Binary quadratic form

In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variables

q(x,y)=ax^{2}+bxy+cy^{2},\,

where a, b, c are the coefficients. Properties of binary quadratic forms depend in an essential way on the nature of the coefficients, which may be real numbers, rational numbers, or in the most delicate case, integers. Arithmetical aspects of the theory of binary quadratic forms are related to the arithmetic of quadratic fields and have been much studied, notably, by Gauss in Section V of Disquisitiones Arithmeticae. The theory of binary quadratic forms has been extended in two directions: general number fields and quadratic forms in n variables.

Brief history

The first works on binary quadratic forms concern the problem of representations of integers by particular binary quadratic forms. Pell's equation is the prime example, and was already considered by the Indian mathematician Brahmagupta in the 7th century CE. In the 17th century, Fermat made several observations concerning representations of integers by particular quadratic forms, including one that later became Fermat's theorem on sums of two squares. Euler provided the first proofs of Fermat's observations and added some new conjectures about representations by specific forms, without proof.

The general theory of quadratic forms was initiated by Lagrange in 1775 in his Recherches d'Arithmétique. Lagrange was the first to realize that proofs of Euler's observations required moving from considering a single form to considering the set of all forms at once. He was the first to recognize the importance of the discriminant and to define the essential notions of equivalence and reduction, which, according to Weil, have "dominated the whole subject of quadratic forms ever since". He showed that there are finitely many equivalence classes of given discriminant, thereby defining for the first time an arithmetic class number. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure. In 1798, Legendre published Essai sur la théorie des nombres, which summarized the work of Euler and Lagrange and added some of his own contributions, including the first glimpse of a composition operation on forms.

The theory was vastly extended and refined by Gauss in Section V of Disquisitiones Arithmeticae. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. He replaced Lagrange's equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a group under the composition operation. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. (Gauss and many subsequent authors wrote 2b in place of b; the modern convention allowing the coefficient of xy to be odd is due to Eisenstein).

These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general number fields. But the impact was not immediate. Section V of Disquisitiones contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader. Combined, the novelty and complexity made Sectoin V notoriously difficult. Dirichlet published simplifications of the theory that made it accessible to a broader audience. The culmination of this work is his text Vorlesungen über Zahlentheorie. The third edition of this work includes two supplements by Dedekind. Supplement XI introduces ring theory, and from then on, especially after the 1897 publication of Hilbert's Zahlbericht, the theory of binary quadratic forms lost its preeminent position in algebraic number theory and became overshadowed by the more general theory of algebraic number fields.

Even so, work on binary quadratic forms continues to the present, including the Shank's introduction of infrastructure, Zagier's introduction of a new reduction algorithm, and Bhargava's reinterpretation of composition through Bhargava cubes.

Main questions

A classical question in the theory of integral quadratic forms (those with integer coefficients) is the representation problem: describe the set of numbers represented by a given quadratic form q. If the number of representations is finite then a further question is to give a closed formula for this number. The notion of equivalence of quadratic forms and the related reduction theory are the principal tools in addressing these questions.

Two integral forms are called equivalent if there exists an invertible integral linear change of variables that transforms the first form into the second. This defines an equivalence relation on the set of integral quadratic forms, whose elements are called classes of quadratic forms. Equivalent forms necessarily have the same discriminant

D(f)=b^{2}-4ac,\quad D(f)\equiv 0,1{\pmod {4}}.

Gauss proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the class number of discriminant D. He described an algorithm, called reduction, for constructing a canonical representative in each class, the reduced form, whose coefficients are the smallest in a suitable sense. One of the deepest discoveries of Gauss was the existence of a natural composition law on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite abelian group called the class group of discriminant D. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each genus consists of finitely many classes of forms.

An integral binary quadratic form is called primitive if a, b, and c have no common factor. If a form's discriminant is a fundamental discriminant, then the form is primitive.^[1]

From a modern perspective, the class group of a fundamental discriminant D is isomorphic to the narrow class group of the quadratic field $\mathbf {Q} ({\sqrt {D}})$ of discriminant D.^[2] For negative D, the narrow class group is the same as the ideal class group, but for positive D it may be twice as big.

Notes

^ Cohen 1993, §5.2
^ Fröhlich & Taylor 1993, Theorem 58

References

Johannes Buchmann, Ulrich Vollmer: Binary Quadratic Forms, Springer, Berlin 2007, ISBN 3-540-46367-4
Duncan A. Buell: Binary Quadratic Forms, Springer, New York 1989
Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206
Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934

External links

Peter Luschny, Positive numbers represented by a binary quadratic form
A.V.Malyshev (2001) [1994], "Binary quadratic form", Encyclopedia of Mathematics, EMS Press

[1] Cohen 1993, §5.2

[2] Fröhlich & Taylor 1993, Theorem 58

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