mathmode, boxsize=0.9em

Congruences for modular forms and applications to crank functions

Hao Zhang1,2 and Helen W.J. Zhang1,2

1School of Mathematics

Hunan University

Changsha 410082, P. R. China

2Hunan Provincial Key Laboratory of

Intelligent Information Processing and Applied Mathematics

Changsha 410082, P. R. China

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, motivated by the work of Mahlburg, we find congruences for a
large class of modular forms. Moreover, we generalize the generating function
of the Andrews-Garvan-Dyson crank on partition and establish several new
infinite families of congruences. In this framework, we showed that both the
birank of an ordered pair of partitions introduced by Hammond and Lewis, and
$k$-crank of $k$-colored partition introduced by Fu and Tang process the same
as the partition function and crank.

AMS Classifications: 05A17, 11P83
Keywords: Modular form, Congruence, Partition function, Crank

##  1 Introduction

The objective of this paper is to present a modular form approach to deriving
congruences of the crank-type generating functions. This work was inspired by
the key construction of Mahlburg [12] that the generating functions of crank
is deeply related to Klein forms and weakly holomorphic modular forms of half-
integral weight on the congruence subgroup $\Gamma_{1}(l^{j})$. This leads to
arithmetic properties of the crank of partitions, which confirmed a conjecture
of Ono [13].

Recall that a partition of a nonnegative integer $n$ is any nonincreasing
sequence of positive integers whose sum is $n$. As usual, let $p(n)$ denote
the number of partitions of $n$. A breakthrough of applying modular forms to
partition functions is due to Ono [14]. Along with this direction, the
congruence properties for partition function are deeply investigated in the
subsequential work . For example, Ahlgren and Ono [1] showed that for prime
$\ell\geq 5$ and $m>0$, there are infinitely many non-nested arithmetic
progressions $An+B$ such that for every integer $n$ we have

$p(An+B)\equiv 0\pmod{\ell^{m}}.$

See [6, 8, 16] for more results.

To give the combinatorial interpretations of the well-known Ramanujan’s
congruences, Dyson [5] conjectured the existence of such a statistic named
crank, and its definition was discovered by Andrews and Garvan [3]. The crank
of a partition was defined as the largest part if the partition contains no
ones, and otherwise as the number of parts larger than the number of ones
minus the number of ones. For more details, please refer to Andrews and Garvan
[3, 9].

In view of partition congruences, Ono further conjectured that the crank
functions enjoy all of the partition congruences. The work of Mahlburg [12,
13] provided an elegant answer to these questions. Let $M(m,n)$ denote the
number of partitions of $n$ with crank is $m$. Then the generating function of
$M(m,n)$ can be written as

$\displaystyle\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}M(m,n)z^{m}q^{n}=\frac{(q;q)_{\infty}}{(zq,z^{-1}q;q)_{\infty}}.$
(1.1)

Mahlburg [12, 13] developed the theory of congruences for crank generating
functions, which are shown to possess the same sort of arithmetic properties
as partition functions. For instance, [12, 13] stated that if $\tau$ is a
positive integer, then there are infinitely many non-nested arithmetic
progressions $An+B$ such that

$\displaystyle M(m,\ell^{j},An+B)\equiv 0\pmod{\ell^{\tau}},$

simultaneously for every $0\leq m\leq\ell^{j}-1$.

Motivated by the work of Mahlburg, we define crank-type generating function as
follows:

$\displaystyle\mathfrak{F}_{r,d,t}(z,q)=\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}M_{r,d,t}(m,n)z^{m}q^{n}=\frac{(q^{r};q^{r})_{\infty}^{d}}{(q;q)_{\infty}^{t}(zq,q/z;q)_{\infty}}.$
(1.2)

Note that taking $r=t=1$ and $d=2$, (1.2) reduces to the ordinary crank
generating function (1.1). When $r=d=t=1$, (1.2) becomes the generating
function of the birank of an ordered pair of partitions given by Hammond and
Lewis [10]. Taking $r=t=1$ and $d=k-1$ in (1.2), we obtain the generating
function introduced by Bringmann and Dousse [4], whose combinatorial
interpretation is given by Fu and Tang [7] as the $k$-crank of $k$-colored
partition.

Our main result shows that the coefficients of crank-type generating function
enjoy the same arithmetic property as partition functions as well as crank.

######  Theorem 1.1.

Let $j,v$ be two positive integers. We fix a prime number $\ell>dr^{2}$ such
that $\left(\frac{g^{2}d-tr}{\ell}\right)$ are the same for all $g|r$. Then
there are infinitely many non-nested arithmetic progressions $An+B$ such that

$\displaystyle M_{r,d,t}(m,\ell^{j},An+B)\equiv 0\pmod{\ell^{v}},$

simultaneously for every $0\leq m\leq\ell^{j}-1$.

Theorem 1.1 provides the congruence of birank and $k$-crank. Recall that the
birank $b(\pi)$ of an ordered pair of partitions $\pi=(\lambda(1),\lambda(2))$
[10] is the number of parts in the first partition minus the number of parts
in the second partition, that is $b(\pi)=\\#(\lambda(1))-\\#(\lambda(2))$. The
number of ordered pairs of partitions of weight $n$ having birank $m$ will be
written as $R(m,n)$, hence the generating function for $R(m,n)$ is

$\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}R(m,n)z^{m}q^{n}=\frac{1}{(zq,q/z;q)_{\infty}}.$

######  Corollary 1.2.

Let $R(m,N,n)$ denote the number of ordered pairs of partitions of weight $n$
with birank congruent to $m$ modulo $N$. Suppose that $\ell\geq 5$ is prime
and that $\tau$ and $j$ are positive integers. Then there are infinitely many
non-nested arithmetic progressions $An+b$ such that

$\displaystyle R(m,\ell^{j},An+B)\equiv 0\pmod{\ell^{\tau}},$

simultaneously for every $0\leq m\leq\ell^{j}-1$.

The $k$-colored partition is a $k$-tuple of partitions
$\lambda=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)})$. For $k\geq 2$,
Fu and Tang [7] defined the $k$-crank of $k$-colored partition as follows:

$k\text{-crank}(\lambda)=\ell(\lambda^{(1)})-\ell(\lambda^{(2)}),$ (1.3)

where $\ell\left(\pi^{(i)}\right)$ denotes the number of parts in $\pi^{(i)}$.
Let $M_{k}(m,n)$ denote the number of $k$-colored partitions of $n$ with
$k$-crank $m$. The generating function of $M_{k}(m,n)$ can be derived by
Bringmann and Dousse [4]:

$\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}M_{k}(m,n)z^{m}q^{n}=\frac{(q;q)_{\infty}^{2-k}}{(zq;q)_{\infty}(z^{-1}q;q)_{\infty}}.$

######  Corollary 1.3.

Let $M_{k}(m,N,n)$ denote the number of $k$-colored partitions of $n$ with
$k$-crank congruent to $m$ modulo $N$. Suppose that $\ell\geq 5$ is prime and
that $\tau$ and $j$ are positive integers. Then there are infinitely many non-
nested arithmetic progressions $An+b$ such that

$\displaystyle M_{k}(m,\ell^{j},An+B)\equiv 0\pmod{\ell^{\tau}},$

simultaneously for every $0\leq m\leq\ell^{j}-1$.

##  2 Preliminaries

In this section, we present some basic proposition of modular forms of half-
integer weight which will be used in the following sections, see [15] for
details. Let $f=\sum_{n\geq 0}a_{n}q^{n}\in M_{k}(\Gamma_{0}(N),\chi)$ be a
modular form of weight $k\in\frac{1}{2}\mathbb{Z}\backslash\mathbb{Z}$ with
Nebentype character $\chi$.

Let $q=e^{2\pi iz}$. If

$f=\sum_{n=0}^{\infty}a(n)q^{n}\in M_{k}(\Gamma_{0}(N),\chi)$

and $\psi$ is a Dirichlet character with modulus $M$, we define the twist of
$f$ by $\psi$ to be

$f(\tau)\otimes\psi=\sum_{n=0}^{\infty}\psi(n)a(n)q^{n}.$

It is well-known that $f(\tau)\otimes\psi\in M_{k}(NM^{2},\chi\psi^{2})$. Fix
a constatn $\epsilon$, we define

$\widetilde{f_{\epsilon,\psi}}(\tau)=f(\tau)-\epsilon f(\tau)\otimes\psi.$

In the following, when $\epsilon$ and the character $\psi$ is specified, we
would write it as $\tilde{f}$.

With the notation above, let $p\nmid N$ be a prime number, then the action of
the half-integra Hecke operator $T_{p^{2}}$ is defined by

$f|T_{p^{2}}=\sum_{n\geq
0}\left(a(p^{2}n)+\chi(p)\left(\frac{(-1)^{k-\frac{1}{2}}n}{p}\right)p^{k-\frac{3}{2}}a(n)+\chi(p^{2})\left(\frac{(-1)^{k-\frac{1}{2}}}{p^{2}}\right)p^{2k-2}a\left(\frac{n}{p^{2}}\right)\right)q^{n}.$

We need the following result from Serre [17].

######  Theorem 2.1.

Let $f_{i}(\tau)\in S_{k_{i}}(\Gamma_{1}(N_{i}))$ be half-integer weight cusp
forms with algebraic integer coefficients where
$k_{i}\in\frac{1}{2}\mathbb{Z}\backslash\mathbb{Z}$ for $i=1,2,\cdots,m$. Then
for any $M\geq 1$, there exists a positive proportion of primes
$p\equiv-1\pmod{N_{1}\cdots N_{m}M}$ such that

$f_{i}|T_{p^{2}}\equiv 0\pmod{M},$

for every $i=1,2,\cdots,m$.

Let $\vec{a}=(a_{1},a_{2})\in\mathbb{Q}^{2}$. We set
$\tau^{\prime}=a_{1}\tau+a_{2}$ and
$q_{\tau^{\prime}}=\textbf{e}(\tau^{\prime})$. The Klein form is given by

$\displaystyle\mathfrak{k}_{\vec{a}}(\tau)=q_{\tau^{\prime}}^{\frac{a_{1}-1}{2}}\frac{(q_{\tau^{\prime}};q)_{\infty}(q/q_{\tau^{\prime}};q)_{\infty}}{(q;q)_{\infty}^{2}}.$
(2.1)

We recall some basic properties of Klein forms, one can find more details in
[11].

######  Proposition 2.2.

The Klein form $\mathfrak{k}_{\vec{a}}(\tau)$ satisfies the transformation
formula, i.e. for any $\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$, we have

$\mathfrak{k}_{\vec{a}}|_{\gamma}(\tau)=\mathfrak{k}_{\vec{a}\cdot\gamma}(\tau).$

If $(n_{1},n_{2})\in\mathbb{Z}^{2}$, then we have

$\mathfrak{k}_{\vec{a}+(n_{1},n_{2})}(\tau)=(-1)^{n_{1}+n_{2}+n_{1}n_{2}}\textbf{e}\left(\frac{a_{1}n_{2}-a_{2}n_{1}}{2}\right)\mathfrak{k}_{\vec{a}}(\tau).$

######  Lemma 2.3.

Let $\ell$ be a prime number, then we have

$\frac{\eta^{\ell}(\ell\tau)}{\eta(\tau)}\in
M_{\frac{\ell-1}{2}}\left(\Gamma_{0}(\ell),\left(\frac{\ast}{\ell}\right)\right).$
(2.2)

######  Lemma 2.4.

For integer $j\geq 2$, let

$E_{j}(\tau)=\frac{\eta^{\ell^{j}}(\tau)}{\eta(\ell^{j}\tau)}.$

Then we have $E_{j}(\tau)\in
M_{\frac{\ell^{j}-1}{2}}(\Gamma_{0}(\ell^{j}),\chi_{j})$ where
$\chi_{j}=\left(\frac{(-1)^{\frac{\ell^{j}-1}{2}}\ell^{j}}{\ast}\right)$.
Moreover, $E_{j}(\tau)$ vanishes at every cusp $\frac{a}{c}$ with $\ell^{j}$
not dividing $c$.

##  3 Proof of Theorem 1.1

The following lemmas play central roles in the proof of Theorem 1.1.

######  Lemma 3.1.

Let $e_{\ell}=\sum_{n=1}^{\ell-1}\left(\frac{n}{\ell}\right)e^{2\pi in/\ell}$
be the Gauss sum. Then the twisted modular form has the following expression.

$\displaystyle
f(z)\otimes\left(\frac{\ast}{\ell}\right)=\frac{e_{\ell}}{\ell}\sum_{n=1}^{\ell-1}\left(\frac{n}{\ell}\right)f(\tau)\left|\begin{pmatrix}1&-\frac{n}{\ell}\\\
0&1\end{pmatrix}\right..$ (3.1)

###### Proof.

The right hand side of (3.1) can be written as

$\displaystyle\frac{e_{\ell}}{\ell}\sum_{n=1}^{\ell-1}\sum_{m\geq
1}a(m)\left(\frac{n}{\ell}\right)\textbf{e}\left(2\pi
im\left(\tau-\frac{n}{\ell}\right)\right)$
$\displaystyle\quad=\frac{e_{\ell}}{\ell}\sum_{m\geq
1}e_{\ell}a(m)\left(\frac{-m}{\ell}\right)q^{m}$
$\displaystyle\quad=\sum_{m\geq 1}a(m)\left(\frac{m}{\ell}\right)q^{m},$

which completes the proof.

Consider a positive integer $N$ and set $\zeta=e^{2\pi i/N}$ in (1.2). For any
residue class $m\pmod{N}$, elementary calculations give the generating
function for the crank

$\displaystyle\sum_{n=0}^{\infty}M_{r,d,t}(m,N,n)q^{n}$
$\displaystyle=\frac{1}{N}\sum_{s=0}^{N-1}\mathfrak{F}_{r,d,t}(\zeta^{s},q)\zeta^{-ms}$
$\displaystyle=\frac{1}{N}\sum_{s=0}^{N-1}\zeta^{-ms}\left(\prod_{n=1}^{\infty}\frac{(1-q^{rn})^{d}}{(1-q^{n})^{t}(1-\zeta^{s}q^{n})(1-\zeta^{-s}q^{n})}\right).$
(3.2)

Substituting the definition of the Klein form (2.1) into (3), we consider the
following series

$\begin{split}g_{m}(\tau)&=\frac{1}{2\pi
i}\sum_{s=1}^{N-1}\frac{\omega_{s}\zeta^{-ms}}{\mathfrak{k}_{(0,\frac{s}{N})}(\tau)}\frac{\eta^{t\ell}(\ell\tau)\eta^{d\ell^{v}}(r\tau)}{\eta^{t}(\tau)}+\frac{\eta^{t\ell}(\ell\tau)\eta^{d\ell^{v}}(r\tau)}{\eta^{t}(\tau)}\\\
&:=G_{m}(\tau)+P(\tau)\end{split}$

where $\omega_{s}=\zeta^{s/2}(1-\zeta^{-s})$.

######  Lemma 3.2.

Let $r,d,t,\ell,v$ defined as before, then we have

$\frac{\widetilde{G_{m}}(24\tau)}{\eta^{t\ell}(24\ell\tau)\eta^{d\ell^{v}}(24r\tau)}E_{j+1}(24\tau)^{\ell^{v}}\in
S_{\lambda+1/2}(\Gamma_{0}(576r\ell^{j+1}),\chi)$ (3.3)

for some integer $\lambda$.

###### Proof.

By the Lemma 2.4 and the Theorem 1.65 of [15], we can show that the order of
$E_{j+1}(\tau)$ at the cusp $\frac{a}{c}$ with $\ell^{j+1}$ not dividing $c$
is at least $\frac{\ell^{2}-1}{24}$. On the other hand, we have

$\operatorname{ord}_{\frac{a}{c}}\eta^{t\ell}(\ell\tau)\eta^{d\ell^{v}}(r\tau)=\frac{t\ell^{3}+dr^{2}\ell^{v}}{24}<\frac{\ell^{2}-1}{24}\ell^{v}.$

Hence it is enough to show that
$\frac{\widetilde{G_{m}}(\tau)}{\eta^{k\ell}(\ell\tau)\eta^{d\ell^{v}}(r\tau)}$
vanishes at each cusp $\frac{a}{c}$ with $\ell^{j+1}|c$. Now we choose
suitable integers $b,d$ such that $A=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{0}(\ell^{j+1})$.

Now we compute the order of $G_{m}$ at the cusp $\frac{a}{c}$. To do that, let
$g=\gcd(c,r)$, $a_{1}=ra/g$, $c_{1}=c/g$. It is easy to see
$\gcd(a_{1},c_{1})=1$, so we can choose integers $b_{1},d_{1}$ such that
$B=\begin{pmatrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z})$. Then

$\displaystyle\Delta(r\tau)|_{12}A=(c\tau+d)^{12}\Delta\left(\frac{ar\tau+br}{c\tau+d}\right)$
$\displaystyle=$
$\displaystyle(c\tau+d)^{12}\Delta\left(B\left(\frac{g\tau+h}{r/g}\right)\right)$
$\displaystyle=$ $\displaystyle\Delta\left(\frac{g\tau+h}{r/g}\right),$

where $h=rbd_{1}-b_{1}d$. On the other hand, by the Lemma 2.3, we see that
$\frac{\eta^{\ell}(\ell\tau)}{\eta(\tau)}$ belongs to
$M_{\frac{\ell-1}{2}}\left(\Gamma_{0}(\ell),\left(\frac{\ast}{\ell}\right)\right)$.
By Proposition 2.2, we have

$\mathfrak{k}_{(0,\frac{s}{N})}|A(\tau)=\textbf{e}\left(\frac{N(cs+ds-\overline{ds})+cds^{2}}{2N^{2}}\right)\mathfrak{k}_{(0,\frac{\overline{ds}}{N})}(\tau),$

where $\overline{ds}$ is the unique integer between $0$ and $N-1$ such that
$N|ds-\overline{ds}$. Hence combining the discussion above, we get

$G_{m}(\tau)|A=\frac{1}{2\pi
i}\sum_{s=1}^{N-1}\frac{w_{s}\zeta^{-ms}}{e_{N}(c,d,s)\mathfrak{k}_{(0,\frac{\overline{ds}}{N})}(\tau)}\left(\frac{d}{\ell}\right)^{t}\frac{\eta^{t\ell}(\ell\tau)}{\eta^{t}(\tau)}\Delta^{(d\ell^{v}+d)/24}\left(\frac{g\tau+h}{r/g}\right),$

where
$e_{N}(c,d,s)=\textbf{e}\left(\frac{N(cs+ds-\overline{ds})+cds^{2}}{2N^{2}}\right)$.
Next we consider $G_{m}\otimes\left(\frac{\ast}{\ell}\right)$. For integer
$0\leq u\leq\ell-1$, we choose $u^{\prime}$ such that $u^{\prime}\equiv
d^{2}u\pmod{\ell}$. Then we have

$\begin{pmatrix}1&-u/\ell\\\ 0&1\end{pmatrix}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}a-cu/\ell&b-cuu^{\prime}/\ell^{2}+(au^{\prime}-du)/\ell\\\
c&d+cu^{\prime}/\ell\end{pmatrix}\begin{pmatrix}1&-u^{\prime}/\ell\\\
0&1\end{pmatrix}$

With the help of Lemma 3.1 we can show that

$G_{m}\otimes\left(\frac{\ast}{\ell}\right)|A=\frac{e_{\ell}}{2\pi
i\ell}\sum_{s=1}^{N-1}\sum_{u=1}^{\ell}\frac{w_{s}\zeta^{-ms}}{e_{N}(c,d^{\prime},s)^{\prime}\mathfrak{k}_{(0,\frac{\overline{d^{\prime}s}}{N})}(\tau)}\left(\frac{d^{\prime
t}u}{\ell}\right)\frac{\eta^{t\ell}(\ell\tau)}{\eta^{t}(\tau)}\Delta^{(d\ell^{v}+d)/24}\left(\frac{g\tau+h_{s}}{r/g}\right)$

where $d^{\prime}=d+cu^{\prime}/\ell$ and $h_{s}$ are integers. We compare the
first nonzero coefficient of $G_{m}(\tau)|A$ and
$G_{m}\otimes\left(\frac{\ast}{\ell}\right)|A$. In fact, if we assume that

$G_{m}(\tau)|A=a_{n}q^{n}+a_{n+1}q^{n+1}+\cdots$

where $n=t\delta_{\ell}+\frac{g^{2}(d\ell^{v}+d)}{24r}$. Then since $r$
divides $d\ell^{v}+d$, the expansion of
$G_{m}\otimes\left(\frac{\ast}{\ell}\right)|A$ is

$a_{n}\left(\frac{n}{\ell}\right)q^{n}+a_{n+1}^{\prime}q^{n+1}+\cdots$

By the assumption on $\ell$, this implies that

$\operatorname{ord}_{\frac{a}{c}}\widetilde{G_{m}}(\tau)\geq
t\delta_{\ell}+\frac{g^{2}(d\ell^{v}+d)}{24r}+1.$

By comparing the order, we see that
$\frac{\widetilde{G_{m}}(\tau)}{\eta^{k\ell}(\ell\tau)\eta^{d\ell^{v}}(r\tau)}$
also vanishes at the cusp $\frac{a}{c}$ when $\ell^{j+1}|c$.

The proof of following lemma is similar to Lemma 3.2, so we omit it.

######  Lemma 3.3.

Let $v$ be an integer large enough, then we have

$\frac{\tilde{P}(24\tau)}{\eta^{t\ell}(24\ell\tau)\eta^{d\ell^{v}}(24r\tau)}E_{j+1}(24\tau)^{\ell^{v}}\in
S_{\lambda^{\prime}+1/2}(\Gamma_{0}(576\ell^{\max\\{3,j+1\\}}),\chi).$ (3.4)

for some integer $\lambda^{\prime}$.

######  Lemma 3.4.

Let $f(q)$ be a formal power series such that $f(q)\equiv\sum_{n\geq
0}a_{n}q^{n\ell}\pmod{\ell^{v}}$. If the series $\sum b_{n}q^{n}$ and $\sum
b_{n}^{\prime}q^{n}$ coincide on the subsequence $\\{\ell
m+d\,|\,m\in\mathbb{Z}\\}$, then the series $\sum b_{n}q^{n}f(q)$ and $\sum
b_{n}^{\prime}q^{n}f(q)$ also coincide on the subsequence $\\{\ell
m+d\,|\,m\in\mathbb{Z}\\}$ modulo $\ell^{v}$.

###### Proof.

By the assumption, $\ell^{v}$ divides the coefficient of $q^{n}$ of $f(q)$
when $\ell\nmid n$. So in the expression of $\sum b_{n}q^{n}f(q)$, the
coefficient of $q^{\ell m+d}$ is congruence to $\sum b_{\ell
n+d}a_{m-n}\pmod{\ell^{v}}$. This implies that $\sum b_{n}q^{n}f(q)$ and $\sum
b_{n}^{\prime}q^{n}f(q)$ coincide on the subsequence $\\{\ell
m+d\,|\,m\in\mathbb{Z}\\}$ modulo $\ell^{v}$.

The following theorem gives the precise congruence relations for
$M_{r,d,t}$,which implies Theorem 1.1.

######  Theorem 3.5.

Let $r,d,t,j,v$ be positive integers and $\ell>dr^{2}$ be a prime number such
that $\left(\frac{g^{2}d-tr}{\ell}\right)$ are the same for all $g|r$. Then
there exists a positive proportion of primes $p\equiv-1\pmod{24\ell}$ such
that

$M_{r,d,t}\left(m,\ell^{j},\frac{np^{3}+k-dr}{24}\right)\equiv
0\pmod{\ell^{v}}$ (3.5)

for all $0\leq m<\ell^{j}$ and $n\equiv k-dr-24\beta\pmod{24\ell}$ with
$p\nmid n$ where $\beta$ is a certain integer between $0$ and $\ell$.

###### Proof.

We first assume that $v\geq\max\\{5,t\\}$ satisfies
$\ell^{v}\equiv-1\pmod{24r}$. We set
$\epsilon=\left(\frac{24(dr-t)}{\ell}\right)$,
$\alpha=\frac{t(\ell^{2}-1)+dr(\ell^{v}+1)}{24}$ and $0\leq\beta<\ell$ such
that $\left(\frac{\alpha+\beta}{\ell}\right)=0$ or $-\epsilon$. By the
definition of $\widetilde{g_{m}}(\tau)$, it is easy to that $g_{m}(\tau)$
coincides with $\widetilde{g_{m}}(\tau)$ on the subsequent $\\{\ell
n^{\prime}+\alpha+\beta\\}$. Moreover, we have

$\eta^{d\ell^{v}}(24r\tau)=q^{dr\ell^{v}}\prod_{n\geq
1}(1-q^{24rn})^{d\ell^{v}}\equiv q^{dr\ell^{v}}\prod_{n\geq
1}(1-q^{24rd\ell^{v}n})\pmod{\ell^{v}}$

Hence by the Lemma 3.4, we see that the subsequence of
$\frac{\widetilde{g_{m}}(24\tau)}{\eta^{t\ell}(24\ell\tau)\eta^{d\ell^{v}}(24r\tau)}$
with the indices $\\{\ell n^{\prime}+\alpha+\beta\\}$ coincides with
$\frac{g_{m}(24\tau)}{\eta^{t\ell}(24\ell\tau)\eta^{d\ell^{v}}(24r\tau)}$ on
the indices $\\{\ell n^{\prime}+\alpha+\beta\\}$ which is exactly

$\sum_{n^{\prime}\equiv\alpha+\beta\pmod{\ell}}\ell^{j}M_{r,d,t}(m,\ell^{j},n^{\prime}-\alpha)q^{24n^{\prime}-k\ell^{2}-dr\ell^{v}}\pmod{\ell^{v}}$

By changing the variable, we get

$\sum_{n\equiv
24\beta+dr-k\pmod{24\ell}}\ell^{j}M_{r,d,t}\left(m,\ell^{j},\frac{n+k-dr}{24}\right)q^{n}.$

Combining Lemma 3.2 and Lemma 3.3, we see that there exist two modular forms
$f_{m}$ and $f$ such that

$\sum_{n\equiv
24\beta+dr-k\pmod{24\ell}}\ell^{j}M_{r,d,t}\left(m,\ell^{j},\frac{n+k-dr}{24}\right)q^{n}\equiv
f_{m}(\tau)+f(\tau)\pmod{\ell^{v}}.$

Moreover, by applying Theorem 2.1, we can find a positive propotion of primes
$p\equiv-1\pmod{24\ell}$ such that

$f_{m}|T_{p^{2}}=f|T_{p^{2}}\equiv 0\pmod{\ell^{v}}.$

This implies that

$\begin{split}&\ell^{j}M_{r,d,t}\left(m,\ell^{j},\frac{p^{2}n+k-dr}{24}\right)+\ell^{j}\chi(p)\left(\frac{(-1)^{\lambda}n}{p}\right)p^{\lambda-1}M_{r,d,t}\left(m,\ell^{j},\frac{n+k-dr}{24}\right)\\\
&+\ell^{j}\chi(p^{2})\left(\frac{(-1)^{\lambda}}{p^{2}}\right)p^{2\lambda-1}M_{r,d,t}\left(m,\ell^{j},\frac{n/p^{2}+k-dr}{24}\right)\equiv
0\pmod{\ell^{v}}.\end{split}$

Finally, replacing $n$ by $pn^{\prime}$ with $p\nmid n^{\prime}$, we get

$\ell^{j}M_{r,d,t}\left(m,\ell^{j},\frac{p^{3}n^{\prime}+k-dr}{24}\right)\equiv
0\pmod{\ell^{v}}$

where $n^{\prime}\equiv k-dr-24\beta\pmod{24\ell}$. This gives

$M_{r,d,t}\left(m,\ell^{j},\frac{p^{3}n^{\prime}+k-dr}{24}\right)\equiv
0\pmod{\ell^{v-j}}$

But we note that there are infinite many $v$ satisfies
$\ell^{v}\equiv-1\pmod{24r}$, so the congruence relation $\eqref{eq:mrdt}$
holds for all $v\in\mathbb{N}$.

We end this article with a short list of questions arising from this project.
A natural question is that can one establish the combinatorial interpretations
of $M_{r,d,t}(m,n)$? Moreover, can we find any other kind of congruence of
$M_{r,d,t}(m,n)$?

Acknowledgements

The first author would like to acknowledge that the research was supported by
Fundamental Research Funds for the Central Universities (Grant No.
531118010622). The second author would like to acknowledge that the research
was supported by the National Natural Science Foundation of China (Grant Nos.
12001182 and 12171487), the Fundamental Research Funds for the Central
Universities (Grant No. 531118010411) and Hunan Provincial Natural Science
Foundation of China (Grant No. 2021JJ40037).

## References

  * [1] S.Ahlgren and K. Ono, _Congruence properties for the partition function_ , Proc. Natl. Acad. Sci. USA 98 (23) (2001) 12882–12884.
  * [2] G.E. Andrews and K. Ono, _Ramanujan’s congruences and Dyson’s crank_ , Proc. Natl. Acad. Sci. USA 102 (43) (2005) 15277.
  * [3] G.E. Andrews and F.G. Garvan, _Dyson’s crank of a partition_ , Bull. Amer. Math. Soc. 18 (2) (1988) 167–171.
  * [4] K. Bringmann and J. Dousse, _On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions_ , Trans. Amer. Math. Soc. 368 (5) (2016) 3141–3155.
  * [5] F.J. Dyson, _Some guesses in the theory of partitions_ , Eureka (Cambrige) 8 (1944) 10–15.
  * [6] A. Folsom, _Mock modular forms and $d$-distinct partitions_, Adv. Math. 254 (2014) 682–705.
  * [7] S. Fu and D. Tang, _On a generalized crank for $k$-colored partitions_, J. Number Theory 184 (2018) 485–497.
  * [8] S.A. Garthwaite and M. Jameson, _Incongruences for modular forms and applications to partition functions_ , Adv. Math. 376 (2021) 107448 17 pp.
  * [9] F.G. Garvan, _New combinatorial interpretations of Ramanujan’s partition congruences mod $5$, $7$ and $11$_, Trans. Amer. Math. Soc. 305 (1) (1988) 47–77.
  * [10] P. Hammond and R. Lewis, _Congruences in ordered pairs of partitions_ , Int. J. Math. Math. Sci. (45-48) (2004) 2509–2512.
  * [11] S. Lang, _Elliptic Functions_ , Springer-Verlag, New York, 1987.
  * [12] K. Mahlburg, _Partition congruences and the Andrews-Garvan-Dyson crank_ , Proc. Natl. Acad. Sci. USA 102 (43) (2005) 15373–15376.
  * [13] K. Mahlburg, _Congruences for the coefficients of modular forms and applications to number theory_ , Thesis (Ph.D.)-The University of Wisconsin- Madison. 2006. 55 pp.
  * [14] K. Ono, _Distribution of the partition function modulo $m$_, Ann. of Math. (2) 151 (1) (2000) 293–307.
  * [15] K. Ono, _The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series_. Conference Board of Mathematical Sciences 102 (2003).
  * [16] C.-S. Radu, _A proof of Subbarao’s conjecture_ , J. Reine Angew. Math. 672 (2012) 161–175.
  * [17] J.-P. Serre, _Divisibilitè de certaines fonctions arithmètiques_ , Enseign. Math. (2) 22 (3-4) (1976) 227–260.