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\large
\textbf{18.782 Introduction to Arithmetic Geometry\hfill Spring~2023}\\\vspace{4pt}
\textbf{Problem Set \#5\hfill Due: 03/16/2023}
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These problems are related to the material covered in Lectures 4-15 (unlike previous problem sets, you may need to refer to material covered in lectures and/or problem sets from earlier weeks).
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.
\medskip

\noindent
The problem set is due at 11:59pm Eastern time on 03/16/2023. 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 remember to identify any collaborators or sources that you consulted that are not listed in the syllabus.

\medskip

\subsection*{Problem 1. Hilbert symbol over $\Q$ is not bilinear (10 points)}
\begin{enumerate}[(a)]
    \item Now that we have proved the Hasse-Minkowski theorem, give an explicit description of the Hilbert symbol $(a,b)_{\Q}$ for $a,b \in \Q^{\times}$ in terms of local Hilbert symbols.
    \item Use part (a) to produce elements $a_1,a_2,b \in \Q^{\times}$ such that $(a_1a_2,b)_{\Q}$ is not equal to the product $(a_1,b)_{\Q}(a_2,b)_{\Q}$.
\end{enumerate}

\subsection*{Problem 2. Extensions and Norm subgroups: A peek into local class field theory (10 points)}
This problem gives another proof of bilinearity of the Hilbert symbol for $\Q_p$ without using the explicit formula.\\[10pt]
Over any field $k$, we have $(a_1a_2,b)_k = 1 \implies (a_1,b)_k = (a_2,b)_k$. Let $H_b$ denote the image of the norm map $\Norm : k(\sqrt{b})^{\times} \rightarrow k^{\times}$. To turn the one-way implication into an equivalence and thus get bilinearity, we need that the index of $H_b$ as a subgroup of $k^{\times}$ is $1$ or $2$.\\[10pt]
Let $p$ be an odd prime. Use Problem 3(a) from Pset 4 to show that this is indeed the case for $k = \Q_p$.\\
(\textbf{Hint:} There are three distinct quadratic extensions of $\Q_p$. List them, and for each of them, show that the norm subgroup is strictly bigger than $(\Q_p^{\times})^2$.)\\[10pt]

\medskip

In Problems 3-5 we will recapitulate much of the material covered over the past month, replacing the field $\Q$ with the field $\Fq(t)$,
the field of rational functions in $t$ with coefficients in a finite field $\Fq$.
The fields $\Q$ and $\Fq(t)$ have much in common and are the two essential sources of global fields; by definition, every global field is a finite extension of either $\Q$ or $\Fq(t)$.

Recall that $\Fq(t)$ is the fraction field of the polynomial ring $\Fq[t]$.
The ring $\Fq[t]$ is a principal ideal domain,  hence a unique factorization domain, and its unit group is $\Fq^\times$.
In particular, every irreducible polynomial in $\Fq[t]$ is prime, and (up to scaling by units) the prime elements of $\Fq[t]$ are precisely the monic irreducible polynomials. We will use $\pi$ to denote such a prime (and also $\infty$, see below).

\subsection*{Problem 3. Absolute values on $\Fq(t)$ (25 points)}

For each prime $\pi\in\Fq[t]$ and any nonzero $f\in \Fq[t]$, let $v_\pi(f)$ be the largest integer for which $\pi^n|f$,
equivalently, the largest $n$ for which $f\in(\pi^n)$.
For each $f/g\in\Fq(t)^\times$ in lowest terms (meaning $f$ and $g$ have no common factor in $\Fq[t]$), we define
\[
v_\pi(f/g)=v_\pi(f)-v_\pi(g),
\]
and we let $v_\pi(0)=\infty$ (with the usual conventions that $n+\infty=\infty$, $\max(n,\infty)=\infty$, and $n^{-\infty}=0$ if $n\ne 0$).
We also define
\[
\deg(f/g) = \deg f -\deg g,
\]
and $\deg 0 = -\infty$.


\begin{enumerate}
\item[\textbf{(a)}] For each prime $\pi\in \Fq[t]$, define
\[
|r|_\pi = (q^{\deg\pi})^{-v_\pi(r)}
\]
for all $r\in \Fq(t)$.  Show that $|\ |_\pi$ is a nonarchimedean absolute value on $\Fq(t)$.

\item[\textbf{(b)}] Let
\[
|r|_\infty = q^{\deg r},
\]
for all $r\in\Fq(t)$.  Prove that $|\ |_\infty$ is a nonarchimedean absolute value on $\Fq(t)$.\footnote{You may be surprised that $|\ |_\infty$ is nonarchimedean, but recall from Corollary 5.4 of Lecture 5 that this is necessarily the case in a field of positive characteristic.}
\end{enumerate}
\noindent
Recall from Problem Set 3 (problem 4e) that in a field $k$ with nonarchimedean absolute value $\Vert\ \Vert$, 
\[
R=\{x\in k: \Vert x\Vert \le 1 \}
\]
is a local ring with unique maximal ideal $\m=\{x\in k:\Vert x\Vert < 1\}$.
The field $R/\m$ is called the \emph{residue field} of $R$ (and of $k$, with respect to $\Vert\ \Vert$).

\begin{enumerate}
\item[\textbf{(c)}] Determine the residue field of $\Fq(t)$ with respect to $|\ |_\pi$, and with respect to $|\ |_\infty$.

\item[\textbf{(d)}] Prove Ostrowski's theorem for $\Fq(t)$: every nontrivial absolute value on $\Fq(t)$ is equivalent to $|\ |_\infty$ or $|\ |_\pi$ for some prime $\pi\in \Fq[t]$.

More precisely, show that if $\Vert\ \Vert$ is a nontrivial absolute value on $\Fq(t)$, either $\Vert t\Vert >1$ and $\Vert\ \Vert\sim|\ |_\infty$, or $\Vert t \Vert\le 1$ and $\Vert\ \Vert\sim|\ |_\pi$ for some prime $\pi\in\Fq[t]$.
\end{enumerate}

\noindent
In view of (d) we now regard $\infty$ as a prime of $\Fq(t)$ and let $\pi$ range over both monic irreducible polynomials in $\Fq[t]$ and $\infty$.

\begin{enumerate}
\item[\textbf{(e)}] Prove the product formula for absolute values on $\Fq(t)$: for every $r\in\Fq(t)^\times$ we have
\[
\prod_{\pi}|r|_\pi = 1.
\]
\end{enumerate}

\subsection*{Problem 4. Completions of $\Fq(t)$ (30 points)}
Let $\Fq(t)_\pi$ denote the completion of $\Fq(t)$ with respect to the absolute value $|\ |_\pi$.
Recall that for any field $k$, ring of formal power series $\sum_{n\ge 0} a_nT^n$ with $a_n\in k$ is denoted $k[[T]]$, and its fraction field $k((T))$ is the field of formal Laurent series $\sum_{n\ge n_0} a_nT^n$, where $n_0\in\Z$.

\begin{enumerate}
\item[\textbf{(a)}] Prove that every $\pi$-adic field $\Fq(t)_\pi$ is isomorphic to $\F_\pi((T))$, where $\F_\pi$ denotes the residue field of $\Fq(t)$ with respect to $|\ |_\pi$.
For each $\pi$ determine the field $\F_\pi$ and describe this isomorphism by identifying the element of $\Fq(t)_\pi$ that corresponds to $T$ (this element is called a uniformizing parameter, or simply a \emph{uniformizer}).
\end{enumerate}

\noindent
It follows from (a) that if we fix a uniformizer (i.e., choose an isomorphism from $\Fq(t)_\pi$ to $\F_\pi((T))$), then
every element $a$ of $\Fq(t)_\pi$ has a unique $\pi$-adic expansion of the form $\sum_{n\ge n_0}a_nT^n$, where $n_0=v_\pi(a)$ is the $\pi$-adic valuation of $a$ (extended from $\Fq(t)$ to its completion $\Fq(t)_\pi$).  We can also write $a$ as $T^{v_\pi(a)}u$, where $u\in\F_\pi[[T]]^\times$ is a $\pi$-adic unit.

\begin{enumerate}
\item[\textbf{(b)}] Determine for which $\pi$ we have $\F_\pi=\Fq$.
Conclude that, unlike the $p$-adic fields~$\Q_p$, distinct $\pi$-adic fields $\Fq(t)_\pi$ may be isomorphic as fields, even though they have inequivalent absolute values.

\item[\textbf{(c)}] Determine the group of square classes $\Fq(t)_\pi^\times/\Fq(t)_\pi^{\times 2}$ (your answer may depend on $q$ and $\pi$, be sure to consider all possibilities).
\end{enumerate}


\subsection*{Problem 5. Hilbert symbols over $\Fq(t)$ (25 points)}
We now assume $q$ is odd.
For $a,b\in\Fq(t)_\pi^\times$, we define the Hilbert symbol
\[
(a,b)_\pi = \begin{cases}
1&\text{if $ax^2+by^2=1$ has a solution in $\Fq(t)_\pi$},\\
-1&\text{otherwise}.
\end{cases}
\]
As noted in class, the proofs of Lemma 10.2 and Corollary 10.3 work over any field, so you can assume they apply to $(a,b)_\pi$.

\begin{enumerate}
\item[\textbf{(a)}] For $a,b\in \Fq(t)_\pi^\times\simeq \F_\pi((T))^\times$, write $a=T^\alpha u$ and $b=T^\beta v$, with $u,v\in\F_\pi[[T]]^\times$.  Prove that
\[
(a,b)_\pi = (-1)^{\alpha\beta\frac{\#\F_\pi-1}{2}}\kron{u}{\pi}^\beta\kron{v}{\pi}^\alpha,
\]
where $\inkron{u}{\pi}$ is the Legendre symbol of the residue field $\F_\pi$ applied to the image of $u$ in~$\F_\pi$ (so $\inkron{u}{\pi}=1$ if and only if the first $\pi$-adic digit of $u$ is a square in $\F_\pi^\times$).

\item[\textbf{(b)}] Prove that the Hilbert symbol $(\cdot,\cdot)_\pi$ is bilinear and nondegenerate.

\item[\textbf{(c)}] Prove that for all $a,b\in\Fq(t)^\times$ we have $(a,b)_\pi=1$ for all but finitely many $\pi$.
Then prove the Hilbert reciprocity law:
\[
\prod_\pi (a,b)_\pi = 1.
\]
You may assume the quadratic reciprocity law for $\Fq[t]$, which states that for any monic irreducible polynomials $\pi_1,\pi_2\in\Fq[t]$ we have
\[
\kron{\pi_1}{\pi_2}=(-1)^{\deg(\pi_1)\deg(\pi_2)\frac{q-1}{2}}\kron{\pi_2}{\pi_1},
\]
where $\inkron{\pi_1}{\pi_2}$ is defined to be $1$ if $\pi_1$ is a square modulo $\pi_2$ and $-1$ otherwise.
\end{enumerate}


\subsection*{Problem 6. 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
Problem 3 & & & \\\hline
Problem 4 & & & \\\hline
Problem 5 & & & \\\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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