Local-global principle (number theory)

The local-global principle refers to various principles in number theory with which in some cases the solvability of the original equation can be inferred from the solvability of Diophantine equations modulo all prime numbers.

Reduction of Diophantine equations and Chinese remainder theorem

A Diophantine equation is an equation of form

${\displaystyle f(x_{1},x_{2},x_{3},\dotsc ,x_{n})=0}$,

where is a given polynomial function with integer coefficients and only integer solutions are sought. ${\displaystyle f}$

If is an integer solution, then obviously for every integer the remainder classes are also modulo ${\displaystyle (x_{1},x_{2},x_{3},\ldots ,x_{n})}$ ${\displaystyle p}$ ${\displaystyle p}$

${\displaystyle \left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right],\ldots ,\left[x_{n}\right]\in \mathbb {Z} /p\mathbb {Z} }$

Solutions of the modulo "reduced" equation ${\displaystyle p}$

${\displaystyle f(\left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right],\dotsc ,\left[x_{n}\right])=\left[0\right]\in \mathbb {Z} /p\mathbb {Z} .}$

It is even an integral solution if and only if the reduced equation modulo holds for all prime numbers . With the help of the Chinese remainder theorem one also gets that every natural number is solvable if and only if there is a solution for every prime and every natural number . ${\displaystyle (x_{1},x_{2},x_{3},\ldots ,x_{n})}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\textstyle f\equiv 0\mod m}$ ${\displaystyle m\in \mathbb {N} }$ ${\textstyle f\equiv 0\mod p^{k}}$ ${\displaystyle p}$ ${\displaystyle k\in \mathbb {N} }$

In general, however, it is not true that the solvability of the equations modulo every prime number or even prime power also implies solvability in whole numbers. For example, the equation has

${\displaystyle (x^{2}-2)(x^{2}-3)(x^{2}-6)=0}$

no integer solution, but it can be solved modulo every prime number because at least one of the numbers is always a quadratic remainder . ${\displaystyle p}$ ${\displaystyle 2,3,6}$

Today, local-global principles are usually formulated using the completion of the rational numbers , i.e. the p-adic numbers (for all prime numbers ) and the real numbers . One then says that an equation , where is a polynomial function with rational coefficients, satisfies the local-global principle if the solvability of the original equation in follows from the solvability in and in for all prime numbers . Poonen and Voloch have shown that the brewer-manin obstruction is the only obstruction for the local-global principle. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f(x_{1},x_{2},x_{3},\dotsc ,x_{n})=0}$ ${\displaystyle f}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Q} }$

Local-global principle for quadratic forms (Hasse-Minkowski theorem)

The Hasse-Minkowski theorem says that the local-global principle for the problem of representing zero by a given square form over the field of rational numbers (that is the original theorem of Minkowski ) or more generally over a number field (that proved Hasse 1921 in his dissertation) applies.

If so

${\displaystyle f(x_{1},x_{2},x_{3},\dotsc ,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}}}$

is a quadratic form with coefficients in a number field (for example the field of rational numbers ), then the existence of nontrivial zeros in and in all p-adic completions already follows the existence of a nontrivial zero in the number field. ${\displaystyle a_{ij}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

This principle cannot be generalized to cubic polynomials . The equation has nontrivial solutions in and in all but not in ( Ernst Sejersted Selmer ). The Fermat equation also has solutions in all and , but not in the rational numbers. ${\displaystyle 3x^{3}+4y^{3}+5z^{3}=0}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle x^{n}+y^{n}=z^{n}}$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {R} }$

The Hasse principle for algebraic groups is closely related to the local-global principle for quadratic forms. This means that for a simply connected algebraic group over a number field one has an isomorphism of the Galois cohomology ${\displaystyle k}$

${\displaystyle H^{1}(k,G)\to \Pi _{k_{s}}H^{1}(k_{s},G)}$

with all completions of iterating over. This principle was used to prove the Weil conjecture for Tamagawa numbers and the strong approximation theorem in algebraic groups. ${\displaystyle k_{s}}$ ${\displaystyle k}$

literature

• Gerhard Frey : Elementary number theory. Vieweg studies: Basic course in mathematics, 56. Friedr. Vieweg & Sohn, Braunschweig, 1984. ISBN 3-528-07256-3

Individual evidence

1. ↑ Jürgen Neukirch : Algebraic Number Theory . Unchanged reprint of the first edition. Springer, Berlin, Heidelberg, New York 2007, ISBN 3-540-37547-3 , pp. 108-109 .
2. ↑ Martin Kneser : Hasse principle for H ¹ of simply connected groups. 1966 Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 159-163 Amer. Math. Soc., Providence, RI