Questions tagged [modular-group]
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93 questions
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Computing cusps of Γ(N) [closed]
I am currently reading Shimura's "Introduction to Arithmetic Theory of Automorphic Functions" and having a hard time trying to understand why Γ(N) has exactly [Γ(1):Γ(N)]/N equivalence ...
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48
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Fundamental domain of $PSL_2(\mathbb{Z})$
Serre, in the book A Course in Arithmetic, shows that $D=\{ z\in H;\ |z|\geq 1\ \wedge \ |Re(z)|\leq 1/2 \}$ is a fundamental domain for the action of $PSL_2(\mathbb{Z})$ in the complex upper half ...
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GL2 action on torsion sections of elliptic curves
I am reading Kato's $p$-adic Hodge theory paper and I am confused about a rather embarrassing and admittedly very minor thing, but it is nevertheless important.
In Section 1.6 of Kato, it is claimed ...
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Meaning of this Eisenstein series notation from Gross–Zagier paper [duplicate]
In equation (2.14) of their paper [1], Gross–Zagier define the following Eisenstein series
\begin{align}
E_N(z, s) = \sum_{\gamma \in \bigg(\begin{matrix} \ast & \ast \\ 0 & \ast \end{matrix}\...
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Sources of non-congruence representations of the modular group
I'd be interested in any references that provide a source of non-congruence representations of the modular group or that discuss such representations. In my own field, vertex algebras, they have ...
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108
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Orbit of $z$ under $SL_2({\mathbb Z})$ action
The modular group $\mbox{SL}_2({\mathbb Z})=\left\lbrace\begin{pmatrix} a & b\\ c & d\end{pmatrix}: a,b,c,d \in {\mathbb Z}, ad-bc=1 \right\rbrace$ acts on the upper half plane ${\mathbb H}=\...
- 1,148
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Family of lattices defined by $\langle 1,\tau_n \rangle$ with vanishing imaginary part as $n\to\infty$
I am considering a family of lattices $\Lambda_n$ in $\mathbb{C}$ generated by $\langle 1,\tau_n\rangle$ for $n\geq 2$ with
$$\tau_n = -\left(\frac{n^2}{4}+1\right)^{-1} \left(\frac{n^2}{2}+1-\frac{n}{...
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65
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Finite index subgroup of congruence subgroup $\Gamma_0(4)$ and $\Gamma$
It is standard notation but I define them anyway to avoid ambiguity: $$\Gamma_0(N) := \left\{\begin{pmatrix} a & b \\
c & d \end{pmatrix} \in \Gamma : c \equiv 0 \text{(mod $...
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45
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Almost holomorphic functions
I am trying to compute an integral which goes as:
$$I=\int_{\mathcal{F}}\left(\sqrt{Im(z)}\eta(z)\overline{\eta(z)}\right)^{-k}\frac{dzd\bar{z}}{Im(z)^2}$$
Where $\eta$ is Dedekind's $\eta$, $\mathcal{...
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43
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Modular symbols, Manin symbols, two and three term relation
Maybe someone could help me out. I consider a $SL_2(\mathbb{Z})$-module $\Omega$. We set
\begin{align}
S:=\left(\begin{array}{rr} 0 & 1\\ -1 & 0\end{array}\right) \ \mathrm{and} \ U:=\left(\...
1
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1
answer
238
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Isomorphism of lattices/complex tori
This is essentially a reference request (apologies if it is a duplicate): it is known that every lattice $\Lambda$ in $\mathbb{C}$ is isomorphic to one of the form $\mathbb{Z} \oplus \mathbb{Z}[\tau]$ ...
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Is $\text{SL}(2, \mathbf Z)\simeq\text{Aut}(\mathbf P^1(\mathbf Z))$?
In the section "Over discrete rings" in this Wikipedia page, it is remarked that
Similarly, a homography of P(Q) corresponds to an element of the modular group, the automorphisms of P(Z).
...
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what is the exact relation between Dedekind tessellation and modular group
In the wiki for the Modular Group the group ($\Gamma$) is described as a $(2,3,\infty)$.
The page has a diagram for "a typical fundamental domain for the action of $\Gamma$ on the upper half-...
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Equivalence between parabolic 1-cocyles and right representations of modular group (from a paper of Zaiger)
I am reading the following paper of Zagier, and I am confused as to his identification of parabolic 1-cocyles and representations.
We will let $\Gamma=PSL_2(\mathbb{Z})$, and we shall denote $S=\begin{...
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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,
\begin{equation} \label{...
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59
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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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Pure mapping classes
I am a new guy in mapping class groups and I am very confused about pure mapping classes tonight. From the Primer by Farb and Margalit, I learned the definition of pure mapping class groups PMod$(S_{g,...
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On Rudin's proof of Picard's little theorem
I got a question in the proof of picard little theorem by W. Rudin in the textbook "Real & Complex Analysis(third edition)". 16.19 Theorem tries to show that $\phi$ in modular group ...
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Unique factorization of elements in $SL_2(\mathbb{Z})$
It is well know that the group $SL_2(\mathbb{Z})$ is generated by the matrices $S= \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$ and $T= \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$. ...
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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=...
- 658
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124
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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 ...
user1097270
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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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75
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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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1
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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
$$ f(\frac{az+b}{cz+d}) =f(z) $$...
- 8,576
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1
answer
133
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Consistency condition of multiplier system
I have recently started reading modular forms and while reading I got to know that we are interested in holomorphic functions F on the upper half plane which satisfies the transformation law
$$
F(Mz)=...
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1
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1
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Family of generators for congruence subgroup $\Gamma_0(11)$
Consider the congruence subgroup
$$\Gamma_0(11)=\left\{\begin{pmatrix}a&b\\11c&d\end{pmatrix}\in M(2,\mathbb{Z}): ad-11bc=1\right\}$$
I want to prove that the family
$$\Omega=\{-\text{Id}\}\...
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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
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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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1
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The Norm of a Differential Form on $\Gamma\backslash\mathbb{H}$
Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
Let $\mathbb{H}$ be the upper half-plane.
Let $\phi$ be a cusp form of weight $2$ for $\Gamma$. Then $\omega=...
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1
answer
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Meaning of $\Psi$ will 'go out' for points on $\mathbf R \cup \{i\infty\}$
$SL_2(\mathbf Z)$ has a fundamental domain $\Omega$ for which it “acts’’ on the upper half-plane $\mathbf H$. Suppose that $f (z)$ is a modular form for a fixed congruence group $\Gamma$. We only know ...
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Differential Forms of Degree $1$ on Riemann Surfaces
Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
Let $\mathbb{H}$ be the upper half-plane.
Let $f$ be a holomorphic complex function defined on the upper ...
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Show this property of primitive quadratic forms
Let $f(x,y),g(x,y)$ be two primitive quadratic forms of discriminant $D < 0$. Then the following statements are equivalent:
(i) $f(x,y)$ and $g(x,y)$ are in the same genus i.e. they represent the ...
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129
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Harada's Conjecture
I am reading a paper written by Masao Kiyota and Tetsuro Okuyama on "A Note on a Conjecture of K. Harada" and my question is regarding a statement in this paper. A link to the paper is ...
user911589
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Normalised Liouville Measure on $\Gamma / PSL(2,\mathbb{R})$
Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2, \mathbb{Z})$.
What does Normalised Liouville Measure on $\Gamma / PSL(2,\mathbb{R})$ mean?
Suggesting a reference is ...
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1
answer
167
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Harmonic $1-$ form on the upper half-plane $\mathbb{H}$
Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
A function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire modular form for the subgroup $\Gamma$ ...
- 2,566
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1
answer
331
views
The Space of Modular Forms and Riemann - Roch Theorem
Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
I think that it is well-known that a function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire ...
- 2,566
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1
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290
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Differential and Modular Form on a Compact Riemann Surface
Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
Let $\mathbb{H}$ be the upper half-plane.
Let $C$ be the set of all cusps of $\Gamma$.
Let $R = (\mathbb{H}\...
- 2,566
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0
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58
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The modular and entire modular forms for a subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$
Let $PSL(2,\mathbb{Z})$ be the modular group and $\Gamma$ be a subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
I think that it is well-known that a function $f:\mathbb{H}\rightarrow ...
- 2,566
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0
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Highly Recommended References for the Theory of Modular Forms for Normal Subgroups of Finite Index of $PSL(2, \mathbb{Z})$
I am looking for a highly recommended reference for the Theory of Modular Forms for Normal Subgroups of Finite Index of $PSL(2, \mathbb{Z})$.
Many references, that have been mentioned before on this ...
- 2,566
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0
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Regarding Congruence Subgroups and Normal Subgroups of Finite Index of the Modular Group $PSL(2,\mathbb{Z})$ ($SL(2,\mathbb{Z})$)
Let $PSL(2,\mathbb{Z})$ ($SL(2,\mathbb{Z})$) be the modular group.
It is too easy to see that every congruence subgroup of the modular group $PSL(2,\mathbb{Z})$ ($SL(2,\mathbb{Z})$) is a normal ...
- 2,566
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1
answer
150
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The action of the Modular Group $PSL(2, \mathbb{Z})$ on $\mathbb{Q} \cup \infty$
How to prove that the action of the Modular Group $PSL(2, \mathbb{Z})$ on $\mathbb{Q} \cup \infty$ is transitive, i.e., for every $q_1,q_2 \in \mathbb{Q} \cup \infty$, there is $g\in PSL(2, \mathbb{Z})...
- 2,566
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0
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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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