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\begin{document}
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\textbf{18.786 Number Theory II\hfill Spring~2024}\\\vspace{2pt}
\textbf{Problem Set \#4\hfill Due: 3/23/2024}\vspace{-8pt}
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\subsection*{Description}

These problems are related to the material covered in Lectures 9-12.
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

\textbf{Instructions}: Pick any combination of problems that sum to 100 points, then complete the survey problem.

\subsection*{Problem 1. Modular forms on $\boldsymbol{\SL_2(\Z)}$ (50 points)}

Throughout this problem $\Gamma:=\SL_2(\Z)$ denotes the full modular group.  Let $\Gamma_\infty$ be the stabilizer of $\infty$, and let
\[
\mathcal F:=\{z\in \H: |z|\ge 1\text{ and }|\Re(z)|\le 1/2\}
\]
denote the standard fundamental domain for $\Gamma$.
For even integers $k\ge 4$ we define the \defi{Eisenstein series} $G_k(z)$ as the Poincar\'e series (see (2.6.2) in \cite{miyake}):
\[
G_k(z):=F_k(z;\phi_0,\chi_0,\Gamma_\infty,\Gamma):=\sum_{\gamma\in \Gamma_\infty\backslash\Gamma} \overline{\chi_0(\gamma)} (\phi_0|_k\gamma)(z),
\]
where $\phi_0=1$ is the identity function and $\chi_0$ is the trivial character.
We also define\footnote{Note that this is \textbf{not} the $E_k$ defined in \cite[\S 4.1]{miyake}.}
\[
E_k(z) := -\frac{B_k}{2k}G_k(x),
\]
where $B_k:= \frac{(-1)^{k/2+1}2\cdot k!}{(2\pi)^k}\zeta(k) = 2k(-1)^{k/2+1}\int_0^\infty \frac{t^{k-1}}{e^{2\pi t}-1}dt\in \Q$ is the $k$th \defi{Bernoulli number},
with $B_2=\tfrac{1}{2}$, $B_4=-\tfrac{1}{30}$, $B_6=\tfrac{1}{42}$ (see Problem 3 for more on Bernoulli numbers).

\begin{enumerate}[{\bf(a)}]
\item Let $k\ge 4$ be even. Prove that $E_k (z)$ has the following $q$-expansion at $\infty$:
\[
E_k(z)=\frac{-B_k}{2k}+q + \sum_{n=2}^\infty \sigma_{k-1}(n)q^n,
\]
where $q:=e^{2\pi iz}$ and $\sigma_e(n):=\sum_{d|n}d^e$.
\item Let $\rho=e^{\pi i/3}$. Prove that for any $k\in \Z$ and nonzero $f\in M_k(\Gamma)$ we have
\[
\ord_\infty(f)+\frac{1}{2}\ord_i(f)+\frac{1}{3}\ord_{\rho}(f)+\!\!\!\!\sum_{\tau\in \mathcal F-\{i,\rho\}}\!\!\!\!\!\!\ord_\tau(f)=\frac{k}{12}.
\]
Conclude that $\dim M_k =0$ unless $k\ge 4$ is even.
\item Let $\Delta:=(G_4^3-G_6^2)/1728$.  Prove that for even $k\ge 4$ the map $f\mapsto \Delta f$ defines an isomorphism $M_{k-12}\overset{\sim}\longrightarrow S_k$.
Conclude that $\dim M_k=1$ and $\dim S_k = 0$ for $k=4,6,8,10,14$.
\item Prove that if $4a+6b =k\equiv 0\bmod 12$ then $3|a$ and $G_4^aG_6^b/G_6^{k/6}=(G_4^3/G_6^2)^{a/3}$.
\item Prove that for all even integers $k\ge 2$ we have
\[
\dim M_k(\Gamma)=\begin{cases}
0 &\text{if $k$ is odd or negative},\\
\lfloor k/12\rfloor &\text{if }k\equiv 2\bmod 12\text{ is positive and even.}\\
\lfloor k/12\rfloor+1 &\text{if }k\not\equiv 2\bmod 12\text{ is positive and even.}
\end{cases}
\]
(this is Corollary 4.1.4 in \cite{miyake}, you are being asked to give an alternative proof).
\item Prove that $\{G_4^aG_6^b:a,b\in \Z_{\ge 0}\text{ with }4a+6b=k\}$ is a basis for $M_k(\Gamma)$ for all $k\in \Z$.
(this is Theorem 4.1.8 in \cite{miyake}, you are being asked to give an alternative proof).
\item Prove that the space $S_k(\Gamma)$ has a unique basis $\{f_1,\ldots, f_d\}$ such that $a_i(f_j)=\delta_{ij}$ for $1\le i,j \le d$, where $f_j$ has $q$-expansion $f_j=\sum_{i\ge 0}a_i(f_j)q^i$ at $\infty$, and show that $f_j\in \Z[[q]]$ (hint: use the $E_k$ and $\Delta$).
This basis for $S_k(\Gamma)$ is called the \defi{Miller basis}.
\item Let $\{f_1,f_2\}$ be the Miller basis for $S_{38}(\Gamma)$.  Compute the integers $a_3(f_i)$ for $1\le i\le 2$ and express each $f_i$ in terms of the basis for $M_{38}(\Gamma)$ given by part {\bf (f)}.
\item Let $\T:= \Z[\Gamma\backslash\Delta/\Gamma]$ be the Hecke algebra for $\Gamma=\SL_2(\Z)$ over $\Z$, with $\Delta$ the set of integer matrices in $\GL_2^+(\Q)$.  For $n\in \Z_{\ge 1}$ let $T(n)\in \mathbb T$ denote the $n$th Hecke operator $T(n):=\sum_{\det\alpha=n} \Gamma\alpha\Gamma$, where the sum is over double cosets of $\Gamma$ in $\Delta$.  Show that for every even integer $k\ge 4$ and integer $n\ge 1$ we have
\[
T(n)(E_k):=E_k|T(n)=\sigma_{k-1}(n)E_k,
\]
and compute the matrix of $T(2)$ on the Miller basis for $S_{38}(\Gamma)$.
\end{enumerate}

\subsection*{Problem 2. Hecke operators on $\boldsymbol{\Gamma_1(N)}$ (50 points)}

Fix $N\in \Z_{>0}$.
In lecture, we considered the Hecke algebra
\[
\T(\Gamma_0(N),\Delta_0(N)):=\Z[\Gamma_0(N)\backslash\Delta_0(N)/\Gamma_0(N)],
\]
where
\begin{align*}
\Gamma_0(N) &:=\{ \abcd\in \SL_2(\Z): c\equiv 0\bmod N, a\perp N\},\\
\Delta_0(N) &:=\{ \abcd\in M_2^+(\Z): c\equiv 0\bmod N, a\perp N\},
\end{align*}
see \cite[4.5]{miyake}.  We now want to consider the Hecke algebra $\T(\Gamma_1(N),\Delta_1(N))$, where
\begin{align*}
\Gamma_1(N) &:=\{ \abcd\in \SL_2(\Z): c\equiv 0\bmod N, a\equiv d\equiv 1\bmod N \},\\
\Delta_1(N) &:=\{ \alpha=\abcd\in M_2^+(\Z): c\equiv 0\bmod N, a\equiv1\bmod N\}.
\end{align*}
We also define the \defi{diamond operator} $\langle d\rangle$ for each $d\in (\Z/N\Z)^\times$, which acts on modular forms of weight $k$ via $f\mapsto f|_k\sigma_d$, where $\sigma_d\in \SL_2(\Z)$ satisfies
\[
\sigma_d\equiv \begin{pmatrix}1/d &0\\0 & d\end{pmatrix}\bmod N.
\]
Throughout this problem $\chi$ denotes a Dirichlet character of modulus $N$, and for any integer $m$ we write $m|N^\infty$ to indicate that every prime divisor of $m$ is also a prime divisor of $N$.

\begin{enumerate}[{\bf(a)}]
\item Show that
\[
M_k(\Gamma_0(N),\chi)=\{f\in M_k(\Gamma_1(N)):\langle d\rangle f = \chi(d)f\text{ for all }d\in (\Z/N\Z)^\times\},
\]
so $\langle d\rangle$ acts on $M_k(\Gamma_0(N),\chi)$ via $f\mapsto \chi(d)f$.
Conclude that $\langle d\rangle$ acts on $M_k(\Gamma_1(N))$ and this action does not depend on the choice of $\sigma_d$.
\end{enumerate}

\noindent
We now define Hecke operators $T(n),T(a,d)\in \T(\Gamma_1(N),\Delta_1(N))$, with $a|d\perp N$, via
\[
T(n):=\sum_{\det\alpha = n} \Gamma_1(N)\alpha\Gamma_1(N),\qquad T(a,d):= \Gamma_1(N)\sigma_a\smat{a}{0}{0}{d}\Gamma_1(N),
\]
where the sum for $T(n)$ is over double cosets of $\Gamma_1(N)$ in $\Delta_1(N)$.
\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{1}
\item Show that $\T(\Gamma_1(N),\Delta_1(N))$ is commutative and for any $\alpha\in\Delta_1(N)$ we can choose $\alpha_1,\ldots,\alpha_n\in \Delta_1(N)$ so that $\Gamma_1(N)\alpha\Gamma_1(N)=\sqcup_i \Gamma_1(N)\alpha_i = \sqcup_i \alpha_i\Gamma_1(N)$.
\item Prove that every Hecke operator $\Gamma_1(N)\alpha\Gamma_1(N)$ with $\alpha\in \Delta_1(N)$ can be uniquely expressed in the form
\[
T(m)T(a,d)=T(a,d)T(m),
\]
with $m|N^\infty$ and $a|d\perp N$.
\item Show that $\T(\Gamma_1(N),\Delta_1(N))=\langle T(p),T(q,q):p,q\text{ prime}, q\perp N\rangle$ and that we also have $\T_\Q(\Gamma_1(N),\Delta_1(N))=\langle T(n)\rangle$.
\item For each $n\in \Z_{>0}$ let $\Delta^n:=\{\alpha\in \Delta_1(N):\det\alpha=n\}$ so that $\Delta_1(N)=\sqcup_{n\ge 0}\Delta^n$.
For each positive integer $a\perp N$ fix a choice of $\sigma_a\in \SL_2(\Z)$.  Show that for each $n\in \Z_{>0}$ we have
\[
\Delta^n=\bigsqcup_{\substack{ad=n\\a\perp N}}\ \bigsqcup_{0\le b< d}\Gamma_1(N)\sigma_a\smat{a}{b}{0}{d}.
\]
Conclude that for each $f\in M_k(\Gamma_0(N),\chi)$ we have
\[
f|_kT(n) = n^{k-1}\sum_{\substack{ad=n\\a\perp N}}\ \sum_{0\le b<d}\chi(a)d^{-k}f\left(\tfrac{az+b}{d}\right),
\]
and
\[
f|_kT(d,d) = d^{k-2}\chi(d)f.
\]
\item Show that when we restrict the action of $\T(\Gamma_1(N),\Delta_1(N))$ to $M_k(\Gamma_0(N),\chi)$ we have
\[
T(m)T(n)=\sum_{d|\gcd(m,n)}d^{k-1}\chi(d)T(mn/d^2),
\]
and derive the identity
\[
\sum_{n\ge 1}T(n)n^{-s} = \prod_p\left(1-T(p)p^{-s}+\chi(p)p^{k-1-2s}\right)^{-1}
\]
where both of the equalities above are identities of operators on $M_k(\Gamma_0(N),\chi)$.
\item Prove the following identity of operators on $M_k(\Gamma_1(N))$:
\[
p^{k-1}\langle p\rangle = T(p)^2-T(p^2),
\]
valid for all primes $p$.
\end{enumerate}


\subsection*{Problem 3. Computing Bernoulli numbers (50 points)}

For integers $n\ge 0$, the \defi{Bernoulli polynomials} $B_n(x)\in \Q[x]$ are defined as the coefficients of the exponential generating function
\[
E(t,x):=\frac{te^{tx}}{e^t-1} = \sum_{n\ge 0}\frac{B_n(x)}{n!}t^n.
\]
The $n$th Bernoulli number is then defined as $B_n=B_n(0)$.

\begin{enumerate}[{\bf(a)}]
\item Prove that $B_0(x)=1$, $B_n'(x)=nB_{n-1}(x)$, and $B_n(1)=B_n(0)$ for $n\ne 1$, and that these properties uniquely determine the Bernoulli polynomials.
\item Prove that $B_n(x+1)-B_n(x)=nx^{n-1}$ and
\[
B_n(x+y)=\sum_{k=0}^n\binom{n}{k}B_k(x)y^{n-k}.
\]
Use this to show that $B_n$ can alternatively be defined by the recurrence $B_0=1$ and
\[
B_n = -\frac{1}{n+1}\sum_{k=0}^{n-1}\binom{n+1}{k}B_k
\]
for all $n>0$, and show that $B_n=0$ for all odd $n>1$.
\end{enumerate}

\noindent
\textbf{(optional)} This part is not required but you can use the result in your solution to \textbf{(c)}.
Let $n\ge 2$ be an even integer.  Generalize \textbf{(b)} to show that for all $y\in \Z_{\ge 1}$ we have
\[
\sum_{m=0}^{y-1}(m+x)^{n-1} = \frac{B_n(x+y)-B_n(x)}{n},
\]
and use this to deduce a formula for sums of $n$th powers using Bernoulli numbers
\[
\sum_{m=1}^{y-1}m^n=\frac{1}{n+1}\sum_{k=0}^n\binom{n+1}{k}B_ky^{n+1-k}.
\]
Now derive a formula for $B_n$ in terms of $n$th powers
\[
B_n = \sum_{k=0}^n \frac{1}{k+1}\sum_{m=0}^k(-1)^m\binom{k}{m}m^n,
\]
and also in terms of the \defi{Stirling numbers} $\stirling{n}{k}:=\frac{1}{k!}\sum_{m=0}^k(-1)^{k-m}\binom{k}{m}m^n$ that count the number of ways to partition a set of $n$ labelled objects into $k$ nonempty subsets:
\[
B_n = \sum_{k=0}^n\frac{k!}{k+1}(-1)^kS(n,k).
\]

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{2}
\item Let $n\ge 2$ be an even integer.
Prove that $v_p(B_n)\ge -1$ for all primes $p$ and that $v_p(B_n)=-1$ if and only if $p-1$ divides $n$.  In other words, when written in lowest terms the denominator of the rational number $B_n$ is $\prod_{(p-1)|n}p$.
\end{enumerate}
\noindent
For a nonzero rational number $r=a/b$ with $a,b\in \Z$ coprime we define its \defi{height} to be $h(r):=\max(\log|a|,\log|b|)\in \R_{\ge 0}$.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{3}
\item Let $n\ge 2$ be an even integer.  Using the identity $\zeta(n)=(-1)^{n/2+1}\frac{(2\pi)^nB_n}{2\cdot n!}$ and the bounds
\[
\sqrt{2\pi} n^{n+\tfrac{1}{2}}e^{-n}e^{\frac{1}{12n+1}} < n! < \sqrt{2\pi} n^{n+\tfrac{1}{2}}e^{-n}e^{\tfrac{1}{12n}}
\]
arising from Stirling's approximation, determine an effective upper bound for $h(B_n)$ as a function of $n$ (you are not required to prove the identity but are free to do so!).
\item Let $\M(b)$ denote the time to multiply two $b$-bit integers (the bound $M(b)=O(b\log b)$ was recently proved \cite{HvdH}, but in practice the Sch\"onhage-Strassen algorithm \cite{SS} with complexity $O(b\log b\log\log b)$ is used).  Estimate the complexity of computing $B_n$ using the recursive formula above, assuming that it takes $\M(b)\log b$ time to reduce a ratio of $b$-bit integers to a rational number in lowest terms (this is achieved by the fast Euclidean algorithm).  Your answer should be an asymptotic upper bound (in big $O$-notation) on the number of bit operations required, as a function of $n$.

\item Describe an algorithm to compute $B_n$ using Newton iteration to compute $1/f$ modulo $x^{n+1}$, where $f(x)=(e^x-1)/x=\sum_{n\ge 0} \tfrac{x^n}{(n+1)!}$, and estimate its complexity.
Note that you can use Kronecker substitution to multiply two polynomials in $\Z[x]$ of degree at most $d$ with coefficients of height $b$ in time $O(\M(bd\log d))$ by replacing~$x$ with a suitable power of $2$.

\item Let $n\ge 2$ be an even integer, let $d=\prod_{(p-1)|n}p$ be the denominator of $B_n$ and let $A$ be an approximation to $2\cdot n!/(2\pi)^n$ to $b$-bits of precision, such that $2^b > e^{h(B_n)}$.
Let $P:=\lceil (Ad)^{1/(n-1)}\rceil$ and let $r:=\prod_{p\le P}(1-p^{-n})^{-1}$.  Show that if $a=(-1)^{n/2+1}\lceil drA\rceil$ then $B_n=a/d$.

\item Estimate the complexity of computing $B_n$ using the algorithm suggested by {\bf (e)}.

\item Illustrate the 3 algorithms to compute $B_n$ considered in this problem for $n=8$.
\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
3/11 & Poincare and Eisenstein series & & & & \\\hline
3/13 & Hecke algebras & & & & \\\hline 
3/18 & Hecke algebras for modular groups & & & & \\\hline 
3/20 & Eigenforms and L-functions  & & & & \\\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{HvdH} D. Harvey and J. van der Hoeven, \href{https://doi.org/10.4007/annals.2021.193.2.4}{\textit{Integer multiplication in time $O(n\log n)$}}, Annals of Mathematics \textbf{193} (2021), 563--617.
\bibitem{miyake} T. Miyake, \href{https://doi.org/10.1007/3-540-29593-3}{\textit{Modular forms}}, Springer, 2006.
\bibitem{SS} A. Sch\"onhage and V. Strassen, \href{https://link-springer.com/article/10.1007/BF02242355}{\textit{Schnelle multiplikation gro\ss er Zahlen}}, Computing \textbf{7} (1971), 281--292.
\end{thebibliography}

\end{document}