Hasse principle

In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. Mathematically, this is handled by examining the equation in the completions of the rational numbers: the real numbers and the "p"-adic numbers. A more formal version of the Hasse principle states that certain types of equation have a rational solution if and only if they have a solution in the real numbers "and" in the "p"-adic numbers for each prime "p".

Intuition

Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a "p"-adic solution, as the rationals embed in the reals and "p"-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and "p"-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution?

One can ask this for other rings or fields: integers, for instance, or number fields. For number fields, rather than reals and "p"-adics, one uses complex embeddings and $mathfrak\; p$-adics, for prime ideals $mathfrak\; p$.

Forms representing 0

Quadratic forms

The Hasse-Minkowski theorem states that the local-global principle holds for the problem of representing 0 by quadratic forms over the rational numbers (which is Minkowski's result); and more generally over any number field (as proved by Hasse), when one uses all the appropriate local field necessary conditions. Hasse's theorem on cyclic extensions states that the local-global principle applies to the condition of being a relative norm for a cyclic extension of number fields.

Cubic forms

A counterexample by Ernst S. Selmer shows that the Hasse-Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3"x"³ + 4"y"³ + 5"z"³ = 0 has a solution in real numbers, and in all p-adic fields, but it has no solution in which "x", "y", and "z" are all rational numbers. [cite journal | author=Ernst S. Selmer | title=The Diophantine equation "ax"³+"by"³+"cz"³=0 | journal=Acta Mathematica | volume=85 | pages=203–362 | year=1957 | doi=10.1007/BF02395746 ]

Heath-Brown showed [cite journal | author=D.R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in 14 variables | journal=Invent. Math. | volume=170 | pages=199–230 | year=2007 | doi=10.1007/s00222-007-0062-1] that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport [cite journal | author=H. Davenport | title=Cubic forms in sixteen variables | journal=Proceedings of the Royal Society of London Series A | volume=272 | pages=285–303 | year=1963 | doi=10.1098/rspa.1963.0054 ] . Hence the local-global principle holds trivially for cubic forms over the rationals in at least 14 variables.

If we confine ourselves to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0, [cite journal | author=D. R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in ten variables | journal=Proceedings of the London Mathematical Society (3)| volume=47 | pages=225–257 | year=1983 | doi=10.1112/plms/s3-47.2.225 ] thus trivially establishing the Hasse principle for this class of forms. It is known that Heath-Brown's result is best possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that don't represent zero. [cite journal | author=L. J. Mordell | authorlink=Louis Mordell | title=A remark on indeterminate equations in several variables | journal=Journal of the London Mathematical Society | volume=12 |pages=127–129 | year=1937 ] However, Hooley showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables. [cite journal | author=C. Hooley | authorlink=Christopher Hooley | title=On nonary cubic forms | journal=J. Für die reine und angewandte Mathematik | volume=386 |pages=32–98 | year=1988 ] Davenport, Heath-Brown and Hooley all used the Hardy-Littlewood circle method in their proofs. According to an idea of Manin, the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the Brauer group; this is the Brauer-Manin obstruction, which accounts completely for the failure of the Hasse principle for some classes of variety. However, Skorobogatov has shown that this is not the complete story [cite journal | author=Alexei N. Skorobogatov | title=Beyond the Manin obstruction | journal=Invent. Math. | volume=135 | issue=2 | pages=399–424 | year=1999 | doi=10.1007/s002220050291 ] .

Forms of higher degree

Counterexamples by Fujiwara and Sudo show that the Hasse-Minkowski theorem is not extensible to forms of degree 10"n" + 5, where "n" is a non-negative integer. [cite journal | author=M. Fujiwara | authorlink=Masahiko Fujiwara | coauthors=M. Sudo | title=Some forms of odd degree for which the Hasse principle fails | journal=Pacific Journal of Mathematics | volume=67 | year=1976 | issue=1 | pages=161–169]

On the other hand, Birch's theorem shows that if "d" is any odd natural number, then there is a number "N"("d") such that any form of degree "d" in more than "N"("d") variables represents 0: the Hasse principle holds trivially.

ee also

* Local analysis
* Hasse condition

Notes

References

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External links

* [http://planetmath.org/encyclopedia/HassePrinciple.html PlanetMath article]
* Swinnerton-Dyer, "Diophantine Equations: Progress and Problems", [http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf online notes]