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\begin{document}
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\large
\textbf{18.782 Introduction to Arithmetic Geometry\hfill Fall~2020}\\\vspace{4pt}
\textbf{Problem Set \#9\hfill Due: 11/6/2020}
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These problems are related to the material covered in Lectures 22-25.
I have made every effort to proof-read them, but some errors may remain.
The first person to spot each error will receive 1-5 points of extra credit.

The problem set is due by 11:59pm on 11/6/2020 and should be submitted electronically as a pdf-file e-mailed to \url{zzyzhang@mit.edu} and \url{roed@mit.edu} (please include ``18.782'' in the subject of the email).
You can use the latex source for this problem set as a template for writing up your solutions;
be sure to include your name in your solutions and to identify collaborators and any sources not listed in the syllabus.

Recall that we have defined a \emph{curve} as a smooth projective variety of dimension one.

\subsection*{Problem 1. Divisor identities (50 points)}
Let $k$ be a perfect (but not necessarily algebraically closed) field.
Let $\phi\colon C_1\to C_2$ be a morphism of curves defined over $k$, and let $\phi^*\colon k(C_2)\to k(C_1)$ be the corresponding embedding of function fields.
We also use $\phi^*$ to denote the pullback map $\Div_k C_2\to \Div_k C_1$, and $\phi_*$ denotes the pushforward map $\Div_k C_1\to \Div_k C_2$, as defined in Definition 19.19.


\begin{enumerate}
\item[\textbf{(a)}] Prove that $\phi_*\circ\phi^*$ acts as multiplication by $\deg \phi$ on $\Div_k C_2$.
\item[\textbf{(b)}] Show that $\phi^*\circ\phi_*$ is not multiplication by $\deg\phi$ on $\Div_k C_1$, in general, but that we do have
\[
\deg(\phi^*\phi_* D) = (\deg\phi)(\deg D)
\]
for all $D\in \Div_k C_1$.
\item[\textbf{(c)}] Prove that $\ord_P(\phi^*f) = e_\phi(P)\ord_{\phi(P)}(f)$ for all closed points $P$ of $C_1/k$ and functions $f$ in $ k(C_2)^\times$.  Then use this to show $\phi^*(\div f)=\div(\phi^* f)$ for all $f\in k(C_2)^\times$.
\item[\textbf{(d)}] For any curve $C/k$ and $f\in k(C)^\times$ and $D\in \Div_k(C)$ such that $\div f$ and $D=\sum n_P P$ have disjoint supports, define
\[
f(D) = \prod f(P)^{n_P\deg P},
\]
where $P$ ranges over the closed points of $C/k$.
Prove that $f(\phi_* D)=(\phi^*f)(D)$ for all $f\in k(C_2)^\times$ and $D\in \Div_k(C_1)$ where both sides are defined.
\item[\textbf{(e)}] Prove that the sequence
\[
1\to k^\times\to k(\P^1)^\times\overset{\div}{\longrightarrow} \Div_k\P^1\overset{\deg}{\longrightarrow}\Z\to 0
\]
is exact.
\end{enumerate}

\subsection*{Problem 2. Weil reciprocity (50 points)}
Let $C/k$ be a curve over an algebraically closed field $k$.
For $f\in k(C)^\times$ and $D\in \Div_k C$ such that $\div f$ and $D$ have disjoint support, define $f(D)$ as in part (d) of Problem 1.
Your goal in this problem is to prove the \emph{Weil reciprocity law}, which states that for all $f,g\in k(C)^\times$ such that $\div f$ and $\div g$ have disjoint support,
\[
f(\div g) = g(\div f).
\]
\begin{enumerate}
\item[\textbf{(a)}] Prove that for $C=\P^1$ we have
\[
\prod_P (-1)^{\ord_P(g)\ord_P(f)}\left(\frac{f^{\ord_P(g)}}{g^{\ord_P(f)}}\right)(P) = 1,
\]
for all $f,g\in k(C)^\times$, whether or not $\div f$ and $\div g$ have disjoint support.
\item[\textbf{(b)}] Use (a) to prove the Weil reciprocity law in the case $C=\P^1$.
\end{enumerate}

\noindent
Associated to any morphism of curves $\phi\colon C_1\to C_2$ we have defined three maps
\begin{center}
\begin{tabular}{ccc}\vspace{8pt}
$\phi^*\colon k(C_2)\to k(C_1)$ && $\phi^*\colon \Div_k C_2\to \Div_k C_1$\\
? &&$ \phi_*\colon \Div_k C_1\to \Div_k C_2$\\
\end{tabular}
\end{center}
It is natural to ask whether there is a fourth map $\phi_*\colon k(C_1)\to k(C_2)$ that completes the table above, and indeed there is.
It is defined by
\[
f\mapsto (\phi^*)^{-1} N(f),
\]
where $N\colon k(C_1)\to \phi^*(k(C_2))$ is the norm map associated to the extension $k(C_1)/\phi^*(k(C_2))$.\footnote{Recall from Definition 7.1 that the norm map of a finite extension $L/K$ sends an element of $L$ with characteristic polynomial $F\in K[x]$ to $(-1)^{[L:K]}F(0)$ which is an element of $K$.}
It is obviously not a morphism of fields (unless $\phi$ is an isomorphism), but it is a morphism of their multiplicative groups.

We then have an identities analogous to those proved in (c) and (d) of Problem 1:
\begin{equation}\label{id1}
\phi_*(\div f)=\div(\phi_*f),
\end{equation}
for all $f\in k(C_1)^\times$, and
\begin{equation}\label{id2}
f(\phi^* D)=(\phi_* f) (D),
\end{equation}
for all $f\in k(C_1)^\times$ and $D\in \Div_k C_2$ where both sides are defined.

We now turn to the general case of the Weil reciprocity law, where $f$ and $g$ are elements of $k(C)$ with disjoint support,
where $C$ is now any curve.
Your objective is to prove Weil's reciprocity law in the general case using Weil reciprocity for $\P^1$, which you proved in (b).

As noted in lecture, any function in $k(C)$ defines a morphism to $\P^1$, so we may view both $f$ and $g$ as morphisms, and we then have associated pullback maps $f^*,g^*$ and pushforward maps $f_*,g_*$ that we can apply both to the divisor groups and function fields of $C$ and~$\P^1$.  We also need the identity morphism $i\colon \P^1\to \P^1$.

\begin{enumerate}
\item[\textbf{(c)}] Prove that $\div g = g^*\div i$ for any $g\in k(C)^\times$.
\item[\textbf{(d)}] Use (b) and (c), parts (c) and (d) of Problem 1, and identities (1) and (2) to prove the Weil reciprocity law in the general case.
\end{enumerate}

\subsection*{Problem 3. Survey}
Complete the following survey by rating each problem on a scale of 1 to~10 according to how interesting you found the problem (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found the problem (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
\end{tabular}
\end{center}

\noindent
Feel free to record any additional comments you have on the problem sets or lectures; in particular, how you think they might be improved.

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