Questions tagged [modular-group]
96 questions
9
votes
1 answer
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…
Seewoo Lee
- 15,670
6
votes
1 answer
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
- 117
6
votes
1 answer
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.
I've…
Klein Four
- 61
6
votes
1 answer
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
- 325
6
votes
1 answer
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…
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
- 529
5
votes
1 answer
Explicit equations for $Y(N)$ for small $N$
Consider the congruence subgroup
$$\Gamma(N) = \left\{\left(\begin{array}{cc}
a & b \\
c & d\end{array} \right) \in SL_2(\mathbb{Z})\ ;\ \left(\begin{array}{cc}
a &…
user806056
5
votes
2 answers
Elliptic Points of Modular Group in Upper Half Plane
This is a very small question.
Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain.
I (probably) don't…
k.stm
- 19,187
4
votes
1 answer
Decomposition of modular group elements
The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by
$$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})$$
with $ad-bc=1$. The modular group is generated by two elements $S$…
ArbiterKC
- 248
4
votes
1 answer
Asymptotics of $j$-invariant at elliptic fixed points of $\text{PSL}(2,\mathbb Z)$
Let $j=12^3 E_4^3/(E_4^3-E_6^2)$ be the modular $j$-invariant
$$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
- 660
4
votes
0 answers
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\{
…
aahlback
- 157
4
votes
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 instances…
El Rafu
- 660
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
- 1,595
4
votes
2 answers
$SL(2, \Bbb Z)$ has only one cusp
Let $\Gamma$ be a congruence subgroup of $SL(2, \Bbb Z)$. A cusp is an equivalence of $\Bbb Q\cup\{\infty\}$ under $\Gamma$-action.
What's the meaning of "equivalence $\Bbb Q\cup\{\infty\}$ under $\Gamma$-action?
How to use the above definition to…
mathbeginner
- 1,941
4
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
- 4,541