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\begin{document}
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\textbf{18.786 Number Theory II\hfill Spring~2024}\\\vspace{2pt}
\textbf{Problem Set \#1\hfill Due: 2/16/2024}\vspace{-8pt}
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\subsection*{Description}

These problems are related to material covered in Lectures 1--3.
Your solutions should be written up in latex (please do not submit handwritten solutions) and submitted as a pdf-file on \href{https://www.gradescope.com/courses/730173}{Gradescope} before midnight on the date due.  You can use the latex source for this problem set as a template, and are welcome to use the latex environment of your choice (I currently use \href{https://www.overleaf.com/}{Overleaf} and \href{https://www.xm1math.net/texmaker/}{Texmaker}, but there are \href{https://en.wikipedia.org/wiki/Comparison_of_TeX_editors}{many other options}).

Collaboration is permitted/encouraged, but you must identify your collaborators or the name of your group on \href{https://psetpartners.mit.edu}{pset partners}, as well any references you consulted that are not listed in the course syllabus. If there are none write ``\textbf{Sources consulted: none}'' at the top of your solution.  Each student is expected to write their own solutions; it is fine to discuss the problems with others, but your work must be your own.
\smallskip

\noindent
\textbf{Instructions:} Pick a combination of problems to solve that sum to 100 and write up your answers in latex, then complete the survey problem 4.

\medskip
\noindent
As in \cite{miyake}, we use $\H:=\{z\in \C: \Im(z)>0\}$ to denote the complex upper half plane.  

\subsection*{Problem 1. Geodesic spaces (50 points)}

Let $(X,d)$ be a metric space. A \defi{path} in $X$ is a continuous function $\gamma\colon [a,b]\to X$ we may also denote $\gamma\colon x\to y$, where $x=\gamma(a)$ and $y=\gamma(b)$.  The \defi{length} of a path is
\[
l(\gamma):= \sup_{a=t_0<\cdots<t_n=b}\sum_{i=1}^n d(\gamma(t_{i-1}),\gamma(t_i)).
\]
where the supremum is taken over all finite subdivisions of $[a,b]$.
A metric space is a \defi{length space} if for all $x,y\in X$ we have $d(x,y)=\inf_{\gamma\colon x\to y} \ell(\gamma)$.

\begin{enumerate}[{\bf(a)}]
\item For a piecewise smooth path $\gamma\colon [a,b]\to \H$ we define the \defi{hyperbolic length}
\[
\ell(\gamma):= \int_a^b \frac{|\gamma'(t)|}{\Im(\gamma(t))}dt.
\]
Show that $d(x,y):=\inf_{\gamma\colon x\to y}\ell(\gamma)$ is a metric on $\H$ that makes $(\H,d)$ a length space, with $\ell=l$ (in other words, the length function $\ell$ induces the metric $d$, which induces the length function $l$).
We call $d$ the \defi{hyperbolic metric} on $\H$.
\end{enumerate}

\begin{definition}
A \defi{geodesic path} $\gamma\colon [a,b]\to X$ is a distance preserving path; that is, we have $d(\gamma(s),\gamma(t))=|s-t|$ for all $s,t\in [a,b]$.  A \defi{geodesic line} (resp. \defi{geodesic ray}) is a distance preserving continuous map $\R\to X$ (resp. $\R_{\ge 0}\to X$).\footnote{N.B.: This differs from the definition of a \defi{geodesic segment} as a shortest path definition used in lecture, and from the two definitions most commonly used in Riemannian geometry; it is slightly stronger.}  By a geodesic path/line/ray in~$X$ we mean the image of a geodesic path/line/ray.
A \defi{geodesic space} is a metric space in which every pair of points can be connected by a geodesic path.  A geodesic space is \defi{straight} if every geodesic path lies in a geodesic line, and \defi{uniquely geodesic} if every $x,y\in X$ are connected by a unique geodesic path.  A subset of a geodesic space is \defi{convex} if it contains all geodesic paths between points in the set.
\end{definition}

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{1}
\item Show that $\H$ is a geodesic space that is a uniquely geodesic space under both the Euclidean metric $|z_1-z_2|$ and the hyperbolic metric defined above, but that it is only straight for the hyperbolic metric, and automorphisms of~$\H$ (as a Riemann surface) are isometries only for the hyperbolic metric.

\item Show that the topologies on $\H$ induced by the Euclidean metric and the hyperbolic metric coincide (in both cases the open balls $B_{<r}(x):=\{z\in \H:d(x,z)<r\}$ of radius $r>0$ about some $x\in \H$ are a basis for the topology).

\item Show that Euclidean and hyperbolic open balls are convex under both the Euclidean and hyperbolic metric, and give examples of subsets of $\H$ that are convex under one metric but not the other (in both directions).

\item Show that under the hyperbolic metric (i) for any distinct $z_1,z_2\in\H$ the set
\[
C_{z_1,z_2}:=\{z\in \H:d(z,z_1)=d(z,z_2)\}
\]
is a geodesic line $\gamma\colon\R\to \H$, (ii) $\H-C_{z_1,z_2}$ is the union of two disjoint convex sets that can meaningfully be called left and right half-planes (relative to $\gamma$), (iii) every geodesic line arises in this way.
Does this apply to all Riemann surfaces that are straight uniquely geodesic spaces for a given metric?

\end{enumerate}

\subsection*{Problem 2. Fuchsian groups (50 points)}
For any subgroup $\Gamma$ of $\SL_2(\R)$ we call $z\in \P^1(\C) = \C\cup \{\infty\}$ a \defi{limit point} of $\Gamma$ if there is a $w\in \P^1(\C)$ and an infinite sequence of distinct $\gamma_n\in \Gamma$ such that $\lim \gamma_n w = z$,  Let $\Lambda(\Gamma)$ to denote the set of of all limit points of $\Gamma$, and let $Z(\Gamma):= \Gamma \cap \{\pm 1\}$.

\begin{enumerate}[{\bf(a)}]
\item Let $\Gamma$ be a subgroup of $\SL_2(\R)$ and let $\H$ be the upper half plane equipped with the hyperbolic metric defined in Problem 1.
Show that the following are equivalent.
\begin{enumerate}[(1)]
\item Every $z\in\H$ lies in an open $U\subseteq\H$ for which $\{\gamma\in \Gamma: U\cap \gamma U\ne \emptyset\}$ is finite.
\item $\Gamma$ is discrete.
\item Any sequence in $\Gamma$ converging to $1$ is eventually constant.
\item For every real $B>0$ the subset of $\Gamma$ with matrix entries in $[-B,B]$ is finite.
\item $\Lambda(\Gamma)$ is a closed subset of $\P^1(\R) =\R\cup \{\infty\}$.
\end{enumerate}
Such a group is called a \defi{Fuchsian group}.
\item Let $\Gamma=\langle\gamma\rangle$ be a cyclic subgroup of $\SL_2(\R)$ with $\gamma\ne\pm 1$ and consider the three cases where $\gamma$ is elliptic, parabolic, or hyperbolic.  Compute $\Lambda(\Gamma)$ in each case and determine whether (or under what conditions) $\Gamma$ is Fuchsian.
\item Show that if a Fuchsian group $\Gamma$ is abelian then $\Gamma/Z(\Gamma)$ is cyclic.
\item Show that the $\SL_2(\R)$-normalizer of a nonabelian Fuchsian group is Fuchsian.
\item Let $\Gamma$ be a Fuchsian group.  Show that the following are equivalent.
\begin{enumerate}[{(1)}]
\item $\#\Lambda(\Gamma)\le 2$
\item There is a finite $\Gamma$-orbit in $\H\cup \RP^1$.
\item $\Gamma/Z(\Gamma)$ is abelian or conjugate to $\langle D, S\rangle$, where $D$ is diagonal and $S:=(\begin{smallmatrix} 0 & -1\\1 & 0\end{smallmatrix})$.
\end{enumerate}
Such a group is called an \defi{elementary Fuchsian group}.
\end{enumerate}


\subsection*{Problem 3. Dirichlet domains (100 points)}

Throughout this problem $\Gamma\subseteq \SL_2(\R)$ is a Fuchsian group.  Recall that given any $z_0\in \H$ that is not an elliptic point we can construct a Dirichlet domain
\[
F_{z_0}:=\{z\in \H:d(z,z_0)\le d(z,\gamma z_0)\text{ for all }\gamma\in \Gamma\},
\]
where $d$ is the hyperbolic distance metric on $\H$.  We proved in class that $F_{z_0}$ is a convex locally finite fundamental domain for $\Gamma$ whose boundary $\partial F$ is a union of geodesic segments $L_\gamma:= F_{z_0}\cap \gamma F_{z_0}$ called \defi{sides}, whose intersections in $\H$ are \defi{vertices}; points in $\partial \H = \R\cup \{\infty\}$ that are the common endpoint of two sides of $F$ that are geodesic rays are \defi{ideal vertices} and are not considered vertices for the purpose of this problem.  Recall that sides are required to have nonzero (possibly infinite) length. 
\defi{Locally finite} means that every compact $A\subseteq \H$ intersects only finitely many $\Gamma$-translates of $F_{z_0}$.

\begin{enumerate}[{\bf(a)}]
\item Let $F$ be a Dirichlet domain for $\Gamma$ and let $T:=\{\gamma\in \Gamma: F\cap \gamma F\ne \emptyset\}$.  Show that $T$ generates $\Gamma$ and includes a conjugate of every elliptic element of $\Gamma$.
\end{enumerate}

We now refine our notion of sides and vertices as follows.  For sides $L_\gamma$ with $\gamma^2=\pm 1$ we have $\gamma L_\gamma = L_\gamma$ and $\gamma$ thus fixes the midpoint of $L_\gamma$ (and swaps its end points).  In this case it will be convenient to view $L_\gamma$ as two distinct sides that intersect at the midpoint of $L_\gamma$, which we now consider a vertex.

Each vertex of $F$ lies in two sides of $F$, one of which we unambiguously distinguish as \defi{the clockwise side of $v_1$} (the one on the right if we look at $v_1$ from the interior of $F$).

Let $S$ denote the set of sides of a Dirichlet domain $F$, and define
\[
P:= \{(\gamma,L,L'): L'=\gamma L\}\subseteq \Gamma\times (S\times S).
\]
Let $G(P)$ be the projection of $P$ to $\Gamma$ modulo $Z(\Gamma)$ (keep only one of $\gamma$ and $-\gamma$).

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{1}
\item Define a relation $\sim$ on $S$ by letting $L\sim L'$ if and only if $(\gamma,L,L')\in P$ for some $\gamma\in \Gamma$.
Show that this is an equivalence relation that partitions $S$ into pairs of sides $\{L,L'\}$ with $L= F\cap \gamma^{-1}F$ and $L'=\gamma L = F\cap \gamma F$.

\item Show that $G(P)$ generates $\Gamma/Z(\Gamma)$ and that if $S$ is finite then $G(P)$ is finite.

\item Let $v_1$ be a vertex of $F$.  Show that there is an open neighborhood $U$ of $v_1$ that contains no other vertices of $F$ and intersects no sides of $F$ that do not contain~$v_1$, and a finite list of elements $1=\delta_0,\delta_1,\ldots,\delta_n\in \Gamma$ such that the sets $\delta_i F$ are all distinct and $U\subseteq \cup_{i=0}^n \delta_i F$ with $v_1\in \delta_i F$ for $0\le i\le n$.

\item Let $v_1$ be a vertex of $F$, let $L_1$ be the clockwise side of $F$, let $L_1'=\gamma_1 L_1$ be the unique side of $F$ paired with $L_1$ via \textbf{(c)} (such a $\gamma_1$ exists, it need not be unique), and let $v_2=\gamma_1 v_1$.  Show that $v_2$ is a vertex of $F$, so we can similarly let $L_2$ be its clockwise side, let $L_2'=\gamma_2 L_2$ be the paired side, and define $v_3=\gamma_2 v_2$.  Continuing in this fashion, we get a sequence of vertices $v_1,v_2,\ldots$ and group elements $\gamma_1,\gamma_2,\ldots$  Show that after finitely many steps we return to $v_{m+1}=v_1$.   

\item With $v_i$ and $\gamma_i$ as in part \textbf{(e)}, let $\delta = (\gamma_m\cdots \gamma_1)^{-1}$.  Show that the value of $m$ in part \textbf{(e)} divides $n+1$, where $n$ is as in part \textbf{(d)}, that $F \cap \Gamma v_1 = \{v_1,\ldots, v_{m-1}\}$ (with the $v_i$ as in part \textbf{(e)}), and that $\delta$ fixes $v_1$ and has order $e:= (n+1)/m$.  We call $\delta^e=1$ the \defi{vertex cycle relation for $v_1$}.  Show that if $v_1'=\gamma v_1$ is a vertex of $F$ then the cycle relation for $v_1'$ is $(\gamma^{-1}\delta\gamma)^e=\gamma^{-1}\delta^e\gamma=1$.  Let $R(P)$ be the set of vertex cycle relations for vertices in $F$.

\item Show that $\Gamma/Z(\Gamma)$ is isomorphic to the free group on $G(P)$ with relations $R(P)$.

\item For $\Gamma=\SL_2(\Z)$, compute a Dirichlet domain using $z_0=2i$ and compute generators and relations for $\Gamma/\Z(\Gamma)=\PSL_2(\Z)$ using the above method.

\item Do the same for  $\Gamma = \Gamma(2):= \{\gamma\in \SL_2(\Z):\gamma\equiv 1\bmod 2\}$.
\end{enumerate}


\subsection*{Problem 4. Survey}
Complete the following survey by rating each problem you attempted on a scale of 1 to~10 according to how interesting you found it (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found it (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem to the nearest half hour.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast", 5=``just right") and the novelty of the material to you (1=``old hat", 10=``all new").

\begin{center}
\begin{tabular}{l|l|r|r|r|r|r}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
2/5 & Automorphisms of $\H$ & & & & \\\hline
2/7 & Fuchsian groups & & & & \\\hline 
2/12 & Dirichlet domains & & & & \\\hline 
2/14 & Quotients by Fuchsian groups & & & & \\\hline 
\end{tabular}
\end{center}

\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, in particular, ways in which they might be improved.

\begin{thebibliography}{99}
% cite any references here, e.g.
\bibitem{miyake} T. Miyake, \href{https://doi.org/10.1007/3-540-29593-3}{\textit{Modular forms}}, Springer, 2006.
\end{thebibliography}

\end{document}