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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Problem Set 11: More Cyclotomic Fields\\
Due in class Wednesday, November 21
\end{center}

Please submit eight of the following problems. (Note: this version includes
known corrections as of class on November 14.)

\head{More on cyclotomic fields}

Let $\zeta_n$ denote a primitive $n$-th root of unity and let
$K = \QQ(\zeta_n + \zeta_n^{-1})$.
\begin{enumerate}
\item
Prove that $K$ is the maximal totally real subfield of $\QQ(\zeta_n)$.
(Recall that a number field is \emph{totally real} if all of its 
embeddings into $\CC$ are real, i.e., have their image in $\RR$.)
\item 
Prove that $\ZZ[\zeta_n + \zeta_{n}^{-1}]$ is the ring of
integers of $K$.
\item
Let $E$ be the group of units of $\ZZ[\zeta_n]$ and let $F$ be the
group of units of $\ZZ[\zeta_n + \zeta_n^{-1}]$. Prove that 
$F$ and the roots of unity of $\QQ(\zeta_n)$ generate a subgroup of $E$
of index at most 2. (Hint: this problem generalizes one of the problems
from Problem Set 5.)
\item
Let $\gotha$ be an ideal of $\gotho_K$ which becomes principal in
$\ZZ[\zeta_n]$. Prove that $\gotha$ is principal. That is, the natural
map from $\Cl(\gotho_K)$ to $\Cl(\ZZ[\zeta_n])$ is injective.
\end{enumerate}

\head{Gauss sums and Stickelberger's theorem}

Reminders: given a character $\chi: \FF_q \to \QQ(\zeta_{m})$,
the Gauss sum $G(\chi)$ is given by
\[
G(\chi) = - \sum_{a \in \FF_q^*} \chi(a) \zeta_p^{\Trace_{\FF_q/\FF_p}(a)}
\]
(note the minus sign here!); given characters
$\chi_1, \chi_2$, the Jacobi sum $J(\chi_1, \chi_2)$ is given by
\[
J(\chi_1, \chi_2) = - \sum_{a \in \FF_q} \chi_1(a) \chi_2(1-a).
\]

\begin{enumerate}
\item
Verify the properties of Jacobi sums not proved in class:
\[
J(\chi_1, \chi_2) = \begin{cases} \chi_1(-1) & \chi_1 \chi_2 = 1 \\
G(\chi_1)G(\chi_2)/G(\chi_1 \chi_2) & \chi_1 \chi_2 \neq 1.
\end{cases}
\]
(Hint: write $G(\chi_1)$ as a sum over $a$, $G(\chi_2)$ as a sum over $b$,
then rewrite $G(\chi_1)G(\chi_2)$ as a double sum in which $a+b$ is one of
the indices.)
\item
Stickelberger's theorem, in full generality, actually says that for any
subfield $M$ of $\QQ(\zeta_m)$ and any fractional ideal $\gotha$,
$\gotha^{\beta \theta}$ is principal, where $\theta = \sum_a \left\{
\frac{a}{m} \right\} \sigma_a^{-1}$ is the Stickelberger element.
Deduce this from the case $M = \QQ(\zeta_m)$ proved in class. (Again, it
suffices to work with $\gotha$ being a prime not dividing $m$.)
\item
Let $p \equiv 3 \pmod{4}$ be a prime, and let $R$ and $N$ be the number
of quadratic residues and nonresidues, respectively, mod $p$ in the
interval $(0,p/2)$. Prove that for every fractional ideal $\gotha$
of $\QQ(\sqrt{-p})$, $\gotha^{R-N}$ is principal. (Hint: use the previous
exercise.)
\end{enumerate}

\head{Interlude: the Weil Conjectures}

For $q$ a prime power and $d$ a positive integer, let $N(d,q)$ denote
the number of solutions of the equation $x^d + y^d = z^d$ with
$x,y,z \in \FF_q$ not all zero, divided by $q-1$. (The result is an integer
by homogeneity.) In these exercises, we reconstruct Weil's motivation for
proposing the powerful \emph{Weil Conjectures} about the number of 
solutions of polynomial equations (i.e., points on algebraic varieties)
over finite fields.

\begin{enumerate}
\item
Suppose $d$ divides $q-1$, and
let $\chi: \FF_q^* \to \CC^*$ be a character of order $d$.
Show that 
\[
N(d,q) = q+1 - \sum_{a,b} J(\chi^a, \chi^b),
\]
where the sum runs over $a,b = 1, \dots, d-1$ but excluding pairs with
$a+b=d$. (Hint: show that the number of $d$-th roots of $u \in \FF_q^*$
is $\sum_{a=1}^d \chi^a(u)$.)
\item
Show that $|N(d,q) - (q+1)| \leq (d-1)(d-2)\sqrt{q}$. (Hint: reduce
to the case where $d$ divides $q-1$.)
\item
(Davenport-Hasse)
Given a character $\chi: \FF_q^* \to \CC^*$, define a character
$\chi': \FF_{q^r}^* \to \CC^*$ by setting $\chi'(a) = 
\chi(\Norm_{\FF_{q^r}/\FF_q}(a))$. Show that $G(\chi') = G(\chi)^r$;
you may assume $p>2$ if it helps. (Hint: first show $G(\chi')/G(\chi)^r$
is a unit and has absolute value 1, so is a root of unity. Then show
it is congruent to 1 modulo primes above $p$. For more details,
see Washington, \textit{Introduction
to Cyclotomic Fields}, Chapter 6.)
\item
Suppose $d$ divides $q-1$, and
let $\chi: \FF_q^* \to \CC^*$ be a character of order $d$.
Show that
\[
N(d,q^r) = q^r+1 - \sum_{a,b} J(\chi^a, \chi^b)^r,
\]
where the sum runs over $a,b$ as above.
\item
Show that $|J(\chi^a, \chi^b)| = \sqrt{q}$ for all $a,b$ as above.
This is the ``Riemann hypothesis'' for the zeta function of the curve
$x^d + y^d = z^d$ in the projective plane.
(Hint: consider the power series $\sum_i (1-\alpha_i z)^{-1}$ in $z$,
where $\alpha_i$ runs over the $J(\chi^a, \chi^b)$, and use the earlier
exercises to bound its coefficients. Or use the fact that
$|G(\chi)| = \sqrt{q}$ if $\chi$ is a character on $\FF_q$.)
\end{enumerate}





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