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\section*{Problem Set 9}

\begin{enumerate}
\item Given any $p,q\in\{0,1,\ldots, N-1\}$ (not both $0$), show that
$$f(\tau):=\sum_{(m,n)\underset{(N)}{\equiv}(p,q)} \frac{1}{(m\tau+n)^k}$$
belongs to $M_k(\Gamma(N))$.\vspace{5mm}
\item Consider the Legendre elliptic curve $Y^2=X(X-1)(X-\lambda)$. This \emph{Legendre form} contains slightly more information than the isomorphism class (of elliptic curve), since the 2-torsion points are essentially ordered (or ``marked''): $(0,0)$, $(1,0)$, and $(\lambda,0)$. Passing to the isomorphism class of the elliptic curve \emph{without} marked 2-torsion, by taking the $j$-invariant, forgets the ordering of these 3 points. So $\lambda\mapsto j(\lambda)$ should be a 6-to-1 map, since $\mathcal{S}_3$ has order 6. In this problem you'll find that map explicitly, thereby obtaining a conceptual derivation of the $\phi(\lambda)$ from Lecture 27.\vspace{1mm}

\noindent (a) By the affine change of coordinates $Y=y/2$, $X=x+\tfrac{\lambda+1}{3}$, put the Legendre curve in Weierstrass form $y^2=4x^3-g_2x-g_3$. In so doing, you get $g_2$ and $g_3$ as explicit functions of $\lambda$.\vspace{1mm}

\noindent (b) These functions aren't really well-defined: there are further changes of coordinates that will transform $(g_2,g_3)\mapsto (\xi^4 g_2,\xi^6 g_3)$, as you can check. But we know the $j$-invariant $\tfrac{g_2^3}{g_2^3-27g_3^2}$ \emph{is} well-defined. Compute it as a function of $\lambda$ using your result from (a); this should coincide with $\phi(\lambda)$.\vspace{5mm}

\item Show that the $\lambda$-function, which we defined on $\mathfrak{H}$, actually does not. analytically continue along any path meeting the real axis. (Hence $\mathfrak{H}$ is truly its ``natural domain''.) To do this, use part of the idea of the proof of little Picard.\vspace{5mm}
\item Consider the polylogarithm functions $$Li_n(z):=\sum_{k\geq 1}\frac{z^k}{k^n},$$ which are defined \emph{a priori} in $D_1$. (Of course $Li_1(z)=-\log(1-z)$.)\vspace{1mm}

\noindent (a) Write $Li_2$ as an integral and use this to continue it to a holomorphic function on $\CC\setminus [1,\infty)$. (You need to use the monodromy theorem here.) Iterate the procedure for the other $Li_n$, $n>2$.\vspace{1mm}

\noindent (b) What is the ``monodromy'' of $Li_n$ about $z=1$, i.e.~what analytic function in (part of) the disk does it yield when continued around this point once counterclockwise? (Use the integral expressions from part (a).) 
\end{enumerate}

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