Hasse principle
Encyclopedia
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
. This is handled by examining the equation in the completions
of the rational number
s: the real number
s and the p-adic 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 number
s and in the p-adic numbers for each prime p.
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
-adics, for prime ideal
s
.
states that the local-global principle holds for the problem of representing 0 by quadratic form
s over the rational number
s (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.
Roger Heath-Brown
showed that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport
. 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, 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. 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. 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.
and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
by using the Chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
to piece together solutions modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
powers of each different prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. This is handled by examining the equation in the completions
Completion (ring theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s: the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s and the p-adic numbers
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
they have a solution in the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s 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
-adics, for prime idealPrime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s
.Quadratic forms
The Hasse–Minkowski theoremHasse–Minkowski theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field...
states that the local-global principle holds for the problem of representing 0 by quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s over the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s (which is Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
's result); and more generally over any number field (as proved by Hasse), when one uses all the appropriate local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
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 3x3 + 4y3 + 5z3 = 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.Roger Heath-Brown
Roger Heath-Brown
David Rodney "Roger" Heath-Brown F.R.S. , is a British mathematician working in the field of analytic number theory. He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker...
showed that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport
Harold Davenport
Harold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...
. 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, 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. However, Hooley
Christopher Hooley
Christopher Hooley FLSW FRS is a British mathematician, emeritus professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in 1973. He was elected a Fellow of the Royal Society in 1983...
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. 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
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
; this is the Brauer–Manin obstruction, which accounts completely for the failure of the Hasse principle for some classes of variety. However, Skorobogatov
Alexei Skorobogatov
Alexei Nikolaievich Skorobogatov is a mathematician and currently a Professor in Pure Mathematics at Imperial College London. He specialises in algebraic geometry....
has shown that this is not the complete story.
Forms of higher degree
Counterexamples by FujiwaraMasahiko Fujiwara
Masahiko Fujiwara is a Japanese mathematician, who is best known as an essayist....
and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer.
On the other hand, Birch's theorem
Birch's theorem
In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.-Statement of 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.
External links
- PlanetMath article
- Swinnerton-Dyer, Diophantine Equations: Progress and Problems, online notes