##

The Local-to-Global Principle of Hasse and Minkowski

Section 16.2.3 might have convinced you that
is a
bizarre pathology. In fact,
is omnipresent in number theory,
as the following two fundamental examples illustrate.
In the statement of the following theorem, a
to a homogeneous polynomial equation is a solution where not all
indeterminates are 0.

This theorem is very useful in practice because the
-adic condition turns out to be easy to check. For more details,
including a complete proof, see
[Ser73, IV.3.2].
The analogue of Theorem 16.2.14
for cubic equations is false.
For example, Selmer proved that the cubic

has a solution other than in
and in
for all primes
but has no solution other than in
(for a proof
see [Cas91, §18]).

**Open Problem. **
Give an algorithm that decides whether or not a cubic

has a nontrivial solution in
.

This open problem is closely related to the Birch and Swinnerton-Dyer
Conjecture
for elliptic curves.
The truth of the conjecture would
follow if we knew that ``Shafarevich-Tate Groups''
of elliptic curves were finite.

William Stein
2004-05-06