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\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lecture 10: Fun Facts about Class Numbers \\
September 19, 2001
\end{center}

\head{Reference}
Neukirch, Section 1.6.

\head{Outline of lecture}

\begin{enumerate}
\item
General warning: if $K \subset L$ are number field, there is a map
$\Cl(K) \to \Cl(L)$ induced by the map of ideal groups sending 
an integral ideal $\gotha$ to $\gotha \gotho_L$. But the map on class
group is not injective!
E.g., $\QQ(\sqrt{-5}) \subset \QQ(\sqrt{-1}, \sqrt{-5})$, the former
has class number 2, the latter has class number 1.
See also the homework.
\item
The class number of the imaginary quadratic field $\QQ(\sqrt{-D})$ is bounded
above by $O(\sqrt{D})$, and is bounded below by $O(D^{1/2-\epsilon})$---but
the lower bound is not effective. (Siegel)
The best effective known lower bound is $O((\log D)^{1-\epsilon})$ (Goldfeld).
\item
The only imaginary quadratic fields $\QQ(\sqrt{-D})$ of class number
1 are for $D = -1, -2, -3$, $-7, -11, -19$, $-43, -67, -163$. The complete
list of class number $N$ is known for some other small values of $N$,
using Goldfeld's lower bound.
\item
On the other hand,
it is conjectured that there are infinitely many real quadratic fields
$\QQ(\sqrt{D})$ of class number 1. The upper bound $O(\sqrt{D})$ is thus
much further from the truth in this case.
\item
There are conjectures (Cohen-Lenstra) about the probability that the class
number of a quadratic field (real or imaginary) is divisible by a prime $p$.
For $p=2$, this is understood by work of Gauss (see homework). For $p=3$,
this is basically understood by Davenport-Heilbronn. Beyond that, we
know almost nothing.
\item
For cyclotomic fields $\QQ(\zeta_n)$, the class number grows with $n$, but
is much harder to compute than in the cyclotomic case. However...
\item
The prime $p$ is called \emph{regular} if $p$ does not divide the class number
of $\QQ(\zeta_p)$. A theorem of Kummer states that $p$ is
regular if and only if $p$ does not divide any of the numerators of the
Bernoulli numbers $B_2, B_4, \dots, B_{p-3}$. (A more precise result is
due to Herbrand and Ribet.)
\item
Kummer proved Fermat's Last Theorem for regular primes. The basic idea is that
if $y^p = z^p - x^p = (z-x)(z-\zeta x) \cdots (z-\zeta^{p-1}x)$, then
each of the ideal classes $(z-x), (z-\zeta x), \dots, (z-\zeta^{p-1}x)$ is
a $p$-th power in the class group. But if $p$ is regular, then an ideal
whose $p$-th power is principal is itself principal!
\item
It is conjectured (Vandiver's conjecture) that the class number of the maximal
real subfield of $\QQ(\zeta_p)$ is \emph{never} divisible by $p$.
\end{enumerate}

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