac12}\left(\phi_{k,m}(F)\right)\hspace{-0.7pt}. \end{align*}$$

Proof Suppose that F is in $\hat {J}_{k,m}$ . Then, it has the theta decomposition

$$\begin{align*}F(\tau,z) =\sum_{j=1}^{2m} F_j(\tau)\theta_{m,j}(\tau,z). \end{align*}$$

Moreover, $(\alpha _k F)|_{m} X =\alpha _k F$ for all $X\in \mathbb {Z}^2$ and $\alpha _k F$ is holomorphic in $z\in \mathbb {C}$ . Therefore, $\alpha _k F$ also has the theta expansion

$$\begin{align*}(\alpha_k F)(\tau,z) =\sum_{j=1}^{2m} (\alpha_k F)_j (\tau)\theta_{m,j}(\tau,z). \end{align*}$$

Note that

$$\begin{align*}L_m(\theta_{m,j}) = 0, \quad \frac{\partial}{\partial\bar{\tau}} (\theta_{m,j}) = 0 \end{align*}$$

for every $1\leq j\leq 2m$ . Therefore, we have

$$\begin{align*}(\alpha_k F)_j =\delta_{k-\frac12}(F_j) \end{align*}$$

for each $1\leq j\leq 2m$ .

Suppose that $F\in \hat {J}_{k,m}$ . Let $\mathcal {F}_{\alpha _k F}$ be the space of functions $\varphi \in C(\mathbb {R},\mathbb {C})$ such that

1. (1) $(\mathcal {L}\varphi )(s)$ converges absolutely for all s with $\mathrm {Re}(s)\geq -2\pi n_0$ ,

2. (2) $(\mathcal \varphi _{\frac 72-k})(s)$ converges absolutely for all s with $\mathrm {Re}(s)>0$ ,

3. (3) for each $1\leq j\leq m$ the series

   $$ \begin{align*} &\sum_{n\gg-\infty\atop n+j^2\equiv 0\pmod{4m}}\left|c^+_{F}\left(\frac{n+j^2}{4m}, j\right)(n/(4m))\right| (\mathcal{L} |\varphi_2|)(\pi n/(2m))\\ &\quad +\sum_{n+\kappa_j<0\atop n+j^2\equiv 0\pmod{4m}}\left|c^-_{F}\left(\frac{n+j^2}{4m}, j\right)(n/(4m))\right| (\pi |n|/m)^{\frac32-k} \\ &\quad \times\int_0^\infty\frac{(\mathcal{L}|\varphi_{\frac72-k}|)(-\pi n(2t+1)/(2m))}{(1+t)^{k-\frac12}} dt \end{align*} $$
   converges.

For $\varphi \in \mathcal {F}_{\alpha _k F}$ , we define $L_{\alpha _k F}(\varphi )$ by

$$ \begin{align*} L_{\alpha_k F}(\varphi) &:=\frac{2k-1}{4}L_F(\varphi) -2\pi\sum_{j=1}^{2m}\sum_{n\gg-\infty\atop n+j^2\equiv 0\pmod{4m}} c^+_{f,j}(n) (n/2m) (\mathcal{L}\varphi_2) (\pi n/(2m))\quad \mathbf{e}_j\\ &\quad -2\pi\sum_{j=1}^{2m}\sum_{n<0\atop n+j^2\equiv 0\pmod{4m}} c^-_{F}\left(\frac{n+j^2}{4m}, j\right)(n/(4m))(-\pi n/m)^{\frac32-k}\\ &\quad\times\int_0^\infty\frac{(\mathcal{L}\varphi_{\frac72-k})(-\pi n(2t+1)/(2m))}{(1+t)^{k-\frac12}} dt\quad \mathbf{e}_j. \end{align*} $$

We prove the converse theorem in the case of harmonic Maass Jacobi forms using a similar argument as in the proof of Theorem 3.5.

Theorem 4.4 Let $(c^+_{F}(l,r))_{4ml-r^2\geq -D_0}$ and $(c^-_{F}(l,r))_{4ml-r^2<0}$ be sequences of complex numbers such that $c^+_F(l,r), c^-_{F}(l,r) = O(e^{C\sqrt {|4ml-r^2|}})$ as $|4ml-r^2|\to \infty $ , for some $C>0$ . For each $\tau \in \mathbb {H}$ and $z\in \mathbb {C}$ , set

$$\begin{align*}F(\tau,z) &:=\sum_{l,r\in\mathbb{Z}\atop 4ml-r^2\gg -\infty} c^+_{F}(l,r)e^{2\pi il\tau} e^{2\pi irz} \\ &\quad +\sum_{l,r\in\mathbb{Z}\atop 4ml-r^2>0} c^-_{F}(l,r)\Gamma\left(1-k,\frac{(4ml-r^2)\pi v}{m}\right)e^{2\pi il\tau} e^{2\pi irz}. \end{align*}$$

Suppose that for each $\varphi \in S_c(\mathbb {R}_+)$ , the functions $L_F(\varphi )$ and $L_{\alpha _k F}(\varphi )$ satisfy

$$\begin{align*}L_F(\varphi) = i^{k-\frac12}\overline{\rho_m}(S) L_f(\varphi|_{2-k+\frac12,\chi_\eta} S) \end{align*}$$

and

$$\begin{align*}L_{\alpha_k F}(\varphi) = -i^{k-\frac12}\overline{\rho_m}(S) L_{\alpha_k f}(\varphi|_{2-k+\frac12,\chi_\eta}S). \end{align*}$$

Then, F is a harmonic Maass Jacobi form in $\hat {J}_{k, m}$ .

5 L-series of harmonic weak Maass forms in the Kohnen plus space

In this section, we consider the case of half-integral weight modular forms in the Kohnen plus space and prove a converse theorem for this case. Let k be a positive even integer. By [Reference Eichler and Zagier11, Theorem 5.4], there is an isomorphism $\psi _k$ between $S_{k,1}$ and $S^+_{k-\frac 12}$ , where $S^+_{k-\frac 12}$ denotes the space of cusp forms in the plus space of weight $k-\frac 12$ on $\Gamma _0(4)$ .

Let f be a cusp form in $S^+_{k-\frac 12}$ with Fourier expansion

$$\begin{align*}f(\tau) =\sum_{\substack{n>0\\ n\equiv 0,3\pmod{4}}} a_f(n)e^{2\pi in\tau}. \end{align*}$$

Then, the L-function of f is defined by

$$\begin{align*}L(f,s) :=\sum_{\substack{n>0\\ n\equiv 0,3\pmod{4}}}\frac{a_f(n)}{n^s}. \end{align*}$$

For $1\leq j\leq 2$ , let $c_j$ be defined by

(5.1) $$ \begin{align} a_{f,j}(n) := \begin{cases} a_f(n) &\text{if } n\equiv -j^2\pmod{4},\\ 0 &\text{otherwise}. \end{cases} \end{align} $$

Then, $a_f(n) = a_{f,1}(n) + a_{f,2}(n)$ for all n. With this, we consider partial sums of $L(f,s)$ by

$$\begin{align*}L(f,j,s) :=\sum_{\substack{n>0\\ n\equiv 0,3\pmod{4}}}\frac{c_j(n)}{n^s} \end{align*}$$

for $1\leq j\leq 2$ . Suppose that F is a Jacobi cusp form in $S_{k,1}$ . By the theta decomposition, we have a corresponding vector-valued modular form $(F_1(\tau ), F_2(\tau ))$ . Then, the isomorphism $\psi _k$ from $S_{k,1}$ to $S^+_{k-\frac 12}$ is given by

$$\begin{align*}\psi_k(F)(\tau) =\sum_{j=1}^2 F_j(4\tau). \end{align*}$$

From this, we see that

$$\begin{align*}L(\psi_k(F),j,s) =\frac{1}{4^s} L(F,j,s). \end{align*}$$

Let $H^+_{k-\frac 12}$ denote the space of harmonic weak Maass forms in the plus space of weight $k-\frac 12$ on $\Gamma _0(4)$ . Then, by [Reference Cho and Choie6], we see that $\psi _k$ can be extended to an isomorphism between $\hat {J}_{k,1}$ and $H^+_{k-\frac 12}$ as follows. Suppose that $F\in \hat {J}_{k,1}$ . Then, F has the theta decomposition

$$\begin{align*}F(\tau,z) =\sum_{j=1}^2 F_j(\tau)\theta_{m,j}(\tau,z). \end{align*}$$

Then, $\psi _k(F)(\tau ) =\sum _{j=1}^2 F_j(4\tau )$ .

Suppose that $f\in H^+_{k-\frac 12}$ with its Fourier expansion

$$ \begin{align*} f(\tau) &=\sum_{n\gg -\infty}c^+_{f}(n) e^{2\pi i n\tau} +\sum_{n< 0}c^-_{f}(n)\Gamma\left(\frac32-k,-4\pi nv\right) e^{2\pi i n\tau}. \end{align*} $$

For $1\leq j\leq 2$ , we define $c^{\pm }_{f,j}(n)$ as in (5.1), and

$$\begin{align*}f_j(\tau) =\sum_{n\gg -\infty}c^+_{f,j}(n) e^{2\pi i n\tau} +\sum_{n< 0}c^-_{f,j}(n)\Gamma\left(\frac32-k,-4\pi nv\right) e^{2\pi i n\tau}. \end{align*}$$

For a vector-valued function $F(\tau ) :=\sum _{j=1}^{2} f_j(\tau /4)\mathbf {e}_j$ , we define the L-series $L_F(\varphi )$ and $L_{\delta _{k-\frac 12} F}(\varphi )$ as in (3.3) and (3.5), respectively.

We now have the following converse theorem for half-integral weight harmonic weak Maass forms in the Kohnen plus space.

Theorem 5.1 Let $(c^+_{f}(n))_{n\geq -n_0}$ and $(c^-_{f}(n))_{n<0}$ be sequences of complex numbers such that

$$\begin{align*}c^+_f(n), c^-_{f}(n) = O\left(e^{C\sqrt{|n|}}\right) \end{align*}$$

as $|n|\to \infty $ , for some $C>0$ . For each $\tau \in \mathbb {H}$ , set

$$\begin{align*}f(\tau) :=\sum_{n\gg -n_0}c^+_{f}(n) e^{2\pi i n\tau} +\sum_{n< 0}c^-_{f}(n)\Gamma\left(\frac32-k,-4\pi nv\right) e^{2\pi i n\tau}. \end{align*}$$

Suppose that for each $\varphi \in S_c(\mathbb {R}_+)$ , the functions $L_F(\varphi )$ and $L_{\alpha _k F}(\varphi )$ satisfy

$$\begin{align*}L_F(\varphi) = i^{k-\frac12}\overline{\rho_2}(S) L_f(\varphi|_{2-k+\frac12,\chi_\eta} S) \end{align*}$$

and

$$\begin{align*}L_{\alpha_k F}(\varphi) = -i^{k-\frac12}\overline{\rho_2}(S) L_{\alpha_k f}(\varphi|_{2-k+\frac12,\chi_\eta}S). \end{align*}$$

Then, f is a harmonic weak Maass form in $H^+_{k-\frac 12}$ .