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\textbf{18.782 Introduction to Arithmetic Geometry\hfill Fall~2020}\\\vspace{4pt}
\textbf{Problem Set \#11\hfill Due: 11/20/2020}
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These problems are related to the material covered in Lectures 27-29.
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 11/20/2020 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.
\bigskip

As usual, a \emph{curve} is a smooth projective (irreducible) variety of dimension one.

\subsection*{Problem 1. A genus 1 curve with no rational points (30 points)}

Consider the homogeneous polynomial
\[
f(x,y,z)=x^3+2y^3+4z^3.
\]

\begin{enumerate}
\item[\textbf{(a)}] Prove that the zero locus of $f$ is a plane curve $C/\Q$.
\item[\textbf{(b)}] Prove that $C$ has genus one.
\item[\textbf{(c)}] Prove that $C$ has no $\Q$-rational points (so it is not an elliptic curve over $\Q$).
\end{enumerate}

\subsection*{Problem 2. Hyperelliptic curves (70 points)}

A \emph{hyperelliptic curve} $C/k$ is a curve of genus $g\ge 2$ whose function field is a separable quadratic extension of the rational function field $k(x)$.  The non-trivial element of $\Gal(k(C)/k(x))$ is called the \emph{hyperelliptic involution}.
In this problem we consider hyperelliptic curves over a perfect field $k$ whose characteristic is not $2$ (so every quadratic extension of $k(x)$ is separable).

\begin{enumerate}
\item[\textbf{(a)}] Let $C/k$ be a hyperelliptic curve of genus $g$.
Prove that $C$ can be defined by an affine equation of the form $y^2=f(x)$, where $f\in k[x]$ is a polynomial of degree $2g+1$ or $2g+2$ (so $C$ is the desingularization of the projective closure of this affine variety).
(hint: consider the Riemann-Roch spaces $\L(nD)$ where $D$ is the pole divisor of $x$, and proceed along the lines of the first part of the proof of Theorem 23.3;  as a first step, figure out what the degree of $D$ must be).

\item[\textbf{(b)}] Prove that the polynomial $f$ in part (a) can be made squarefree, and that $y^2-f(x)$ is irreducible in $\kbar[x,y]$.
Then show that if $k$ is algebraically closed one can make $f$ monic and of degree $2g+1$.

\item[\textbf{(c)}] Let $f$ be any squarefree polynomial in $k[x]$ of degree $d\ge 5$.
Prove that the curve defined by $y^2=f(x)$ is a hyperelliptic curve of genus $g\le (d-1)/2$.

\item[\textbf{(d)}] Let $C/k$ be a hyperelliptic curve of genus $g$ defined by $y^2=f(x)$ with $f$ squarefree of degree $d$, where $k$ is algebraically closed.
Prove that there are at least $d$ distinct places of $k(C)$ that are fixed by the hyperelliptic involution, but not every place of $k(C)$ is fixed by the hyperelliptic involution.

\item[\textbf{(e)}] Let $C/k$ be a hyperelliptic curve of genus $g$ over an algebraically closed field $k$, and let $\sigma$ be an automorphism of $k(C)$ that fixes $k$.
Prove that if $\sigma$ does not fix every place of $k(C)$ then it fixes at most $2g+2$ places.
(hint: show that there is a nonconstant function $x\in \L((g+1)P)$, where $P$ is a place not fixed by $\sigma$, and then show that every place fixed by $\sigma$ corresponds to a zero of $\sigma(x)-x$).

\item[\textbf{(f)}] Using (b-e), prove that every equation of the form $y^2=f(x)$ with $f\in k[x]$ a squarefree polynomial of degree $d\ge 5$ defines a hyperelliptic curve $C/k$ of genus $g = \lfloor \frac{d-1}{2}\rfloor$.
Your proof should work whether or not $k$ is algebraically closed.

\item[\textbf{(g)}] Prove that every curve of genus 2 is hyperelliptic (hint: first show there exists an effective canonical divisor $W$, then consider a non-constant $x\in \L(W)$).
\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.

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& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
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\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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