Hasse principle

From Wikipedia, the free encyclopedia

In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the assertion that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p.

Contents

[edit] Quadratic forms

The Hasse-Minkowski theorem states that the local-global principle holds for 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.

[edit] Forms of higher degree

For algebraic forms of degree greater than two the situation is more complicated.

Counterexamples show that the Hasse-Minkowski theorem is not extendable to forms of degree 3 or 10n + 5, where n is a non-negative integer. In the case of degree three Ernst S. Selmer provided the cubic form 3x3+4y3+5z3, which represents 0 over all p-adic fields, but not over Q.[1] For forms of degree congruent to 5 modulo 10, Fujiwara and Sudo gave counterexamples.[2]

On the other hand, Birch's theorem and Brauer's theorem show 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 obeys the Hasse principle.

[edit] Cubic forms

D. J. Lewis showed that every cubic form over a p-adic field represents zero if it has more than 10 variables.[3] Davenport demonstrated that every cubic form over the integers in at least 16 variables represents 0.[4] Hence the local-global principle holds for cubic forms over the rationals in at least 16 variables. Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0,[5] thus, in connexion with Lewis' result, 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.[6] However, Hooley showed that the Hasse principle holds for non-singular cubic forms over the rational numbers in at least nine variables.[7] 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; it is only recently that it has been shown that this setting isn't the complete story (Alexei Skorobogatov, 1999).

[edit] See also

[edit] References

  1. ^ Ernst S. Selmer, The Diophantine equation ax3+by3+cz3=0, Acta Mathematica, 85, pages 203-362, (1957)
  2. ^ M. Fujiwara, M. Sudo, Some forms of odd degree for which the Hasse principle fails, Pacific Journal of Mathematics, 67 (1976), No. 1, pages 161-169
  3. ^ D. J. Lewis, Cubic homogeneous polynomials over p-adic number fields, Annals of Mathematics, 56, pages 473-478, (1952)
  4. ^ H. Davenport, Cubic forms in sixteen variables, Proceedings of the Royal Society London Series A, 272, pages 285-303, (1963)
  5. ^ D. R. Heath-Brown, Cubic forms in ten variables, Proceedings of the London Mathematical Society, 47(3), pages 225-257, (1983)
  6. ^ L. J. Mordell, A remark on indeterminate equations in several variables, Journal of the London Mathematical Society, 12, pages 127-129, (1937)
  7. ^ C. Hooley, On nonary cubic forms, Journal für die reine und angewandte Mathematik, 386, pages 32-98, (1988)

[edit] External links

Retrieved from "http://en.wikipedia.org../../../h/a/s/Hasse_principle.html"
In other languages