Congruence relations for $r$-colored partitions

Let $n\geq 1$ be an integer. A partition of $n$ is any non-increasing sequence of positive integers whose sum is $n$. The partition function $p(n)$ counts the number of partitions of $n$. We agree that $p(0)=1$ and that $p(n)=0$ if $n\not\in\{0,1,2,\dots\}.$ It is well-known that we have

$$\sum^{\infty}_{n=0}p(n)q^{n}=\prod^{\infty}_{n=1}(1-q^{n})^{-1}.$$

(1.1)

The study of the arithmetic properties of $p(n)$ has a long history. For example, Ramanujan [Ram21] proved the following congruences for $p(n)$:

$$p(\ell n+\beta_{\ell})\equiv 0\pmod{\ell}\ \ \ \ \ \text{ for $\ell=5,7,11,$}$$

(1.2)

where $\beta_{\ell}:=\frac{1}{24}\pmod{\ell}$. Since the work of Ramanujan, many authors found more examples of congruences for $p(n)$. These take the form

$$p(\ell Q^{m}n+\beta)\equiv 0\pmod{\ell},$$

(1.3)

where $\ell\geq 5$ is prime, $Q$ is a prime distinct from $\ell$ and $m\geq 1$ is an integer. In his breakthrough work, Ono [Ono00] showed that for every $\ell\geq 5$, there are infinitely many primes $Q$ for which (1.3) holds with $m=4$. Such arithmetic phenomena also occur for the coefficients of a wide class of weakly holomorphic modular forms (see e.g. [Tre06] and [Tre08]). In contrast to this, Ahlgren and Boylan [AB03] showed that if (1.3) holds for $m=0$, then $\ell=5,7,\text{ or }11$. For $m=1,2$, Ahlgren, Beckwith, and Raum [ABR22] have shown that congruences of the form (1.3) are scarce in a precise sense.

In the 1960s, Atkin discovered examples of congruences (1.3) with $m=3$. In recent work, Ahlgren, Allen, and Tang [AAT21] study the existence of such congruences. These come from two families which we describe below.

For primes $13\leq\ell\leq 31$, Atkin [Atk68a, eq. $52$] gave examples of primes $Q$ for which

$$p\left(\frac{\ell Q^{2}n+1}{24}\right)\equiv 0\pmod{\ell}\ \ \ \text{ if }\left(\frac{n}{Q}\right)=\epsilon_{Q},$$

(1.4)

where $\epsilon_{Q}\in\{\pm 1\}$. By fixing $n$ in one of the allowable residue classes modulo $24Q$, we obtain a congruence (1.3) with $m=3$. We refer to these as type I congruences. For each of $\ell=5,7,\text{ and }13$, Atkin [Atk68a, Thm. $1$, $2$] showed that for $Q\equiv-2\pmod{\ell}$, we have

$$p\left(\frac{Q^{2}n+1}{24}\right)\equiv 0\pmod{\ell}\ \ \text{ if }\left(\frac{-n}{\ell}\right)=-1\text{ and }\left(\frac{-n}{Q}\right)=-1.$$

(1.5)

We refer to these as type II congruences. By using modular Galois representations, Ahlgren, Allen and Tang proved that there are infinitely many type I congruences for every prime $\ell\geq 5$ and that there are infinitely many type II congruences for $\geq 17/24$ of the primes $\ell$.

For a positive integer $r$, define the $r$-colored partition function by

$$\sum^{\infty}_{n=0}p_{r}(n)q^{n}:=\prod^{\infty}_{n=1}(1-q^{n})^{-r}=1+rq+\cdots.$$

We agree that $p_{r}(n)=0$ if $n\not\in\{0,1,2,...\}$. Let $\ell\geq 5$ be prime. Many congruences have been proven for $p_{r}(n)$ (see e.g. [Atk68b, Thm. $1$] and [Gor83, Thm. $2$]). In this paper, for a wide range of $c\in\mathbb{F}_{\ell}$, we prove congruences of the form $p_{r}(\ell Q^{3}n+\beta_{0})\equiv c\cdot p_{r}(\ell Qn+\beta_{1})\pmod{\ell}$ for infinitely many primes $Q$. In particular, when $c=0$, we get congruences analogous to (1.4). Our first result relies on the assumption that $r$ and $\ell$ are compatible (we choose to delay the precise definition to Section $2$ for ease of exposition). Moreover, we will prove in Section $5$ that $r$ and $\ell$ are compatible if we have $2^{r+2}\not\equiv 2^{\pm 1}\pmod{\ell}$ and $\ell>5r+19$. We will also prove in Section $5$ that if $1\leq r\leq 39$ and $\ell>r+4$, then $r$ and $\ell$ are compatible whenever $2^{r+2}\not\equiv 2^{\pm 1}\pmod{\ell}$. Stating the result requires some notation. Define

$$\chi^{(r)}:=\begin{cases}\left(\frac{-4}{\bullet}\right)&\text{if }3\mid r,\\ \left(\frac{12}{\bullet}\right)&\text{if }3\nmid r.\end{cases}$$

Our first result states that we have type I congruences modulo $\ell$ for the functions $p_{r}(n)$ when $r$ and $\ell$ are compatible. We state these theorems under the assumption that $r<\ell-4$. When $r=\ell-4$, the situation is much simpler and will be discussed in the last section.

Thus, by choosing $n$ in any of the allowable residue classes modulo $24Q$, we obtain congruences of the form

$$p_{r}(\ell Q^{3}n+\beta)\equiv 0\pmod{\ell}.$$

We prove a more general result from which Theorem 1.1 follows by setting $\alpha=1$.

Thus, by choosing $n$ in any of the allowable residue classes modulo $24Q$ we obtain congruences of the form

$$p_{r}(\ell Q^{3}n+\beta_{0})\equiv(\alpha-1)\chi^{(r)}(Q)p_{r}(\ell Qn+\beta_{1})\pmod{\ell}.$$

The next result gives congruences analogous to $(1.4)$ and $(1.5)$ involving primes $Q$ in a different residue class modulo $\ell$. Our result does not rely on the assumption of compatibility.

The organization of this paper is as follows. In Section $2$, we give some background on modular forms and Galois representations. In Section $3$, we construct for each $r$ and $\ell$ important modular forms whose coefficients capture the relevant values of $p_{r}(n)$ modulo $\ell$. In the remaining sections (save for Section $7$), our main tool is the theory of modular Galois representations. In Section $4$, we prove that a wide range of $r$ and $\ell$ are compatible. In Section $5$, we prove the main technical result which we will need in order to prove Theorem 1.2. In Section $6$, we prove Theorems $1.2-1.3$. Finally, in Section $7$, we discuss congruences for $p_{r}(n)$ when $r=\ell-4$; in particular, we give a short proof of [Boy04, Thm. $2.1$, $(2)$].

We follow the exposition in [ABR22] and [AAT21]. Throughout, let $\ell\geq 5$ be prime and set $q:=e^{2\pi iz}$. Suppose that $k\in\frac{1}{2}\mathbb{Z}$ and that $N$ is a positive integer. For a function $f(z)$ on the upper half plane and

$$\gamma=\left(\begin{matrix}a&b\\ c&d\end{matrix}\right)\in\operatorname{GL}_{2}^{+}(\mathbb{Q}),$$

we have the weight $k$ slash operator

$$f(z)\big{|}_{k}\gamma:=\det(\gamma)^{\frac{k}{2}}(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right).$$

Let $A\subseteq\mathbb{C}$ be a subring. If $\nu$ is a multiplier system on $\Gamma_{0}(N)$, we denote by $M_{k}(N,\nu,A)$ (resp. $S_{k}(N,\nu,A)$) the space of modular forms (resp. cusp forms) of weight $k$ and multiplier $\nu$ on $\Gamma_{0}(N)$ whose Fourier coefficients are in $A$. When $\nu=1$ or $A$ is the subring of algebraic numbers that are integral at all of the primes above $\ell$, we omit them from the notation. Forms in these spaces satisfy the transformation law

$$f\big{|}_{k}\gamma=\nu(\gamma)f\ \ \ \text{ for }\ \ \ \gamma=\left(\begin{matrix}a&b\\ c&d\end{matrix}\right)\in\Gamma_{0}(N)$$

and the appropriate conditions at the cusps of $\Gamma_{0}(N)$.

If $k$ is even, we denote by $S^{\text{new}}_{k}(N,\mathbb{C})$ the new subspace. Let $S^{\text{new}}_{k}(N,A)$ be defined by $S_{k}(N,A)\cap S^{\text{new}}_{k}(N,\mathbb{C})$. When $N$ is square-free, there is an Atkin-Lehner involution $W_{p}$ on $S_{k}(N,\mathbb{C})$ for every prime divisor $p$ of $N$. Given a tuple $\epsilon=(\epsilon_{p})_{p\mid N}$ (where each $\epsilon_{p}\in\{\pm 1\}$), let $S^{\operatorname{new}}_{k}(N,\mathbb{C},\epsilon)$ be the subspace of forms $f$ with $f\big{|}_{k}W_{p}=\epsilon_{p}f$ for $p\mid N$. If $\ell$ and $N$ are coprime, then we let $S^{\operatorname{new}}_{k}(N,\epsilon)$ be the subspace of $S^{\operatorname{new}}_{k}(N)$ attached to the tuple $\epsilon$ (for such $\ell$, the operator $W_{p}$ acts on $S^{\operatorname{new}}_{k}(N)$). We define the eta function by

$$\eta(z):=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})$$

and the theta function by

$$\theta(z):=\sum_{n=-\infty}^{\infty}q^{n^{2}}.$$

The eta function has a multiplier $\nu_{\eta}$ satisfying

$$\eta(\gamma z)=\nu_{\eta}(\gamma)(cz+d)^{\frac{1}{2}}\eta(z),\ \ \ \ \ \gamma=\left(\begin{matrix}a&b\\ c&d\end{matrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z});$$

throughout, we choose the principal branch of the square root. For $c>0$, we have the formula [Kno70,  $\mathsection$$4.1$]

$$\nu_{\eta}(\gamma)=\begin{cases}\left(\frac{d}{c}\right)e\left(\frac{1}{24}((a+d)c-bd(c^{2}-1)-3c)\right)&\text{if }c\text{ is odd,}\\ \left(\frac{c}{d}\right)e\left(\frac{1}{24}((a+d)c-bd(c^{2}-1)+3d-3-3cd)\right)&\text{if }c\text{ is even}.\end{cases}$$

(2.1)

For the multiplier of the theta function we have

$$\nu_{\theta}(\gamma):=(cz+d)^{-\frac{1}{2}}\frac{\theta(\gamma z)}{\theta(z)}=\left(\frac{c}{d}\right)\epsilon_{d}^{-1},\ \ \ \ \ \gamma=\left(\begin{matrix}a&b\\ c&d\end{matrix}\right)\in\Gamma_{0}(4),$$

where

$$\epsilon_{d}=\begin{cases}1&\text{if }d\equiv 1\pmod{4},\\ i&\text{if }d\equiv 3\pmod{4}.\end{cases}$$

For odd values of $d$, we have the formula

$$e\left(\frac{1-d}{8}\right)=\left(\frac{2}{d}\right)\epsilon_{d}.$$

(2.2)

For $r\in\mathbb{Z}$, we have

$$M_{k}(1,\nu^{r}_{\eta})=\{0\}\ \ \ \text{if}\ \ \ 2k-r\not\equiv 0\pmod{4}.$$

(2.3)

If $f\in M_{k}(1,\nu^{r}_{\eta})$, then $\eta^{-r}f$ is a weakly holomorphic form on $\operatorname{SL}_{2}(\mathbb{Z})$. This implies that each $f\in M_{k}(1,\nu^{r}_{\eta})$ has a Fourier expansion of the form

$$f=\sum_{n\equiv r\pmod{24}}a(n)q^{\frac{n}{24}}.$$

(2.4)

We have $\eta^{-r}f\in M_{k-\frac{r}{2}}(1)$ when $0<r<24$.

We next recall the $U$ and $V$ operators. For a positive integer $m$, we define them on Fourier expansions by

$$\left(\sum_{n=1}^{\infty}a(n)q^{\frac{n}{24}}\right)\big{|}U_{m}:=\sum_{n=1}^{\infty}a(mn)q^{\frac{n}{24}},$$

$$\left(\sum_{n=1}^{\infty}a(n)q^{\frac{n}{24}}\right)\big{|}V_{m}:=\sum_{n=1}^{\infty}a(n)q^{\frac{mn}{24}}.$$

If $k\in\frac{1}{2}\mathbb{Z}\backslash\mathbb{Z}$, then a computation using (2.1), (2.2) and (7.1) implies that

$$f\in S_{k}(1,\nu^{r}_{\eta})\implies f\big{|}V_{24}\in S_{k}\left(576,\left(\frac{12}{\bullet}\right)\nu^{r}_{\theta}\right)\ \ \ \text{ if $3\nmid r$}$$

(2.5)

and

$$f\in S_{k}(1,\nu^{r}_{\eta})\implies f\big{|}V_{8}\in S_{k}\left(64,\nu^{r}_{\theta}\right)\ \ \ \text{ if $3\mid r$.}$$

(2.6)

If $k\in\frac{1}{2}\mathbb{Z}\backslash\mathbb{Z}$, then for each prime $Q\geq 3$ we have the Hecke operator

$$T_{Q^{2}}:S_{k}(1,\nu^{r}_{\eta})\rightarrow S_{k}(1,\nu^{r}_{\eta}).$$

For $f=\displaystyle\sum a(n)q^{\frac{n}{24}}\in S_{k}(1,\nu^{r}_{\eta})$ and $Q\geq 5$, we have

$$f\big{|}T_{Q^{2}}=\sum\left(a(Q^{2}n)+Q^{k-\frac{3}{2}}\left(\frac{-1}{Q}\right)^{k-\frac{1}{2}}\left(\frac{12n}{Q}\right)a(n)+Q^{2k-2}a\left(\frac{n}{Q^{2}}\right)\right)q^{\frac{n}{24}}.$$

(2.7)

(see e.g. [Yan14, Proposition $11$]). Yang only states the result for $r$ such that $3\nmid r$. The result when $3\mid r$ follows from (2.6) and [Shi73, Thm. $1.7$]. When $3\mid r$, we also have (2.7) for $Q=3$.

For $k\in\frac{1}{2}\mathbb{Z}\backslash\mathbb{Z}$ such that $k\geq\frac{5}{2}$ and each odd squarefree $t$, if $3\nmid r$ (resp. $3\mid r$), then we have a Shimura lift on $S_{k}(1,\nu^{r}_{\eta})$ defined via (2.5) (resp. (2.6)) and the usual Shimura lift [Shi73] on $S_{k}\left(576,\left(\frac{12}{\bullet}\right)\nu^{r}_{\theta}\right)$ (resp. $S_{k}\left(64,\nu^{r}_{\theta}\right)$). Define

$$\psi^{(r)}:=\begin{cases}\left(\frac{4}{\bullet}\right)&\text{if }3\mid r,\\ \left(\frac{12}{\bullet}\right)&\text{if }3\nmid r.\end{cases}$$

and

$$w_{r}:=\begin{cases}3&\text{if }3\mid r,\\ 1&\text{if }3\nmid r.\end{cases}$$

The action on Fourier expansions is given by

$$\text{Sh}_{t}\left(\sum a(n)q^{\frac{n}{24}}\right)=\sum A_{t}(n)q^{n},$$

where

$$A_{t}(n)=\sum_{d\mid n}\left(\frac{-1}{d}\right)^{k-\frac{1}{2}}\psi^{(r)}(d)\left(\frac{t}{d}\right)d^{k-\frac{3}{2}}a\left(\frac{w_{r}tn^{2}}{d^{2}}\right).$$

We have (see e.g. [ABR22, $(2.13)$])

$$f\equiv 0\pmod{\ell}\iff\text{Sh}_{t}(f)\equiv 0\pmod{\ell}\ \ \ \text{ for all squarefree $t$}.$$

(2.8)

They only state their result for when $3\nmid r$, but the same argument applies when $3\mid r$. From [Yan14, Thm. $1$, $2$], it follows that

$$\text{Sh}_{t}:S_{k}(1,\nu^{r}_{\eta})\rightarrow S^{\text{new}}_{2k-1}\left(6,-\left(\frac{8}{r}\right),-\left(\frac{12}{r}\right)\right)\otimes\left(\frac{12}{\bullet}\right)\ \ \text{if $3\nmid r$}$$

and

$$\text{Sh}_{t}:S_{k}(1,\nu^{r}_{\eta})\rightarrow S_{2k-1}^{\text{new}}\left(2,\left(\frac{8}{r}\right)\right)\otimes\left(\frac{-4}{\bullet}\right)\ \ \text{if $3\mid r$}$$

(note that Yang uses $3r$ in place of $r$). Moreover, we have

$$\operatorname{Sh}_{t}\left(f\big{|}T_{Q^{2}}\right)=\left(\operatorname{Sh}_{t}f\right)\big{|}T_{Q},$$

where $T_{Q}$ is the Hecke operator of index $Q$ on the integral weight space.

We now summarize some facts about modular Galois representations. See [Hid00] and [Edi92] for more details. We begin with some notation. Recall that $\ell\geq 5$ is prime. Let $k$ be an even integer and $N\in\mathbb{Z}^{+}$ with $\ell\nmid N$. Let $\overline{\mathbb{Q}}\subseteq\mathbb{C}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. If $p$ is prime, let $\overline{\mathbb{Q}}_{p}$ be a fixed algebraic closure of $\mathbb{Q}_{p}$ and fix an embedding $\iota_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$. The embedding $\iota_{\ell}$ allows us to view the coefficients of forms in $S_{k}(N)$ as elements of $\overline{\mathbb{Q}}_{\ell}$, and for each prime $p$, the embedding $\iota_{p}$ allows us to view $G_{p}:=\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ as a subgroup of $G_{\mathbb{Q}}:=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. If $I_{p}\subseteq G_{p}$ is the inertia subgroup, we denote the coset of absolute Frobenius elements above $p$ in $G_{p}/I_{p}$ by $\text{Frob}_{p}$. For any finite extension $K/\mathbb{Q}$, denote by $\text{Frob}_{p}|_{K}$ the restrictions to $K$ of elements in $\text{Frob}_{p}$. For representations $\rho:G_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\overline{\mathbb{Q}}_{\ell})$ which are unramified at $p$ (which is to say that $I_{p}\subseteq\ker{\rho}$), the image of any $\sigma\in G_{p}$ under $\rho$ only depends on $\sigma I_{p}\in G_{p}/I_{p}$ (the same conclusion holds for representations of $\operatorname{Gal}(K/\mathbb{Q})$ and $\sigma|_{K}$); for any such representation, we denote by $\rho(\text{Frob}_{p})$ the image of any element in $\text{Frob}_{p}$ under $\rho$ and make a similar definition for representations of $\operatorname{Gal}(K/\mathbb{Q})$ and $\operatorname{Frob}_{p}|_{K}$.

We denote by $\chi:G_{\mathbb{Q}}\rightarrow\mathbb{Z}^{*}_{\ell}$ and $\omega:G_{\mathbb{Q}}\rightarrow\mathbb{F}^{*}_{\ell}$ the $\ell$-adic and mod $\ell$ cyclotomic characters, respectively. We let $\omega_{2},\omega^{\prime}_{2}:I_{\ell}\rightarrow\mathbb{F}^{*}_{\ell^{2}}$ denote Serre’s fundamental characters of level $2$ (see [Ser87,  $\mathsection$$2.1$]). Both characters have order $\ell^{2}-1$, and we have $\omega^{\ell+1}_{2}=\omega^{\prime\ell+1}_{2}=\omega$.

The following theorem is due to Deligne, Fontaine, Langlands, Ribet, and Shimura (see [AAT21, Thm. 2.1]).

We now define what it means for $r$ and $\ell$ to be compatible. Define

$$N_{r}:=\begin{cases}2&\text{if }3\mid r,\\ 6&\text{if }3\nmid r\end{cases}$$

and

$$\epsilon_{r}:=\begin{cases}\left(\frac{8}{r\ell}\right)&\text{if }3\mid r,\\ (-\left(\frac{8}{r\ell}\right),-\left(\frac{12}{r\ell}\right))&\text{if }3\nmid r.\end{cases}$$

We now discuss filtrations. Recall that $\ell\geq 5$ is prime. If $f\in M_{k}(1,\mathbb{Z})$ is given by $f=\displaystyle\sum^{\infty}_{n=0}a(n)q^{n}$, then we define

$$\overline{f}:=\sum^{\infty}_{n=0}\overline{a(n)}q^{n}\in\mathbb{F}_{\ell}[[q]]$$

and

$$w_{\ell}(\overline{f}):=\inf\{k^{\prime}:\ \ \ \text{there exists $g\in M_{k^{\prime}}(1,\mathbb{Z})$ with $\overline{f}=\overline{g}$}\}.$$

We also define

$$\Theta:=\frac{1}{2\pi i}\frac{d}{dz}=q\frac{d}{dq}.$$

(2.9)

We require the following facts (see e.g. [Ser73, $\mathsection$$2.2$] and [Joc82, $\mathsection$$1$]).

Finally, for an even integer $k$, denote the weight $k$ Eisenstein series on $\operatorname{SL}_{2}(\mathbb{Z})$ by $E_{k}$.

Let

$$\Delta:=q\prod^{\infty}_{n=1}(1-q^{n})^{24}$$

be the unique normalized cusp form of weight $12$ on $\operatorname{SL}_{2}(\mathbb{Z})$. Set

$$\delta_{\ell}:=\frac{\ell^{2}-1}{24}.$$

By studying the filtration $w_{\ell}(\Delta^{r\delta_{\ell}}\big{|}U_{\ell})$, we prove the following result.