# 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 $\bar{K}$ be an algebraic closure of $K$. For a valuation $\nu$ of $K$, we write $K_{\nu}$ for the completion of $K$ at $\nu$. In this case, we can also consider $V$ defined over $K_{\nu}$ and talk about $V(K_{\nu})$.

###### Definition 1.
1. 1.

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

2. 2.

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

3. 3.

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

The Hasse Principle is the idea (or desire) that an everywhere 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 condition but have no rational points.

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

## References

• 1 Swinnerton-Dyer, Diophantine Equations: Progress and Problems, http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdfonline notes.
Title Hasse principle HassePrinciple 2013-03-22 13:50:39 2013-03-22 13:50:39 alozano (2414) alozano (2414) 7 alozano (2414) Definition msc 14G05 HasseMinkowskiTheorem Hasse principle Hasse condition locally soluble