Questions tagged [modular-group]

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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}=\...
Math101's user avatar
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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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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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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{...
Isolated Pole's user avatar
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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(\...
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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). ...
node196884's user avatar
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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-...
unknown's user avatar
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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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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 ...
WHy ILoveY's user avatar
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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 $T= \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$. ...
Giacomo Bascapè's user avatar
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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=...
El Rafu's user avatar
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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 ...
Klein Four's user avatar
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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 ...
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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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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 ...
ReiReiRei's user avatar
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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 ...
ReiReiRei's user avatar
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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 ...
Geet Thakur's user avatar
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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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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 ...
Fan Yun Hung's user avatar
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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 $$ f(\frac{az+b}{cz+d}) =f(z) $$...
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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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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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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 ...
Neil hawking's user avatar
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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=...
Neil hawking's user avatar
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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 ...
M R James 2017's user avatar
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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 ...
Neil hawking's user avatar
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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 ...
MathIsNice1729's user avatar
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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 ...
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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 ...
Neil hawking's user avatar
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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 $\Gamma$ ...
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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 ...
Neil hawking's user avatar
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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}\...
Neil hawking's user avatar
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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 ...
Neil hawking's user avatar
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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 ...
Neil hawking's user avatar
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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 ...
Neil hawking's user avatar
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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})...
Neil hawking's user avatar
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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 ...
Neil hawking's user avatar
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Cusps of the modular forms

According to the definition of the cusps attached above the set of cusps is infinite and to be more precise it is $Q \cup \infty$ since $G$ is a subgroup of $SL(2,Z)$ so the identity matrix (that is a ...
Neil hawking's user avatar
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The Riemann Surface $G/H$

Let $G$ be a subgroup of $SL(2,Z)$ that is of finite index and $H$ be the upper half-plane. How is the quotient topological space $G/H$ defined (understood)?
Neil hawking's user avatar
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Identification of quotient involving the Hilbert Modular Group

$K$ is a real quadratic number field with ring of integers $\mathcal O_K$. In Zagiers Modular Forms Associated to Real Quadratic Fields from 1975 at page 6 he introduces a surjective map $$m : S\to T$$...
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2 answers
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Modified weight 2 Eisenstein series is a modular form for $\Gamma_0(N)$

I'm doing exercise 1.2.8(e) in Diamond & Shurman's A First Course in Modular Forms. The problem is to show that $G_{2,N}(\tau) := G_2(\tau)-NG_2(N\tau)$ is in $M_2(\Gamma_0(N))$. To show this, I ...
Nico's user avatar
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3 votes
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Proving if $\gamma\in SL(2,\mathbb{Z})$ has order $6$, then $\gamma$ is conjugate to $\begin{bmatrix}0 & -1\\ 1 & 1\end{bmatrix}^{\pm1}$

I am reading Diamond's book on modular forms, and I came across the proof that the stabilizers of elliptic points on the upper half-plane are cyclic. To prove that, Diamond starts to show that a ...
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