Hasse principle

Let $V$ be an algebraic variety defined over a field $K$. By$V(K)$ we denote the set of points on $V$ defined over $K$. Let$\overline{K}$ be an algebraic closure of $K$. For a valuation $\nu$ of$K$, we write $K_{\nu }$ for the completion of $K$ at $\nu$. Inthis case, we can also consider $V$ defined over $K_{\nu }$ andtalk about $V(K_{\nu })$.

Definition 1.

1. If $V(K)$ is not empty we say that $V$ is soluble in$K$.

2. If $V(K_{\nu })$ is not empty then we say that $V$ islocally soluble at $\nu$.

3. If $V$ is locally soluble for all $\nu$ then we say that $V$satisfies the Hasse condition, or we say that $V/K$ iseverywhere locally soluble.

The Hasse Principle is the idea (or desire) that aneverywhere locally soluble variety $V$ must have a rational point,i.e. a point defined over $K$. Unfortunately this is not true,there are examples of varieties that satisfy the Hasse conditionbut have no rational points.

Example: A quadric (of any dimension) satisfies the Hassecondition. This was proved by Minkowski for quadrics over$\mathbb{Q}$ and by Hasse for quadrics over a number field.