Non-Vanishing of L-Functions of Vector-Valued Modular Forms

Subong Lim; Wissam Raji

arXiv:2302.12576·math.NT·May 12, 2023

Non-Vanishing of L-Functions of Vector-Valued Modular Forms

Subong Lim, Wissam Raji

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TL;DR

This paper proves that for vector-valued modular forms, there are non-zero average L-values and guarantees the existence of at least one form with a non-vanishing L-function, advancing understanding of their analytic properties.

Contribution

It establishes non-vanishing results for L-functions of vector-valued modular forms and demonstrates the existence of non-vanishing L-functions within a basis.

Findings

01

Non-vanishing of average L-values for vector-valued modular forms

02

Existence of at least one basis element with non-zero L-function

03

Conditions under which non-vanishing is guaranteed

Abstract

We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose L-function does not vanish under certain conditions.

Equations178

l = 1 \sum d_{k} \frac{< L ^{*} ( f _{k, l} , s ) , e _{i} >}{( f _{k, l} , f _{k, l} )} b_{k, l, i} (n_{i, 0})

l = 1 \sum d_{k} \frac{< L ^{*} ( f _{k, l} , s ) , e _{i} >}{( f _{k, l} , f _{k, l} )} b_{k, l, i} (n_{i, 0})

χ (γ_{3}) (c_{3} τ + d_{3})^{k} = χ (γ_{1}) χ (γ_{2}) (c_{1} γ_{2} τ + d_{1})^{k} (c_{2} τ + d_{2})^{k},

χ (γ_{3}) (c_{3} τ + d_{3})^{k} = χ (γ_{1}) χ (γ_{2}) (c_{1} γ_{2} τ + d_{1})^{k} (c_{2} τ + d_{2})^{k},

(f ∣_{k, χ, ρ} γ) (τ) := χ^{- 1} (γ) (c τ + d)^{- k} ρ^{- 1} (γ) f (γ τ) .

(f ∣_{k, χ, ρ} γ) (τ) := χ^{- 1} (γ) (c τ + d)^{- k} ρ^{- 1} (γ) f (γ τ) .

f_{i} (τ) = n + κ_{j} \geq 0 \sum a_{j} (n) e^{2 π i (n + κ_{j}) τ} .

f_{i} (τ) = n + κ_{j} \geq 0 \sum a_{j} (n) e^{2 π i (n + κ_{j}) τ} .

L (f, s) = j = 1 \sum m n + κ_{j} > 0 \sum \frac{a _{j} ( n )}{( n + κ _{j} ) ^{s}} e_{j} .

L (f, s) = j = 1 \sum m n + κ_{j} > 0 \sum \frac{a _{j} ( n )}{( n + κ _{j} ) ^{s}} e_{j} .

\frac{Γ ( s )}{( 2 π ) ^{s}} L (f, s) = \int_{0}^{\infty} f (i y) y^{s} \frac{d y}{y} .

\frac{Γ ( s )}{( 2 π ) ^{s}} L (f, s) = \int_{0}^{\infty} f (i y) y^{s} \frac{d y}{y} .

L^{*} (f, s) = i^{k} χ (S) ρ (S) L^{*} (f, k - s),

L^{*} (f, s) = i^{k} χ (S) ρ (S) L^{*} (f, k - s),

L^{*} (f, s) = \frac{Γ ( s )}{( 2 π ) ^{s}} L (f, s)

L^{*} (f, s) = \frac{Γ ( s )}{( 2 π ) ^{s}} L (f, s)

p_{s, l} (τ) := τ^{- s} e_{l} .

p_{s, l} (τ) := τ^{- s} e_{l} .

R_{k, s, l} := γ_{k} (s) γ \in SL_{2} (Z) \sum p_{s, l} ∣_{k, χ, ρ} γ,

R_{k, s, l} := γ_{k} (s) γ \in SL_{2} (Z) \sum p_{s, l} ∣_{k, χ, ρ} γ,

< j = 1 \sum m λ_{j} e_{j}, j = 1 \sum m μ_{j} e_{j} >= j = 1 \sum m λ_{j} \overline{μ_{j}} .

< j = 1 \sum m λ_{j} e_{j}, j = 1 \sum m μ_{j} e_{j} >= j = 1 \sum m λ_{j} \overline{μ_{j}} .

(f, g) := \int_{F} < f (τ), g (τ) > v^{k} \frac{d u d v}{v ^{2}}

(f, g) := \int_{F} < f (τ), g (τ) > v^{k} \frac{d u d v}{v ^{2}}

(f, R_{k, \overset{s}{ˉ}, l}) = c_{k} < L^{*} (f, s), e_{l} >,

(f, R_{k, \overset{s}{ˉ}, l}) = c_{k} < L^{*} (f, s), e_{l} >,

(R_{k, s, l})_{j} (τ) = γ_{k} (s) (a c b d) \in SL_{2} (Z) \sum χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d))_{j, l} (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s},

(R_{k, s, l})_{j} (τ) = γ_{k} (s) (a c b d) \in SL_{2} (Z) \sum χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d))_{j, l} (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s},

(a c b d) \in SL_{2} (Z) \sum χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d))_{j, l} (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s} \leq (a c b d) \in SL_{2} (Z) \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s}

(a c b d) \in SL_{2} (Z) \sum χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d))_{j, l} (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s} \leq (a c b d) \in SL_{2} (Z) \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s}

(a c b d) \in SL_{2} (Z) \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s}

(a c b d) \in SL_{2} (Z) \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d})^{- s}

γ_{k} (s) (c, d) \in Z^{2} (c, d) = 1 \sum n \in Z \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d} + n)^{- s} χ^{- 1} ((a c b d)) χ^{- 1} ((10 n 1)) ρ^{- 1} ((a c b d)) ρ^{- 1} ((10 n 1)) e_{l},

γ_{k} (s) (c, d) \in Z^{2} (c, d) = 1 \sum n \in Z \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d} + n)^{- s} χ^{- 1} ((a c b d)) χ^{- 1} ((10 n 1)) ρ^{- 1} ((a c b d)) ρ^{- 1} ((10 n 1)) e_{l},

(R_{k, s, l})_{j} (τ) = γ_{k} (s) (c, d) \in Z^{2} (c, d) = 1 \sum n \in Z \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d} + n)^{- s} e^{- 2 π in κ_{l}} χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d))_{j, l} .

(R_{k, s, l})_{j} (τ) = γ_{k} (s) (c, d) \in Z^{2} (c, d) = 1 \sum n \in Z \sum (c τ + d)^{- k} (\frac{a τ + b}{c τ + d} + n)^{- s} e^{- 2 π in κ_{l}} χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d))_{j, l} .

\frac{Γ ( s )}{( - 2 π i ) ^{s}} k \in Z \sum \frac{e ^{2 π iak}}{( k + τ ) ^{s}} = n = 1 \sum \infty \frac{e ^{2 π i τ (n - a)}}{( n - a ) ^{1 - s}} .

\frac{Γ ( s )}{( - 2 π i ) ^{s}} k \in Z \sum \frac{e ^{2 π iak}}{( k + τ ) ^{s}} = n = 1 \sum \infty \frac{e ^{2 π i τ (n - a)}}{( n - a ) ^{1 - s}} .

(R_{k, s, l})_{j} (τ)

(R_{k, s, l})_{j} (τ)

R_{k, s, l} (τ) = (2 π)^{s} Γ (k - s) n + κ_{l} > 0 \sum (n + κ_{l})^{s - 1} P_{k, n, l} (τ),

R_{k, s, l} (τ) = (2 π)^{s} Γ (k - s) n + κ_{l} > 0 \sum (n + κ_{l})^{s - 1} P_{k, n, l} (τ),

P_{k, n, l} (τ) = \frac{1}{2} (c, d) \in Z^{2} (c, d) = 1 \sum χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d)) (c τ + d)^{- k} e^{2 π i (n + κ_{l}) (a τ + b) / (c τ + d)} e_{l}

P_{k, n, l} (τ) = \frac{1}{2} (c, d) \in Z^{2} (c, d) = 1 \sum χ^{- 1} ((a c b d)) ρ^{- 1} ((a c b d)) (c τ + d)^{- k} e^{2 π i (n + κ_{l}) (a τ + b) / (c τ + d)} e_{l}

(f, P_{k, n, l}) = a_{n} (l) \frac{Γ ( k - 1 )}{( 4 π ( n + κ _{l} ) ) ^{k - 1}} .

(f, P_{k, n, l}) = a_{n} (l) \frac{Γ ( k - 1 )}{( 4 π ( n + κ _{l} ) ) ^{k - 1}} .

(f, R_{k, \overset{s}{ˉ}, l})

(f, R_{k, \overset{s}{ˉ}, l})

R_{k, s, l} (τ) = j = 1 \sum m n + κ_{j} > 0 \sum r_{k, s, l, j} (n) e^{2 π i (n + κ_{j}) τ},

R_{k, s, l} (τ) = j = 1 \sum m n + κ_{j} > 0 \sum r_{k, s, l, j} (n) e^{2 π i (n + κ_{j}) τ},

r_{k, s, l, j} (n)

r_{k, s, l, j} (n)

2 γ_{k} (s) n \in Z \sum (z + n)^{- s} e^{- 2 π n κ_{l}} e_{l}

2 γ_{k} (s) n \in Z \sum (z + n)^{- s} e^{- 2 π n κ_{l}} e_{l}

(2 π)^{s} Γ (k - s) n + κ_{l} > 0 \sum (n + κ_{l})^{s - 1} e^{2 π i (n + κ_{l}) z} e_{l} .

(2 π)^{s} Γ (k - s) n + κ_{l} > 0 \sum (n + κ_{l})^{s - 1} e^{2 π i (n + κ_{l}) z} e_{l} .

2 γ_{k} (s) n \in Z \sum (- z)^{- s} (z + n)^{s - k} χ^{- 1} ((10 n 1)) χ^{- 1} ((01 - 1 0)) ρ^{- 1} ((10 n 1)) ρ^{- 1} ((01 - 1 0)) e_{l}

2 γ_{k} (s) n \in Z \sum (- z)^{- s} (z + n)^{s - k} χ^{- 1} ((10 n 1)) χ^{- 1} ((01 - 1 0)) ρ^{- 1} ((10 n 1)) ρ^{- 1} ((01 - 1 0)) e_{l}

= 2 γ_{k} (s) (- z)^{- s} j = 1 \sum m χ^{- 1} ((01 - 1 0)) ρ^{- 1} ((01 - 1 0))_{j, l} n \in Z \sum e^{- 2 π in κ_{j}} (z + n)^{s - k} e_{j} .

(- 1)^{k /2} (2 π)^{k - s} Γ (s) j = 1 \sum m χ^{- 1} ((01 - 1 0)) ρ^{- 1} ((01 - 1 0))_{j, l} n + κ_{j} > 0 \sum (n + κ_{j})^{k - s - 1} e^{2 π i (n + κ_{j}) z} e_{j} .

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TopicsAnalytic Number Theory Research · Advanced Algebra and Geometry · Mathematical Dynamics and Fractals

Full text

Non-Vanishing of L-Functions of Vector-Valued Modular Forms

Subong Lim and Wissam Raji

Abstract.

Kohnen proved a non-vanishing result for $L$ -functions associated to Hecke eigenforms of integral weights on the full group. In this paper, we show a non-vanishing result for the averages of $L$ -functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms of weight $k\in\frac{1}{2}\mathbb{Z}$ on the full group. We also show the existence of at least one basis element whose L-function does not vanish under certain conditions. As an application, we generalize the result of Kohnen to $\Gamma_{0}(N)$ and prove the analogous result for Jacobi forms.

1. Introduction

Vector-Valued modular forms have played a crucial role in the theory of modular forms. In particular, Selberg used these forms to give an estimation for the Fourier coefficients of the classical modular forms [14]. Moreover, vector-valued modular forms arise naturally in the theory of Jacobi forms, Siegel modular forms, and Moonshine. Some applications of vector-valued modular forms stand out in high-energy physics by mainly providing a method of differential equations in order to construct the modular multiplets, and also revealing the simple structure of the modular invariant mass models [5]. Other applications concerning vector-valued modular forms of half-integer weight seem to provide a simple solution to the Riemann-Hilbert problem for representations of the modular group [2].

In [8],[9], and [10], Knopp and/or Mason gave a systematic development of the theory of vector-valued modular forms where they introduced the foundation of the space of these forms mainly through the introduction of vector-valued Poincaré series and vector-valued Eisenstein series leading to a better understanding of the space of vector-valued modular forms. More recently, several algorithms for computing Fourier coefficients of vector-valued modular forms were determined in connection to Weil representations due to their importance in the Moonshine applications [13].

On the other hand, $L$ -functions of vector-valued modular forms play important role in the above-mentioned computations as well so it is natural to study them. In this paper, we show a non-vanishing result for averages of $L$ -functions associated with vector-valued modular forms. To illustrate, we let $\{f_{k,1},\ldots,f_{k,d_{k}}\}$ be an orthogonal basis of the space of cusp forms $S_{k,\chi,\rho}$ with Fourier coefficients $b_{k,l,j}(n)$ , where $\chi$ is a multiplier system of weight $k\in\frac{1}{2}\mathbb{Z}$ on $\mathrm{SL}_{2}(\mathbb{Z})$ and $\rho:\mathrm{SL}_{2}(\mathbb{Z})\to\mathrm{GL}_{m}(\mathbb{C})$ is an $m$ -dimensional unitary complex representation. We let $t_{0}\in\mathbb{R},\epsilon>0$ , and $1\leq i\leq m$ . Then, there exists a constant $C(t_{0},\epsilon,i)>0$ such that for $k>C(t_{0},\epsilon,i)$ the function

[TABLE]

does not vanish at any point $s=\sigma+it_{0}$ with $\frac{k-1}{2}<\sigma<\frac{k}{2}-\epsilon$ , where $<L^{*}(f_{k,l},s),\mathbf{e}_{i}>$ denotes the $i$ th component of $L^{*}(f_{k,l},s)$ . By using the integral weight case, we generalize a result of Kohnen in [11] to $\Gamma_{0}(N)$ in Section 5. On the other hand, by using the half-integral weight case, we prove the analogous result for Jacobi forms in Section 6.

2. Preliminaries

Let $k\in\frac{1}{2}\mathbb{Z}$ and $\chi$ a unitary multiplier system of weight $k$ on $\Gamma$ , i.e. $\chi:\mathrm{SL}_{2}(\mathbb{Z})\to\mathbb{C}$ satisfies the following conditions:

(1)

$|\chi(\gamma)|=1$ for all $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ . 2. (2)

$\chi$ satisfies the consistency condition

[TABLE]

where $\gamma_{3}=\gamma_{1}\gamma_{2}$ and $\gamma_{i}=\left(\begin{smallmatrix}a_{i}&b_{i}\\ c_{i}&d_{i}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ for $i=1,2$ , and $3$ .

Let $m$ be a positive integer and $\rho:\mathrm{SL}_{2}(\mathbb{Z})\to\mathrm{GL}_{m}(\mathbb{C})$ a $m$ -dimensional unitary complex representation. We assume that $\rho(-I)$ is the identity matrix, where $I$ denotes the identity matrix. 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 $\gamma\in\Gamma$ , define a slash operator by

[TABLE]

Definition 2.1.

A vector-valued modular form of weight $k$ and multiplier system $\chi$ with respect to $\rho$ on $\mathrm{SL}_{2}(\mathbb{Z})$ is a sum $f=\sum_{j=1}^{m}f_{j}\mathbf{e}_{j}$ of functions holomorphic in $\mathbb{H}$ satisfying the following conditions:

(1)

$f|_{k,\chi,\rho}\gamma=f$ * for all $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ .* 2. (2)

For each $1\leq j\leq m$ , each function $f_{j}$ has a Fourier expansion of the form

[TABLE]

Here and throughout the paper, $\kappa_{j}$ is a certain positive number with $0\leq\kappa_{j}<1$ .

We write $M_{k,\chi,\rho}$ for the space of vector-valued modular forms of weight $k$ and multiplier system $\chi$ with respect to $\rho$ on $\mathrm{SL}_{2}(\mathbb{Z})$ . There is a subspace $S_{k,\chi,\rho}$ of vector-valued cusp forms for which we require that each $a_{j}(n)=0$ when $n+\kappa_{j}$ is non-positive.

From the condition (2) in Definition 2.1, we see that $\chi\left(\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)\right)\rho\left(\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)\right)$ is a diagonal matrix whose $(j,j)$ entry is $e^{2\pi i\kappa_{j}}$ . If $f\in S_{k,\chi,\rho}$ is a vector-valued cusp form, then $a_{j}(n)=O(n^{k/2})$ for every $1\leq j\leq m$ , as $n\to\infty$ by the same argument for classical modular forms (for example, see [10, Section 1]). For a vector-valued cusp form $f(z)=\sum_{j=1}^{m}\sum_{n+\kappa_{j}>0}a_{j}(n)e^{2\pi i(n+\kappa_{j})z}\mathbf{e}_{j}$ we define the $L$ -series

[TABLE]

This series converges absolutely for $\mathrm{Re}(s)\gg 0$ .

The following theorem for vector-valued modular forms follows from the same argument used for classical modular forms.

Theorem 2.2.

Let $k\in\frac{1}{2}\mathbb{Z}$ . If $f\in S_{k,\rho}$ is a vector-valued cusp form, then

[TABLE]

Furthermore, $L(f,s)$ has an analytic continuation and functional equation

[TABLE]

where

[TABLE]

and $S=\left(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right)$ .

3. The Construction of the Kernel Function

In what follows, we define the kernel function $R_{k,s,l}$ which will play a crucial role in determining the Fourier coefficients of the orthogonal basis of the space of vector-valued cusp forms using Petersson’s scalar product. Moreover, we determine the Fourier coefficients of this kernel function using the Lipshitz summation formula.

Let $l$ be an integer with $1\leq l\leq m$ . Define

[TABLE]

For $\tau\in\mathbb{H}$ and $s\in\mathbb{C}$ with $1<\mathrm{Re}(s)<k-1$ , we define

[TABLE]

where $\gamma_{k}(s):=\frac{1}{2}e^{\pi is/2}\Gamma(s)\Gamma(k-s)$ .

We write $<\cdot,\cdot>$ for the standard scalar product on $\mathbb{C}^{m}$ , i.e.

[TABLE]

Then, for $f,g\in M_{k,\rho}$ , we define the Petersson scalar product of $f$ and $g$ by

[TABLE]

if the integral converges, where $\mathcal{F}$ is the standard fundamental domain for the action of $\mathrm{SL}_{2}(\mathbb{Z})$ on $\mathbb{H}$ .

Lemma 3.1.

Let $k\in\frac{1}{2}\mathbb{Z}$ with $k>2$ , and let $s\in\mathbb{C}$ with $1<\mathrm{Re}(s)<k-1$ .

(1)

The series $R_{k,s,l}$ converges absolutely uniformly whenever $\tau=u+iv$ satisfies $v\geq\epsilon,u\leq 1/\epsilon$ for a given $\epsilon>0$ , and $s$ varies over a compact set. 2. (2)

The series $R_{k,s,l}$ is a vector-valued cusp form in $S_{k,\chi,\rho}$ . 3. (3)

For $f\in S_{k,\chi,\rho}$ , we have

[TABLE]

where $c_{k}:=\frac{(-1)^{k/2}\pi(k-2)!}{2^{k-2}}$ .

Proof.

For the first part, note that for each $1\leq j\leq m$ , we have

[TABLE]

where $\rho^{-1}\left(\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\right)_{j,l}$ denotes the $(j,l)$ -th entry of the matrix $\rho^{-1}\left(\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\right)$ . Then, we have

[TABLE]

since $\rho$ is a unitary representation. It is known that the series

[TABLE]

converges absolutely uniformly whenever $\tau=u+iv$ satisfies $v\geq\epsilon,u\leq 1/\epsilon$ for a given $\epsilon>0$ , and $s$ varies over a compact set (see [11, Section 4]). Therefore, the series $R_{k,s,l}$ converges absolutely uniformly whenever $\tau=u+iv$ satisfies $v\geq\epsilon,u\leq 1/\epsilon$ for a given $\epsilon>0$ , and $s$ varies over a compact set. The second part follows readily from that.

For the last part, we follow the argument in [11, Lemma 1]. It is enough to consider the case when $1<\mathrm{Re}(s)<\frac{k-1}{2}$ . Note that for each $(c,d)\in\mathbb{Z}^{2}$ with $(c,d)=1$ , we can find $(a,b)\in\mathbb{Z}^{2}$ such that $ad-bc=1$ . Then, we see that $R_{k,s,l}(\tau)$ is equal to

[TABLE]

where for each coprime pair $(c,d)\in\mathbb{Z}^{2}$ , one chooses a fixed pair $(a,b)\in\mathbb{Z}^{2}$ such that $ad-bc=1$ . Therefore, we have

[TABLE]

Next, we will use the Lipschitz summation formula [12]: For $0\leq a\leq 1,\mathrm{Re}(s)>1$ and $\tau\in\mathbb{H}$ , we have

[TABLE]

Therefore, we have

[TABLE]

From this, we have

[TABLE]

where

[TABLE]

is a vector-valued Poincaré series. By following the argument as in [10, Theorem 5.3], we have

[TABLE]

Therefore, we see that

[TABLE]

∎

We now compute the Fourier expansion of $R_{k,s,l}$ by following a similar argument as in [11, Lemma 2].

Lemma 3.2.

Let $k\in\frac{1}{2}\mathbb{Z}$ with $k>2$ . The function $R_{k,s,l}$ has the Fourier expansion

[TABLE]

where $r_{k,s,l,j}(n)$ is given by

[TABLE]

where ${}_{1}F_{1}(\alpha,\beta;z)$ is Kummer’s degenerate hypergeometric function.

Proof.

First, we consider the contribution of the terms where $ac=0$ . The contribution of the terms $\pm\left(\begin{smallmatrix}1&n\\ 0&1\end{smallmatrix}\right)$

[TABLE]

can be written as follows by the Lipschitz summation formula in (3.1)

[TABLE]

Note that $\left(\begin{smallmatrix}0&-1\\ 1&n\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right)\left(\begin{smallmatrix}1&n\\ 0&1\end{smallmatrix}\right)$ . Therefore, the contribution of the terms $\pm\left(\begin{smallmatrix}0&-1\\ 1&n\end{smallmatrix}\right)$ is equal to

[TABLE]

By the similar computation as in the case of the terms $\pm\left(\begin{smallmatrix}1&n\\ 0&1\end{smallmatrix}\right)$ , we see that (3) is equal to

[TABLE]

The contribution of the terms with $ac\neq 0$ at the $j$ -th component is given by

[TABLE]

for any fixed positive real number $C$ . By the change of variables $z\mapsto z+m(m\in\mathbb{Z})$ , we see that (3.3) is equal to

[TABLE]

By the change of variables $z\mapsto z-d/c$ , we see that (3.4) is equal to

[TABLE]

If $ac>0$ , then we have

[TABLE]

Therefore, by the change of variable $z\mapsto(c/a)it$ , we see that the integral in (3.5) is equal to

[TABLE]

Note that for $\mathrm{Re}(\mu),\mathrm{Re}(\nu)>0,p\in\mathbb{C}$ , we have

[TABLE]

(see [7]). Therefore, (3.6) can be written as

[TABLE]

From this, we see that the contribution of the terms with $ac>0$ at the $j$ -th component is equal to

[TABLE]

The contribution of the terms with $ac<0$ at the $j$ -th component is obtained by the same argument if we replace $(a,c)$ by $(-a,c)$ . ∎

4. The Main Result

In this section, we give the main result where we determine the non-vanishing of the averages of L-functions associated with the orthogonal basis of the space of cusp forms. We also show the existence of at least one basis element whose L-function does not vanish under certain conditions. Let

[TABLE]

Theorem 4.1.

Let $k\in\frac{1}{2}\mathbb{Z}$ with $k>2$ . Let $\{f_{k,1},\ldots,f_{k,d_{k}}\}$ be an orthogonal basis of $S_{k,\chi,\rho}$ with Fourier expansions

[TABLE]

Let $t_{0}\in\mathbb{R},\epsilon>0$ , and $1\leq j\leq m$ . Then, there exists a constant $C(t_{0},\epsilon,j)>0$ such that for $k>C(t_{0},\epsilon,j)$ the function

[TABLE]

does not vanish at any point $s=\sigma+it_{0}$ with $\frac{k-1}{2}<\sigma<\frac{k}{2}-\epsilon$ .

Remark 4.2.

Note that

[TABLE]

is the $n_{j,0}$ th Fourier coefficient of $R_{k,\bar{s},j}$ . Therefore, it is equal to a nonzero constant multiple of

[TABLE]

Therefore, Theorem 4.1 implies the nonvanishing of $<L^{*}(P_{k,n_{j,0},j},s),\mathbf{e}_{j}>$ .

Proof.

By Lemma 3.1, we have

[TABLE]

If we take the first Fourier coefficients of both sides at the $j$ th component, then by Lemma 3.2 we have

[TABLE]

where

[TABLE]

Suppose that

[TABLE]

If we divide (4) by $(2\pi)^{s}\Gamma(k-s)\kappa_{j}^{s-1}$ , we have

[TABLE]

where

[TABLE]

Let $s=\frac{k}{2}-\delta-it_{0}$ , where $\epsilon<\delta<\frac{1}{2}$ . Then, we have

[TABLE]

For $\mathrm{Re}(\beta)>\mathrm{Re}(\alpha)>0$ , we have

[TABLE]

By [1, 13.21], for $\mathrm{Re}(\alpha)>1,\mathrm{Re}(\beta-\alpha)>1$ , and $|z|=1$ , we have

[TABLE]

If we take absolute values in (4.3), then we have

[TABLE]

By [1, 6.147], we have

[TABLE]

as $k\to\infty$ . Therefore, (4.4) becomes $1\leq 0$ as $k\to\infty$ , which is a contradiction. ∎

We now give a corollary that is a direct consequence of Theorem 4.1 which basically demonstrates the existence of a basis element of the space of vector-valued cusp forms whose $L$ -function does not vanish.

Corollary 4.3.

Let $k\in\frac{1}{2}\mathbb{Z}$ with $k>2$ . Let $\{f_{k,1},\ldots,f_{k,d_{k}}\}$ be an orthogonal basis of $S_{k,\chi,\rho}$ with Fourier expansions

[TABLE]

Let $t_{0}\in\mathbb{R}$ and $\epsilon>0$ .

(1)

For any $k>C(t_{0},\epsilon,j)$ , any $1\leq j\leq m$ , and any $s=\sigma+it_{0}$ with

[TABLE]

there exists a basis element $f_{k,l}\in S_{k,\chi,\rho}$ such that

[TABLE] 2. (2)

There exists a constant $C(t_{0},\epsilon)>0$ such that for any $k>C(t_{0},\epsilon)$ , and any $s=\sigma+it_{0}$ with

[TABLE]

there exists a basis element $f_{k,l}\in S_{k,\chi,\rho}$ such that

[TABLE]

5. The Case of $\Gamma_{0}(N)$

In what follows, we consider the case of a scalar-valued modular form on the congruence subgroup $\Gamma_{0}(N)$ . By using Theorem 4.1, we can extend Kohnen’s result in [11] to the case of $\Gamma_{0}(N)$ . To illustrate, let $N$ be a positive integer and let $\Gamma=\Gamma_{0}(N)$ . Let $S_{k}(\Gamma)$ be the space of cusp forms of weight $k$ on $\Gamma$ . Let $\{\gamma_{1},\ldots,\gamma_{m}\}$ be the set of representatives of $\Gamma\setminus\mathrm{SL}_{2}(\mathbb{Z})$ with $\gamma_{1}=I$ . For $f\in S_{k}(\Gamma)$ , we define a vector-valued function $\tilde{f}:\mathbb{H}\to\mathbb{C}^{m}$ by $\tilde{f}=\sum_{j=1}^{m}f_{j}\mathbf{e}_{j}$ and

[TABLE]

where $(f|_{k}\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right))(z):=(cz+d)^{-k}f(\gamma z)$ . Then, $\tilde{f}$ is a vector-valued modular form of weight $k$ and the trivial multiplier system with respect to $\rho$ on $\mathrm{SL}_{2}(\mathbb{Z})$ , where $\rho$ is a certain $m$ -dimensional unitary complex representation such that $\rho(\gamma)$ is a permutation matrix for each $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ and is an identity matrix if $\gamma\in\Gamma$ . Then, the map $f\mapsto\tilde{f}$ induces an isomorphism between $S_{k}(\Gamma)$ and $S_{k,\rho}$ , where $S_{k,\rho}$ denotes the space of vector-valued cusp forms of weight $k$ and trivial multiplier system with respect to $\rho$ on $\mathrm{SL}_{2}(\mathbb{Z})$ .

Suppose that $f,g\in S_{k}(\Gamma)$ . Then, we have

[TABLE]

where $(f,g)$ denotes the Petersson inner product. Therefore, if $f,g\in S_{k}(\Gamma)$ such that $f$ and $g$ are orthogonal, then $\tilde{f}$ and $\tilde{g}$ is also orthogonal.

Corollary 5.1.

Let $k$ be a positive even integer with $k>2$ . Let $N$ be a positive integer and $\Gamma=\Gamma_{0}(N)$ . Let $\{f_{k,1},\ldots,f_{k,e_{k}}\}$ be an orthogonal basis of $S_{k}(\Gamma)$ . Let $t_{0}\in\mathbb{R},\epsilon>0$ . Then, there exists a constant $C(t_{0},\epsilon)>0$ such that for $k>C(t_{0},\epsilon)$ there exists a basis element $f_{k,l}\in S_{k}(\Gamma)$ satisfying

[TABLE]

at any point $s=\sigma+it_{0}$ with

[TABLE]

6. The case of Jacobi forms

Let $k$ be a positive even integer and $m$ be a positive integer. Let $J_{k,m}$ be the space of Jacobi forms of weight $k$ and index $m$ on $\mathrm{SL}_{2}(\mathbb{Z})$ . From now, we use the notation $\tau=u+iv\in\mathbb{H}$ and $z=x+iy\in\mathbb{C}$ . We review basic notions of Jacobi forms (for more details, see [6]). 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

[TABLE]

and

[TABLE]

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

With these notations, we introduce the definition of a Jacobi form.

Definition 6.1.

A Jacobi form of weight $k$ and index $m$ on $\mathrm{SL}_{2}(\mathbb{Z})$ is a holomorphic function $F$ on $\mathbb{H}\times\mathbb{C}$ satisfying

(1)

$F|_{k,m}\gamma=F$ * for every $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ ,* 2. (2)

$F|_{m}X=F$ * for every $X\in\mathbb{Z}^{2}$ ,* 3. (3)

$F$ * has the Fourier expansion of the form*

[TABLE]

We denote by $J_{k,m}$ the vector space of all 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 $4ml-r^{2}>0$ , then it is called a Jacobi cusp form. We denote by $S_{k,m}$ the vector space of all Jacobi cusp forms of weight $k$ and index $m$ on $\mathrm{SL}_{2}(\mathbb{Z})$ .

For $1\leq j\leq 2m$ , we consider the theta series

[TABLE]

Suppose that $F(\tau,z)$ is a holomorphic function of $z$ and satisfy

[TABLE]

Then we have

[TABLE]

with uniquely determined holomorphic functions $F_{a}:\mathbb{H}\to\mathbb{C}$ . Furthermore, if $F$ is a Jacobi form in $J_{k,m}$ with the Fourier expansion (6.1), then functions in $\{F_{j}|\ 1\leq j\leq 2m\}$ have the Fourier expansions

[TABLE]

In [3], it is proved that the Petersson inner product of skew-holomorphic Jacobi cusp forms can be expressed as the sum of partial $L$ -values of skew-holomorphic Jacobi cusp forms. Similarly, for a Jacobi cusp form $F\in J_{k,m}$ with its Fourier expansion (6.1), we define partial $L$ -functions of $F$ by

[TABLE]

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

We write $\mathrm{Mp}_{2}(\mathbb{R})$ for the metaplectic group. The elements of $\mathrm{Mp}_{2}(\mathbb{R})$ are pairs $(\gamma,\phi(\tau))$ , where $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{R})$ , and $\phi$ denotes a holomorphic function on $\mathbb{H}$ with $\phi(\tau)^{2}=c\tau+d$ . The product of $(\gamma_{1},\phi_{1}(\tau)),(\gamma_{2},\phi_{2}(\tau))\in\mathrm{Mp}_{2}(\mathbb{R})$ is given by

[TABLE]

The map

[TABLE]

defines a locally isomorphic embedding of $\mathrm{SL}_{2}(\mathbb{R})$ into $\mathrm{Mp}_{2}(\mathbb{R})$ . Let $\mathrm{Mp}_{2}(\mathbb{Z})$ be the inverse image of $\mathrm{SL}_{2}(\mathbb{Z})$ under the covering map $\mathrm{Mp}_{2}(\mathbb{R})\to\mathrm{SL}_{2}(\mathbb{R})$ . It is well known that $\mathrm{Mp}_{2}(\mathbb{Z})$ is generated by $\widetilde{T}$ and $\widetilde{S}$ .

We define a $2m$ -dimensional unitary complex representation $\widetilde{\rho}_{m}$ of $\mathrm{Mp}_{2}(\mathbb{Z})$ by

[TABLE]

and

[TABLE]

Let $\chi$ be a multiplier system of weight $\frac{1}{2}$ on $\mathrm{SL}_{2}(\mathbb{Z})$ . We define a map $\rho_{m}:\mathrm{SL}_{2}(\mathbb{Z})\to\mathrm{GL}_{2m}(\mathbb{C})$ by

[TABLE]

for $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ . The map $\rho_{m}$ gives a $2m$ -dimensional unitary representation of $\mathrm{SL}_{2}(\mathbb{Z})$ .

Let $\{\mathbf{e}_{1},\ldots,\mathbf{e}_{2m}\}$ denote the standard basis of $\mathbb{C}^{2m}$ . For $F\in S_{k,m}$ , we define a vector-valued function $\tilde{F}:\mathbb{H}\to\mathbb{C}^{2m}$ by $\tilde{F}=\sum_{j=1}^{2m}F_{j}\mathbf{e}_{j}$ , where $F_{j}$ is defined by the theta expansion in (6.2). Then, the map $F\mapsto\tilde{F}$ induces an isomorphism between $S_{k,m}$ and $S_{k-\frac{1}{2},\bar{\chi},\rho_{m}}$ (for more details, see [6, Section 5] and [4, Section 3.1]).

Suppose that $F,G\in S_{k,m}$ . The Petersson inner product of $F$ and $G$ by

[TABLE]

where $\mathrm{SL}_{2}(\mathbb{Z})^{J}=\mathrm{SL}_{2}(\mathbb{Z})\ltimes\mathbb{Z}^{2}$ . Then, by Theorem 5.3 in [6], we have

[TABLE]

Note that $\rho_{m}(-I)$ is not equal to the identity matrix in $\mathrm{GL}_{2m}(\mathbb{C})$ . Instead, we have

[TABLE]

Then, the corresponding kernel function $R_{k,s,l}$ has the Fourier expansion

[TABLE]

where $r_{k,s,l,j}(n)$ is given by

[TABLE]

By the similar argument, we prove the same result as in Corollary 4.3 for the representation $\rho_{m}$ . From this, we have the following corollary.

Corollary 6.2.

Let $k$ be a positive even integer with $k>2$ . Let $\{F_{k,m,1},\ldots,F_{k,m,d}\}$ be an orthogonal basis of $S_{k,m}$ . Let $t_{0}\in\mathbb{R}$ and $\epsilon>0$ .

(1)

For any $k>C(t_{0},\epsilon,j)$ , any $1\leq j\leq 2m$ , and any $s=\sigma+it_{0}$ with

[TABLE]

there exists a basis element $F_{k,m,l}\in S_{k,m}$ such that

[TABLE] 2. (2)

There exists a constant $C(t_{0},\epsilon)>0$ such that for any $k>C(t_{0},\epsilon)$ , and any $s=\sigma+it_{0}$ with

[TABLE]

there exist a basis element $F_{k,m,l}\in S_{k,m}$ and $j\in\{1,\ldots,2m\}$ such that

[TABLE]

Bibliography14

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TL;DR

Contribution

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Abstract

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Non-Vanishing of L-Functions of Vector-Valued Modular Forms

Abstract.

1. Introduction

2. Preliminaries

Definition 2.1**.**

Theorem 2.2**.**

3. The Construction of the Kernel Function

Lemma 3.1**.**

Proof.

Lemma 3.2**.**

Proof.

4. The Main Result

Theorem 4.1**.**

Remark 4.2**.**

Proof.

Corollary 4.3**.**

5. The Case of Γ0(N)\Gamma_{0}(N)Γ0​(N)

Corollary 5.1**.**

6. The case of Jacobi forms

Definition 6.1**.**

Corollary 6.2**.**

Definition 2.1.

Theorem 2.2.

Lemma 3.1.

Lemma 3.2.

Theorem 4.1.

Remark 4.2.

Corollary 4.3.

5. The Case of $\Gamma_{0}(N)$

Corollary 5.1.

Definition 6.1.

Corollary 6.2.