In mathematics, Helmut Hasse's localglobal 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. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the padic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the padic numbers for each prime p.
Contents

Intuition 1

Forms representing 0 2

Quadratic forms 2.1

Cubic forms 2.2

Forms of higher degree 2.3

Albert–Brauer–Hasse–Noether theorem 3

Hasse principle for algebraic groups 4

See also 5

Notes 6

References 7

External links 8
Intuition
Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a padic solution, as the rationals embed in the reals and padics: 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 padics 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 padics, one uses complex embeddings and \mathfrak padics, for prime ideals \mathfrak p.
Forms representing 0
Quadratic forms
The Hasse–Minkowski theorem states that the localglobal 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 localglobal 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 3x^{3} + 4y^{3} + 5z^{3} = 0 has a solution in real numbers, and in all padic fields, but it has no nontrivial solution in which x, y, and z are all rational numbers.^{[1]}
Roger HeathBrown showed^{[2]} that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport.^{[3]} Hence the localglobal principle holds trivially for cubic forms over the rationals in at least 14 variables.
If we confine ourselves to nonsingular forms, one can do better than this: HeathBrown proved that every nonsingular cubic form over the rational numbers in at least 10 variables represents 0,^{[4]} thus trivially establishing the Hasse principle for this class of forms. It is known that HeathBrown's result is best possible in the sense that there exist nonsingular cubic forms over the rationals in 9 variables that don't represent zero.^{[5]} However, Hooley showed that the Hasse principle holds for the representation of 0 by nonsingular cubic forms over the rational numbers in at least nine variables.^{[6]} Davenport, HeathBrown 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.^{[7]}
Forms of higher degree
Counterexamples by Fujiwara and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a nonnegative integer.^{[8]}
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.
Albert–Brauer–Hasse–Noether theorem
The Albert–Brauer–Hasse–Noether theorem establishes a localglobal principle for the splitting of a central simple algebra A over an algebraic number field K. It states that if A splits over every completion K_{v} then it is isomorphic to a matrix algebra over K.
Hasse principle for algebraic groups
The Hasse principle for algebraic groups states that if G is a simplyconnected algebraic group defined over the global field k then the map from

H^1(k,G)\rightarrow\prod_s H^1(k_s,G)
is injective, where the product is over all places s of k.
The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.
Kneser (1966) and several others verified the Hasse principle by casebycase proofs for each group. The last case was the group E_{8} which was only completed by Chernousov (1989) many years after the other cases.
The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.
See also
Notes

^ Ernst S. Selmer (1957). "The Diophantine equation ax^{3} + by^{3} + cz^{3} = 0". Acta Mathematica 85: 203–362.

^

^ H. Davenport (1963). "Cubic forms in sixteen variables".

^

^

^

^ Alexei N. Skorobogatov (1999). "Beyond the Manin obstruction". Invent. Math. 135 (2): 399–424.

^
References

Chernousov, V. I. (1989), "The Hasse principle for groups of type E8", Soviet Math. Dokl. 39: 592–596,

Kneser, Martin (1966), "Hasse principle for H¹ of simply connected groups", Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.:


Alexei Skorobogatov (2001). Torsors and rational points. Cambridge Tracts in Mathematics 144. Cambridge: Cambridge Univ. Press. pp. 1–7,112.
External links

Hazewinkel, Michiel, ed. (2001), "Hasse principle",

PlanetMath article

SwinnertonDyer, Diophantine Equations: Progress and Problems, online notes
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