# Questions tagged [modular-group]

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### Riemann Surfaces Associated to Subgroups of the Modular Group

If $G$ is a subgroup of the modular group of index $\mu$, then the triangulation of $\Gamma\backslash\mathbb{H}^\ast$ (where $\mathbb{H}^\ast$ is the extended upper half plane) induces a triangulation ...
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### Reference request: Universal central extension of $\operatorname{PSl}_2(\mathbb Z)$ is the braid group $\mathcal B_3$.

$\DeclareMathOperator{\PSl}{PSl}$ According to Wikipedia, the universal central extension of $\PSl_2(\mathbb Z)$ is given by the braid group $\mathcal B_3$ on three strands, \label{...
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### Quotient of modular group and principal congruence subgroup

The principal congruence subgroup $\Gamma(n)$ of $SL(2,\mathbb Z)$ is a normal subgroup of $SL(2,\mathbb Z)$ with index $$[SL(2,\mathbb Z):\Gamma(n)]=n^3\prod_{p|n}(1-\tfrac{1}{p^2}),$$ where the ...
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### What's the maximum order of an element in $SL_2(\mathbb{Z} /p\mathbb{Z})$ for $p>2$ prime?

I know the answer is $2p$ as I've checked it for $p=3,5$ and $71$. The characteristic polynomial of a matrix $A\in$ $SL_2(\mathbb{Z} /p\mathbb{Z})$ is $P_A(x)=x^2-tr(A)x+1$, so if this polynomial has ...
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### What's the maximum order of $A \in SL_2(\mathbb{Z}/p\mathbb{Z})$ if $tr(A)+2$ and $tr(A)-2$ are both either quadratic residue modp or non quad. res??

I have already proved that $SL_2(\mathbb{Z}/p\mathbb{Z})$ is a group with the product of matrices. I also proved that $|SL_2(\mathbb{Z}/p\mathbb{Z})|=p^3-p$ using the first isomorphism theorem and the ...
1 vote
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### Are matrices in $\text{PSL}(2, \mathbb{Z})$ conjugate to their inverses?

As I understand it, this comes down to calculating the slope of the expanding eigenvector of each matrix... but I am having trouble with the details. I feel that the fact that we have identified every ...
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### Modular group with a finite order $T$

Let $G$ be the modular group. We know this can be described by the relations (in terms of the $S$ and $T$ transformations) given by $S^4 = I, (ST)^3 = S^2$. In my work matrix representations of $G$ ...
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### Boundary of the fundamental domain of the modular group

Wikipedia states that the boundary of any fundamental domain should have some restrictions, such as smoothness of polyhedrality. Further, this thesis on Page 8, imposes the condition that the boundary ...
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### Irreducible representations of modular group $SL(2,Z)$ with finite image

The classification of irreducible representations of modular group $SL(2,Z)$ is difficult. But for the representations with finite image (the 1-dimensional representation is simple, with only 12), the ...
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### How to obtain normal subgroups of SL(2,Z) from normal subgroups of PSL(2,Z)?

When I look for a classification of the normal subgroups of $SL(2,Z)$, I found that people tend to only study $PSL(2,Z)=SL(2,Z)/\{1,-1\}$'s subgroups or normal subgroups, as far as I know Morris ...
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### What are examples of modular form of level $1$ (i.e. modular form on $SL_2(\mathbb{Z})$) with poles?

I am thinking of Eisenstein series, $$G_k(\tau) = \sum_{(c,d)\in {\mathbb{Z}}^2-\{(0,0)\}}\frac{1}{(c\tau+d)^k}, \tau \in \mathbb{H}$$ because we don't sum over $(0,0)$, so I'd like to call $(0,0)$ a ...
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### Decompose elements in $\Gamma_0(N)$

Consider the groups \begin{align*} \Gamma_0(N) \; &:= \; \biggl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) \;:\; c \equiv 0 \mod N \...
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### Proving Diamond & Shurman Exercise 3.7.1, about conjugacy class of $\Gamma_0^{\pm}(N)$.

I am reading chapter 3 of A First Course in Modular Forms but have troubles in Exercise 3.7.1 (c) and (d). (c) Show that the $\Gamma_0^{\pm}(N)$-conjugacy class of $\gamma \in \Gamma_0(N)$ is the ...
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### Fourier series analogue for Modular Group?

is there a fourier series analogue to the modular group ?=? $$z\mapsto\frac{az+b}{cz+d},$$ for example we could expand any function that satisfies the modular equation $$f(\frac{az+b}{cz+d}) =f(z)$$...
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### Question on an exercise involving the modular group

I‘m working on an exercise of a book over modular forms (Henri Cohen, Modular Forms, a classical approach) and I’m confused by the frasing of the question: Show that the map $\gamma \mapsto \gamma$i ...
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1 vote
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### The Branch Schema of a Subgroup of the Modular Group

Let $\Gamma$ be a subgroup of the modular group $PSL(2, \mathbb{Z})$. What is the best and easiest way to grasp the notion of the Branch Schema of the subgroup $\Gamma$. Why do we have only four cases ...
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### Regarding the definition of the entire modular forms

I was reading the Wiki page of the Modular Forms https://en.m.wikipedia.org/wiki/Modular_form In the definition, the function is assumed to be holomorphic at all cusps, then the entire modular form is ...
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### Cusps of the modular forms

According to the definition of the cusps attached above the set of cusps is infinite and to be more precise it is $Q \cup \infty$ since $G$ is a subgroup of $SL(2,Z)$ so the identity matrix (that is a ...
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1 vote
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### The Riemann Surface $G/H$

Let $G$ be a subgroup of $SL(2,Z)$ that is of finite index and $H$ be the upper half-plane. How is the quotient topological space $G/H$ defined (understood)?
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### Identification of quotient involving the Hilbert Modular Group

$K$ is a real quadratic number field with ring of integers $\mathcal O_K$. In Zagiers Modular Forms Associated to Real Quadratic Fields from 1975 at page 6 he introduces a surjective map $$m : S\to T$$...
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### Modified weight 2 Eisenstein series is a modular form for $\Gamma_0(N)$

I'm doing exercise 1.2.8(e) in Diamond & Shurman's A First Course in Modular Forms. The problem is to show that $G_{2,N}(\tau) := G_2(\tau)-NG_2(N\tau)$ is in $M_2(\Gamma_0(N))$. To show this, I ...
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