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.

Title	Hasse principle
Canonical name	HassePrinciple
Date of creation	2013-03-22 13:50:39
Last modified on	2013-03-22 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