The Diophantine equation $x^2 - 34y^2 = -1$ has no integer solutions,
even though it has solutions in ${\bf Z}_p$ for all $p$ (including
$p = \infty$ if we understand "${\bf Z}_\infty$" as $\bf R$).
This is the first example of the failure of the Hasse principle for
the minus case of the Fermat-Pell equation $x^2 - \Delta y^2 = \pm 1$
(with $\Delta $ a fixed positive integer that is not a square), or
equivalently for the existence of units of norm $-1$ in ${\bf Z}[\sqrt{\Delta}]$.
It can also be regarded as the first example of a nontrivial element of the
"Tate-Šafarevič group" for the torus $x^2 - \Delta y^2 = +1$
(since $x^2 - \Delta y^2 = -1$ is a principal homogeneous space
for that torus).

[NB the equation $x^2 - 34y^2 = -1$ does have *rational* solutions,
such as $(x,y) = (5/3,1/3)$. Indeed Minkowski already showed that
a quadratic equation in any number of variables has a rational solution
**iff** it has a solution in each ${\bf Q}_p$ and in ${\bf R}$;
Hasse generalized this from ${\bf Q}$ to an arbitrary number field.]

*[Added later:]* In general
$x^2 - \Delta y^2 = -1$ has solutions in every ${\bf Z}_p$
**iff** $\Delta$ is either a product of primes congruent to $1 \bmod 4$
or twice such a product; equivalently, **iff** $\Delta$ is the sum of
two coprime squares. If such $\Delta$ is of the form $n^2 \pm 2$
then $(n + \sqrt\Delta)^2 / 2$ is a unit of norm $+1$, and is fundamental
unless $\Delta=2$. This accounts for infinitely many examples, including
the first two, $\Delta = 34 = 5^2 + 3^2 = 6^2 - 2$ and
$\Delta = 146 = 11^2 + 5^2 = 12^2 + 2$ (see
OEIS sequence A031398).
The infinitude may be shown with a polynomial identity such as
$$
(2t^2+2t+1)^2 + (2t+1)^2 = (2t^2+2t+2)^2 - 2
$$
which recovers $\Delta = 34$ for $t=1$. It's then a natural question
to ask: as $M \rightarrow \infty$, among those positive $\Delta < M$ that are
sums of two coprime squares, for what fraction does
$x^2 - \Delta y^2 = -1$ have solutions? I guess that it is conjectured,
but not known, that there is a positive limit strictly smaller than $1$.

rationalselliptic curves with the same $L$-series are isogenous. Another possibility is abelian varieties with a finite Tate-Shafarevich group of nonsquare size; for ell. curves the size must be a square, but in higher dimensions that is false. See mathoverflow.net/questions/9924/…. $\endgroup$ – KConrad Mar 18 '12 at 17:39