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# Conjugacy growth series for wreath product finitary symmetric groups

- Ian Wagner
^{1}Email author

**Received:**15 December 2016**Accepted:**4 April 2017**Published:**1 August 2017

## Abstract

In recent work, Bacher and de la Harpe define and study conjugacy growth series for finitary permutation groups. In two subsequent papers, Cotron, Dicks, and Fleming study the congruence properties of some of these series. We define a new family of conjugacy growth series for the finitary alternating wreath product that are related to sums of modular forms of integer and half-integral weights, the so-called *mixed weight modular forms*. The previous works motivate the study of congruences for these series. We prove that congruences exist modulo powers of all primes \(p \ge 5\). Furthermore, we lay out a method for studying congruence properties for sums of mixed weight modular forms in general.

## Keywords

- Modular forms
- Group theory

## Mathematics Subject Classification

- 11F33
- 11F37
- 20B30
- 20C32

## 1 Introduction and statement of results

In the recent paper [1], Bacher and de la Harpe develop the theory of conjugacy growth series. This theory uses the minimum word length statistics and the conjugacy classes of a group to produce the conjugacy growth series. In particular, this theory ties together infinite permutation groups with finite support and the usual number theoretic partition function.

*G*be a group and

*S*a set that generates

*G*. Then for each \(g \in G\) define the

*word length*, \(\ell _{G,S}(g)\), to be the smallest nonnegative integer

*n*for which there are \(s_{1}, s_{2}, \ldots , s_{n} \in S \cup S^{-1}\) such that \(g = s_{1}s_{2}\cdot \cdot \cdot s_{n}\). Define the

*conjugacy length*, \(\kappa _{G,S}(g)\), as the smallest integer

*n*for which there exists

*h*in the conjugacy class of

*g*such that \(\ell _{G,S}(h) = n\). For \(n \in \mathbb {N}\) define \(\gamma _{G,S}(n) \in \mathbb {N}\cup \{0\} \cup \{\infty \}\) as the number of conjugacy classes of

*G*which contain elements

*g*with \(\kappa _{G,S}(g) = n\). Whenever \(\gamma _{G,S}(n)\) is finite for all

*n*, we can define the

*conjugacy growth series*:

*G*.

*finitary symmetric group*of \(\mathbb {N}\). It is the group of permutations of \(\mathbb {N}\) with finite support. Let the

*finitary alternating group*of \(\mathbb {N}\), Alt(\(\mathbb {N}\)), be the subgroup of Sym(\(\mathbb {N}\)) of permutations with even signature. Define the two generating sets of Sym(\(\mathbb {N}\)), \(S_{\mathbb {N}}^{Cox} = \{(i, i+1) : i \in \mathbb {N}\}\) and \(T_{\mathbb {N}} = \{(x,y) : x, y \in {\mathbb {N}}\) are distinct\(\}\). Let \(S \subset \) Sym(\(\mathbb {N}\)) be a generating set such that \(S_{\mathbb {N}}^{Cox} \subset S \subset T_{\mathbb {N}}\). Then Bacher and de la Harpe prove that (see Proposition 1 of [1])

*m*into an even number of parts.

*M*is a positive integer, let

Let *H* be a group and let \(H^{(\mathbb {N})}\) be the group of functions from \(\mathbb {N}\) to *H*. Then \(W := H \wr _{\mathbb {N}}\)Sym(\(\mathbb {N}\)) \(=H^{(\mathbb {N})} \rtimes \) Sym(\(\mathbb {N}\)) is called a *permutation wreath product*. Sym(\(\mathbb {N}\)) has a natural action on \(H^{(N)}\); \(\sigma \in \) Sym(\(\mathbb {N}\)) acts on \(\phi \in H^{(\mathbb {N})}\) by \(\sigma (\phi ) = \phi \circ \sigma ^{-1}\). One can also think of \(H^{(\mathbb {N})}\) as \(|\mathbb {N}|\) copies of *H*, and so an element of \(H^{(\mathbb {N})}\) can be thought of as \(|\mathbb {N}|\) elements of *H* indexed by \(\mathbb {N}\). In particular, Sym(\(\mathbb {N}\)) acts naturally on these indices. For \(\sigma , \tau \in \) Sym(\(\mathbb {N}\)) and \(\phi , \psi \in H^{(\mathbb {N})}\), the multiplication in the wreath product is given by \((\phi , \sigma )(\psi , \tau ) = (\phi \sigma (\psi ), \sigma \tau )\). The alternating wreath product, \(W^{'} := H \wr _{\mathbb {N}}\)Alt(\(\mathbb {N}\)), can be defined analogously.

Cotron, Dicks, and Fleming use the theory of modular forms to obtain their results. Therefore, one expects to use modular forms to study the conjugacy growth series (1.5). However, a difficulty arises; these functions are *mixed weight modular forms*, finite sums of modular forms with different weights. Therefore, we must first obtain a general theorem about congruences for coefficients of mixed weight modular forms. Let \(q := e^{2 \pi i z}\) with \(z \in \mathbb {H}\), and let \(\mathcal {M}_{k}(\Gamma _{0}(N), \chi )\) denote the space of weakly holomorphic modular forms of weight *k*, level *N*, and with character \(\chi \). *Weakly holomorphic modular forms* are those meromorphic modular forms whose poles (if any) are supported at cusps. Then the following offers such a theorem.

### Theorem 1.1

*K*be an algebraic number field with ring of integers \(\mathcal {O}_{K}\). Suppose \(f_{i}(z) = \sum _{n=0}^\infty a_{i}(n) q^{n} \in \mathcal {M}_{k_{i}}(\Gamma _{0}(N_{i}), \chi _{i}) \cap \mathcal {O}_{K}((q))\), \(g_{j}(z) = \sum _{m=0}^\infty b_{j}(m) q^{m} \in \mathcal {M}_{\lambda _{j} + \frac{1}{2}}(\widetilde{\Gamma _{0}(M_{j})}, \chi _{j}) \cap \mathcal {O}_{K}((q))\) where \(4 \mid M_{j}\) for every

*j*, and let

*N*be minimal such that \(N_{i} \mid N\) and \(M_{j} \mid N\) for every

*i*and

*j*, and let

*p*be prime such that \((N, p) = 1\). If

*r*is a sufficiently large integer, then for each positive integer

*j*, a positive proportion of primes \(Q \equiv -1 \pmod {N p^j}\) have the property that

*t*is a nonnegative integer.

### Remark

If there are any half integral weight forms in the sum of forms above, then we will have \(4 \vert N\). In this case, it is clear that *p* must be an odd prime.

Applying Theroem 1.1 to the conjugacy growth series leads to the following theorem.

### Theorem 1.2

*j*be a positive integer. If

*r*is a sufficiently large integer, then for a positive proportion of primes \(Q \equiv -1 \pmod {576p^{j}}\), we have that

*n*coprime to

*Qp*, and for all nonnegative integers

*t*.

*Dedekind’s eta-function*. From this we find the following congruence example.

### Example

To prove Theorems 1.1 and 1.2 we will make use of the theory of modular forms. The relevant generating functions for Theorem 1.2 turn out to be mixed weight modular forms. We will use the work of Treneer in [12] on weakly holomorphic modular forms and a famous theorem of Serre (see [6]). We will show that Theorem 1.1 follows from a proposition of Ono and Skinner in [8] which allows us to use the theory of Galois representations attached to modular forms for a finite set of modular forms simultaneously. Section 3.1 will cover basic facts about modular forms, operators for modular forms, and cusps. In Sect. 3.2 we will state important propositions of Treneer, Serre, Ono and Ahlgren, and others which are vital to our proofs. We will also discuss the interplay between Galois representations and Hecke operators in Sect. 3.2. Theorem 1.1 will be proved in Sect. 4 and Theorem 1.2 will be proved in Sect. 5. The paper will conclude with a short explanation of the example at the end of Sect. 1 in Sect. 6.

## 2 Conjugacy growth series for the finitary alternating wreath product

It is natural to ask if there are other subgroups of the finitary symmetric group that produce interesting conjugacy growth series. Recall the wreath product \(W = H \wr _{\mathbb {N}}\)Sym(\(\mathbb {N}\)).

For \(a \in H{\setminus }\{1\}\) and \(m \in \mathbb {N}\), let \(\phi _{m}^{a} \in W\) be the permutation that maps \((h, m) \in H \times \mathbb {N}\) to (*ah*, *m*) and fixes (*h*, *n*) if \(n \ne m\). Note that \((\phi _{m}^{a})_{a \in H{\setminus }\{1\}, m \in \mathbb {N}}\) generates the subgroup \(H^{(\mathbb {N})}\) and that \(\phi _{m}^{a}\) and \(\phi _{k}^{b}\) are conjugate in *W* if and only if *a* and *b* are conjugate in *H*. For \(m \in \mathbb {N}\), let \(H_{m} = \{\phi _{m}^{a} : a \in H{\setminus }\{1\}\}\) and let \(T_{H} = \bigcup _{m \in \mathbb {N}} H_{m}\) be a subset of \(H^{(\mathbb {N})}\). Recall that \(T_{\mathbb {N}}\) is the subset of all transpositions in Sym(\(\mathbb {N}\)). We consider subsets \(S_{H} \subset T_{H}\) and \(S_{\mathbb {N}} \subset T_{\mathbb {N}}\) and define \(S_{*}\) to be the disjoint union \(S_{H} \sqcup S_{\mathbb {N}}\). If \(S_{H} = \{\phi _{m_{1}}^{a_{1}}, \ldots , \phi _{m_{r}}^{a_{r}}\}\) where \(\{a_{1}, \ldots , a_{r}\}\) generate *H*, then \(S_{*}\) generates *W*. Define \(S_{*}^{'}\) analogously using subsets of \(T_{H}^{A}\) and \(T_{\mathbb {N}}^{A}\); then \(S_{*}^{'}\) generates \(W^{'} = H\wr _{\mathbb {N}}\)Alt(\(\mathbb {N}\)). This leads to the following proposition.

### Proposition 2.1

*H*be a finite group; denote by

*M*the number of conjugacy classes of

*H*. If \(W_{M}^{'} = H\wr _{\mathbb {N}}\)

*Alt*(\(\mathbb {N}\)) and \(S_{*}^{'}\) is a generating set satisfying (PCwr) in [1], then

###
*Proof of Proposition *2.1

*H*to the family of partitions

Let \(\mathbb {N}^{(w)}\) be the finite subset of \(\mathbb {N}\) that is the union of the supports of \(\phi \) and \(\sigma \) and let \(\sigma \) be the product of the disjoint cycles \(c_{1}, \ldots , c_{k}\) where \(c_{i} = (x_{1}^{(i)}, x_{2}^{(i)}, \ldots , x_{v_{i}}^{(i)})\) with \(x_{j}^{(i)} \in \mathbb {N}^{(w)}\) and \(v_{i} = \text { length}(c_{i})\). We include cycles of length 1 for each \(n \in \mathbb {N}\) such that \(n \in \) sup\((\phi )\) and \(n \notin \) sup\((\sigma )\) so that \(\mathbb {N}^{(w)} = \bigsqcup _{1\le i \le k}\) sup\((c_{i})\). Define \(\eta _{*}^{w}(c_{i}) \in H_{*}\) to be the conjugacy class of \(\phi (x_{v_{i}}^{(i)}) \phi (x_{v{i}-1}^{(i)}) \cdot \cdot \cdot \phi (x_{1}^{(i)}) \in H\). For \(\eta \in H_{*}\) and \(\ell \ge 1\), let \(m_{\ell }^{w,\eta }\) denote the number of cycles *c* in \(\{c_{1}, \ldots , c_{k} \}\) that are of length \(\ell \) and such that \(\eta _{*}^{w}(c) = \eta \). Let \(\mu ^{w, \eta } \vdash n^{w, \eta }\) be the partition with \(m_{\ell }^{w, \eta }\) parts equal to \(\ell \), for all \(\ell \ge 1\). Note that \(\sum _{\eta \in H_{*}, \ell \ge 1} n^{w, \eta } = \sum _{\eta \in H_{*}, \ell \ge 1} \ell m_{\ell }^{w, \eta } = |\mathbb {N}^{(w)}|\). Also observe that the partition \(\mu ^{w, 1}\) does not have parts of size 1 because if \(v_{i} = 1\) then \(\eta _{*}^{w}(c_{i}) \ne 1\). Using the same notation as above, let \(\lambda ^{w, 1}\) be the partition with \(m_{\ell }^{w, 1}\) parts equal to \(\ell - 1\). Because we are working in Alt(\(\mathbb {N}\)), we can write \(\sigma = \sigma _{e} \sigma _{o}\); and so this method actually splits to map to two partitions, one of which has an even number of parts. Define the *type* of *w* as the family \(\Big (\lambda ^{(1)}, \nu ^{(1)}; (\mu ^{(\eta )}, \gamma ^{(\eta )})_{\eta \in H_{*} {\setminus } 1} \Big )\). Then two elements in \(W_{M}^{'}\) are conjugate if and only if they have the same type. Thus, each \(H_{*}\)-indexed family of partitions, \(\Big (\lambda ^{(1)}, \nu ^{(1)}; (\mu ^{(\eta )}, \gamma ^{(\eta )})_{\eta \in H_{*} {\setminus } 1} \Big )\), is the type of one conjugacy class in \(W_{M}^{'}\).

Consider an \(H_{*}\)-indexed family of partitions \(\Big (\lambda ^{(1)}, \nu ^{(1)}; (\mu ^{(\eta )}, \gamma ^{(\eta )})_{\eta \in H_{*} {\setminus } 1} \Big )\) and the corresponding conjugacy class in \(W_{M}^{'}\). Let \(u^{(1)}, v^{(1)}, u^{(\eta )}, v^{(\eta )}\) be the sums of the parts of \(\lambda ^{(1)}, \nu ^{(1)}, \mu ^{(\eta )}, \gamma ^{(\eta )}\) and let \(k^{(1)}, t^{(1)}, k^{(\eta )}, t^{(\eta )}\) be the be the number of parts of \(\lambda ^{(1)}, \nu ^{(1)}, \mu ^{(\eta )}, \gamma ^{(\eta )}\) respectively.

## 3 Preliminaries

Here we will recall basic properties of modular forms. For more information see [6].

### 3.1 Modular forms, operators, and cusps

A large part of the proof of Theorem 1.2 involves understanding cusps of congruence subgroups. A *cusp* of \(\Gamma \subset SL_{2}(\mathbb {Z})\) is an equivalence class of \(\mathbb {Q}\cup \{\infty \}\) under the action of \(\Gamma \). We will divide the rest of the section into subsections on integer weight modular forms, half-integral weight modular forms, and then a section on the modularity of eta-quotients.

#### 3.1.1 Integer weight modular forms

*k*be an integer which denotes the weight of an integer weight modular form. For each meromorphic function

*f*on the upper half complex plane \(\mathbb {H}\) and each integer

*k*, define the slash operator, \(\vert _{k}\), by

*f*is a

*meromorphic modular form*of weight

*k*on \(\Gamma \) if

*f*a

*weakly holomorphic modular form*if its poles are supported at the cusps. If

*f*is holomorphic at the cusps we say it is a

*holomorphic modular form*, and if it vanishes at the cusps we say it is a

*cusp form*. We denote the spaces of weakly holomorphic, holomorphic, and cusp forms with character \(\chi \) by \(\mathcal {M}_{k}(\Gamma , \chi ), M_{k}(\Gamma , \chi ),\) and \(S_{k}(\Gamma , \chi )\), respectively. If \(\chi \) is a Dirichlet character modulo

*N*, then we say that a form \(f(z) \in M_{k}(\Gamma _{1}(N))\) (resp. \(S_{k}(\Gamma _{1}(N))\) or \(\mathcal {M}_{k}(\Gamma _{1}(N))\)) has

*Nebentypus character*\(\chi \) if

*f*has a Fourier expansion \(f(z) = \sum a(n) q^n\), where \(q := e^{2 \pi i z}\). If \(f(z) = \sum _{n \ge n_{0}} a(n)q^n\), the

*U*-operator, \(U_{t}\), on

*f*(

*z*) is defined by

*V*-operator, \(V_{t}\), is defined by

*k*is an integer.

- 1.If
*t*is a positive integer, then$$\begin{aligned} f(z) \vert V_{t} \in M_{k}(\Gamma _{0}(Nt), \chi ). \end{aligned}$$ - 2.If \(t \mid N\), then$$\begin{aligned} f(z) \vert U_{t} \in M_{k}(\Gamma _{0}(N), \chi ). \end{aligned}$$

*f*(

*z*) is a cusp form, then so are \(f(z) \vert U_{t}\) and \(f(z) \vert V_{t}\). If \(f(z) \in M_{k}(\Gamma _{0}(N), \chi )\), for each prime \(p \not \mid N\), the integer weight Hecke operator \(T_{p,k,\chi }\) preserves the space \( M_{k}(\Gamma _{0}(N), \chi )\) and acts on \(f(z) = \sum _{n=0}^\infty a(n) q^{n}\) by

#### 3.1.2 Half-integral weight modular forms

*f*is a

*weakly holomorphic modular form*of weight \( \lambda +\frac{1}{2}\) on \(\tilde{\Gamma }\) if it is holomorphic on \(\mathbb {H}\), meromorphic at the cusps, and satisfies

*f*is holomorphic at the cusps we say it is a

*holomorphic modular form*, and if it vanishes at the cusps we say it is a

*cusp form*. We denote the spaces of weakly holomorphic, holomorphic, and cusp forms with character \(\chi \) by \(\mathcal {M}_{\lambda +\frac{1}{2}}(\Gamma , \chi ), M_{\lambda +\frac{1}{2}}(\Gamma , \chi ),\) and \(S_{\lambda +\frac{1}{2}}(\Gamma , \chi )\). If \(\chi \) is a Dirichlet character modulo 4

*N*, then we say \(g(z) \in M_{\lambda + \frac{1}{2}}(\Gamma )\) (resp. \(S_{\lambda + \frac{1}{2}}(\Gamma )\) or \(\mathcal {M}_{\lambda + \frac{1}{2}}(\Gamma )\)) has

*Nebentypus character*\(\chi \) if

*U*-operator, \(U_{t}\), on

*g*(

*z*) is defined by

*V*-operator, \(V_{t}\), is defined by

- 1.If
*t*is a positive integer, then$$\begin{aligned} g(z) \vert V_{t} \in M_{\lambda + \frac{1}{2}} \left( \Gamma _{0}(4Nt), \left( \frac{4t}{\bullet } \right) \chi \right) . \end{aligned}$$ - 2.If \(t \mid N\), then$$\begin{aligned} g(z) \vert U_{t} \in M_{\lambda +\frac{1}{2}} \left( \Gamma _{0}(4N), \left( \frac{4t}{\bullet } \right) \chi \right) . \end{aligned}$$

*g*(

*z*) is a cusp form, then so are \(g(z) \vert U_{t}\) and \(g(z) \vert V_{t}\). If \(g(z) \in M_{ \lambda + \frac{1}{2}}(\Gamma _{0}(4N), \chi )\), where \(\lambda \) is an integer, then for each prime \(\ell \not \mid 4N\) the half-integral weight Hecke operator preserves the space \( M_{\lambda + \frac{1}{2}}(\Gamma _{0}(4N), \chi )\) and acts on \(g(z) = \sum _{m=0}^\infty b(m) q^m\) by

#### 3.1.3 Modularity of eta-quotients

*Dedekind’s eta-function*is a weight \(\frac{1}{2}\) modular form defined as

*eta-quotient*is a function

*f*(

*z*) of the form

### Proposition 3.1

*f*(

*z*) satisfies

### 3.2 Theorems of Treneer, Serre, and Ono

In order to study congruence properties, we turn to a result from Serre on the action of the Hecke operator on integral weight modular forms.

### Proposition 3.2

*K*, and

*M*is a positive integer. Furthermore, suppose \(k >1\). Then a positive proportion of the primes \(p \equiv -1 \pmod {MN}\) have the property that

There is an analogous proposition for half-integral weight modular forms due to Ono and Ahlgren which is proved using Proposition 3.2 and Shimura’s correspondence between half-integral weight modular forms and even integer weight modular forms.

### Proposition 3.3

*K*, and

*M*is a positive integer. Furthermore, suppose \(\lambda >1\). Then a positive proportion of the primes \(\ell \equiv -1 \pmod {4MN}\) have the property that

It is natural to ask for a generalization of Propositions 3.2 and 3.3 where a Hecke operator for a prime *p* could simultaneously annihilate a finite set of modular forms. In order to tackle this problem we will now turn our attention to modular Galois representations. Let \(\overline{\mathbb {Q}}\) be an algebraic closure of \(\mathbb {Q}\), and for each rational prime \(\ell \), let \(\overline{\mathbb {Q}}_{\ell }\) be an algebraic closure of \(\mathbb {Q}_{\ell }\). Fix an embedding of \(\overline{\mathbb {Q}}\) into \(\overline{\mathbb {Q}}_{\ell }\). This fixes a choice of decomposition group \(D_{\ell } = \{ \sigma \in \mathrm {Gal}(K/\mathbb {Q}) : \sigma (\mathfrak {p}_{\ell , K}) = \mathfrak {p}_{\ell , K} \}\). Specifically, if *K* is any finite extension of \(\mathbb {Q}\) and \(\mathcal{O}_{K}\) is the ring of integers of *K*, then for each \(\ell \) this fixes a choice of prime ideal \(\mathfrak {p}_{\ell ,K}\) of \(\mathcal{O}_{K}\) dividing \(\ell \). Let \(\mathbb {F}_{\ell ,K} = \mathcal{O}_{K}/\mathfrak {p}_{\ell ,K}\) be the residue field of \(\mathfrak {p}_{\ell , K}\) and let \(\vert \cdot \vert _{\ell }\) be an extension to \(\overline{\mathbb {Q}}_{\ell }\) of the usual \(\ell \)-adic absolute value on \(\mathbb {Q}_{\ell }\).

### Theorem 3.4

*a*(

*n*) and values of \(\chi \) to \(\mathbb {Q}\). If

*K*is any finite extension of \(\mathbb {Q}\) containing \(K_{f}\) and \(\ell \) is any prime, then due to work of Eichler, Shimura, Deligne, and Serre there is a continuous semisimple representation

- 1.
\(\rho _{f, \ell }\) is unramified at all primes \(p \not \mid N \ell \).

- 2.
\(Tr (\rho _{f, \ell }(Frob _{p})) \equiv a(p) \pmod {\mathfrak {p}_{\ell , K}}\) for all primes \(p \not \mid N \ell \).

- 3.
\(det (\rho _{f, \ell }(Frob _{p})) \equiv \chi (p) p^{k-1} \pmod {\mathfrak {p}_{\ell , K}}\) for all primes \(p \not \mid N \ell \).

- 4.
\(det (\rho _{f, \ell }(c)) = -1\) for any complex conjugation

*c*.

### Remark

*f*, and let

*f*does not have complex multiplication and \(\ell \) is sufficiently large, then the image of \(\rho _{f, \ell }\) contains a normal subgroup \(H_{f}\) conjugate to \(SL_{2}(\mathbb {F}_{f, \ell })\). Essentially, this means that the image of \(\rho _{f, \ell }\) is almost always ‘as large as possible’.

Newforms are eigenforms for the Hecke operator \(T_{p,k, \chi }\) with eigenvalues given by the *p*th coefficients of the newform. The fact that the image of \(\rho _{f, \ell }\) is large, along with an application of the Chebotarev Density Theorem, tells us we can choose the image of \(\mathrm {Frob}_{p}\) to have a trace of zero a positive proportion of the time. This determines the *p*th coefficient and thus implies Proposition 3.2 of Serre. The following lemma from [8] extends the idea of these representations having large image and allows us to apply it to sums of modular forms.

### Lemma 3.5

- 1.
the image of \(\rho _{1} \times \cdot \cdot \cdot \times \rho _{v}\) is conjugate to \(SL_{2}(\mathbb {F}_{1}) \times \cdot \cdot \cdot \times SL_{2}(\mathbb {F}_{v})\).

- 2.For each positive integer
*d*and each \(w \in \mathbb {F}_{i}\), a positive density of primes \(p \equiv 1 \pmod {d}\) satisfies$$\begin{aligned} a_{i}(p) \equiv w \pmod {\mathfrak {p}_{\ell , K}}. \end{aligned}$$ - 3.
For each pair of coprime positive integers

*r*,*d*, a positive density of primes \(p \equiv r \pmod {d}\) satisfies \(\vert a_{i}(p) \vert _{\ell } = 1\).

Part (1) of Lemma 3.5 specifically tells us that, with small adjustments, we can apply Propositions 3.2 and 3.3 to a finite set of modular forms simultaneously. This fact is crucial for the proof of Theorem 1.1.

A large portion of this paper will apply work of Treneer in [12] to \(C_{W_{M}^{'},S_{*}^{'}}(q)\). The main result from [12] follows.

### Proposition 3.6

*p*is an odd prime, and that

*k*and

*r*are integers with

*k*odd. Let

*N*be a positive integer with \(4 \mid N\) and \((N, p) = 1\), and let \(\chi \) be a Dirichlet character modulo

*N*. Let

*K*be an algebraic number field with ring of integers \(\mathcal {O}_{K}\), and suppose \(f(z) = \sum a(n) q^n \in \mathcal {M}_{\frac{k}{2}}(\widetilde{\Gamma _{0}(N)}, \chi ) \cap \mathcal {O}_{K}((q))\). If

*r*is sufficiently large, then for each positive integer

*j*, a positive proportion of primes \(Q \equiv -1 \pmod {Np^j}\) have the property that

*n*coprime to

*Qp*.

### Remark

A similar statement is true for integer weight forms which will be apparent in the proof. This should not be surprising due to the fact that Serre has shown that almost all coefficents of integer weight forms are \(0 \pmod {m}\) for any integer \(m > 1\).

## 4 Proof of Theorem 1.1

In order to find congruences for sums of mixed weight modular forms, we must examine where their coefficients overlap. The following lemma will describe this.

### Lemma 4.1

### Remark

If \(p_{i} = \ell _{j}\) for some *i* and *j*, then remove the \(p_{i}\) for the congruence to hold. This is made clear in the following corollary and in the proof of Lemma 4.1.

### Corollary

*p*, then

*t*is a nonnegative integer.

###
*Proof of Lemma *4.1

*n*with \(p^2 r\) with

*r*and

*p*coprime in \( f(z) \vert T_{p, k, \chi } = \sum _{n=0}^\infty \big ( a(pn) + \chi (p) p^{k-1} a(n/p) \big ) q^{n}\) to arrive at

*p*and

*r*coprime. This process can be repeated to show \(a(p^{2t+1}n) \equiv 0 \pmod {Q}\) for

*p*and

*n*coprime.

*m*with \(\ell s\) where \((\ell , s) = 1\) shows that \(b(\ell ^{3} s) \equiv 0 \pmod {Q}\). If \(\ell ^{2} \mid m\), then we can replace

*m*with \(\ell ^{5} s\) with \(\ell \) and

*s*coprime to get

*s*coprime. This process can be repeated to show \(b(\ell ^{4t+3}m) \equiv 0 \pmod {Q}\) for \(\ell \) and

*m*coprime. These two observations combined lead to Lemma 4.1.

## 5 Proof of Theorem 1.2

In this section we will explicitly work out the congruence properties of the conjugacy growth series from Sect. 2 following the work of Treneer in [12].

### Lemma 5.1

*p*is an odd prime and

*r*and \(N_{M, k}\) are integers with \((N_{M,k}, p) = 1\). If

*r*is sufficiently large, then for every positive integer

*j*there exists an integer \(\beta \ge j -1\) and a cusp form

###
*Proof of Lemma *5.1

*r*large enough so that \(F_{M,k}(z) \vert U_{p^{r}} = \sum a_{M,k}(p^{r}n)q^{n}\) is holomorphic at each cusp \(\frac{a}{c}\) with \(p^{2} \mid c\). Then we will define

*s*. Our cusp form will end up being \(F_{M,k,r}(z) \cdot F_{p}(z)^{p^{\beta }}\) for some integer \(\beta \). First we must find an explicit description of the Fourier expansion of \(F_{M,k}(z) \vert U_{p^{r}}\) at a cusp \(\frac{a}{c}\) with \(p^2 \not \mid c\). Note that for the remainder of the paper \(\gamma \) is used with the slash operator, even when the form being hit with the slash operator is a half-integral weight form. It is implied that \(\gamma \) (and other matrices) should be replaced by \(\tilde{\gamma }\) where appropriate. \(\square \)

### Proposition 5.2

###
*Proof of Proposition *5.2

*C*is a constant. We can see from this that \(t_{i}a = \alpha _{i} a_{i}\) and \(c= \alpha _{i} c_{i}\), so \(\alpha _{i} = (t_{i}, c) \le t_{i}\). Taking \(t_{1} = 12, n_{1} = -2(M-k), t_{2} = 24,\) and \(n_{2} = -k\) we arrive at the conclusion that \(F_{M,k}(z) \vert _{\frac{k}{2} - M} A = \sum _{n=n_{0}}^\infty a_{M,k,0}(n) q_{24}^{n}\) where \(n_{0} \ge -24M\). If we define \(\sigma _{v,t} := \left( \begin{array}{cc} 1 &{} v \\ 0 &{} t \end{array} \right) \), \(\widetilde{\sigma _{v,t}} := \left( \left( \begin{array}{cc} 1 &{} v \\ 0 &{} t \end{array} \right) , t^{1/4} \right) \), then notice that

*v*does. Also define

*v*does, therefore

### Proposition 5.3

*r*sufficiently large, \(F_{M,k,r}(z)\) vanishes at each cusp \(\frac{a}{cp^2}\) of \((\Gamma _{0}(N_{M,k}p^2)\) with \(ac > 0\).

###
*Proof of Proposition *5.3

*r*sufficiently large, \(-p^r < -24M \le n_{0}\). In the Fourier expansion if \(a_{M,k,0}(n) \ne 0\), in order for \(n \equiv 0 \pmod {p^r}\) to be true, \(n \ge 0\) must be true. Therefore, we have

*p*as

*v*does, so

*s*. Let

*r*be sufficiently large, and fix

*j*. If \(\beta \ge j - 1\) is sufficiently large, then

### Lemma 5.4

- (1)If
*k*is even, then \(F_{M,k,p,j}(z)\) is an integer weight cusp form, so for a positive proportion of primes \(Q \equiv -1 \pmod {N_{M,k} p^j}\), we havefor all nonegative integers$$\begin{aligned} a_{M,k}(Q^{2t+1} p^{r} n) \equiv 0 \pmod {p^j} \end{aligned}$$*t*, and*n*coprime to*Qp*. - (2)If
*k*is odd, then \(F_{M,k,p,j}(z)\) is a half-integral weight cusp form, so for a positive proportion of primes \(Q \equiv -1 \pmod {N_{M,k} p^j}\), we havefor all nonegative integers$$\begin{aligned} a_{M,k}(Q^{4t+3} p^r n) \equiv 0 \pmod {p^j} \end{aligned}$$*t*, and*n*coprime to*Qp*.

###
*Proof of Lemma *5.4

If *k* is even (resp. odd), then \(F_{M,k,p,j}(z)\) is an integral (resp. half-integral) weight cusp form. Thus, by Proposition 3.2 (resp. Proposition 3.3), for a positive proportion of primes \(Q \equiv -1 \pmod {N_{M,k} p^j}\), we have \(F_{M,k,p,j}(z) \vert T_{Q} \equiv 0 \pmod {p^j}\) (resp. \(F_{M,k,p,j}(z) \vert T(Q^2) \equiv 0 \pmod {p^j}\)). If we let \(F_{M,k,p,j}(z) = \sum _{n=1}^\infty c_{M,k}(n)q^{n}\), then by Lemma 4.1 we have \(c_{M,k}(Q^{2t+1}n) \equiv 0 \pmod {p^j}\) (resp. \(c_{M,k}(Q^{4t+3}n) \equiv 0 \pmod {p^j}\)) for any nonnegative integer *t* and *Q* and *n* coprime. The rest of the proof follows from the fact that \(c_{M,k}(n) \equiv a_{M,k}(p^r n) \pmod {p^j}\).

We will now refer back to part (1) of Lemma 3.5. Using this and Chebotarev’s Density Theorem, we are able to apply Proposition 3.2 or 3.3 simultaneously to each term in a sum of modular forms, which in turn allows us to apply Lemma 4.1 to our entire sum at the same time instead of piece by piece as in Lemma 5.4. As in Theorem 1.1, if we have a sum of modular forms \(f_{i}\) of mixed weights and level \(N_{i}\), we can replace the level *N* in Proposition 3.2 or 3.3 with the smallest \(N'\) such that each \(N_{i}\) divides \(N'\).

*k*even and

*k*odd, for all nonegative integers

*t*, and

*n*coprime to

*Qp*. Theorem 3.4 and Lemma 3.5 together imply that Lemma 5.4 can be applied to each \(F_{M,k}(z)\) simultaneously for a positive proportion of primes \(Q \equiv -1 \pmod {576p^j}\), so \(a_{M,k}(Q^{4t+3}p^r n) \equiv 0 \pmod {p^j}\) for each \(a_{M,k}(n)\). Since the congruence holds for each part of the sum, we also have

## 6 Details of the example

At the end of Sect. 1 the following example was given:

### Example

*Ramanujan’s tau function*. Using a theorem of Sturm in [11], one can verify with a finite computation that

## Declarations

### Acknowledgements

The author would like to thank Ken Ono for his guidance on this project and Maddie Locus for her helpful commments.

### Open Access

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## Authors’ Affiliations

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