\documentstyle[12pt]{report}
\newcommand{\sk}{\vskip 0.2in}
\newcommand{\GL}{{\bf GL}}
\newcommand{\F}{{\bf F}}
\newcommand{\cO}{{\cal O}}
\newcommand{\N}{{\mbox{N}}}
\newcommand{\fp}{{\mbox{FP}}}
\begin{document}
\begin{center}
{\Huge 189-726B: Modular Forms II} \\ \sk
{\Huge Assignment 1} \\ \sk
{\Large Due: Wednesday, January 16 }
\end{center}
\sk
\sk
\noindent
1. Let $F$ be a number field and let
$$ \zeta(F,s) = \sum_a (\N a)^{-s}$$
be its Dedekind zeta-function, where the sum is taken over all integral
ideals of $F$.
If $\lambda$ is an integral ideal, define $\N(\lambda)$ to be the index of
$\lambda$ in $\cO_F$, and extend this definition to fractional ideals in the
obvious way.
To any fractional ideal $\lambda$ we can associate the zeta function
$$ \zeta_\lambda(s) = \sum_{x\in \lambda/\cO_F^\times} |Nx|^{-s},$$
and to an ideal class ${\cal C}$, the zeta-function
$$ \zeta({\cal C},s) = (\N\lambda)^{s} \zeta_\lambda(s),$$
where $\lambda$ is any representative ideal in ${\cal C}$.
Show that
$$ \zeta(F,s) = \sum_{{\cal C}} \zeta({\cal C},s),$$
where the sum is taken over the distinct ideal classes of $F$.
\sk\sk\noindent
2. Fill in the details that are needed to prove the integral representation
for the $L$-function of a cusp form $f$ of weight $k$ on
$\Gamma_0(N)$:
$$ \Lambda(f,s) = \int_{1/\sqrt{N}}^\infty f(it) t^s \frac{dt}{t}
+ i^k N^{k-s} \int_{1/\sqrt{N}}^\infty g(it) t^{k-s} \frac{dt}{t},$$
where $g:= f|_k S_N$.
\sk\sk\noindent
3. Use the modular transformation formula
for the Riemann theta-function
$$ \theta(\tau) = \sum_{n=1}^\infty e^{2\pi i n^2\tau}$$
which you saw last semester to derive the functional
equation for the Riemann zeta-function.
\end{document}