Questions tagged [modular-group]
The modular-group tag has no usage guidance.
42
questions
3
votes
1
answer
190
views
Irreducible unitary representation of PSL(2,Z)
Do we already know the classification of the finite-dimensional irreducible unitary representations of the modular group $PSL(2,\mathbb{Z})=\mathbb{Z}/2*\mathbb{Z}/3$?
I'm particularly interested in ...
- 449
2
votes
1
answer
402
views
Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?
Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...
- 83
2
votes
1
answer
137
views
Fractional group index?
The Hecke group of level two, $\Gamma_{0}(2)$, is an index-$2$ subgroup of the Fricke group of level two, $\Gamma_{0}^{+}(2)$, i.e. $\left[\Gamma_{0}^{+}(2):\Gamma_{0}(2)\right] = 2$. The index of $\...
- 389
1
vote
0
answers
69
views
Fourier series with fractional linear transformation of argument
Set as usual $e(z):=\exp(2\pi iz)$. Having two series that are related for all $z$ in the upper halfplane via a transformation formula
$$
\sum_{n=1}^{\infty}{\alpha_n e\Big(n\frac{az+b}{cz+d}\Big)}=p(...
- 386
1
vote
0
answers
88
views
Different modular data with same T-matrix
Modular data (MD) is an invariant of a modular fusion category $\mathcal{C}$. It is a couple of symmetric matrices $S, T \in M_r(\mathbb{C})$ with:
$r$ the rank of $\mathcal{C}$,
$S$ invertible,
$T$ ...
- 24.7k
1
vote
0
answers
68
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 ...
- 315
26
votes
4
answers
992
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-...
- 261
6
votes
1
answer
351
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 ...
- 295
3
votes
2
answers
275
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)$ ...
- 171
3
votes
0
answers
97
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 $\...
user35360
2
votes
1
answer
215
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 ...
- 1,712
3
votes
1
answer
249
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).$$
First question
What can we say in general about the factor $j(\...
- 2,295
3
votes
1
answer
111
views
Representatives for the action on an unknown set of matrices by an unknown modular subgroup
Let $\Gamma$ be the modular group and $\mathcal M_n$ the set of all primitive matrices with determinant $n\geq1$. Recall that a primitive matrix has relatively prime entries.
The modular group $\...
- 2,295
6
votes
1
answer
422
views
Behavior of a modular form in the lower strip
Let $f$ be an (elliptic) modular form of weight $k>0$, and consider the vertical strip $S_m=\{x+iy\in\mathbb{C}:|x|\le 1/2, y>m$}. For every $m\ll 1$, the fundamental domain for $SL_2(Z)$ is ...
- 253
3
votes
1
answer
374
views
The degree of the cube root of the $j$-invariant
I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...
- 2,295
5
votes
0
answers
178
views
Modular functions of the type $\mathfrak f(\cdot)^{k}\mathfrak f(\cdot)^{23nk}$
Let $\eta(\omega)$ be the Dedekind eta function and let $$\mathfrak f(w)=e^{-\pi i/24}\frac{\eta((\omega+1)/2)}{\eta(\omega)}.$$
In his paper On the “gap” in a theorem of Heegner, Stark fills the gap ...
- 2,295
6
votes
1
answer
212
views
Current interest in geometric properties of Hilbert fundamental domains
Harvey Cohn published several articles in the 1960's analyzing geometric properties of fundamental domains for Hilbert modular surfaces.
H. Cohn, "On the shape of the fundamental domain of the ...
- 2,428
10
votes
1
answer
2k
views
Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?
The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can ...
- 1,584
6
votes
1
answer
465
views
Fourth cohomology of the modular group
Is $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known? I ask this in response to the recent calculation of the same cohomology group for $\mathrm{Co}_0$ and $\mathrm{Co}_1$.
- 32.2k
14
votes
4
answers
1k
views
Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup
Let $G\subset \mathrm{SL}_2(\mathbb R)$ be a subgroup such that $\mathrm{SL}_2(\mathbb Z)\subset G$.
What are the possible groups such that $\mathrm{SL}_2(\mathbb Z)\subset G$ is of finite index? Is $...
- 185
2
votes
1
answer
103
views
Compute the automaton for the modular group
The modular group $\mathrm{PSL}_2(\mathbb{Z})$ has 3 generators $A,B,C$, where
$$A:z\to z+1,\quad B:z\to z-1,\quad C:z\to -1/z.$$
I want to compute the automaton that recognize the words of the ...
- 143
3
votes
3
answers
504
views
Finite subgroups (not finite index, just finite) of the modular group
The modular group is commonly described as the group of linear fractional transforms $z \mapsto \displaystyle \frac{az+b}{cz+d}$ with $a,b,c,d$ integers and $ad-bc = 1$. Of course, a great deal is ...
14
votes
2
answers
548
views
Non-congruence normal subgroups of $SL_2(\mathbb{Z}[1/2])$
Let $G=SL_2(\mathbb{Z}[1/2])$, i.e., the modular group (if you wish) over the ring $\mathbb{Z}[1/2]$ consisting of rationals whose denominators are powers of $2$. Unlike $SL_2(\mathbb{R})$, $G$ is ...
- 18.6k
5
votes
1
answer
254
views
Upper bound on level of a congruence subgroup of the modular group
Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of $\...
- 53
19
votes
0
answers
729
views
In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?
First let's recall some definitions. Let $G$ be a perfect group, so that
$$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$
for all abelian groups $A$ by universal coefficients. This means that when $A = ...
- 112k
2
votes
0
answers
78
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 ...
- 305
1
vote
2
answers
199
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 &...
user76479
6
votes
3
answers
713
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 \...
- 4,079
11
votes
1
answer
521
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?
- 11.1k
4
votes
2
answers
234
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{...
- 18.6k
3
votes
2
answers
687
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 ...
- 125
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
2k
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) ...
- 9,408
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 ...
- 71
3
votes
1
answer
345
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 ...
- 383
8
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.) ...
- 383
1
vote
2
answers
394
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 ...
- 383
2
votes
1
answer
940
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 ...
- 1,431
9
votes
1
answer
686
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 ...
- 2,100
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\...
- 12.7k
12
votes
4
answers
798
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 ...
- 14.7k
18
votes
2
answers
2k
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
$\Gamma=\mathrm{SL}_2(\mathbb{Z})$ ...
- 20.4k