Tagged Questions

2
votes
2answers
45 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
2
votes
1answer
31 views

Need help with finding length of sides and angles of a triangle in upper half plane model

The given points are $i, 3i, 1 + 2i$ I know that the distance for points on a vertical line can be found by using the formula $$\ln\left|\frac{y_2}{y_1}\right|$$ So the distance between points $ i$ ...
1
vote
1answer
63 views

Finding an angle of a triangle in the upper half plane model given three points

I've been given three points in the upper half plane $(i, 3i, 1 + 2i$), and one of the homework questions that asked is to find the angles of the given triangle. A previous problem asks to find the ...
1
vote
0answers
96 views

Calculating hyperbolic distance between two points

I was looking for a formula to calculate the hyperbolic distance between two planes and came across this formula $$\ln\left(\csc b-\cot b\over \csc a - \cot a\right)$$ where $a$ and $b$ are the ...
3
votes
1answer
60 views

Möbius Transformations are Orientation Preserving?

This question is truly stupid, but is driving me crazy. I just need an outside viewpoint to sort out what's going on. In my textbook: "Show that every linear fractional (LF) transformation of ...
0
votes
0answers
40 views

Fréchet mean of the hyperbolic shape space

The Fréchet mean of a general subspace is defined as $$F(x)=\int_Mdist(x,y)^2d\mu(y),$$ where $\mu$ is the probability measure on a general metric space $(M,dist)$. I understand that the Fréchet mean ...
2
votes
2answers
335 views

How does a conformal mapping preserve angles in hyperbolic geometry?

Suppose I have a sector $D = \{0 < \arg z < \alpha\}$ where $\alpha \leq 2\pi$. If I apply the function $w = \frac{\zeta - i}{\zeta + i}$ from the upper half plane to the unit disc ($\zeta = ...
0
votes
1answer
142 views

Geodesic distance on complex upper half plane

Let $z$ be a point in a fundamental domain of $\Gamma(2)\subset \mathrm{SL}_2(\mathbf{Z})$ in the complex upper half plane. Does there exist an $\epsilon >0$ such that the geodesic distance ...
1
vote
1answer
102 views

Why is this fundamental group a discrete subgroup in $\operatorname{SL}_2(\mathbf{R})$ of finite volume?

Let $B$ be a finite set in $\mathbf{P}^1(\mathbf{C})$. Let $G$ be the fundamental group of $\mathbf{P}^1(\mathbf{C}) - B$. We can view $G$ as a subgroup of $\mathrm{SL}_2(\mathbf{R})$. Why is $G$ ...
0
votes
1answer
71 views

Holomorphic function on an open subset of the complex upper-half plane

Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $U$ be a bounded open subset in $\mathbf{H}$ contained in $$\{\tau \in \mathbf{H}: \mathrm{Im}(\tau) ...
0
votes
1answer
159 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
0
votes
0answers
81 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
2
votes
1answer
96 views

Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...
1
vote
0answers
63 views

For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset

Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve. Let $P$ be the identity element of $E$. Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
2
votes
1answer
126 views

Möbius Transforms that preserve $\mathbb{H}$

I know that every möbius transform that preserves the upper half plane is of the form $m(z) = \frac{az+b}{cz+d}$, where $a,b,c,d \in \mathbb{R}$, or $m(z) = \frac{a\bar{z} + b}{c\bar{z} + d}$, where ...
2
votes
1answer
822 views

Möbius Transforms that preserve the unit disk

Say I wish to prove that every möbius transformation of the unit disk onto itself can be written in the form $A(z) = e^{i\theta}\frac{z+a}{1+\bar{a}z}$, where $\theta$ is a real number and $a$ is a ...
1
vote
1answer
223 views

Hyperbolic area and $SL_2$

Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that: a. The measure $\mu$ is invariant under all $g \in ...
4
votes
1answer
230 views

Circle preserving homeomorphisms in the closure of $\mathbb{C}$ and Möbius Transformations

I am presently a learner of Hyperbolic Geometry and am using J. W. Anderson's book $Hyperbolic$ $Geometry$. Now the author presents a sketch proof of why every circle preserving homeomorphism in ...