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%% William G. McCallum and Pavlos Tzermias
%% On Shafarevich--Tate groups and the arithmetic of Fermat curves
%% For SwDyer 75th birthday book, CUP
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 \markboth{\qquad On Shafarevich--Tate groups and the arithmetic of Fermat
curves \hfill}{\hfill William G. McCallum and Pavlos Tzermias \qquad}

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\begin{document}
\title{On Shafarevich--Tate groups and the arithmetic of Fermat curves}
\date{\em To Sir Peter Swinnerton-Dyer on his 75th birthday.}
\author{William G. McCallum \and Pavlos Tzermias}

\maketitle
\addcontentsline{toc}{chapter}{William G. McCallum and Pavlos Tzermias,
On Shafarevich--Tate groups and the arithmetic of Fermat curves}
% \tableofcontents

\section{Introduction}\label{sec!introduction}
Let $\Q$ denote the field of rational numbers and $\overline{\Q}$ a
fixed algebraic closure of $\Q$.  For a fixed prime $p$ such that $p
\geq 5$, choose a primitive $p$th root of unity $\zeta$ in
$\overline{\Q}$ and let $K=\Q(\zeta)$.  If $a$, $b$ and $c$ are
integers such that $0<a,b,a+b<p$ and $a+b+c=0$, let
$F_{a,b,c}$ denote a smooth projective model of the affine curve
\begin{equation}\label{eq!1}
y^{p} = x^{a} (1-x)^{b}
\end{equation}
and let $J_{a,b,c}$ be the Jacobian of $F_{a,b,c}$.  Then $J_{a,b,c}$
has complex multiplication induced by the birational automorphism
$(x,y) \mapsto (x, \zeta y)$ of $F_{a,b,c}$.  Let $\la$ denote the
endomorphism $\zeta - 1$ of $J_{a,b,c}$.  Note that $\la^{p-1}$ is, up
to a unit in $\Z[\zeta]$, multiplication by $p$ on $J_{a,b,c}$.

We are interested in the Shafarevich--Tate group of $J_{a,b,c}$ over
$K$, which we denote simply by $\Sh$.  In \cite{mccallum!1988}, the
first author studied the restriction of the Cassels--Tate pairing
\begin{equation}\label{eq!2}
\Sh[\la] \times \Sh[\la] \longrightarrow \Q/\Z
\end{equation}
and showed that $\Sh[\la]$ is nontrivial in certain cases depending
on the reduction type of the minimal regular model of $F_{a,b,c}$ over
$\Z_p[\zeta]$.  The purpose of this paper is to extend those results
by carrying out higher descents, and to derive some consequences for
the arithmetic of Fermat curves using the techniques of the second
author.

First we recall the main result of \cite{mccallum!1988}.  The possible
reduction types for $F_{a,b,c}$ are as shown in
Figure~\ref{fig!reductiontype} \cite{mccallum!1982}, with the proper
transform of the special fiber of the model~(\ref{eq!1}) indicated.
\begin{figure}\label{fig!reductiontype}
\begin{picture}(350,140)(-30,0)
\put(85,55){\footnotesize multiplicity 2}
\put(90,110){Tame}
\put(259,110){Wild}
\thicklines
\put(6,77){\footnotesize proper}
\put(6,67){\footnotesize transform}
\put(30,50){\line(1,0){150}}
\thinlines
\put(50,40){\line(0,1){50}}
\put(70,40){\line(0,1){50}}
\put(100,75){\circle*{2}}
\put(115,75){\circle*{2}}
\put(130,75){\circle*{2}}
\put(160,40){\line(0,1){50}}
\put(44,35){$\underbrace{\kern4.3cm}$}
\put(95,20){\footnotesize $p+1$}
\qbezier{(250,100)(300,55)(250,10)}
\qbezier{(300,100)(250,55)(300,10)}
\put(227,55){\line(1,0){100}}
\put(220,46){\footnotesize proper}
\put(220,36){\footnotesize transform}
\end{picture}
\caption{Reduction types of $F_{a,b,c}$}
\end{figure}
The wild type is further divided into split and nonsplit, according
to whether the two tangent components are defined over the finite
field $\F_p$ or conjugate over a quadratic extension.  The reduction
type can be computed as follows.  For a rational number $x$ of
$p$-adic valuation $0$, let $q(x)=(x^{p-1}-1)/p$, viewed as an element
of $\F_p$. Then $F_{a,b,c}$ is
$$
\begin{cases}
\text{tame}& \text{if $-2 a b c q(a^a b^b c^c) = 0$,}\\
\text{wild split} & \text{if 
$-2 a b c q (a^a b^b c^c) \in \F_p^{\times 2}$,} \\
\text{wild nonsplit} & \text{if  
$-2 a b c q(a^a b^b c^c) \notin \F_p^{\times 2}$.}
\end{cases}
$$
Let $M_K$ be the set of finite places of $K$ and let $w$ denote the
unique place of $K$ above $p$. Define
\begin{align}
U & = \bigl\{ x \in K^{\times}/K^{\times p}:\mbox{$v(x) \equiv 0
  \pmod{p}$ for all $v \in M_K$} \bigr\}, \notag \\[4pt]
V &= K_w^{\times}/K_w^{\times p}.\label{eq!36}
\end{align}
Let $\pi$ be the uniformizer of $K_w$
defined by 
\begin{equation}
 \pi^{p-1}=-p \quad\text{and}\quad \frac{\pi}{1-\zeta} \equiv 1\pmod{w}.
\label{eq!35}
\end{equation}
If $\kappa\colon\Gal(K/\Q) \to \Z_p^{\times}$ is the Teichm\"{u}ller
character, let $V(i)$ denote the intersection of the $\kappa^{i}$th
eigenspace of $V$ with the subgroup of $V$ generated by units
congruent to $1$ modulo $\pi^i$. Thus $V(i)$ is one-dimensional if $2
\le i \le p$.

\begin{theorem}[\cite{mccallum!1988}]\label{thm!old}
  Suppose that $F_{a,b,c}$ is wild split, $p \equiv 1 \pmod{4}$, and
  the image of $U$ is nontrivial in both $V((p-1)/2)$ and
  $V((p+3)/2)$.  Then $$\Sh[\lambda]/\lambda \Sh[\lambda^2] \iso
  \Z/p\Z\oplus \Z/p\Z.$$
\end{theorem}

The condition on $U$ is satisfied if $p\nmid B_{(p-1)/2}B_{(p+3)/2}$,
where $B_k$ is the $k$th Bernoulli number.  As noted in
\cite{mccallum!1988}, the technique used to prove
Theorem~\ref{thm!old} applies to the pairing
\begin{equation}\label{eq!3}
\Sh[\la^2] \times \Sh [\la] \longrightarrow \Q/\Z
\end{equation}
and yields information about $\Sh[\la^2]$. 
\begin{theorem}\label{thm!free}
  Suppose that either of the following conditions is satisfied:
  \renewcommand{\theenumi}{\alph{enumi}}
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $F_{a,b,c}$ is wild split and $p \equiv 3
  \pmod{4}$\label{item!1};
 \item $F_{a,b,c}$ is wild nonsplit or tame and the image of $U$ in
  either $V((p+1)/2)$ or $V((p+3)/2)$ is trivial. \label{item!2}
\end{enumerate}
Then the pairing~\eqref{eq!3} is trivial. Thus $\Sh[\lambda^2]/\lambda
\Sh[\lambda^3] = 0$, that is, $\Sh[\la^3]$ is a free module over
$\Z[\zeta]/(\la^3)$.
\end{theorem}

As discussed in \cite{mccallum!1988}, the hypothesis on $U$ in
condition~(\ref{item!2}) of the theorem is quite mild, since for $U$
to be nontrivial in $V(k)$ with $k > 1$ and odd requires that $p$
divides $B_{p-k}$.

\begin{corollary}\label{thm!corollary}
  If one of conditions~(\ref{item!1}) or (\ref{item!2}) of
  Theorem~\ref{thm!free} is satisfied, and if $|\Sh[p^{\infty}]| <
  p^{3}$, then $\Sh[p^{\infty}] = 0$.
\end{corollary}

Under the conditions of Theorem~\ref{thm!free}, it is natural to ask
which occurs more often: $|\Sh[p]|=0$ or $|\Sh[p]|\ge p^3$. To explore
this question, we compute
\begin{equation}\label{eq!4}
\Sh[\la^3] \times \Sh [\la] \longrightarrow \Q/\Z.
\end{equation}
\begin{theorem}\label{thm!nontrivial}
  Suppose that $p\ge19$ is regular, $p \equiv 3 \pmod{4}$, $F_{a,b,c}$
  is tame or wild nonsplit and
  \begin{equation}\label{eq!39}
    q(a^ab^bc^c)^3+ a b c B_{p-3} \not\equiv 0 \pmod{p}.
  \end{equation}
  Then the pairing~\eqref{eq!4} is nontrivial. Thus $\Sh[\la^3] \neq
  0$ (and hence, by Corollary~\ref{thm!corollary}, $|\Sh[p^\infty]|
  \ge p^3$).
\end{theorem}

For example, the curve $y^{19} = x^2(1-x)$ satisfies the conditions of
the theorem. Modest numerical experiments suggest that about half the
curves satisfy the conditions.
More precisely, there are about $p/6$ isomorphism classes of curves
$F_{a,b,c}$ for a given prime $p$, and heuristically about half of
them are tame or wild nonsplit. The incongruence~\eqref{eq!39} is
usually satisfied for these curves; for example, it is
satisfied for all such curves if $p < 100$ (and $p \equiv 3
\pmod{4}$).


The next result shows that, in certain cases, one can combine
Theorems~\ref{thm!free} and~\ref{thm!nontrivial} to describe the exact
structure of $\Sh[p^{\infty}]$:

\begin{theorem}\label{prop!exact-p-primary} 
  Suppose that $p$, $a$, $b$ and $c$ are chosen to satisfy the hypotheses
  of both Theorems~\ref{thm!free} and~\ref{thm!nontrivial}.
  If, in addition, the free $\Z[\zeta]/(\la^3)$-module
  $\Sh[\la^3]$ has rank $2$, then
  $$\Sh[p^{\infty}]=\Sh[\la^3] \iso (\Z[\zeta]/(\la^3))^2. $$
\end{theorem}

\noindent In Section~\ref{sec!tame-reduction} we establish the following
application of the above results:

\begin{theorem}\label{cor!diophant}
  Let $p=19$, $a=7$, $b=1$. Then
\begin{enumerate}
\item $\Sh[p^{\infty}] \iso (\Z[\zeta]/(\la^3))^2$. \label{diophant!1}
\item The Mordell--Weil rank of $J_{7,1,-8}$ over $\Q$ equals $1$.
           \label{diophant!2}
\item The only quadratic points (i.e.\ algebraic points whose field of
  definition is a quadratic extension of $\Q$) on the Hurwitz--Klein
  curve $F_{7,1,-8}$ and also on the Fermat curve
  $X^{19}+Y^{19}+Z^{19}=0$ are those described by Gross and
  Rohrlich in \cite{gross-rohrlich!1978}.
           \label{diophant!3}
\end{enumerate} 
\end{theorem}

\noindent 
We also note that, by combining Theorem~\ref{thm!nontrivial} with
Faddeev's bounds in \cite{faddeev:1961}, one gets that the
Mordell--Weil rank (over $\Q$) of any tame or wild nonsplit quotient
of the Fermat curve $F_{19}$ or $F_{23}$ is at most 2.

Lim \cite{lim!1995} has also stated a result attempting to improve on
\cite{mccallum!1988} in certain cases. However, in
Section~\ref{sec!tame-reduction}, we show that the hypotheses of
Propositions A and B of \cite{lim!1995} are never simultaneously
satisfied.
% In particular, the Mordell--Weil rank estimates given in
% the last section of \cite{lim!1995} are incorrect.


\section{Formulas for the pairings}\label{sec!formulas-pairings}
We recall the situation and notation of~\cite{mccallum!1988}. For $\phi
\in {\mathcal O}_K$ and $F$ a field containing $K$, we write
$\delta=\delta_{\phi,F}$ for the coboundary map $J(F) \to H^1(F,
J[\phi])$.  The $\phi$-Selmer group $S_\phi \subset H^1(K,J[\phi])$ is
defined to be the subgroup whose specialization to each
completion $K_v$ of $K$ lies in the image of $\delta_{\phi,K_v}$. It sits
in an exact sequence
\begin{equation*}
  0 \to J(K)/\phi J(K) \to S_\phi \to \Sh[\phi] \to 0.
\end{equation*}
For $\phi, \psi \in \End(J)$, we have a pairing
 \begin{equation}
 \label{eq!SelmerPairing}
S_\phi \times S_{\hat\psi} \to
\Q/\Z,
\end{equation}
described in \cite{mccallum!1988}, which is a lift of the restriction
of the Cassels pairing to $\Sh[\phi]\times\Sh[\hat\psi]$.  An
expression for the pairing~\eqref{eq!SelmerPairing} is given in
\cite{mccallum!1988}, under a certain splitting hypothesis.

We use\cite{mccallum!1988} to derive formulas for the pairings~(\ref{eq!3})
and~(\ref{eq!4}). The formula for~(\ref{eq!3}) is a straightforward
consequence of Theorem 2.6 in \cite{mccallum!1988}; the formula
for~(\ref{eq!4}) takes more work. The point is that $J[\lambda^3]
\subset J(K)$ (Greenberg \cite{greenberg!1981}), so that it is possible to
express
the pairings (\ref{eq!2}) and (\ref{eq!3}) as purely local pairings at
$w$, as explained in \cite{mccallum!1988}. However, by
\cite{greenberg!1981} and Kurihara \cite{kurihara:1992}, the
$\lambda^4$-torsion on $J_{a,b,c}$ is not in general defined over $K$,
introducing an essentially global aspect to the calculation
of~\eqref{eq!4}.

For technical reasons, it is convenient to replace $\lambda$ with an
endomorphism (which we also denote by $\lambda$) that is congruent
modulo $\lambda^5$ to the uniformizer $\pi$ defined by~\eqref{eq!35}, 
since then
\begin{equation*}
  \mclambda^\delta \equiv \kappa(\delta) \mclambda 
  \pmod{\mclambda^5},\quad \delta \in
  \Gal(K/\Q). 
\end{equation*}
In particular, we have $\hat\mclambda \equiv -\mclambda$ modulo
$\mclambda^5$, and we will often replace $\hat\mclambda$ with
$-\mclambda$ without mention in what follows, in cases where we are
dealing with a module killed by $\mclambda^5$. Furthermore, it
suffices to prove Theorems~\ref{thm!free} and~\ref{thm!nontrivial}
with this new choice of $\mclambda$.  Since $\mclambda/\hat\mclambda$
is a unit, $\Sh[\mclambda] = \Sh[\hat\mclambda]$, and we can proceed
by computing the pairing $\langle\ ,\,\rangle_k$ mentioned in 
\eqref{eq!SelmerPairing} with $\phi =
\mclambda^k$ and $\psi = \hat\mclambda$.

The local formula for the Cassels--Tate pairing is expressed in terms
of certain local descent maps as follows.  Given a $p$-torsion point
$Q$ in $J(\overline K)$ we denote by $D_Q$ a divisor defined over
$K(Q)$ representing $Q$ and by $f_Q$ a function on $F_{a,b,c}$ whose
divisor is $pD_Q$. Evaluating $f_Q$ on divisors induces a map
$\iota_Q\colon J(F) \to F^\times/F^{\times p}$ for any field $F$ containing
$K(Q)$.

By \cite{greenberg!1981},
\begin{equation}
K(J[\mclambda^3]) = K \quad\hbox{and}\quad
K(J[\mclambda^4]) = L
= K(\eta_{p-3}^{1/p}), \label{eq!5}
\end{equation}
where $\eta_{p-3}$ is a generator for the $\kappa^{p-3}$-eigenspace of
the cyclotomic units in $K$. Let $\tilde \Delta \subset
\Gal(L/\Q)$ be a subgroup projecting isomorphically to
$\Gal(K/\Q)$.  For $i=1,2,3,4$, we choose points $P_{i}$
of order $\mclambda^{i}$ on $J$ and a generator $\sigma$ for
$G=\Gal(L/K)$ such that
\begin{enumerate}
\item $P_1$ is the point represented by the divisor $(0,0)-\infty$;
\item $\mclambda P_i = P_{i-1}$ for $i = 2, 3, 4$;
\item $P_i$ is an eigenvector for the action of $\tilde \Delta$ with
  character $\kappa^{1-i}$;
\item $\sigma P_4 = P_4 + P_1$.
\end{enumerate}
For $i \leq 4$, let $e_{\lambda^i}(P,Q)$ be the $\lambda^i$ Weil pairing on
$J[\lambda^i]$. We have an 
isomorphism $J[\mclambda^i] \iso \mu_p^i$ defined over
$K(P_i)$ (and thus over $K$ for $i \leq 3$), given by 
\begin{equation}
  \label{eq!37}
  Q \mapsto (e_{\lambda^i}(Q, P_1), \dots, e_{\lambda^i}(Q, P_i)).
\end{equation}
With this identification, by \cite[Lemma 2.2]{mccallum!1988}, we have
\begin{equation*}
  \delta_i = \delta_{\mclambda^i, K(P_i)} = 
   \iota_{P_1} \times \cdots \times \iota_{P_i}.
\end{equation*}
Since
$J$ has good reduction outside $p$ and $\mclambda$ has degree $p$, we
can regard $S_{\mclambda^i}$ as a subgroup of $H^1(K(p)/K,
J[\mclambda^i])$, where $K(p)/K$ is the maximal extension of $K$
unramified outside $p$. As explained in Section 7 of \cite{mccallum!1988}, 
we can also regard $S_{\mclambda^i}$ as
a subgroup of $U^i$ for $i \leq 3$,
where $U$ is as defined in~\eqref{eq!36}. 
For $a, b \in K_w^\times$, denote by $(a,b)$ the Hilbert symbol.
\begin{proposition}\label{thm!pairingformula}
  Let $a \in S_{\mclambda^2}$, $b \in S_{\hat\mclambda}$, $a_w =
  \delta(x_w)$, $x_w \in J(K_w)$. Then
  $$\zeta^{p\langle a, b \rangle_2}
  =(\iota_{P_3}(x_w), b_w).$$
\end{proposition}

\begin{proof}
  This follows from \cite[Theorem~2.6]{mccallum!1988}, with $\phi =
  \mclambda^2$ and $\psi = \mclambda$.
\end{proof}
For a number field $F$ we denote by $\Oh_F^{\prime}$ the ring
of $p$-integers in $F$. Suppose $F \subset K(p)$ and let $C$ be the ideal
class group of $\Oh_F^{\prime}$. Since the group
$\Oh_{K(p)}^{\prime \times}$ is $p$-divisible, we have an
exact sequence
\begin{equation*}
0 \to \mu_p \to \Oh_{K(p)}^{\prime \times} 
\xrightarrow{\,p\,} \Oh_{K(p)}^{\prime \times} \to 0,
\end{equation*}
which induces a long exact sequence of Galois cohomology
\begin{align*}
\cdots \to 
H^{i-1}(K(p)/F, \Oh_{K(p)}^{\prime \times}) \xrightarrow{\,p\,} 
H^{i-1}(K(p)/F, \Oh_{K(p)}^{\prime \times}) & \to
 \\
 H^i(K(p)/F, \mu_p) \to 
H^{i}(K(p)/F, \Oh_{K(p)}^{\prime \times}) \xrightarrow{\,p\,}
H^{i}(K(p)/F, \Oh_{K(p)}^{\prime \times}) & \to \cdots 
\end{align*}
If $i=1$ then, since $H^1(K(p)/F, \Oh_{K(p)}^{\prime \times})$ is isomorphic
to $C$, we obtain the exact sequence 
\begin{equation}
  \label{eq!6}
  0 \to \Oh_F^{\prime
  \times}/\Oh_F^{\prime \times p} \to H^1(K(p)/F,
  \mu_p) \to C[p] \to 0. 
\end{equation}
Also, by \cite[VIII.3]{neukirch-schmidt-wingberg:2000}, it follows that
$H^2(K(p)/F, \Oh_{K(p)}^{\prime \times})[p^\infty]$ can be identified 
with the subgroup $\Br(K(p)|F)[p^\infty]$
of $\Br(F)[p^\infty]$.
Setting $i=2$ in the above long exact cohomology sequence gives another 
exact sequence  
\begin{equation}\label{eq!7}
  0 \to C/pC \to H^2(K(p)/F, \mu_p) \to
  \Br(K(p)/F)[p] \to 0. 
\end{equation}

\begin{lemma}\label{lemma:lifting}
  Every element of $H^1(K(p)/K, J[\mclambda^k])$ lifts to $H^1(K,
  J[\mclambda^{k+1}])$. Moreover, if $p$ is regular, it lifts to
  $H^1(K(p)/K, J[\mclambda^{k+1}])$.
\end{lemma}
\begin{proof}
  Let $a \in H^1(K(p)/K, J[\mclambda^k])$, and let $\delta a \in
  H^2(K(p)/K, J[\mclambda])$ be  the coboundary of $a$ for
  the sequence
\begin{equation}
0 \to J[\mclambda] \to J[\mclambda^{k+1}] \to J[\mclambda^k] \to
0.
\end{equation}
Then the inflation of $\delta a$ in $H^2(K, J[\mclambda]) \iso H^2(K,
\mu_p) = \Br(K)[p]$ has zero invariant at every place not dividing
$p$. Thus it is zero by the Brauer--Hasse--Noether theorem (since there
is only one place of $K$ dividing $p$). For the second statement, we
argue in the same way, using~\eqref{eq!7}.
\end{proof}

We recall the definition of $\langle\ ,\,\rangle_3$. Let $a\in
S_{\mclambda^3}$ and $b \in S_{\hat \mclambda}$. Lift $a$ to an
element $a_1$ of $H^1(K, J[\mclambda^{4}])$ (which is possible by
Lemma~\ref{lemma:lifting}). For each place $v$ of $K$, lift $a_v$ to
an element $a_{v,1}$ that is in the image of $\delta$. Then $a_{1,v} -
a_{v,1}$ is the image of an element $c_v \in H^1(K_v, J[\mclambda])$,
and
\begin{equation*}
 \langle a, b \rangle = \sum_v c_v \cup b_v
\end{equation*}
where the cup product is with respect to the local pairing
$$H^1(K_v, J[\mclambda]) \times H^1(K_v, J[\hat\mclambda]) \to
\Q/\Z.$$
If $p$ is regular, $L/K$ is totally ramified at $w$, and
there is a unique extension of $w$ to $L$, that we also denote by
$w$.  Let $N' = \sum_{i=1}^{p-1} i\sigma^i$.

\begin{proposition}\label{thm!prop-supp-p}
Let $a \in S_{\mclambda^3}$, $b \in
   S_\mclambda$, $a_w = \delta(x_w)$, $x_w
   \in J(K_w)$.  Suppose that $\mclambda^2_*a$, regarded as an element of
   $\Oh_K^\times/\Oh_K^{\times p}$, can be written
as $N_{L/K}\epsilon$ for
   some $\epsilon \in\Oh_L^\times$.  Then there exists a
   $\mclambda^4$-torsion point $P_4$, and an element $c_w \in
   K_w^\times$ such that
   $$
   \zeta^{p \langle a, b \rangle_3} = (c_w, b_w),
   $$
   and the image of $c_w$ in $L_w^{\times}/L_w^{\times p}$ satisfies
   $$
   c_w = \iota_{P_4}(x_w)^{-1}\eta N'\epsilon,
   $$
where $\eta \in H^1(K(p)/L, \mu_p)^G$. In addition, if $a$ and $b$ are
eigenvectors for the action of $\Delta$, we may assume that $c_w$ is also.
\end{proposition}

\begin{proof}
Consider the sequence
\begin{equation} 
\label{eq!8}
0 \to J[\mclambda] \to J[\mclambda^4] \to J[\mclambda^3] \to 0
\end{equation}
and the commutative diagram with exact rows
  \[
\renewcommand{\arraycolsep}{.25em}
\begin{matrix}
 0 \to & H^1(K(p)/L, J[\mclambda])^G &
\xrightarrow{\hphantom{\mclambda_*}} &
 H^1(K(p)/L,J[\mclambda^4])^G & \xrightarrow{\mclambda_*} &
 H^1(K(p)/L,J[\mclambda^3])^G \\[6pt]
 & {\scriptstyle\res_{L/K}}\Big\uparrow\kern1cm &&
 {\scriptstyle\res_{L/K}}\Big\uparrow\kern1cm &&
 {\scriptstyle\res_{L/K}}\Big\uparrow\kern1cm \\
 & H^1(K(p)/K, J[\mclambda]) & \xrightarrow{\hphantom{\mclambda_*}} & 
 H^1(K(p)/K, J[\mclambda^4])
 & \xrightarrow{\mclambda_*} &
 H^1(K(p)/K,J[\mclambda^3]).
 \end{matrix}
  \]
%%%%%%%%%%%%%%%%%%%%
%  \begin{equation*}
% \xymatrix{0 \ar[r] & H^1(K(p)/L, J[\mclambda])^G \ar[r] & H^1(K(p)/L,
% J[\mclambda^4])^G
%   \ar[r]^{\mclambda_*} & H^1(K(p)/L, J[\mclambda^3])^G \\ 
%  & H^1(K(p)/K, J[\mclambda]) \ar[r] \ar[u]^{\res_{L/K}} & 
% H^1(K(p)/K, J[\mclambda^4])
%   \ar[u]^{\res_{L/K}}\ar[r]^{\mclambda_*} & H^1(K(p)/K, 
% J[\mclambda^3])\ar[u]^{\res_{L/K}}}
%   \end{equation*}
%%%%%%%%%%%%%%%%%%%%
  The top row is exact because (\ref{eq!8}) splits over $L$, and hence
  the sequence
  $$
  0 \to H^1(K(p)/L, J[\mclambda]) \to H^1(K(p)/L, J[\mclambda^4])
  \to H^1(K(p)/L, J[\mclambda^3])
  $$
  is exact.  By Lemma~\ref{lemma:lifting}, $a$ lifts to an element
  $a_1 \in H^1(K(p)/K, J[\mclambda^4])$. Let $a'_1 \in H^1(K(p)/L,
  J[\mclambda^4])^G$ be any lift of $\res_{L/K} a$ ($\res_{L/K} a_1$
  itself is one such).  Then
\begin{equation}\label{eq!9}
 \res_{L/K} a_1 = a'_1 \eta \quad\hbox{and}\quad
 \eta \in H^1(K(p)/L,\mu_p)^G.
\end{equation}
We now construct a candidate for $a'_1$. Under the
identification~\eqref{eq!37} between $J[\lambda^i]$ and $\mu_p^i$, the
map $\lambda^{i-1}: J[\lambda^i] \to J[\lambda]$ corresponds to
projection on the first component. Hence under the identification $H^1(K,
J[\mclambda^3]) = (K^\times/K^{\times p})^3$, $a$ corresponds to an
element $(x_1, x_2, x_3) \in (K^\times/K^{\times p})^3$, and
$\mclambda^2_*a = x_1$.  Moreover, in the identification
$$
H^1(L, J[\mclambda^4]) \iso (L^\times/L^{\times p})^4,
$$
the action of $\sigma$ on $H^1(L, J[\mclambda^4])$ is
intertwined with
$$
(t_1, t_2, t_3, t_4) \mapsto (t_1^\sigma, t_2^\sigma,
t_3^\sigma, t_4^\sigma t_1^\sigma), \quad t_i \in L^\times/L^{\times p}.
$$
Thus $(t_i)$ is fixed by $G$ if
$$
t_i^\sigma = t_i, \quad i = 1, 2, 3, \quad \text{and} \quad
t_4^{\sigma-1} = t_1^{-1}.
$$
By hypothesis, $x_1 = N_{L/K}\epsilon$, $\epsilon \in {\mathcal
  O}_L^\times$. Then
\begin{equation}\label{eq!10}
a'_1 = (x_1,
  x_2, x_3, N'\epsilon)
\end{equation}
is an equivariant lift of $(x_1, x_2, x_3)$.

Now let $a_{w,1}$ be the local lift of $a$ given by $a_{w,1} =
\delta_4(x_w)$.  Then
\begin{equation}\label{eq!11}
\res_{L_w/K_w} a_{w,1} = (x_1,  x_2, x_3, \iota_{P_4}(x_w)).
\end{equation}
Thus, from equations~\eqref{eq!9}, \eqref{eq!10}, and \eqref{eq!11},
we get
\begin{equation*}
\res_{L_w/K_w}(c_w) = \res_{L_w/K_w}(a_{1,w} - a_{w,1}) =
  \iota_{P_4}(x_w)^{-1}\eta N' \epsilon.
\end{equation*}
The last statement of the proposition is clear, since at each stage in
the calculation we can choose eigenvectors, and the maps $\lambda$ and
$\iota_{P_k}$ are also eigenvectors for the action of
$\Delta$, by the choices we have made of $\lambda$ and $P_k$. 
\end{proof}

 \section{The local approximation}\label{sec!doing-local-appr}
 Let $P_i$ be as in the previous section, $i = 1, 2, 3, 4$, let $D_i$
 be a divisor on $F_{a,b,c}$ representing $P_i$, and let $f_i$ be a
 function whose divisor is $pD_i$. Take $D_i$ and $f_i$ to be defined
 over $K = \Q(\zeta)$ if $i = 1, 2, 3$ and over $L =
 K(\eta_{p-3}^{1/p})$ if $i = 4$.  The maps $\iota_{P_i}$ in
 Propositions~\ref{thm!pairingformula} and~\ref{thm!prop-supp-p} are
 computed by evaluating $f_i$ on certain divisors.  We use the
 approximation technique in \cite{mccallum!1988} to find expansions
 for $f_i$ on $p$-adic discs in $F_{a,b,c}$. Given a function $f$
 whose divisor is divisible by $p$ we approximate $f$ on an affinoid
 $Y$ in $F_{a,b,c}$ using the fact that
\begin{equation}
  \label{eq!12}
  \frac{df}{f} \equiv \omega \quad \affmd{Y}{p}.
\end{equation}
for some holomorphic differential $\omega$ on $F_{a,b,c}$
(\cite{mccallum!1988}, Theorem 5.2).  For general facts about rigid
analysis, we refer the reader to \cite{bosch-guentzer-remmert!1984}.

We recall the notion of congruence used in~\eqref{eq!12}.  If $Y$ is a
one-dimensional affinoid defined over an extension $F$ of $\Q_p$ with
uniformizer $\pi_F$, we let $A(Y)$ be the ring of rigid analytic
functions on $Y$, $M(Y)$ the quotient field of $A(Y)$, and $D(Y)$ the
module of K\"ahler differentials of $M(Y)$. We define
sub-$\Oh_F$-modules
\begin{eqnarray*}
  A^0(Y) &=& \{ f \in A(Y): \text{$|f(x)| \leq  1$ for all $x \in
  Y(\C_p)$}\}\\
  M^0(Y) &=& \{ f/g: f \in A^0(Y), g \in A^0(Y) \setminus \pi_F
  A^0(Y)\}\\
  D^0(Y) &=& \{ \sum_i f_i\,dg_i: f_i, g_i \in M^0(Y)\}.
\end{eqnarray*}
If $f, g \in A(Y)$, $c \in F$, we say that $f \equiv g \affmd{Y}{c}$ if
$(f-g) \in c A^0(Y)$, and similarly we define the notion of congruence
 on $Y$ in $M(Y)$ and $D(Y)$.  In order to deduce
from~\eqref{eq!12} information about power series expansions of $f$ on
closed discs in $Y$, we need the following lemmas.

\begin{lemma}\label{thm!lemma-if-z}
  Suppose that $Y$ is a one-dimensional affinoid over a finite
  extension $F$ of $\Q_p$, $Y$ has good reduction, and $Z$ is
  an affinoid contained in $Y$, isomorphic to a closed disc. If
  $\omega \in D^0(Y)$ is a differential with at worst simple poles on
  $Y$ that is regular on $Z$, then $\omega \in D^0(Z)$.
\end{lemma}

\begin{proof}
  Since $Z$ is isomorphic to a closed disc, it is contained in a
  residue class $U$ of $Y$ (or is equal to $Y$, in which case there is
  nothing to prove). It is clear from the definitions that $D^0(Y)|_U
  = D^0(U)$, hence $\omega \in D^0(U)$. Furthermore, since $Y$ has
  good reduction, $U$ is isomorphic to an open disc. Choose a
  parameter $t$ for $U$ such that $Z$ is the disc $|t|\leq |c| < 1$
  for some $c \in F $, and write
  \[
   \omega = g\,dt + \sum_{i=1}^n \frac{a_i}{t-b_i}\,dt,\quad g \in
   \Oh_F[[t]], \quad a_i, b_i \in \Oh_F, \quad |c| < |b_i| < 1.
  \]
  Expanding the polar terms in powers of $t/b_i$ and setting $t = cs$,
  we get $\omega = f \, ds$ for some $f \in \Oh_F[[s]]$. Since
  $s$ is a parameter on $Z$, this proves the lemma.
\end{proof}

\begin{lemma}\label{thm!YtoX}
  Suppose that $f$ is a function whose divisor is divisible by $p$.
  Let $Y$ be an affinoid with good reduction contained in $F_{a,b,c}$
  and let $Z$ be a $p$-adic disc contained in $Y$ such that there is a
  function on $F_{a,b,c}$ restricting to a parameter on $Z$.  If
  $\omega$ satisfies the congruence (\ref{eq!12}) $\affmd{Y}{p}$,
  then it satisfies the same congruence $\affmd{Z}{p}$.
\end{lemma}

\paragraph{Proof}
  With notation as in \cite{mccallum!1988}, we have
  $$
  \frac{df}{f} = \omega + p \eta, \quad \eta \in D^0(Y).
  $$
  Let $g$ be a function on $F_{a,b,c}$ such that $f/g^p$ is regular
  on $Z$ (we can construct $g$ using a parameter on $Z$ as in the
  hypotheses).  Since a suitable scalar multiple of $g$ is in
  $M^0(Y)$, $\dlog g \in D^0(Y)$. Thus $\eta - \dlog g \in D^0(Y)$ and
  is also regular on $Z$, and hence is in $D^0(Z)$ by
  Lemma~\ref{thm!lemma-if-z}. Thus
  $$
  \frac{df}{f} \equiv \frac{df}{f} - p\frac{dg}{g} = \omega +
  p\Bigl(\eta - \frac{dg}{g}\Bigr) \equiv \omega \quad \affmd{Z}{p}. \qed
  $$
\par\medskip

We apply these considerations to the affinoid $Y$ introduced in
\cite{mccallum!1988} and defined as follows. Choose $\pi_K=\pi$ as 
the 
uniformizer  
for $K_w$. Let $s$ and $t$ be
the functions on $F_{a,b,c}$ defined by
\begin{eqnarray}
  \label{eq!13}
  x &=& -\frac{a}{c}(1 + \pi^{(p-1)/2}s)\\
y &=& (-1)^ca^ab^bc^c(1+\pi t).
\end{eqnarray}
Let $Y$ be the affinoid defined over $L_w$ by the inequalities
\begin{equation*}
  |t| \leq |\pi_L^{-1}|,\quad |s| \leq 1.
\end{equation*}
A basis of holomorphic differentials on $F_{a,b,c}$ is
\begin{equation*}
  \omega_k = 
E_k \frac{x^{\left[\frac{ka}{p}\right]}(1-x)^{\left[\frac{kb}{p}\right]}}{y^k}
  \, dx, \quad k \in H_{a,b,c},
\end{equation*}
for some constants $E_k$ and where $H_{a,b,c}$ is a certain subset of
$\{1, 2, \dots, p-1\}$ of cardinality $(p-1)/2$ (we can identify
$H_{a,b,c}$ with the CM-type of $F_{a,b,c}$). We can and do choose
the constants
$E_k$ so that $\omega_k$ has expansion
\begin{equation}
  \label{eq!14}
  \omega_k \equiv ds \quad \affmd{Y}{\pi_L},
\end{equation}
(note that this normalization is different from that of
\cite{mccallum!1988}).  Now $P_1$ is the $\mclambda$-torsion point
represented by the divisor $(0,0) - \infty$, and we choose $f_1 = x$.
In \cite{mccallum!1988} it was shown that
\begin{equation}
  \label{eq!15}
\frac{df_1}{f_1} \equiv  \pi^{(p-1)/2} \sum_{k \in H_{a,b,c}} b_k
\omega_k \quad \affmd{Y}{p}
\end{equation}
for some $p$-adic integers $b_k$ satisfying
\begin{equation}
  \label{eq!16}
  \sum_{k \in H_{a,b,c}} b_k k^i \equiv
   \begin{cases}
     F & i=0\\
     0 & 1 \le i \le (p-3)/2
  \end{cases} \pmod{\pi_K}, \quad F \in \Z/p\Z^\times. 
\end{equation}
Note that although it was assumed that $F_{a,b,c}$ is wild split in
Section~5 of \cite{mccallum!1988}, there is nothing in the definition
of $Y$ or the calculation showing~(\ref{eq!15}) and~(\ref{eq!16}) that
uses this assumption. It is only at the end of that section that the
assumption comes in.

\begin{lemma}
  \label{thm!congruencelemma}
  If $$
  \sum_{k \in H_{a,b,c}} u_k \omega_k \equiv \sum_{k \in
    H_{a,b,c}} v_k \omega_k \affmd{Y}{\pi^{n+ (p-3)/2}}$$
  then
  $$u_k \equiv v_k\pmod{\pi^n}, \quad k \in H_{a,b,c}.$$
\end{lemma}


\begin{proof}
  See pages 658--659 of \cite{mccallum!1988}.
\end{proof}

\begin{proposition}\label{thm!approximation}
  We have
  \begin{equation*}
\frac{df_{3}}{f_{3}} \equiv \sum_{k \in H_{a,b,c}}c_k \omega_k 
\affmd{Y}{p}, \quad \text{$c_k \equiv 0 \pmod{\pi^{(p-5)/2}}$}
  \end{equation*}
  and
\begin{equation}
  \label{eq!17}
  \frac{df_{4}}{f_{4}} \equiv \sum_{k\in H_{a,b,c}} d_k
  \omega_k \affmd{Y}{p}, \quad \text{$d_k \equiv
  -\pi^{(p-7)/2}\frac{b_k}{k^3} \pmod{\pi^{(p-5)/2}}$},
\end{equation}
where the $b_k$ are as in equation~\eqref{eq!15}.
\end{proposition}

\begin{proof}
  We have
  $$
  \mclambda^2_* \frac{df_{3}}{f_{3}} \equiv \frac {df_1}{f_1}
  \affmd{Y}{p}.
  $$
  Since $\zeta_* \omega_k = \zeta^{-k}\omega_k$, we have $\lambda_*
  \omega_k = \lambda^{\sigma} \omega_k$, for some $\sigma \in
  \Gal(K_w/\Q_p)$. Hence it follows from
  Lemma~\ref{thm!congruencelemma} and \eqref{eq!15} that
  $$\lambda^{2\sigma} c_k \equiv \pi^{(p-1)/2}b_k
  \pmod{\pi^{(p+1)/2}}.$$
  Thus $c_k \equiv 0 \pmod{\pi^{(p-5)/2}}$, as
  claimed.  Similarly, we have $\mclambda_*^3 (df_{4}/f_{4}) \equiv
  (df_1/f_1)$, so $\lambda^{3\sigma} d_k \equiv \pi^{(p-1)/2}b_k
  \pmod{\pi^{(p+1)/2}}$. Furthermore, since $\zeta^\sigma =
  \zeta^{-k}$, it follows from our choice of $\lambda$ that
  $\lambda^\sigma/\pi \equiv -k \pmod{\pi}$ for $1 \le k \le p-1$, so
  we get equation~\eqref{eq!17}.
\end{proof}


\begin{lemma}
  \label{thm!calculation}
  $$
  -\sum_{k \in H_{a,b,c}} \frac{b_k}{k^3} \equiv F( q(a^ab^bc^c)^3+
  a b c B_{p-3})\pmod{\pi},
  $$
  where $F$ is as in~\eqref{eq!16}.
\end{lemma}

\begin{proof}
  Let $n = (p-1)/2$.  Define
  $$
  \Gamma_k(x_1, \dots, x_n) = \det \left[ \begin{array}{cccc} 1 & 1 & \dots & 1 \\
      x_1 & x_2 & \dots & x_n \\
      x_1^2 & x_2^2 & \dots & x_n^2 \\
      \vdots & \vdots & \ddots & \vdots \\
      x_1^{n-2} & x_2^{n-2} & \dots & x_n^{n-2}\\
      x_1^{n-1+k} & x_2^{n-1+k} & \dots & x_n^{n-1+k}
\end{array}
\right].
$$
Then an elementary
linear algebra calculation using~\eqref{eq!16} gives
\begin{equation*}
  \sum_{k \in H_{a,b,c}} \frac{b_k}{k^3} \equiv 
F \Gamma_3(H_{a,b,c}^{-1})/\Gamma_0(H_{a,b,c}^{-1})
\pmod{\pi}.
\end{equation*}
Let $S_i(x_1, \dots, x_n)$ be the $i$th symmetric function. Then
$$
\Gamma_3 = \Gamma_0(S_1^3 - 2 S_1 S_2 + S_3).
$$
This can be proved by the usual method: the determinant vanishes if
$x_i = x_j$ for $i\neq j$, or if there is a polynomial of degree $n+2$
vanishing on the $x_i$, and with no term of degree $n-1$, $n$, or
$n+1$. Thus, if the roots of the polynomial are $x_1, \dots, x_n,
\alpha, \beta$, then
\begin{eqnarray*}
  \alpha + \beta + S_1 &=&0,\\
S_2 + (\alpha+\beta) S_1 + \alpha \beta &=&0,\\
\alpha \beta S_1 + \alpha S_2 + \beta S_2 + S_3 &=& 0.
\end{eqnarray*}
Eliminating $\alpha$ and $\beta$ gives the condition $S_1^3 - 2S_1S_2
+ S_3 = 0$. Now, we have
\begin{eqnarray}
  S_1(H_{a,b,c}^{-1})  &\equiv & -q(a^ab^bc^c),\label{eq!18}\\
  S_2(H_{a,b,c}^{-1}) &\equiv& 0, \label{eq!19}\\
  S_3(H_{a,b,c}^{-1}) & \equiv & -\frac{B_{p-3}}{3}(a^3+b^3+c^3) \equiv -a b c B_{p-3}.\label{eq!20}
\end{eqnarray}
It is explained in \cite{mccallum!1988}, Lemma 5.24, how
equation~\eqref{eq!18} follows from \cite[17]{vandiver:1919};
equation~\eqref{eq!19} follows from parity considerations; and
equation~\eqref{eq!20} follows from \cite[16]{vandiver:1919},
in exactly the same way as \eqref{eq!18} follows from
\cite[17]{vandiver:1919}.
\end{proof}


We now define $p$-adic discs in $Y$, to which we apply
Lemma~\ref{thm!YtoX}. Let $X$ be the sub-affinoid of $Y$ defined by
$|t| \leq 1$ in the wild case and by $|s| \leq |\pi_K|$ in the tame
case.  Let $E_w$ be the quadratic unramified extension of $K_w$.  If
$F_{a,b,c}$ is wild, $X$ is isomorphic to a union of two closed discs,
which are defined over $K_w$ in the split case and over $E_w$ in the
nonsplit case.  Furthermore, $T = t$ is a parameter on each disc. If
$F_{a,b,c}$ is tame, then $X$ is isomorphic to a union of $p$ closed
discs defined over $K_w$, and $T = s/\pi_K$ is a parameter on each
disk. For proofs of these facts we refer the reader
to\cite{mccallum!1988} (where $T = s'$ in the tame case).  We denote
by $Z$ any of the discs that are components of $X$, with parameter
$T$. We can write
\begin{equation*}
  f_i|_X = C_i u_i(T) v_i(T^{p}) g_i(T)^p, \quad i = 1, 2, 3, 4,
\end{equation*}
where $u_i$ and $v_i$ are unit power series with constant term 1 and
integer coefficients, $u_i$ has no terms in $T^{p}$, and $g_i$ is a
monic polynomial with integer coefficients. Furthermore, these
conditions uniquely determine the $u_i$, $v_i$ and $g_i$. Then
\begin{equation}
  \label{eq!21}
  \frac{df_i}{f_i} \equiv \frac{du_i}{u_i} \affmd{Z}{p}.
\end{equation}
For a $p$-adic field $H$ we denote by $U_H[[T]]$ the power series in
$\Oh_H[[T]]$ which are congruent to 1 modulo the maximal ideal
in $\Oh_H[[T]]$.

\begin{theorem}\label{thm!proposition-let-d}
  Let $Z$ be any of the discs that are components of $X$ and let $T$
  be a parameter on $Z$. Then
  \begin{equation}
    \label{eq!22}
    u_i = 1 + \pi^{(p+3)/2-i}D_i T + O(\pi^{(p+5)/2-i}T),
  \end{equation}
  where $|D_i| \leq 1$, $i = 1, 2, 3, 4$. Moreover, $|D_1| = 1$,
  and under the hypotheses of Theorem~\ref{thm!nontrivial}, $|D_4| =
  1$ and
\begin{equation*}
  \frac{D_4}{D_1} \equiv 
    q(a^ab^bc^c)^3 + abc B_{p-3}.
\end{equation*}
Finally, $u_i$ for $i = 1, 2, 3$ are defined over $E_w$, and
\begin{equation*}
  u_4 \in 1 + \pi_K^{(p-5)/2}D_4 T U_{E_w}[[T]] + \pi_K^{(p+1)/2}\pi_L^{-3}ED_1
  T U_{F_w}[[T]],
\end{equation*}
where $E \in \Z/p\Z^\times$ is independent of the triple
$(a,b,c)$ and $F_w$ is the quadratic unramified extension of $L_w$.
\end{theorem}

\begin{proof}
  In both wild and tame cases we have $\omega_k \equiv \pi D \, dT$
  $\affmd{Z}{\pi^2}$ for all $k \in H_{a,b,c}$, with $D \in
  \Z/p\Z^\times$ independent of $k$. This
  follows from our normalization~\eqref{eq!14}, since in the tame case
  we have
  \begin{equation}
 s = \pi T,\label{eq!23}
 \end{equation}
 and in the wild case it follows from \cite[(5.6)]{mccallum!1988},
 where it is shown that the expansion of $s$ in terms of $t$ on either
 of the discs in $X$ is
\begin{equation}
  \label{eq!24}
  s^2 = \frac{-q(a^ab^bc^c)2b}{ac} + \pi \frac{2b}{ac}(t^p - t) + O(\pi^2).
\end{equation} 
The statements about $D_1$ follow from~\eqref{eq!13}, \eqref{eq!23} (in
the tame case) and \eqref{eq!24} (in the wild case), since $f_1 = x$.
The statement about $D_2$ was proved in
\cite[Theorem~5.13]{mccallum!1988}.  Although this theorem is stated
only for the wild split case, the consequence~\eqref{eq!22} is easily
seen to hold also in the other cases (the part of Theorem~5.13
specific to the wild split case translates into the statement $|D_2| =
1$ in the current notation, and we do not need it here). The
statements about $D_3$ and $D_4$ follow from
Proposition~\ref{thm!approximation}, Lemma~\ref{thm!calculation},
and~\eqref{eq!21}. The statement about the ratio $D_4/D_1$ follows
from~\eqref{eq!15}, the case $i=0$ of~\eqref{eq!16}, \eqref{eq!17},
and Lemma~\ref{thm!calculation}, taking note of the
normalization~\eqref{eq!14} and the fact that $ds \equiv
\mathrm{unit}\times \pi dT \affmd{Y}{\pi^2}$. The statements about
the fields of definition follow from the fact that $f_i$ is defined
over $K$ for $i = 1, 2, 3$ and $f_4$ is defined over $L$, and that the
discs $Z$ are always defined over $E_w$. The final statement follows
from considerations of ramification theory. Locally, we have
$\eta_{p-3} = 1 + a \pi^{p-3}$ modulo $p$th powers, so the (upper)
conductor of $L_w/K_w$ is 3. Now, it follows from the properties of
the $P_i$ that $u_4^{\sigma-1} \equiv u_1$ modulo
$\Oh_{F_w}[[T]]^{\times p}U_{F_w}[[T^p]]$, and, since $u_1 \in 1 +
\pi^{(p+1)/2}TD_1 U_{F_w}[[T]]$, this implies the final statement with
$E$ such that $(\sigma -1)\pi_L^{-3} \equiv E^{-1} \pmod{\pi_L}$.
\end{proof}


\section{Computation of the Cassels pairing}
Recall the local descent maps
\begin{equation*}
\delta_i = \iota_{P_1}\times \cdots\times \iota_{P_i}\colon J(K_w) \to  
(K_w^\times/K_w^{\times p})^i
\end{equation*}
described in Section~\ref{sec!formulas-pairings}.  We start by
noting a couple of properties that follow from the choice of $P_i$
made in Section~\ref{sec!formulas-pairings}. First, we have
\begin{equation}
  \label{eq!25}
  \iota_{P_i}\circ \mclambda = \iota_{P_{i-1}}, \quad i = 2, 3, 4.
\end{equation}
Second, for $i = 1, 2, 3$ we have, from eigenspace considerations,
\begin{equation}
\iota_{P_i}(J(K_w)(k)) \subset V(k-i+1). \label{eq!26}
\end{equation}
Let $A \subset J(K_w)$ be the subgroup generated by divisors supported
on the discs $|T| \leq |\pi_K|$ in $X$.  Let
$$
V[i,j] = \bigoplus_{i \le k \le j} V(k).
$$
Note that $V(i) = 0$ for $i>p$.
\begin{proposition}\label{thm!proposition-we-have}
  We have
\begin{equation}
 \iota_{P_i}(A)
  \subset V\bigl[(p+5)/2-i,p\bigr],\quad i = 1, 2, 3.\label{eq!27}
  \end{equation}
  If $F_{a,b,c}$ is wild split, we have
\begin{equation}
  \label{eq!28}
  \iota_{P_i}(J(K_w)) \subset V\bigl[(p+1)/2-i,p\bigr],\quad i = 1, 2, 3.
\end{equation}
If $F_{a,b,c}$ is wild nonsplit or tame, we have
\begin{equation}
\label{eq!29}
  \iota_{P_i}(J(K_w)) \subset V\bigl[(p+3)/2-i,p\bigr],\quad i = 1, 2, 3.
\end{equation}
Furthermore, in the case $i=1$, the inclusions in~\eqref{eq!27}
and~\eqref{eq!29} are equalities.
\end{proposition}

\begin{proof}
  The inclusions in~\eqref{eq!27} follow immediately from
  Theorem~\ref{thm!proposition-let-d}, as does the claim that the
  inclusion is equality in the case $i=1$. Now
  Faddeev \cite{faddeev:1961} proved that
  \begin{equation*}
    \im \iota_{P_1} = 
    \begin{cases}
      V((p-1)/2) \oplus V[(p+3)/2, p] & \text{if $F_{a,b,c}$ is wild
        split,}\\
      V[(p+1)/2,p] & \text{otherwise.}
    \end{cases}
  \end{equation*}
  This implies the statements~\eqref{eq!28} and~\eqref{eq!29} in the
  case $i=1$, and also that the image of $A$ in $J(K_w)/\mclambda
  J(K_w)$ has codimension 1, and the eigenvalue of the quotient is
  $\kappa^{(p-1)/2}$ in the wild split case and $\kappa^{(p+1)/2}$
  in the other cases. The remaining statements now follow from
  \eqref{eq!25}, \eqref{eq!26}, and~\eqref{eq!27}.
\end{proof}

\begin{proposition}\label{thm!proposition-if-f_a-1}
  If $F_{a,b,c}$ is wild nonsplit or tame, then $$\delta_2(J(K_w)) =
  V\bigl[(p+1)/2,p\bigr]^2.$$
\end{proposition}

\begin{proof}
  {From}~\eqref{eq!25} with $i=2$ we have
  \begin{equation}
\im \delta_2 \cap (K^\times/K^{\times p}) \times 1  =  \im
  \delta_1 \times 1 = V\bigl[(p+1)/2,p\bigr] \times 1\label{eq!30}
  \end{equation}
  Furthermore, given $u \in V[(p+3)/2, p]$, we can find $a \in A$ such
  that $\iota_{P_1}(a) = u$, and $\iota_{P_2}(a) \in V[(p+1)/2,p]
  \subset \im \iota_{P_1}$.  Thus, modifying $a$ by $\mclambda
  J(K_w)$, we can choose it so that $\iota_{P_2}(a) = 1$. Hence
  \begin{equation}
    \label{eq!31}
\im  \delta_2 \supset 1 \times V\bigl[(p+3)/2,p\bigr]. 
  \end{equation}
  Now it follows from local duality that $\im \delta_2$ must be
  maximal isotropic with respect to the cup product pairing on
  $(K^\times/K^{\times p})^2 = H^1(K, J[\mclambda^2])$ induced by the
  Weil pairing on $J[\mclambda^2]$. Since $\lambda^2 = \hat\lambda^2$,
  the Weil pairing is skew symmetric. Thus the pairing on
  $(K^\times/K^{\times p})^2$ is a nonzero multiple of $((a_1, b_1),
  (a_2, b_2)) \mapsto (a_1, b_2)_w(b_1, a_2)_w^{-1}$, where $(\ ,\,)_w$
  denotes the Hilbert symbol at $w$. The only maximal isotropic
  subgroup satisfying~\eqref{eq!30} and~\eqref{eq!31} is the one given
  in the statement of the proposition.
\end{proof}
Define a subspace $V_{\mathrm{global}} \subset V$ by 
\begin{equation*}
  V_{\mathrm{global}} = \bigoplus_{\substack{2\le i\le p-3 \\
\text{$i$ even}}} V(i).
\end{equation*}

\begin{proposition}\label{thm!proposition-assume-p}
  Assume $p \geq 11$ and $F_{a,b,c}$ is wild nonsplit or tame.  There
  exists a point $x \in A$ such that $\iota_{P_1}(x)$ generates $V((p+5)/2)$
  and $\iota_{P_i}(x) \in V_{\mathrm{global}}$ for $i = 2, 3$.
\end{proposition}

\begin{proof}
  It follows from Proposition~\ref{thm!proposition-we-have} that $A$,
  regarded as a $\Z_p[\zeta]$-submodule of $J(K_w)$, has codimension at
  most $1$. Hence $\delta_3(A)$ has codimension $\le3$ as a
  $\F_p$-vector space in $\delta_3(J(K_w))$. By
  Proposition~\ref{thm!proposition-we-have} we can choose $x \in A$
  such that $\iota_{P_1}(x)$ generates $V((p+5)/2)$. This condition 
 leaves freedom to modify $x$
  by anything in $\mclambda J(K_w)$, which would change $\delta_3(x)$
  by anything in $\im \delta_2$. Thus, modifying $x$ as needed, we can
  ensure that $\iota_{P_i}(x) \in V_{\mathrm{global}}$, $i = 2, 3$.
  The number of degrees of freedom in performing this modification is
  equal to the dimension of $\im \delta_2 \cap V_{\mathrm{global}}$,
  which is at least 4 if $p\geq 11$, by
  Proposition~\ref{thm!proposition-if-f_a-1}. Thus we can ensure that
  $x$ remains in $A$ when making the modification.
\end{proof}

\paragraph{Computation of the Cassels pairing for Theorem~\ref{thm!free}}
  We now use Proposition~\ref{thm!pairingformula} to show 
that the pairing $\langle\ ,\,\rangle_2$ is trivial under
  the hypotheses of Theorem~\ref{thm!free}. In the next section we
  explain how this implies the theorem. 

Denote by $\ell_i\colon
  S_{\mclambda^i} \to J(K_w)/\mclambda^i J(K_w)$ the localization map.
  We claim that, under the hypotheses of Theorem~\ref{thm!free},
  $\iota_{P_1}(\ell_1(S_\mclambda)) \subset V[(p+3)/2,p]$ or
  $\iota_{P_1}(\ell_1(S_\mclambda)) \subset V((p+1)/2) \oplus
  V[(p+5)/2,p]$. Now, $V(i)$ pairs nontrivially with $V(j)$ under the
  Hilbert pairing if and only if $i + j \equiv p \pmod{p-1}$. Thus, it
  follows from our claim and from~\eqref{eq!26} that
  $\iota_{P_3}(\ell_3(J(K_w))$ pairs trivially with
  $\iota_{P_1}(\ell_1(J(K_w))$.

  To see the claim, note that if hypothesis (\ref{item!1}) of
  Theorem~\ref{thm!free} is satisfied, namely that $F_{a,b,c}$ is
  wild split and $p \equiv 3 \pmod{4}$, then, by \cite{faddeev:1961},
  $\iota_{P_1}(\ell_1(S_\mclambda)) \subset V((p-1)/2)\oplus V[(p+3)/2,p]$.
  Furthermore, we can eliminate $V((p-1)/2)$ as a possibility, because
  $\ell_1$ factors through $H^1(K(p)/K, \mu_p) \to H^1(K_w,
  \mu_p)$. Since $(p-1)/2$ is odd, it follows from~\eqref{eq!6} that
  $\ell_1$ can have nontrivial image in $V((p-1)/2)$ only if
  $C((p-1)/2)$ is nontrivial, which would imply $p \mid B_{(p+1)/2}$.
  This never happens if $p \equiv 3 \pmod{4}$.

  If hypothesis (\ref{item!2}) of Theorem~\ref{thm!free} is
  satisfied, namely that $F_{a,b,c}$ is wild nonsplit or tame and the
  image of $U$ in either $V((p+1)/2)$ or $V((p+3)/2)$ is trivial, then
  the claim follows immediately from~\eqref{eq!29}.
\par\medskip

The proof of Theorem~\ref{thm!nontrivial} uses the following
lemma.
\begin{lemma}\label{thm!lemma-suppose-that}
  Suppose $p\geq 5$ is regular, and let $L$ be as in~\eqref{eq!5}. Then
  \begin{enumerate}
  \item  the
  map $H^1(K(p)/L, \mu_p) \to H^1(L_w, \mu_p)$ is injective
\item the norm map $N_{L/K}\colon
  \Oh_L^\times \to \Oh_K^\times$ is
  surjective
\item 
$H^1(K(p)/L, \mu_p)^G(i) = 0$ if $i$ is odd and $i \neq 1$, or if $i=p-1$.
  \end{enumerate}
\end{lemma}

\begin{proof}
  Let $H_K$ (resp.\ $H_L$) be the Hilbert class field of $K$ (resp.\
$L$). Since $L/K$ is unramified outside $p$, and there is only prime
  of $L$ above $p$, it follows that $\Gal(H_L/L)/(\sigma-1) \iso
  \Gal(H_K/K)$. Therefore $p$ does not divide the order of
the class group   $C_L \iso \Gal(H_L/L)$. The
  injectivity statement follows, since anything in the kernel would
  generate an unramified Kummer extension of $L$ of degree $p$. 
Furthermore, every unit of
  $K$ is a local norm everywhere except possibly at the prime above
  $p$, and therefore is a local norm there also by the product
  formula. Thus it is a global norm. The surjectivity of the norm map
  follows by a standard argument using
  $\Gal(L/K)$ cohomology of the sequences
 \[
1 \to \Oh_L^\times \to L^\times \to
  P_L \to 1
 \quad\hbox{and}\quad
  1 \to P_L \to I_L \to C_L \to 1,
 \]
where $I_L$ and $P_L$ are the groups of ideals and principal ideals
respectively.
 Finally, by~\eqref{eq!6},
 \[
  H^1(K(p)/K, \mu_p) = \Oh_K^{\prime
    \times}/\Oh_K^{\prime \times p}
 \quad\hbox{and}\quad
 H^1(K(p)/L,\mu_p)=\Oh_L^{\prime \times}/\Oh_L^{\prime \times p}.
 \] 
Moreover, the
  cokernel of $\Oh_K^\times/\Oh_K^{\times p}$ in
  $(\Oh_L^\times/\Oh_L^{\times p})^G$ is $H^2(L/K,
  \mu_p) \iso \Z/p\Z$, with $\Gal(K/\Q)$ acting via
  $\kappa^{p-3}$, since it acts on $G$ via $\kappa^{3}$. Since $p-3$
  is even and $(\Oh_K^\times/\Oh_K^{\times p})(i) = 0$
  if $i$ is odd and $i\neq 1$, or if $i = p-1$, the third statement of the lemma
  follows.
\end{proof}

\begin{pfof}{Theorem~\ref{thm!nontrivial}}
  We exhibit $a \in S_{\lambda^3}$ and $b \in S_\lambda$ which pair
  nontrivially under the Cassels pairing. 

  Since $p$ is regular, the exact sequence \cite[7.3]{mccallum!1988}
identifies $U$ with  
  $\Oh_K^\times/\Oh_K^{\times p}$. Thus $S_{\lambda^i}
  \subset (\Oh_K^\times/\Oh_K^{\times p})^i$ for $i \leq 3$.
  The Selmer group is the subgroup obtained by imposing the local
  conditions at $w$. Since $(p+1)/2$ is even, we can choose an element
  $b \in \Oh_K^\times/\Oh_K^{\times p}$ which
  generates $V((p+1)/2)$, and $b$ satisfies
  the local condition by Proposition~\ref{thm!proposition-we-have}, so
  $b \in S_\lambda$. 

  As for $a$, by Proposition~\ref{thm!proposition-assume-p} there
  exists $a_w = (a_{w,1}, a_{w,2}, a_{w,3}) = \delta_3(x)$, $x \in A$,
  such that $a_{w,1}$ generates $V((p+5)/2)$ and $a_{w,2}, a_{w,3} \in
  V_{\mathrm{global}}$.  Using a suitable projector, we may further
  assume that $x$ is an eigenvector for the action
  of $\Delta$. Choose eigenvectors $a_i \in
  \Oh_K^\times/\Oh_K^{\times p}$ specializing to
  $a_{w,i}$ for $i = 1, 2, 3$ and define $a \in S_{\lambda^3}$ by $a =
  (a_1, a_2, a_3)$. 

  Now, by Lemma~\ref{thm!lemma-suppose-that}, $\mclambda_*^2 a = a_1
  \in V((p+5)/2)$ is the norm of a global unit $\epsilon$ in
  $\Oh_L^\times$, and by Proposition~\ref{thm!prop-supp-p},
 the Cassels pairing of $a$ and $b$ is the Hilbert pairing
  $(c_w, b_w)$, where $c_w \in K^\times/K^{\times p}$ is an eigenvector and 
\begin{equation}
  \label{eq!32}
  c_w = \iota_{P_4}(x)^{-1}  \eta N'\epsilon
  \quad \text{in $L_w^\times/L_w^{\times p}$},
\end{equation}
where $\eta \in H^1(K(p)/L, \mu_p)^G$. We can identify the precise
  eigenspace in which $c_w$ lies as follows. Since $a_{w,1} =
  \iota_{P_1}(x)$, and since $P_1$ is fixed by $\Delta$, $x$ has
  eigenvalue $\kappa^{(p+5)/2}$. Then, since $\lambda$  has eigenvalue
  $\kappa$ (modulo $\lambda^5$), it follows that $\delta_3(x)$,
  and hence $c_w$, have eigenvalue $\kappa^{(p+5)/2-3} =
  \kappa^{(p-1)/2}$. Thus we may assume
without loss of generality that $c_w$, $\eta$, $N'\epsilon$, and
$\iota_{P_4}(x)$ are eigenvectors for a lift $\tilde \Delta$ of
$\Delta = \Gal(K_w/\Q_p)$ to $\Gal(L_w/\Q_p)$, with
eigenvalue $\kappa^{(p-1)/2}$.  
Since $\eta \in H^1(K(p)/L, \mu_p)^G$, its projection onto an
eigenspace $(L_w^\times/L_w^{\times p})(i)$ with $i > 1$ odd is
trivial, by Lemma~\ref{thm!lemma-suppose-that}. This applies in
  particular to $i = (p-1)/2$, so that the image of $\eta$ in
  $L_w^\times/L_w^{\times p}$ is trivial. 

Under the Hilbert pairing the $\kappa^{(p-1)/2}$ and
$\kappa^{(p+1)/2}$ eigenspaces of $K_w^\times/K_w^{\times p}$ pair
nontrivially. Thus, to prove that the  pairing $(c_w,b_w)$ 
is nontrivial, it suffices to show that
$c_w$ is not a $p$th power, and for that it suffices to show that its
image in $L_w^\times/L_w^{\times p}$ is nontrivial. 

Since $x \in A$, we may choose a divisor $D$ supported on $|T|\leq
|\pi|$ such that $a_{w,i} = f_i(D)$, $1 \leq i \leq 3$. Since $D$ is
supported on $|T| \leq |\pi|$, we have
  \begin{equation*}
u = f_4(D) \equiv u_4(D) \pmod{L_w^{\times p}(1 +
  \pi^p\Oh_{L_w})}.
\end{equation*}
{From} the Galois properties of the $P_i$, we have
\begin{equation}\label{eq!33}
u \in (L_w^\times/L_w^{\times
  p})((p-1)/2)), \quad (\sigma-1)u = v,
\end{equation}
where $v$ is the image in $L_w^\times/L_w^{\times p}$ of a generator
of $V((p+5)/2)$.  Since $p \geq 19$, $(p+5)/2$ is less than $p-3$, and
thus $v \neq 0$. Thus the subspace of $L_w^\times/L_w^{\times p}$
satisfying the conditions~\eqref{eq!33} is two-dimensional, with
generators $u_1$ and $u_2$, where $u_1$ is the image of a generator of
$V((p-1)/2)$ with expansion $u_1 = 1 + \pi_K^{(p-1)/2} +
O(\pi_K^{(p+1)/2})$, and $u_2 \in (L_w^\times/L_w^{\times
  p})((p-1)/2)$ has expansion $u_2 = 1 + \pi_K^{(p+5)/2}\pi_L^{-3} +
O(\pi_K^{(p+5)/2})$.  Thus $u = u_1^{\alpha} u_2^{\beta}$ for some
$\alpha, \beta \in \Z/p\Z$. Expanding the binomial
series, we get 
\begin{equation}
u=1+\alpha \pi_K^{(p-1)/2} +\beta \pi_K^{(p+5)/2} \pi_L^{-3} +O(\pi_K^{p-1}). 
\end{equation}
We can now use   
Theorem~\ref{thm!proposition-let-d} to evaluate $u_4$ at $D$. Comparing
appropriate coefficients (note that $D$ is supported on $|T| \leq |\pi|$), 
we see that
  \begin{equation}
    \label{eq!34}
\frac{\alpha}{\beta} = \frac{D_4}{E D_1} =\frac{1}{E} \gamma(a,b,c) =
    \frac{1}{E}(q(a^ab^bc^c)^3 + abc B_{p-3}). 
  \end{equation}
  Now, we may replace $(a,b,c)$ by any $(a', b', c') \equiv (ta, tb,
  tc) \pmod{p}$, for $t \in \F_p^\times$. It is easily seen,
  using the property $q(xy) \equiv q(x) + q(y)$, that $\gamma(ta, tb,
  tc) \equiv t^3 \gamma(a,b,c)$. Thus, from~\eqref{eq!34}, we see that
  by varying $t$ appropriately we may ensure that $u$, and hence
  $c_w$, varies in $L_w^\times/L_w^{\times p}$, and, in particular,
  takes on nonzero values. Hence there exists a choice of $t$ such
  that the pairing is nontrivial for the curve $F_{a', b', c'}$.
  However, this curve is isomorphic to $F_{a,b,c}$, and hence the
  pairing must be nontrivial in that case as well.
\end{pfof}

\section{Shafarevich--Tate groups}\label{sec!shaf-tate-groups}
The proofs of Theorem~\ref{thm!free} and
Theorem~\ref{prop!exact-p-primary} follow from the computations of
the Cassels--Tate pairing by means of the following proposition.
\begin{proposition}\label{thm!prop-all-posit}
  For all positive integers $m$ and $n$, the restriction of the
  Cassels--Tate pairing induces a perfect pairing
  $$
 \Bigl(\Sh[\la^m]/(\la^n\Sh[\la^{n+m}])\Bigr) \times
 \Bigl(\Sh[\la^n]/(\la^m
  \Sh[\la^{n+m}])\Bigr) \ \longrightarrow \ \Q/\Z.
 $$
\end{proposition}
\medskip

Let $\Sh_{\div}$ denote the maximal divisible subgroup of $\Sh$, i.e.
$x \in \Sh_{\div}$ if and only if for every nonzero integer $n$ there
exists $y \in \Sh$ such that $x = n y$. Let $\Sh_{\red}$ denote the
quotient group $\Sh/\Sh_{\div}$. Note that:

\begin{lemma}\label{thm!lemma-sh_d-divis}
  $\Sh_{\div}$ is a divisible group in the usual sense that multiplication
by any nonzero $n\in\Z$ is surjective on it.
\end{lemma}

\begin{proof}
  The argument is standard: since $\Sh[m]$ is finite for all nonzero
  $m \in \Z$, the groups $N\Sh[Nm]$, $N>0$, stabilize for sufficiently
  large $N$. Thus for every $m$ there is an integer $N(m)$ such that
  if an element of $\Sh[m]$ is divisible by $N(m)$ it is infinitely
  divisible. Now if $x \in \Sh_{\div}[m]$ and $n>0$, choose $y \in
  \Sh[N(nm)nm]$ such that $N(nm)ny = x$. Then $y' = N(nm)y$ is in
  $\Sh_{\div}[nm]$ and $ny' = x$.
\end{proof}

Note that since $\zeta$ is an automorphism of $\Sh$ it preserves
$\Sh_{\div}$, and hence so does $\Z[\zeta]$. Furthermore, since
$\mclambda^{p-1}$ is a unit times $p$ in $\Z[\zeta]$, $\Sh_{\div}$ is
divisible by $\mclambda^{n}$ for any positive $n$.

\begin{lemma}\label{thm!lemma-exact-sequence}
  The exact sequence
  $$0 \longrightarrow \Sh_{\div} \longrightarrow \Sh \longrightarrow
  \Sh_{\red} \longrightarrow 0 $$
  induces by restriction an exact
  sequence
  $$0 \longrightarrow \Sh_{\div}[\la^n] \longrightarrow \Sh[\la^n]
  \longrightarrow \Sh_{\red}[\la^n] \longrightarrow 0 $$
  for any
  positive integer $n$.
\end{lemma}

\begin{proof}
  Only the surjectivity is in question.  Let $x \in
  \Sh_{\red}[\la^n]$. Lift $x$ to $y \in \Sh$.  Then $\la^n y = z \in
  \Sh_{\div}$. By Lemma~\ref{thm!lemma-sh_d-divis}, we can find $w \in
  \Sh_{\div}$ such that $\la^n w = z = \la^n y$. But then $y-w \in
  \Sh[\la^n]$ and $y-w$ reduces to $x$ in $\Sh_{\red}$.
\end{proof}


It is well known that $\Sh_{\red}[p^{\infty}]$ is a finite group and
that the Cassels--Tate pairing induces a perfect pairing
$$[\cdot \, \cdot]\colon  \Sh_{\red}[p^{\infty}] \ \times \ 
\Sh_{\red}[p^{\infty}] \ \longrightarrow \ \Q/\Z. $$
We now have the
following lemma:

\begin{lemma}\label{thm!lemma-annih-sh_r}
  The annihilator of $\Sh_{\red}[\la^m]$ with respect to the latter
  pairing equals $\la^m \Sh_{\red}[p^{\infty}]$, for all positive
  integers $m$.
\end{lemma}

\begin{proof}
  It is clear from the definition of the pairing given in
  \cite{mccallum!1988}, for example, and from the functoriality
  properties of the Weil pairing, that $[\zeta a, a'] = [a,
  \zeta^{-1} a']$. Hence, if $\hat\la = \zeta^{-1} -1$, then
  $\hat\la^m \Sh_{\red}[p^{\infty}]$ annihilates $\Sh_{\red}[\la^m]$.
  Since $\hat \mclambda/\mclambda$ is a unit in $\Z[\zeta]$, we have
  $\hat\la^m \Sh_{\red}[p^{\infty}] = \la^m \Sh_{\red}[p^{\infty}]$.
  So the kernel $H$ on the right factor of the restricted pairing
  $$\Sh_{\red}[\la^m] \times \Sh_{\red}[p^{\infty}] \longrightarrow
  \Q/\Z $$
  contains $\la^m \Sh_{\red}[p^{\infty}]$. Note that the
  kernel on the left factor of the latter pairing is trivial.
  Therefore, the cardinalities of $\Sh_{\red}[\la^m]$ and
  $\Sh_{\red}[p^{\infty}]/H$ are equal. But
  $$|\Sh_{\red}[p^{\infty}]| = |\Sh_{\red}[\la^m]| \ |\la^m
  \Sh_{\red}[p^{\infty}]|, $$
  hence $H=\la^m \Sh_{\red}[p^{\infty}]$.
\end{proof}


\begin{lemma}\label{thm!lemma-all-positive}
  For all positive integers $m$ and $n$, the restriction of the
  Cassels--Tate pairing induces a perfect pairing
  $$
\Bigl(\Sh_{\red}[\la^m]/(\la^n \Sh_{\red}[\la^{n+m}])\Bigr) \ \times \ 
  \Bigl(\Sh_{\red}[\la^n]/( \la^m \Sh_{\red}[\la^{n+m}])\Bigr) \
\longrightarrow
  \ \Q/\Z. $$
\end{lemma}

\begin{proof}
  By Lemma~\ref{thm!lemma-annih-sh_r}, the annihilator of
  $\Sh_{\red}[\la^m]$ in $\Sh_{\red}[\la^n]$ equals
  $$\la^m \Sh_{\red}[p^{\infty}] \ \cap \ \Sh_{\red}[\la^n] = \la^m
  \Sh_{\red}[\la^{n+m}], $$
  and the assertion follows.
\end{proof}


\begin{pfof}{Proposition~\ref{thm!prop-all-posit}}
  By Lemma~\ref{thm!lemma-all-positive}, it suffices to show that for
  all $m$ and $n$ the groups $\Sh[\la^m]/(\la^n \Sh[\la^{n+m}])$ and
  $\Sh_{\red}[\la^m]/(\la^n \Sh_{\red}[\la^{n+m}])$ are isomorphic.
  By Lemma~\ref{thm!lemma-exact-sequence}, we have a commutative
  diagram
  $$
  \begin{array}{lllllllll} 0 & \to &
    \Sh_{\div}[\la^{n+m}] & \longrightarrow & \Sh[\la^{n+m}] &
    \longrightarrow & \Sh_{\red}[\la^{n+m}] &
    \to & 0 \\[6pt]
    & & \kern0.6cm\Big\downarrow \alpha =\la^n & & \kern0.6cm\Big\downarrow
\beta=\la^n & &
    \kern0.6cm\Big\downarrow  \gamma=\la^n \\[6pt]
    0 & \to & \Sh_{\div}[\la^m] & \longrightarrow &
    \Sh[\la^m] & \longrightarrow & \Sh_{\red}[\la^m] & \to
    & 0
\end{array} $$
where the horizontal sequences are exact.  By the Snake Lemma, we get
an exact sequence
 \begin{align*}
0 &\to\Ker(\alpha) \to\Ker(\beta) \to\Ker(\gamma) \to \\[4pt]
 &\to\Coker(\alpha)
\to\Coker(\beta) \to\Coker(\gamma) \to 0.
 \end{align*}
By Lemma~\ref{thm!lemma-sh_d-divis}, we have $\Coker(\alpha)=0$,
hence $\Coker(\gamma)$ is isomorphic to $\Coker(\beta)$, and this
completes the proof.
\end{pfof}

\begin{pfof}{Theorem~\ref{thm!free}}
  By the structure theorem for torsion modules over Dedekind domains
  we have a $\Z[\zeta]$-module decomposition
  $$\Sh[\la^3] \iso (\Z[\zeta]/(\la))^{t_1} \ \oplus \ 
  (\Z[\zeta]/(\la^2))^{t_2} \ \oplus \ (\Z[\zeta]/(\la^3))^{t_3}, $$
  where $t_1$, $t_2$ and $t_3$ are nonnegative integers.  The
  computations in the previous section show that the pairing (obtained
  by restricting the Cassels--Tate pairing)
  $$\Sh[\la^2] \ \times \ \Sh[\la] \ \to \ \Q/\Z $$
  is trivial.  By
  Proposition~\ref{thm!prop-all-posit} (for $m=2$ and $n=1$), we get
  that the groups $\Sh[\la^2]/(\la\Sh[\la^3])$ and
  $\Sh[\la]/(\la^2\Sh[\la^3])$ are both trivial. But then
  $$(\Z[\zeta]/(\la))^{t_1} \oplus \ (\Z[\zeta]/(\la))^{t_2} \oplus
  (\Z[\zeta]/(\la))^{t_3} \iso \Sh[\la]=\la^2 \Sh[\la^3] \iso
  (\Z[\zeta]/\la)^{t_3} $$
  so $t_1=t_2=0$, which proves the claim.
\end{pfof}

\begin{pfof}{Theorem~\ref{prop!exact-p-primary}}
  Let
  $$\Sh[\la^4] \iso (\Z[\zeta]/(\la))^a \ \oplus \ 
  (\Z[\zeta]/(\la^2))^b \ \oplus \ (\Z[\zeta]/(\la^3))^c \ \oplus \ 
  (\Z[\zeta]/(\la^4))^d. $$
  If we show that $d=0$, then $\la^3$
  annihilates $\Sh[\la^4]$, therefore $\Sh[\la^4]=\Sh[\la^3]$. By
  induction, this implies
  $\Sh[p^{\infty}]=\Sh[\la^{\infty}]=\Sh[\la^3]$. So assume $d \ge 1$.
  Since the Cassels--Tate pairing on $\Sh[\la^3] \times \Sh[\la]$ is
  nontrivial, Proposition~\ref{thm!prop-all-posit} implies that
  $\Sh[\la^3]/(\la \Sh[\la^4])$ has dimension $\ge 2$ over $\F_p$. Now
  $$\la \Sh[\la^4] \iso (\Z[\zeta]/(\la))^b \ \oplus \ 
  (\Z[\zeta]/(\la^2))^c \ \oplus \ (\Z[\zeta]/(\la^3))^d. $$
  Counting
  $\F_p$-dimensions, we get $6-(b+2c+3d) \ge 2$, therefore $b+2c+3d
  \le 4$. This implies $d=1$ and $c=0$. Therefore,
  $$\Sh[\la^4] \iso (\Z[\zeta]/(\la))^a \ \oplus \ 
  (\Z[\zeta]/(\la^2))^b \ \oplus \ (\Z[\zeta]/(\la^4)). $$
  This
  implies that
  $$(\Z[\zeta]/(\la))^2=\la^2(\Z[\zeta]/(\la^3))^2 \iso
  \la^2\Sh[\la^3] \subseteq \la^2\Sh[\la^4] \iso \Z[\zeta]/(\la^2),
  $$
  a contradiction.
\end{pfof}


\section{Tame reduction}\label{sec!tame-reduction}

Although it is not strictly necessary for
Theorem~\ref{cor!diophant}, we take the opportunity to prove a
general lemma on tame reduction, since it clears up some confusion in
the literature. In \cite{lim!1995}, an attempt was made to improve the
result of \cite{mccallum!1988} on the existence of nontrivial
elements in $\Sh[\la]$ in the wild split case, under the additional
hypothesis that the Jacobian of the Fermat curve in question is
nonsimple.  However, as Lemma~\ref{thm!lemma-let-a} shows,
nonsimple Jacobian and wild split reduction over $\Z_p[\zeta]$
are incompatible properties, so  the
Mordell--Weil rank estimates given in the last section of
\cite{lim!1995} are incorrect. As far as we can tell, the problem lies
in the use of the function $q(x)$ which computes the reduction type
(see the introduction).  Here as well as in \cite{mccallum!1988}, $q$
is evaluated on triples $(a,b,c)$ of integers such that $0 < a, b,
a+b< p$ and $a+b+c=0$.  In \cite{lim!1995} however, $q$ is evaluated
on triples $(a,b,c)$ such that $0 < a, b, a+b < p$ and $a+b+c=p$.
While it does not make any difference which of the two types of
triples one chooses to define the curve $F_{a,b,c}$, it does make a
difference which type of triple one uses to evaluate $q$ and hence the
reduction type.  We have the following lemma:

\begin{lemma}\label{thm!lemma-let-a}
  Let $(a,b,c)$ be such that $J_{a,b,c}$ is nonsimple.  Then
  $F_{a,b,c}$ has tame reduction over $\Z_p[\zeta]$.
\end{lemma}


\begin{proof}
  By \cite{koblitz-rohrlich!1978}, $J_{a,b,c}$ is nonsimple if and
  only if $p \equiv 1 \pmod{3}$ and $F_{a,b,c}$ is isomorphic to
  $F_{1,r,-(r+1)}$, where $r^2+r+1=0$ in $\F_p$.  By definition of
  $q(x)$, it therefore suffices to show that $(r+1)^{(r+1)(p-1)} -
  r^{r(p-1)} \equiv 0 \pmod{p^{2}}$.  Since 6 divides $p-1$, it
  suffices to show that
  $$(r+1)^{6(r+1)}- r^{6r} \equiv 0 \pmod{p^2}. $$
  Let $k$ be an
  integer such that $r^2+r+1=pk$. Then $(r+1)^2=pk+r$. Therefore,
  $$(r+1)^6=(pk+r)^3 \equiv r^3+3 r^2 pk \pmod{p^2}. $$
  Hence
  $(r+1)^{6(r+1)} \equiv (r^3+3r^2pk)^{r+1} \equiv (r^{3(r+1)} + 3r^2
  pk(r+1)r^{3r}) \pmod{p^2}$.  Now note that $r^{3r}r^2 (r+1) \equiv -r
\pmod{p}$ since $r$ is a cube root  of unity modulo $p$, so that $3r^2
  pk (r+1)r^{3r} \equiv -3rpk \pmod{p^2}$.  Hence, $(r+1)^{6(r+1)}
  \equiv (r^{3r+3} - 3rpk) \pmod{p^2}$.  Therefore,
  $$(r+1)^{6(r+1)} - r^{6r} \equiv (r^{3r}(r^3-r^{3r})-3rpk)
  \pmod{p^2}. $$
  Since $r^3=pk(r-1)+1$, we get $r^{3r} \equiv
  (rpk(r-1)+1) \pmod{p^2}$, so $r^3-r^{3r} \equiv -pk(r-1)^2
  \pmod{p^2}$. Hence,
  $$(r+1)^{6(r+1)} - r^{6r} \equiv -pk (r^{3r}(r-1)^2 +3r) \pmod{p^2}.
  $$
  Since $r^{3r}(r-1)^2+3r \equiv 0 \pmod{p}$, this proves the
  proposition.
\end{proof}

 \begin{rem}\rm A less computational proof of Lemma~\ref{thm!lemma-let-a}
was suggested to us by Dino Lorenzini. The argument goes as follows: 
To show that the reduction is tame, it suffices, by work of McCallum, to
show that the degree of the 
minimum extension $M/K_w^{unr}$ such that $J_{a,b,c}$  
has good reduction over $M$ is prime to $p$. It is known that 
this minimum degree is at most $2g+1$. Now suppose 
$J_{a,b,c}$ is isogenous
to the product of two abelian varieties of smaller dimension. Then $M$ is 
the compositum of the corresponding minimum extensions for the factors.    
Each of the latter extensions has degree strictly less than $p$, so the
degree of their compositum is prime to $p$. 
 \end{rem}

\begin{pfof}{Theorem~\ref{cor!diophant}}
  By Lemma~\ref{thm!lemma-let-a}, the reduction is tame in this case.
  By Theorem~\ref{thm!nontrivial} and
  Proposition~\ref{thm!prop-all-posit}, the $\F_p$-dimension of
  $\Sh[\la]/(\la^3\Sh[\la^4])$ is $\ge 2$. In particular, the
  $\F_p$-dimension of $\Sh[\la]$ is $\ge 2$.  Since $p$ is regular,
  the results of Faddeev (\cite{faddeev:1961}) show that the Selmer
  group $S_{\la}$ is 3-dimensional over $\F_p$. On the other hand,
  Gross and Rohrlich (\cite{gross-rohrlich!1978}) have shown that the
  Mordell--Weil rank of $J_{7,1,-8}$ over $\Q$ is nonzero. Therefore,
  the rank equals 1 and $\Sh[\la]$ is 2-dimensional over $\F_p$. By
  Theorem~\ref{thm!free} it follows that $\Sh[\la^3]$ has rank 2 over
  $\Z[\zeta]/(\la^3)$. Theorem~\ref{prop!exact-p-primary} then
  implies that $\Sh[p^{\infty}] \iso (\Z[\zeta]/(\la^3))^2$.  The
  statement about quadratic points on $F_{7,1,-8}$ and on the Fermat
  curve $X^{19}+Y^{19}+Z^{19}=0$ follows immediately from Corollary~2.2
and Theorem~1.3 of \cite{tzermias:2002}.
\end{pfof}

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\bigskip
\noindent
William G. McCallum, \\
Department of Mathematics, \\
P.O. Box 210089, 617 N. Santa Rita, \\
The University of Arizona, \\
Tucson, AZ 85721-0089, USA \\
e-mail: wmc@math.arizona.edu

\medskip
\noindent
Pavlos Tzermias, \\
Department of Mathematics, \\
University of Tennessee, \\
Knoxville, TN 37996-1300, USA \\
e-mail: tzermias@math.utk.edu


\end{document}