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Hasse principle (Definition)

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. 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.

Bibliography

1
Swinnerton-Dyer, Diophantine Equations: Progress and Problems, online notes.



"Hasse principle" is owned by alozano.
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See Also: Hasse-Minkowski theorem

Also defines:  Hasse principle, Hasse condition, locally soluble
Keywords:  Hasse principle
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Cross-references: number field, dimension, rational, completion, valuation, algebraic closure, points, field, variety, algebraic
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This is version 4 of Hasse principle, born on 2003-08-12, modified 2003-08-14.
Object id is 4581, canonical name is HassePrinciple.
Accessed 5804 times total.

Classification:
AMS MSC: 14G05 (Algebraic geometry :: Arithmetic problems. Diophantine geometry :: Rational points)
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