Mathematics

Questions tagged [modular-group]

96 questions
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Center of $\pi_{1}(\mathbb{R}^{3} \backslash \text{trefoil knot})$ and $\mathrm{PSL}_{2}(\mathbb{Z})$

One of my friend tell me the following question: What is a center of the group $G = \langle a, b | a^{2} = b^{3}\rangle$ ? It is well-known that this is a fundamental group of a complement of a trefoil knot. However, I don't know whether there is a…
Seewoo Lee
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7
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Why is $\Gamma(N)$ generated by parabolic elements precisely when $X(N)$ has trivial genus?

According to this paper by M. D. Baker and A. W. Reid, if $\Gamma(N)$ denotes the principal congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ of level $N\geq1$ , then it should be a straightforward observation that $$\Gamma(N)\text{ is generated by…
K. Makabre
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6
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Using covering spaces to look at commutator of $G = \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3 \mathbb{Z}$

I'm trying to use covering spaces to show that the commutator $[G,G]$ of $G = \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3 \mathbb{Z}$ is isomorphic to the free group $F_2$ . I can write $G$ as the fundamental group of $X = \mathbb{R}P^2 \vee X_3$ where…
zork zork
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6
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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 a solution, the matrix can be diagonalized.…
Klein Four
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6
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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 union of the $\Gamma_0(N)$ conjugacy classes of…
fyhung88
  • 327
6
votes
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Index of a subgroup of the Modular Group

The subgroup of $SL(2,\mathbb{Z})$ generated by $\begin{pmatrix}1&0\\1&1\end{pmatrix}$ and $\begin{pmatrix}1&5\\0&1\end{pmatrix}$ has come up in a research question in string theory, and I am interested in determining whether or not its index is…
Daniel Longenecker
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6
votes
2 answers

Does every subgroup of finite index contain a power of each element of the group?

Let $G$ be a group, not necessarily finite. If $H$ is a normal subgroup of $G$ of a finite index, say $(G:H)=n$ , then for every $g\in G$ we have $g^n\in H$ . Does this statement remain valid if do not assume $H$ to be normal? In particular let…
Shimrod
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5
votes
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Explicit equations for $Y(N)$ for small $N$

Consider the congruence subgroup $$\Gamma(N) = \left\{\left(\begin{array}{cc} a & b \\ c & d\end{array} \right) \in SL_2(\mathbb{Z})\ ;\ \left(\begin{array}{cc} a & b \\ c & d\end{array} \right) \equiv \left(\begin{array}{cc} 1 & 0 \\ 0 &…
user806056
5
votes
1 answer

Free subgroups of $PSL(2,\mathbb{Z})$ of index 6

There are two "natural" subgroups of $PSL(2,\mathbb{Z})\cong C_2\ast C_3$ of index 6. One is the congruence subgroup $\Gamma_0(2)$ which is the kernel of the map $PSL(2,\mathbb{Z})\to PSL(2,\mathbb{Z}/2\mathbb{Z})$ . The other subgroup $H$ is the…
Thomas Browning
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5
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2 answers

Elliptic Points of Modular Group in Upper Half Plane

This is a very small question. Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain. I (probably) don't…
k.stm
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4
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1 answer

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with $ad-bc=1$. The modular group is generated by two elements $S$…
ArbiterKC
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Asymptotics of $j$-invariant at elliptic fixed points of $\text{PSL}(2,\mathbb Z)$

Let $j=12^3 E_4^3/(E_4^3-E_6^2)$ be the modular $j$ -invariant $$j(\tau)=q^{-1} + 744 + 196884q +...,$$ with $q=e^{2\pi i \tau}$ and $E_k$ the weight $k$ Eisenstein series. At the elliptic points $\tau=i$ and $\tau=\zeta_3=e^{2\pi i/3}$ of…
El Rafu
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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 \biggr\} \\ \Gamma_\infty \; &:= \; \biggl\{ \begin{pmatrix} \pm 1 & n \\ 0 & \pm 1…
aahlback
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4
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1 answer

$N$-th root of modular forms

In my research, I have encountered many $q$ -series of functions that turn out to be the Fourier expansions of roots of modular forms. Examples are $n$ -th roots of Eisenstein series and the $j$ -invariant. These seem to be related in certain…
El Rafu
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4
votes
1 answer

Fundamental domain for congruence subgroup.

Let $\Gamma$ be a congruence subgroup of the modular group $\mathrm{SL}_{2}(\mathbb{Z})$ . Let $R$ be coset representatives of the quotient $\Gamma\setminus\mathrm{SL}_{2}(\mathbb{Z})$ and let $\mathcal{D}$ be the standard fundamental domain for…
richarddedekind
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