Let be an algebraic variety defined over a field . By we denote the set of points on defined over . Let be an algebraic closure of . For a valuation of , we write for the completion of at . In this case, we can also consider defined over and talk about .
If is not empty we say that is soluble in .
If is not empty then we say that is locally soluble at .
If is locally soluble for all then we say that satisfies the Hasse condition, or we say that is everywhere locally soluble.
The Hasse Principle is the idea (or desire) that an everywhere locally soluble variety must have a rational point, i.e. a point defined over . 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 and by Hasse for quadrics over a number field.
- 1 Swinnerton-Dyer, Diophantine Equations: Progress and Problems, http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdfonline notes.
|Date of creation||2013-03-22 13:50:39|
|Last modified on||2013-03-22 13:50:39|
|Last modified by||alozano (2414)|