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\begin{document}
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\large
\textbf{18.786 Number Theory II\hspace{185pt}Spring~2018}\\\vspace{4pt}
\textbf{Problem Set \#1\hspace{225pt}Due: 3/2/2018}
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\subsection*{Description}

These problems are related to the material covered in Lectures 1-5.
Your solutions are to be written up in latex and submitted as a pdf-file with a filename of the form \texttt{SurnamePset1.pdf} via e-mail to \texttt{drew@math.mit.edu} by \textbf{noon} on the date due.
Collaboration is permitted/encouraged, but you must identify your collaborators, and any references consulted.
If there are none, write ``\textbf{Sources consulted:\ none}" at the top of your problem set.
The first person to spot each non-trivial typo/error in any of the problem sets or lecture notes will receive 1-5 points of extra credit.
\medskip

\noindent
\textbf{Instructions:} Pick any combination of problems to solve that sum to 100 and write up your answers in latex, then complete the survey problem 4.

\subsection*{Problem 0. Cohomological triviality (50 points)}
Throughout this problem $G$ is a finite group.  A $G$-module $A$ is said to by \emph{acyclic} if $\hat H^n(G,A)=0$ for all $n\in\Z$.  It is \emph{cohomologically trivial} if $\hat H^n(H,A)=0$ for all subgroups $H\le G$ and all $n\in\Z$.

\begin{enumerate}[{\bf(a)}]
\item Give an example of an acyclic $G$-module $A$ that is not cohomologically trivial (hint: use a cyclic $G$), then show that induced $G$-modules $\Ind^G(A):= \Z[G]\otimes_\Z A$ are cohomologically trivial.
\item Show that for any $\Z$-module $R$ the $G$-module $R[G]$ (with $G$ acting trivially on $R$ and by left-multiplication on $G$) is cohomologically trivial, as is any free $R[G$]-module.

\item Let $G$ be a $p$-group and let $A$ be a $G$-module of exponent dividing $p$.  Show that if $H_1(G,A)=0$ then $A$ is free as an $\Fp[G]$-module, hence cohomologically trivial.  Conclude that if $\hat H^n(G,A)=0$ for any $n\in\Z$ then $A$ is cohomologically trivial.

\item Let $G$ be a $p$-group and $A$ a $G$-module with trivial $p$-torsion.  Show the following are equivalent: (i) $A$ is cohomologically trivial, (ii) $\hat H^n(G,A)=\hat H^{n+1}(G,A)=0$ for some $n\in\Z$, (iii) $A/pA$ is a free $\Fp[G]$-module.

\item Show that a $G$-module $A$ is cohomologically trivial if and only if it is cohomologically trivial as a $G_p$-module for every $p$-sylow subgroup $G_p$, equivalently, if and only if for each prime $p$ dividing $\#G$ we have $\hat H^n(G_p,A)=\hat H^{n+1}(G_p,A)=0$ for some $p$-Sylow subgroup $G_p$ and some integer $n\in\Z$.

\end{enumerate}


\subsection*{Problem 1. The cup product (50 points)}

For $G$-modules $A$ and $B$, let $A\otimes B$ denote the $G$-module $A\otimes_\Z B$ with diagonal $G$-action (so $g(a\otimes b)=ga\otimes gb$).

\begin{definition}
Let $A$ and $B$ be $G$-modules.  The (cochain) \emph{cup product} is the family of maps (indexed by integers $m,n\ge 0$)
\[
C^m(G,A)\times C^n(G,B)\overset{\cup}\longrightarrow C^{m+n}(G,A\otimes B)
\]
defined by
\[
(f_A\cup f_B)(g_1,\ldots,g_{m+n}) = f_A(g_1,\ldots,g_m)\otimes g_1\cdots g_mf_B(g_{m+1},\ldots,g_{m+n})
\]
By Proposition 23.13, we can equivalently define the cup product as a map
\[
\Hom_{\Z[G]}(\Z[G^{m+1}],A)\times \Hom_{\Z[G]}(\Z[G^{n+1}],B)\overset{\cup}\longrightarrow \Hom_{\Z[G]}(\Z[G^{m+n+1}],A\otimes B)
\]
defined by
\[
(\varphi_A\cup \varphi_B)(g_0,\ldots,g_{m+n}) = \varphi_A(g_0,\ldots,g_m)\otimes\varphi_B(g_m,\ldots,g_{m+n})
\]
which you may find easier to work with.
\end{definition}
\begin{enumerate}[{\bf(a)}]
\item Prove that for $f_A\in C(G,A)$ and $f_B\in C(G,B)$ we have
\[
d^{m+n}(f_A\cup f_B) = (d^m(f_A)\cup f_B) + (-1)^m (f_A\cup d^n(f_B)),
\]
where the $d^*$ are coboundary maps.
\item Show that the cochain cup product induces well defined maps in cohomology
\[
H^m(G,A)\otimes_\Z H^n(G,B)\overset{\cup}\longrightarrow H^{m+n}(G,A\otimes B),
\]
and also in Tate cohomology for $m,n\ge 0$, and that for $m=n=0$ this map is induced by the identity $G$-morphism on $A\otimes B$.
\item Prove that the cup product is associative: $(x\cup y)\cup z= x\cup (y \cup z)$, and symmetric when $mn$ is even and skew symmetric when $mn$ is odd, that is $x\cup y=(-1)^{mn}(y\cup x)$.
Hint: use dimension shifting (see Section 29.1).
\item For $n\ge 0$ show that the isomorphism $\hat H^n(G,\Z)\overset{\sim}\longrightarrow \hat H^{n+2}(G,A)$ given by Tate's Theorem (see Theorem 29.22) is the cup product with $\gamma$, where $H^2(G,A)=\langle \gamma\rangle$.
\item Let $G$ be a finite cyclic group and fix an isomorphism $\chi\colon G\overset{\sim}\longrightarrow \Z/n\Z$.
Construct a canonical isomorphism $\delta\colon\Hom(G,\Z/n\Z)\to H^2(G,\Z/n\Z)$ and show that for any $G$-module $A$ and all $n\ge 0$, cup product with $\delta(\chi)$ defines an explicit isomorphism $\hat H^n(G,A)\overset{\sim}\longrightarrow H^{n+2}(G,A)$ that depends only on the choice of $\chi$ (this provides a more explicit version of Theorem 23.33).
\end{enumerate}

\subsection*{Problem 2. The local Hilbert symbol (50 points)}

Let $K$ be a local field whose characteristic is not $2$.\footnote{Note that this excludes only extensions of $\F_2(t)$; extensions of $\Q_2$ are fine.}

\begin{definition}
The \emph{local Hilbert symbol} is the map $(\cdot,\cdot)\colon K^\times/K^{\times 2}\times K^\times/K^{\times 2}\to \{\pm 1\}$
\[
(a,b) := \begin{cases}
1 &\text{if $ax^2+by^2=1$ has a solution in $K$},\\
-1 &\text{otherwise.}
\end{cases}
\]
Here $a,b\in K^\times $ are understood to represent elements of $K^\times/K^{\times 2}$, since we can absorb squares into $x$ and $y$.
\end{definition}
\begin{enumerate}[{\bf(a)}]
\item Prove the Hilbert symbol is symmetric, bimultiplicative (so $(a,bc)=(a,b)(a,c)$) and nondegenerate (left kernel is trivial).
Is this true if $K$ is not a local field?
\item Prove that for $a\not\in K^{\times 2}$ we have $(a,b)=1$ if and only if $b\in \N_{K(\sqrt{a})/K}(K(\sqrt{a})^\times)$.
\item For $a,b,c\in K^\times$ prove $ax^2+by^2=c$ has a solution if and only if $(-ab,c)=(a,b)$.
\item For $a,b\in K^\times$ define the quaternion algebra $H_{a,b}$ as the $K$-algebra $K(i,j)$ with $i^2=a$, $j^2=b$, $ij=-ij$.
Show that $(a,b)=1$ if and only if $H_{a,b}\simeq \mathrm{M}_2(K)$, the $2\times 2$ matrix algebra over $K$ (such quaternion algebras are said to \emph{split}).
\item Show that $H_{a,b}\simeq H_{a,c}$ if and only if $[b]=[c]$ in $K^\times/N(K(\sqrt{a})^\times)$.  Deduce that the isomorphism class of $H_{a,b}$ depends only on the Hilbert symbol $(a,b)$.
\end{enumerate}

\subsection*{Problem 3. Formal groups (100 points)}

Let $A$ be a commutative ring and for $F\in A[[X,Y]]$ consider the properties.
\begin{enumerate}
\item[(1)] $F(X,Y)= X+Y + (\deg \ge 2)$,
\item[(2)] $F(X,F(Y,Z))=F(F(X,Y),Z)$,
\end{enumerate}
that are satisfied by any formal group law over $A$, and in particular by
\[
\mathbb{G}_a(X,Y):=X+Y,\qquad\text{and}\qquad \mathbb{G}_m(X,Y):=X+Y+XY.
\]
The goal of this problem is to precisely characterize the rings $A$ for which (1) and (2) are sufficient to define a formal group law (that is, they imply (3), (4), and (5) below).
You will prove that this is true if and only if $A$ contains no nonzero torsion nilpotents
(which certainly applies to Lubin-Tate group laws, since then $A$ is an integral ideal).

For any $F,G\in A[[X,Y]]$ (whether they are formal group laws or not), we define
\[
\Hom(F,G):= \{f\in TA[[T]]: f(F(X,Y))=G(f(X),f(Y))\},
\]
and say that $\Hom(F,G)$ is nonzero if $\Hom(F,G)\ne \{0\}$.

\begin{enumerate}[{\bf(a)}]
\item Show that if $F\in A[[X,Y]]$ satisfies (1) and (2) then it also satisfies:
\begin{enumerate}
\item[(3)] There is a unique $i_F\in XA[[X]]$ such that $F(X,i_F(X))=0$;
\item[(4)] $F(X,0)=X$ and $F(0,Y)=Y$.
\end{enumerate}
\item Let $A:=\Fp[\epsilon]/(\epsilon^2)$.  Show that $F(X,Y)=X+Y+\epsilon XY^p$ satisfies (1), (2) but not
\begin{enumerate}
\item[(5)] $F(X,Y)=F(Y,X)$.
\end{enumerate}
Now let $A$ be any commutative ring that contains a nonzero torsion nilpotent.  Show that for some $\epsilon\in A$ and prime $p$, we have $p\epsilon = 0 = \epsilon^2$, and $F(X,Y)=X+Y+\epsilon XY^p$ satisfies (1), (2) but not (5).
\item Let $F\in A[[X,Y]]$ satisfy (1), (2).  Show that there is a unique $\omega_F\in A[[T]]$ satisfying
\[
\omega_F(F(T,S))F_X(T,S)=\omega_F(T),\qquad \omega_F(0)=1,
\]
where $F_X$ is the formal partial derivative of $F(X,Y)$ with respect to $X$.

\item Let $A$ be a torsion free ring, put $K:=A\otimes_\Z\Q$, and let $F\in A[[X,Y]]$ satisfy (1), (2).  The \emph{formal logarithm} $\log_F\in K[[T]]$ is the formal integral of $\omega_F$ with $\log_F(0)=0$.  Show that $\log_F(F(X,Y))=\log_F(X)+\log_F(Y)$, and that there exists $\exp_F\in K[[T]]$ such that $\log_F\circ \exp_F=\exp_F\circ \log_F = T$.
Conclude that $F(X,Y)=F(Y,X)$.

\item Let $F\in A[[X,Y]]$ satisfy (1), (2), let $H(X,Y):=F(X,F(Y,i_F(X)))$, and show $H(X,Y)=Y+\sum_{i\ge 1}h_i(X)Y^i$ for some $h_i\in A[[X]]$.
Let $n$ be the least integer for which $h_n\ne 0$ or $0$ if no such $n$ exists.  Prove that one of the following holds:
\begin{itemize}
\item $n=0$ and $F(X,Y)=F(Y,X)$;
\item $n=1$ and $h_1\in \Hom(F,\mathbb{G}_m)$;
\item $n>1$ and $h_n\in \Hom(F,\mathbb{G}_a)$.
\end{itemize}

\item Let $A$ be an integral domain and let $F,G\in A[[X,Y]]$ satisfy (1), (2).  Show that if $\Hom(F,G)$ is nonzero and $G(X,Y)=G(Y,X)$ then $F(X,Y)=F(Y,X)$.

\item Prove that there exists $F\in A[[X,Y]]$ satisfying (1), (2) but not (5) if and only if $A$ contains a nonzero torsion nilpotent (Hint: consider the nilradical of $A$).
\end{enumerate}


\subsection*{Problem 4. Survey}
Complete the following survey by rating each problem you attempted on a scale of 1 to~10 according to how interesting you found it (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found it (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem to the nearest half hour.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 0 & & & \\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast", 5=``just right") and the novelty of the material to you (1=``old hat", 10=``all new").

\begin{center}
\begin{tabular}{l|l|r|r|r|r|r}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
2/7 & Tate's Theorem & & & & \\\hline
2/12 & Local Artin Reciprocity I & & & & \\\hline 
2/14 & Local Artin Reciprocity II & & & & \\\hline 
2/20 & Lubin-Tate formal group laws & & & & \\\hline 
2/21 & Local existence theorem & & & & \\\hline 
\end{tabular}
\end{center}

\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, in particular, ways in which they might be improved.

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