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\large
\textbf{18.786 Number Theory II\hfill Spring~2024}\\\vspace{2pt}
\textbf{Problem Set \#3\hfill Due: 3/13/2024}\vspace{-8pt}
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\subsection*{Description}

These problems are related to the material covered in Lectures 4-8.
Your solutions should be written up in latex and submitted as a pdf-file on \href{https://www.gradescope.com/courses/730173}{Gradescope} before midnight on the date due.
Collaboration is permitted/encouraged, but you must identify your collaborators or the name of your group on \href{https://psetpartners.mit.edu}{pset partners}, and any references you consulted. If there are none write ``\textbf{Sources consulted: none}'' at the top of your solution.
\medskip

\subsection*{Problem 1. Automorphic forms in an adelic setting (50 points)}

Let $\A_\Q$ denote the adele ring of $\Q$, which we recall contains~$\Q$ embedded diagonally as the subring of principal adeles.  Let $\A_\Q^\infty:=\{a\in \A_\Q:\|a\|_\infty = 0\}$ denote the \defi{finite adeles}, and embed each completion of $\Q$ in $\A_\Q$ via $(x\mapsto (0,\ldots,0,x,0,\ldots 0))$.\footnote{See the \href{https://math.mit.edu/classes/18.785/2021fa/lectures.html}{18.785 Lecture notes} for more  about the ring $\A_\Q$.}

Note that under these embeddings the finite adeles $\A_\Q^\infty\subseteq \A_\Q$ and the $\Q$-completions $\Q_p,\R\subseteq \A_\Q$ are $\A_\Q$-ideals that are rings, but they are not subrings of $\A_\Q$ (because they do not contain $1\in \A_\Q$), whereas $\Q\subseteq \A_\Q$ is a subring but not an $\A_\Q$-ideal.

\begin{enumerate}[{\bf(a)}]
\item Show that one can embed $\GL_2(\A_\Q^\infty)$ and $\GL_2(\R)$ in $\GL_2(\A_\Q)$ so that we have isomorphisms of topological groups
\[
\GL_2(\A_\Q)\simeq \GL_2(\A_\Q^{\infty})\times\GL_2(\R)\simeq\GL_2(\A_\Q^\infty)\GL_2(\R)=\GL_2(\R)\GL_2(\A_\Q^\infty),
\]
and thus may view any subgroup of $\GL_2(\R)$ or $\GL_2(\A_\Q^\infty)$ as a subgroup of $\GL_2(\A_\Q)$.
\end{enumerate}

\noindent
Let $\widehat \Z=\prod_p \Z_p\simeq \varprojlim_n \Z/n\Z$ denote the profinite completion of $\Z$, which we view as a subring of $\A_\Q^\infty$ (via the embeddings of $\Z_p\subseteq \Q_p$).
For each positive integer $N$ we define
\begin{align*}
K_0(N) &:=\{\abcd\in\GL_2(\Zhat):c\equiv 0\bmod N\}\subseteq \GL_2(\A_\Q^\infty),\\
\Gamma_0(N)&:= K_0(N)\cap \GL_2^+(\Z)\subseteq \SL_2(\Z).
\end{align*}

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{1}
\item Show that 
\[
\GL_2(\A_\Q)=\GL_2(\Q)\GL_2^+(\R)K_0(N),
\]
thus we can write any $a\in \GL_2(\A_\Q)$ as $g=\gamma \sigma \kappa$ with $\gamma\in \GL_2(\Q)$, $\sigma\in \GL_2^+(\R)$, and  $\kappa\in K_0(N)$, not necessarily uniquely.

\item Show that there is a natural isomorphism of topological coset/double-coset spaces
\[
\Gamma_0(N)\backslash\GL_2^+(\R) \simeq \GL_2(\Q)\backslash \GL_2(\A_\Q)/K_0(N).
\]
\item Let $K_\infty^+\:=\SO_2(\R)$ and let $Z_\infty^+$ denote the subgroup of scalar matrices in $\GL_2^+(\R)$.  Show that we have a natural isomorphism of topological spaces
\[
\Gamma_0(N)\backslash\H \simeq \GL_2(\Q)\backslash\GL_2(\A_\Q)/K_0(N)K_\infty^+Z_\infty^+
\]
and that $\Gamma_0(N)$ is an arithmetic lattice in $\SL_2(\R)$.  Let $X_0(N):= X_{\Gamma_0(N)}$ denote the corresponding compact Riemann surface.
\end{enumerate}

\noindent
Let $k\in \Z_{>0}$ be even.  For any cusp form $f\in S_k(\Gamma_0(N))$, we define the function $\varphi_f\colon \GL_2(\A_\Q) \to \C$
by writing $g\in \GL_2(\A_\Q)$ in the form $g=\gamma \sigma\kappa$ as in \textbf{(b)} and putting
\[
\varphi_f(g):= (f|_k\sigma)(i),
\]
where  $(f|_k \sigma )(z):= \det(\sigma)^{k/2}j(\sigma,z)^{-k}f(\sigma z)$ is the slash operator and $j\bigl(\abcd,z\bigr):=cz+d$ is the automorphy factor.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{4}
\item Show that $\varphi_f$ is well defined, in other words, for all $\gamma,\gamma'\in \GL_2(\Q)$, $\sigma,\sigma'\in \GL_2^+(\R)$, $\kappa,\kappa'\in K_0(N)$ such that $\gamma \sigma\kappa = \gamma'\sigma'\kappa'$, we have $(f|_k\sigma)(i) = (f|_k\sigma')(i)$.
\end{enumerate}

Define $\xi\colon K_\infty^+\to \C$ by $\xi(r_\theta):=e^{i k\theta}$ for $r_\theta=\smat{\cos\theta}{\sin \theta}{-\sin\theta}{\cos\theta}\in K_\infty^+=\SO_2(\R)$.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{5}
\item Let $K:=K_0 (N)K_\infty^+$. Show that $\varphi:=\varphi_f$ satisfies the following:
\begin{enumerate}[(i)]
\setlength{\itemsep}{0pt}
\item \textbf{automorphy}: $\varphi(\gamma g)=\varphi(g)$ for all $\gamma \in \GL_2(\Q)$ and $g\in \GL_2(\A_\Q)$.
\item \textbf{$K$-finite}: $\varphi(g r_\theta \kappa)=\xi(r_\theta)\varphi(g)$ for all $g\in \GL_2(\A_\Q)$, $r_\theta\in K_\infty^+$, $\kappa \in K_0(N)$,  so the $\C$-vector space spanned by $\{\varphi(gh):h\in K\}$ has finite dimension.
\item \textbf{central action}: $\varphi(\lambda g )=\varphi(g)$ for all scalar $\lambda\in \GL_2(\A_\Q)$ and $g\in \GL_2(\A_\Q)$.
\item \textbf{$Z$-finite}: For every $g\in \GL_2(\A_\Q^\infty)$ the map $\H \to \C$ given by
\[
z\mapsto \varphi(g\sigma )j(\sigma ,i)^k\det(\sigma)^{-k/2},
\]
where $\sigma \in \GL_2^+(\R)$ satisfies $\sigma i=z$, is well-defined and holomorphic.
\item \textbf{moderate growth}: There exist $c,r\in \R_{>0}$ such that for all $g\in \GL_2(\A_\Q^\infty)$ the function $\phi_g$ on $\GL_2^+(\R)$ defined by $\sigma \mapsto \varphi(g\sigma)$ satisfies the bound
\[
|\phi_g(\sigma)|\le c\|\sigma\|^r,
\]
with $\|\sigma\|:=a^2+b^2+c^2+d^2+\det(\sigma)^{-2}$ for $\sigma=\abcd$.
\item \textbf{cuspidal}: Show that the function $\phi_g$ defined in (v) is bounded on $\GL_2^+(\R)$.
\end{enumerate}

\item Show that any smooth function $\varphi\colon \GL_2(\A_\Q)\to \C$ satisfying conditions (i)-(vi) above arises as $\varphi_f$ for some $f\in S_k(\Gamma_0(N))$.
Such functions are \defi{cuspidal automorphic forms} on $\A_\Q$ for $K_0(N)$.
\end{enumerate}

\begin{remark}
Our definition of $K$-finiteness depends not only on $K$, but also on the choice of the function $\xi$ (a \defi{fundamental idempotent}).
Our definition of $Z$-finiteness is specific to the $\GL_2$ setting, where holomorphicity is equivalent to being an eigenfunction for the Casimir element $\Delta$, which together with $1$ generates the center $Z(\mathfrak{g})$ of the universal enveloping algebra for the complexified Lie algebra $\mathfrak{g}:=\mathfrak{gl}_2\otimes_\R\C$; in general one specifies a finite index ideal of $Z(\mathfrak g)$ that annihilates $\varphi$.  An automorphic form on $\GL_2(\A_\Q)$ is \defi{cuspidal} if for all $g\in \GL_2(\A_\Q)$ we have
\[
\int_{\Q\backslash \A_\Q} \varphi(\smat{1}{x}{0}{1}g)\,dx = 0.
\]
See \cite[Cor.\,12.4]{KL} for a proof that this is implied by the cuspidal condition above.
\end{remark}

\begin{remark}
The $\GL_2(\A_\Q)$ automorphic forms for $K_0(N)$ described in this problem admit many generalizations, including: (1) replace $\Q$ with a number field $F$ and $\Zhat$ with $\widehat \O_F$,
(2) replace $\GL_2$ with a connected linear algebraic group $G$ and $\Gamma_0(N)$ with any arithmetic subgroup of $G(F)$ (and adjust $K_0(N)$ and $\GL_2^+(\R)$ accordingly), (3) allow a character $\chi$ (and adjust the fundamental idempotent and the central action condition accordingly).
\end{remark}

\subsection*{Problem 2. Subgroups of the modular group (50 points)}

In this problem we consider subgroups of the \defi{modular group} $\Gamma(1):=\SL_2(\Z)$.  We use $\Gamma(N)$ to denote the subgroup of $\Gamma(1)$ congruent to the identity modulo a positive integer~$N$.  For any subgroup $\Gamma$ of $\SL_2(\Z)$ we use $\overline\Gamma$ to denote $\Gamma/Z(\Gamma)\subseteq \PSL_2(\Z)$.

\begin{enumerate}[{\bf(a)}]
\item Show that $X(1):=X_{\Gamma(1)}$ has hyperbolic volume $\pi/3$ and is therefore compact.
\item Show that every elliptic element of $\Gamma(1)$ is conjugate to a power of $R:= \Rmat$ or $S:=\Smat$, every parabolic element is conjugate to a power of $T:=\Tmat$, and that $\overline\Gamma(1)$ admits the presentation $\langle r,s\,|\,r^3=s^2=1\rangle$.
\item Show that the commutator subgroup $\overline \Gamma(1)':=\langle \alpha\beta\alpha^{-1}\beta^{-1}:\alpha,\beta\in \overline\Gamma(1)\rangle\le \overline \Gamma(1)$ is a normal subgroup of index 6 is isomorphic to the free group on two letters.
\item Show that $\overline \Gamma(2)\le \overline \Gamma(1)$ is also a normal subgroup of index 6 isomorphic to the free group on two letters.
\item Are the isomorphic normal subgroups $\overline \Gamma(1)',\overline \Gamma(2)\le \overline \Gamma(1)$ equal?  If not, describe their intersection and determine the least integer $N$ (if any) for which $\overline \Gamma(N)\subseteq \overline \Gamma(1)'$.
\end{enumerate}

For a subgroup $\Gamma$ of $\Gamma(1)$ let $e_n(\Gamma)$ count the $\Gamma$-inequivalent elliptic points of order~$n$, and let $e_\infty(\Gamma)$ count the $\Gamma$-inequivalent cusps. 
\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{5}
\item Prove that for any finite index subgroup $\Gamma$ of $\Gamma(1)$ the following formula holds
\begin{align*}
g(\Gamma):=g(X_\Gamma) = 1 + \frac{[\overline\Gamma(1):\overline\Gamma]}{12} - \frac{e_2(\Gamma)}{4} - \frac{e_3(\Gamma)}{3} - \frac{e_\infty(\Gamma)}{2}.
\end{align*}
Your proof should rely only on material covered in the first two chapters of \cite{miyake}.
\item Compute an explicit (and reasonably sharp) lower bound on $g(\Gamma(N))$ as a function of $N$ and determine its asymptotic growth.  Use this to explicitly determine the sets of integers $\{N:g(\Gamma(N))=g\}$ for $g\le 10$ (you are welcome to use a \href{https://math.mit.edu/classes/18.786/2024/links.html}{computer algebra system} to help, but you must prove your list is complete).
\item Let $\Gamma:=\Gamma(1)'$ be the commutator subgroup of $\Gamma(1)$.  Compute $g(\Gamma)$ using your formula, then compute $\dim S_k(\Gamma)$ for $2\le k\le 6$.
\end{enumerate}

\begin{definition}
A \defi{congruence subgroup} $\Gamma$ is a subgroup of $\SL(2,\Z)$ that contains $\Gamma(N)$ for some positive integer $N$.  The least such $N$ is the \defi{level} of $\Gamma$.
\end{definition}
\begin{definition}
The \defi{width} of a cusp $x$ of a finite index $\Gamma\le \SL(2,\Z)$ is the least integer $h\ge 1$ for which $\pm \sigma \Gamma_x\sigma_{-1} = \pm \langle\bigl(\begin{smallmatrix}1&h\\0&1\end{smallmatrix}\bigr)\rangle$, where $\sigma \in \SL_2(\R)$ satisfies $\sigma x=\infty$.
\end{definition}

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{8}
\item Prove that for all $N\ge 1$, the group $\Gamma(1)/\Gamma(N)\simeq \SL_2(\Z/N\Z)$ contains a subgroup of index 7 only when $7|N$, in which case it projects on to an index 7 subgroup of $\SL(2,\Z/7\Z)$.  Conclude that every congruence subgroup of index 7 has level 7.
\item Consider the subgroup $\Gamma := \left\langle\smat{1}{5}{0}{1}, \smat{0}{-1}{1}{0}, \smat{2}{3}{1}{2}\right\rangle$ of $\SL(2,\Z)$ found in \cite{scholl}, which evidently has a cusp of width 5.  Prove that if $\Gamma$ contains $\Gamma(N)$ then the width of every cusp of $\Gamma$ divides $N$.
Then show that $\Gamma$ has index 7 in $\SL(2,\Z)$ and is therefore not a congruence subgroup, and compute $g(\Gamma)$ using your formula.
\end{enumerate}

\subsection*{Problem 3. 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
\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
2/26 & Meromorphic differentials & & & & \\\hline 
2/28 & Dimension formulas & & & & \\\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}
\bibitem{KL} A. Knightly and C. Li, \href{https://bookstore.ams.org/surv-133}{\textit{Traces of Hecke operators}}, American Mathematical Society, 2006.
\bibitem{miyake} T. Miyake, \href{https://doi.org/10.1007/3-540-29593-3}{\textit{Modular forms}}, Springer, 2006.
\bibitem{scholl} A.J. Scholl, \href{=https://doi.org/10.1112/S0024609396002639}{\textit{On the Hecke algebra of a noncongruence subgroup}}, Bull. London Math. Soc. \textbf{29} (1997), 395--399.
\end{thebibliography}

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