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\begin{document}

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\section*{Moebius transformation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{crossratios}{Cross-ratios}\dotfill \pageref*{crossratios} \linebreak
\noindent\hyperlink{action_on_hyperbolic_space}{Action on hyperbolic space}\dotfill \pageref*{action_on_hyperbolic_space} \linebreak
\noindent\hyperlink{modular_group}{Modular group}\dotfill \pageref*{modular_group} \linebreak
\noindent\hyperlink{abstract_structure_of_modular_group}{Abstract structure of modular group}\dotfill \pageref*{abstract_structure_of_modular_group} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Let $k$ be a [[field]], and let $\mathbb{P}^1(k)$ be the [[projective line]] over $k$. A \emph{M\"o{}bius transformation} (also called a \emph{homography}, a \emph{linear fractional transformation}, or a \emph{fractional linear transformation} which is my own preference -- Todd) is a function $f: \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ defined by the rule

\begin{displaymath}
f(x) = \frac{a x + b}{c x + d}
\end{displaymath}
where $a, b, c, d \in k$ and $a d - bc \in k^\times$. M\"o{}bius transformations form a group under composition, isomorphic to the [[projective linear group]]

\begin{displaymath}
PGL_2(k) \coloneqq GL_2(k)/\{\lambda I: \lambda \in k^\times\}
\end{displaymath}
If, as in the case $k = \mathbb{C}$, each element of $k^\times$ has a square root, then this group is identified with

\begin{displaymath}
PSL_2(k) \coloneqq SL_2(k)/\pm I.
\end{displaymath}
Alternatively, a fractional linear transformation can be considered as synonymous with an automorphism of the field of [[rational functions]] $k(x)$ as a field over $k$ of [[transcendence degree]] 1.

In [[complex analysis]] (which is the usual context when one speaks of M\"o{}bius transformations; otherwise one usually calls them by some combination of ``linear'' and ``fractional''), M\"o{}bius transformations are precisely the [[biholomorphisms]] of the [[Riemann sphere]], hence exactly its bijective [[conformal transformations]].

Often, and particularly when $k$ is the the [[commutative ring]] of integers $\mathbb{Z}$, one considers a \emph{modular group} where the coefficients $a, b, c, d$ are assumed to lie in an integral domain and $a d - b c = 1$. (The [[homotopy quotient]] of the [[upper half-plane]] by the group $PGL_2(\mathbb{Z})$ is the [[moduli stack of elliptic curves]] over the [[complex numbers]].)

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{crossratios}{}\subsubsection*{{Cross-ratios}}\label{crossratios}

\begin{prop}
\label{}\hypertarget{}{}
The action $PGL_2(k) \times \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ is [[transitive action|3-transitive]], i.e., any triplet of distinct points $(a, b, c)$ may be mapped to any other triplet of distinct points $(a', b', c')$ by applying a group element.

\end{prop}
\begin{proof}
It suffices to consider $a' = 0, b' = 1, c' = \infty$ where one applies the transformation $x \mapsto \frac{(x-a)(b-c)}{(x-c)(b-a)}$.

\end{proof}
This motivates the following definition: given a 4-tuple $(a, b, c, d)$ of distinct points in $\mathbb{P}^1(k)$, its \emph{cross-ratio} is

\begin{displaymath}
\gamma(a, b, c, d) = \frac{(d-a)(b-c)}{(d-c)(b-a)}.
\end{displaymath}
It is not hard to see that the group action preserves the cross-ratio, i.e., $\gamma(g \cdot a, g \cdot b, g \cdot c, g \cdot d) = \gamma(a, b, c, d)$. Moreover, the group action is transitive on each cross-ratio-equivalence class of 4-tuples.

In the case $k = \mathbb{C}$ where $\mathbb{P}^1$ is interpreted as the Riemann sphere, it turns out that the cross-ratio of a 4-tuple is a [[real number]] if and only if the four points lie on a [[circle]] (or a line which is a circle passing through $\infty$). Hence M\"o{}bius = conformal transformations take circles to circles.

\hypertarget{action_on_hyperbolic_space}{}\subsubsection*{{Action on hyperbolic space}}\label{action_on_hyperbolic_space}

As explained at [[Poincare group]], the group $PSL_2(\mathbb{C})$ can be identified with those [[linear transformations]] of Minkowski space $\mathbb{R}^4$ that preserve the Minkowski [[quadratic form|form]] $Q$, are [[orientation]]-preserving, and take the forward [[light cone]] $\{v = (\vec{x}, t): Q(v) = 0, t \gt 0\}$ to itself. It follows that $PSL_2(\mathbb{C})$ acts on the hyperboloid sheet

\begin{displaymath}
H^3 = \{v = (\vec{x}, t): Q(v) = 1, t \gt 0\}
\end{displaymath}
which is naturally identified with hyperbolic 3-space. There is a Poincar\'e{} disk model for $H^3$; consider the disk $D^3$ that is the intersection of the [[future cone]] $\{v = (\vec{x}, t): Q(v) \geq 0, t \gt 0\}$ with the hyperplane $t = 1$. Its interior is an open 3-disk $int(D^3)$ which can be placed in perspective with $H^3$ by considering lines through the origin in $\mathbb{R}^4$: each line that passes through a unique point in $H^3$ passes through a unique point of $int(D^3)$. In this way, $D^3$ is viewed as a natural compactification of $H^3$, and the action of $PSL_2(\mathbb{C})$ on $H^3$ induces an action of $PSL_2(\mathbb{C})$ on $D^3$. The restriction of this action to the boundary $S^2 = \partial D^3$ (``the heavenly sphere'') coincides with the action on the Riemann sphere $S^2 = \mathbb{P}^1(\mathbb{C})$.

\hypertarget{modular_group}{}\subsection*{{Modular group}}\label{modular_group}

The \emph{modular group} $\Gamma$ is the subgroup $PSL_2(\mathbb{Z}) \hookrightarrow PSL_2(\mathbb{C})$ consisting of M\"o{}bius transformations $z \mapsto \frac{a z + b}{c z + d}$ where $a, b, c, d \in \mathbb{Z}$ and $a d - b c = 1$.

The group $PSL_2(\mathbb{R})$ acts on the upper half-plane $H = \{z \in \mathbb{C}: Im(z) \geq 0\}$ (or rather $H \cup \{\infty\}$ as a subspace of the Riemann sphere), by restriction of the action of $PSL_2(\mathbb{C})$ on the Riemann sphere. Indeed, the action of $PSL_2(\mathbb{R})$ takes the real line $\mathbb{R} \cup \{\infty\}$ to itself, and any element $f(z) = \frac{a z + b}{c z + d}$ takes $i$ to $(b + a i)(d - c i)/(c^2 + d^2)$, whose imaginary part $(ad - b c)/(c^2 + d^2) = 1/(c^2 + d^2)$ is positive. By continuity it follows that the action preserves the sign of the imaginary part, hence takes the upper-half plane $H$ to itself.

It is illuminating to think of complex numbers $\tau$ such that $Im(\tau) \gt 0$ as representing [[elliptic curves]] $E$. Indeed, the [[field]] of [[meromorphic functions]] on an elliptic curve (i.e., a complex projective curve $E$ of [[genus]] $1$, or a [[torus]] equipped with a structure of complex analytic 1-manifold) can be identified with a field of doubly periodic [[holomorphic functions]] $\mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})$ where $L = \langle \omega_1, \omega_2\rangle$ is a fundamental lattice (a discrete cocompact subgroup of the additive [[topological group]] $\mathbb{C}$) attached to $E$. In essence, this field is generated by [[Weierstrass elliptic functions]] $\wp(z), \wp'(z): \mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})$ (here $\wp'$ is the [[derivative]] of $\wp$) which satisfy a [[cubic curve|cubic algebraic relation]]

\begin{displaymath}
(\wp')^2 = 4\wp^3 - g_2\wp - g_3
\end{displaymath}
where the constants $g_2, g_3$ are expressed as certain [[Eisenstein series]] in the fundamental periods $\omega_1, \omega_2$. The $\mathbb{Z}$-[[linear basis]] elements $\omega_1, \omega_2$ of the [[lattice]] may be arranged so that $\tau = \omega_2/\omega_1$ has positive imaginary part. Of course, if there is a [[homothety]] $z \mapsto \lambda z$ that takes a lattice $L$ to a lattice $L'$, then the elliptic curves $E = \mathbb{C}/L$ and $E' = \mathbb{C}/L'$ are analytically isomorphic, so the map

\begin{displaymath}
\tau \mapsto \mathbb{C}/\langle 1, \tau\rangle
\end{displaymath}
gives a surjection from complex numbers with positive imaginary part to isomorphism classes of smooth elliptic curves. Thus we may restrict attention to lattices of the form $L = \langle 1, \tau \rangle$.

Of course, $L$ admits more than one such basis $(1, \tau)$, but for any other $(1, \tau')$ there is a [[linear transformation]] $\gamma \in \Gamma \coloneqq PSL_2(\mathbb{Z})$ such that $\gamma(\tau) = \tau'$. In summary, the [[orbit space]]

\begin{displaymath}
\{\tau \in \mathbb{C}: Im(\tau) \gt 0\}/\Gamma
\end{displaymath}
is a coarse [[moduli space]] for elliptic curves. In this context, one often says that elliptic curves are paramatrized by the $j$-[[j-invariant|invariant]], a certain [[modular form]] $j(\tau)$ defined on the upper half-plane such that $j(\tau) = j(\tau')$ if and only if $\tau' = \gamma \cdot \tau$ for some $\gamma \in \Gamma$.

Of course, in some cases there may be more than one $\gamma \in \Gamma$ that fixes a given $\tau$; this is notably the case when $\tau$ is a fourth root of unity or a sixth root of unity. A more refined approach then is to consider, instead of the coarse orbit space, the (compactified) [[moduli stack]] $(H \cup \{\infty\})//\Gamma$ for elliptic curves, as a central geometric object of interest.

\hypertarget{abstract_structure_of_modular_group}{}\subsubsection*{{Abstract structure of modular group}}\label{abstract_structure_of_modular_group}

As an abstract [[group]], $\Gamma = PSL_2(\mathbb{Z})$ is a [[free product]] $\mathbb{Z}/(2) \ast \mathbb{Z}/(3)$; explicitly, we may take the generator of order $2$ to be given by the Moebius transformation $z \mapsto -1/z$, and the generator of order $3$ to be given by $z \mapsto (z-1)/z$.

This [[concrete group]] and certain of its subgroups, such as [[congruence subgroups]], are fairly ubiquitous; for example the modular group appears in the theory of \href{http://ncatlab.org/nlab/show/continued+fraction#rational_tangles}{rational tangles} and of [[combinatorial map|trivalent maps]], and these groups frequently crop up in the theory of buildings (stuff on hyperbolic buildings to be filled in).

It is also worth pointing out that $\Gamma$ is a quotient of the [[braid group]] $B_3$. Consider the standard Artin presentation of $B_3$, with two generators $\sigma_1$, $\sigma_2$ subject to the relation

\begin{displaymath}
\sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2.
\end{displaymath}
Then $z \coloneqq (\sigma_1 \sigma_2)^3$ is a [[center|central element]] of $B_3$, and there is a central extension

\begin{displaymath}
1 \to \mathbb{Z} \stackrel{1 \mapsto z}{\to} B_3 \stackrel{q}{\to} \Gamma \to 1
\end{displaymath}
where $q$ is the unique homomorphism mapping $\sigma_1\sigma_2$ to $\lambda z. (z-1)/z$, and $\sigma_1\sigma_2\sigma_1$ to $\lambda z. \frac{-1}{z}$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[SL(2,Z)]]


\item [[conformal group]]


\item [[elliptic fibration]]


\item [[congruence subgroup]]


\item [[level n subgroup]],


\item [[modular equivariant elliptic cohomology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after \emph{[[August Möbius]]}.

[[!redirects Moebius transformations]] [[!redirects moebius transformation]] [[!redirects moebius transformations]] [[!redirects Möbius transformation]] [[!redirects Möbius transformations]] [[!redirects möbius transformation]] [[!redirects möbius transformations]] [[!redirects linear fractional transformation]] [[!redirects linear fractional transformations]] [[!redirects fractional linear transformation]] [[!redirects fractional linear transformations]]

[[!redirects Möbius group]] [[!redirects Moebius group]]

[[!redirects modular group]] [[!redirects modular groups]]



\end{document}