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 $. In this case, we can also consider $V$ defined over ${K}_{\nu}$ and talk 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 $V$ is locally 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 $V/K$ 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 SwinnertonDyer, Diophantine Equations^{}: Progress and Problems, http://swc.math.arizona.edu/notes/files/DLSSwDyer1.pdfonline notes.
Title  Hasse principle 

Canonical name  HassePrinciple 
Date of creation  20130322 13:50:39 
Last modified on  20130322 13:50:39 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  7 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 14G05 
Related topic  HasseMinkowskiTheorem 
Defines  Hasse principle 
Defines  Hasse condition 
Defines  locally soluble 