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\textbf{18.782 Introduction to Arithmetic Geometry\hfill Spring~2023}\\\vspace{4pt}
\textbf{Problem Set \#10\hfill Due: 05/06/2023}
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These problems are related to the material covered in Lectures 30-32.
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 at 11:59pm Eastern time on 05/06/2023. It is to be submitted electronically as a pdf-file through \href{https://www.gradescope.com/courses/510542/}{gradescope}.
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 identify collaborators and any sources not listed in the syllabus.

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

\subsection*{Problem 1. Differentials (30 points)}

Let $F/k$ be a function field of transcendence degree 1, with $\char k = 0$ and $k$ algebraically closed in $F$. Define $\Delta_F$ to be the $F$-vector space generated by the set of formal symbols $\{dx: x\in F\}$, subject to the relations
\[
\textrm{(1) } d(x+y)=dx+dy,\qquad \textrm{(2) } d(xy)=xdy+ydx,\qquad\textrm{(3) } da=0\text{ for } a\in k.
\]
Note that $x$ and $y$ denote elements of $F$ (functions), not free variables, so $\Delta_F$ reflects the structure of $F$ and will be different for different function fields.

\begin{enumerate}
\item[\textbf{(a)}] Prove that $\dim_F\Delta_F=1$, and that any $dx$ with $x\not\in k$ is a basis.
\end{enumerate}

The set $\Delta=\Delta_F$ is often used as an alternative to the set of Weil differentials $\Omega$.
They are both one-dimensional $F$-vector spaces, hence isomorphic (as $F$-vector spaces).
But in order to be useful, we need to associate divisors to differentials in $\Delta$, as we did for $\Omega$.
\smallskip

For any differential $\omega\in \Delta$ and any place $P$, we may pick a uniformizer $t$ for $P$ and write $\omega = wdt$ for some function $w\in F$ that depends on our choice of $t$; note that $t$ is necessarily transcendental over $k$, since it is a uniformizer.
We then define $\ord_P(\omega):= \ord_P(w)$, and the divisor of $\omega$ is then given by
\[
\div \omega := \sum_P \ord_P(\omega)P.
\]
The value $\ord_P(\omega)$ does not depend on the choice of the uniformizer $t$.

\begin{enumerate}
\item[\textbf{(b)}] Prove that $\div udv = \div u + \div dv$ for any $u,v\in F$ with $v$ transcendental over $k$.
Conclude that the set of nonzero differentials in $\Delta$ constitutes a linear equivalence class of divisors.
\item[\textbf{(c)}] Let $F=k(t)$ be the rational function field.
Compute $\div dt$ and prove that it is a canonical divisor.
Conclude that a divisor $D\in \Div_k \P^1$ is canonical if and only if $D=\div df$ for some transcendental $f\in F$.
\end{enumerate}

\noindent
Part (c) holds for arbitrary curves, but you are not asked to prove this.
It follows that the space of differentials $\Delta$ plays the same role as the space of Weil differentials $\Omega$, and it has the virtue of making explicit computations much easier.

\begin{enumerate}
\item[\textbf{(d)}] Prove that the curve $x^2+y^2+z^2$ over $\Q$ has genus $0$ (even though it is not isomorphic to $\P^1$ because it has no rational points) by explicitly computing a canonical divisor.
\end{enumerate}

\subsection*{Problem 2. A genus 1 curve with no rational points (20 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 $C/\Q$. (That is, prove smoothness and deduce irreducibility as in Lemma 23.13.)
\item[\textbf{(b)}] Prove that $C$ has genus one. (\textbf{Hint:} It will be helpful to look at Theorem 22.25, and recall Problem 4 of Pset 1.)
\item[\textbf{(c)}] Prove that $C$ has no $\Q$-rational points (so it is not an elliptic curve over $\Q$).
\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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