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Hasse principle
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(Definition)
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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
- If $V(K)$ is not empty we say that $V$ is soluble in $K$
- If $V(K_{\nu})$ is not empty then we say that $V$ is locally soluble at $\nu$
- 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.
- 1
- Swinnerton-Dyer, Diophantine Equations: Progress and Problems, online notes.
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"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
There are 4 references to this entry.
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 7594 times total.
Classification:
| AMS MSC: | 14G05 (Algebraic geometry :: Arithmetic problems. Diophantine geometry :: Rational points) |
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Pending Errata and Addenda
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