The Local-to-Global Principle of Hasse and Minkowski

Section 16.2.3 might have convinced you that $\mathbf{Q}_N$ is a bizarre pathology. In fact, $\mathbf{Q}_N$ 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.

Theorem 16.2.14 (Hasse-Minkowski)   The quadratic equation

$$a_1x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2 = 0,$$ (16.1)

with $a_i\in\mathbf{Q}^{\times }$, has a nontrivial solution with $x_1,\ldots, x_n$ in $\mathbf{Q}$ if and only if (16.2.1) has a solution in $\mathbf{R}$ and in $\mathbf{Q}_p$ for all primes $p$.

This theorem is very useful in practice because the $p$-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

$$3x^3 + 4y^3 + 5z^3 = 0$$

has a solution other than $(0,0,0)$ in $\mathbf{R}$ and in $\mathbf{Q}_p$ for all primes $p$ but has no solution other than $(0,0,0)$ in  $\mathbf{Q}$ (for a proof see [Cas91, §18]).

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

$$ax^3 + by^3 + cz^3=0$$

has a nontrivial solution in  $\mathbf{Q}$.

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.