# Questions tagged [modular-group]

The modular-group tag has no usage guidance.

38
questions

**0**

votes

**0**answers

87 views

### relationship between two modular automorphism groups associated n.s.f weights

Let $w$ be a normal semifinite faithful weight on a von Neumann algebra $M$. If $w_n$ is a sequence of weights which converge to $w$ in norm topology.
Does there exist relationship between $\{\sigma_t^...

**1**

vote

**0**answers

50 views

### continued fractions and cusp non-excursions

Consider the modular surface $X:=\mathbb{H^2}/PSL_2(\mathbb{Z})$.
Fix a width-of-cusp parameter $w, 0<w<<1$.
Let $B_w$ be the cusp neighborhood of width $w$. (So $w=1$ corresponds to the ...

**25**

votes

**4**answers

853 views

### Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$

Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...

**4**

votes

**1**answer

177 views

### Mapping class group of torus, why is $(ST)^3=S^2$?

In the context of topological quantum field theories, I am interested in the mapping class group of a torus. Here I can consider the torus as a square with identified edges and also decorated with ...

**3**

votes

**2**answers

213 views

### A question about congruence subgroups [closed]

For which $N_1$ and $N_2$ and $N$ be the greatest common divisor of $N_1$ and $N_2$, it is true that a congruence subgroup in $\mathrm{SL}_2(\mathbb{Z})$ generated by $\Gamma(N_1)\cup\Gamma(N_2)$ ...

**3**

votes

**0**answers

85 views

### Coset representatives of a principal congruence subgroup by another principal congruence subgroup

Consider the principal congruence subgroup $\Gamma(N)$, this consists of entries in $\mathrm{SL}_2(\mathbb{Z})$ congruent to the identity modulo $N$. Consider another principal congruence subgroup $\...

**2**

votes

**1**answer

167 views

### When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class?

In the paper p-adic L-functions and p-adic periods of modular forms, Greenberg/Stevens assert that if $\sigma_l:=\begin{pmatrix}l&0\\0&1\end{pmatrix}$ is the usual Hecke operator at $l$ double ...

**3**

votes

**1**answer

210 views

### How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...

**3**

votes

**1**answer

104 views

### Representatives for the action on an unknown set of matrices by an unknown modular subgroup

Let $\Gamma$ be the modular group and $\mathcal M_n$ the set of all primitive matrices with determinant $n\geq1$. Recall that a primitive matrix has relatively prime entries.
The modular group $\...

**6**

votes

**1**answer

415 views

### Behavior of a modular form in the lower strip

Let $f$ be an (elliptic) modular form of weight $k>0$, and consider the vertical strip $S_m=\{x+iy\in\mathbb{C}:|x|\le 1/2, y>m$}. For every $m\ll 1$, the fundamental domain for $SL_2(Z)$ is ...

**3**

votes

**1**answer

308 views

### The degree of the cube root of the $j$-invariant

I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...

**5**

votes

**0**answers

172 views

### Modular functions of the type $\mathfrak f(\cdot)^{k}\mathfrak f(\cdot)^{23nk}$

Let $\eta(\omega)$ be the Dedekind eta function and let $$\mathfrak f(w)=e^{-\pi i/24}\frac{\eta((\omega+1)/2)}{\eta(\omega)}.$$
In his paper On the “gap” in a theorem of Heegner, Stark fills the gap ...

**6**

votes

**1**answer

194 views

### Current interest in geometric properties of Hilbert fundamental domains

Harvey Cohn published several articles in the 1960's analyzing geometric properties of fundamental domains for Hilbert modular surfaces.
H. Cohn, "On the shape of the fundamental domain of the ...

**10**

votes

**1**answer

786 views

### Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?

The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can ...

**6**

votes

**1**answer

430 views

### Fourth cohomology of the modular group

Is $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known? I ask this in response to the recent calculation of the same cohomology group for $\mathrm{Co}_0$ and $\mathrm{Co}_1$.

**14**

votes

**4**answers

899 views

### Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup

Let $G\subset \mathrm{SL}_2(\mathbb R)$ be a subgroup such that $\mathrm{SL}_2(\mathbb Z)\subset G$.
What are the possible groups such that $\mathrm{SL}_2(\mathbb Z)\subset G$ is of finite index? Is $...

**2**

votes

**1**answer

94 views

### Compute the automaton for the modular group

The modular group $\mathrm{PSL}_2(\mathbb{Z})$ has 3 generators $A,B,C$, where
$$A:z\to z+1,\quad B:z\to z-1,\quad C:z\to -1/z.$$
I want to compute the automaton that recognize the words of the ...

**2**

votes

**3**answers

404 views

### Finite subgroups (not finite index, just finite) of the modular group

The modular group is commonly described as the group of linear fractional transforms $z \mapsto \displaystyle \frac{az+b}{cz+d}$ with $a,b,c,d$ integers and $ad-bc = 1$. Of course, a great deal is ...

**14**

votes

**2**answers

438 views

### Non-congruence normal subgroups of $SL_2(\mathbb{Z}[1/2])$

Let $G=SL_2(\mathbb{Z}[1/2])$, i.e., the modular group (if you wish) over the ring $\mathbb{Z}[1/2]$ consisting of rationals whose denominators are powers of $2$. Unlike $SL_2(\mathbb{R})$, $G$ is ...

**5**

votes

**1**answer

199 views

### Upper bound on level of a congruence subgroup of the modular group

Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of $\...

**17**

votes

**0**answers

624 views

### In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?

First let's recall some definitions. Let $G$ be a perfect group, so that
$$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$
for all abelian groups $A$ by universal coefficients. This means that when $A = ...

**2**

votes

**0**answers

70 views

### Action of a Drinfeld modular group on a Drinfeld symmetric space

Let $\Bbb C_\infty$ be functional field case analog of $\Bbb C$, i.e. $\Bbb C_\infty$ is the completion of the algebraic closure of the field of Laurent series $\Bbb F_q((\theta^{-1}))$, where $q$ is ...

**1**

vote

**2**answers

187 views

### Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 &...

**6**

votes

**3**answers

604 views

### Enumerating cosets of the modular group

Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in \...

**11**

votes

**1**answer

488 views

### Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?

**4**

votes

**2**answers

227 views

### Images of the fundamental domain of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ whose Euclidean area is large

Let $S$ be a compact subset of the closure of the upper half plane. (Assume that $S$ is a (Euclidean) rectangular box, if you wish.) Let $D$ be the standard fundamental domain of $\text{SL}_2(\mathbb{...

**3**

votes

**2**answers

632 views

### the Action of $SL_2(\mathbb{R})$ on a fundamental domain of $\Gamma$

Let $F=\{z\in H: |z|\geq1, |Re(z)|\leq 1/2\}$. It is a fundamental domain of the modular group $\Gamma$ acting on the upper half plane $H$. (Strictly speaking, one should take part of the boundary ...

**10**

votes

**4**answers

1k views

### Analytic function avoiding elements of the modular group

A friend recently told me the following two facts, for which he cannot recall a proof or a reference
(but he remembers seeing them in the literature):
Let $f$ be a holomorphic function mapping the ...

**6**

votes

**5**answers

1k views

### Are there noncongruence subgroups (of finite index) of the modular group generated only by 2 or 3 elements?

By the modular group I mean either $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$.
Where can I find examples of these?
Another question: is there a good (ideally analytical, but possibly computer-aided) ...

**7**

votes

**4**answers

2k views

### Does every polynomial diophantine equation have solutions modulo p?

Obviously, this is not exactly true; what I am really asking is whether any diophantine polynomial equation with integer coefficients (let's call them DPEICs) who's solution does not admit ...

**3**

votes

**1**answer

326 views

### Simplifying presentations of modular subgroups

I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - Handbook of Computational Group Theory) to find the presentations of various modular subgroups. For example, this ...

**6**

votes

**3**answers

1k views

### Are congruence subgroups of the modular group finitely presented?

Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) ...

**1**

vote

**2**answers

346 views

### Identifying Subgroups of the Modular Group via Permutation Representations on Cosets

Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to identify the generators ...

**2**

votes

**1**answer

852 views

### A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...

**8**

votes

**1**answer

642 views

### Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...

**11**

votes

**3**answers

1k views

### Congruence subgroups as abstract groups

This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup
$$
\pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:\;a,b,c,d\in\...

**12**

votes

**4**answers

710 views

### Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology

It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff ...

**18**

votes

**2**answers

1k views

### Distinguishing congruence subgroups of the modular group

This question is something of a follow-up to
Transformation formulae for classical theta functions .
How does one recognise whether a subgroup of the modular group
$\Gamma=\mathrm{SL}_2(\mathbb{Z})$ ...