Binary quadratic form: Difference between revisions

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{{Short description|Quadratic homogeneous polynomial in two variables}}

{{About|binary quadratic forms with [[integer]] coefficients|binary quadratic forms with other coefficients|quadratic form}}

{{more footnotes|date=July 2009}}

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Another instance of quadratic forms is [[Pell's equation]] <math>x^2-ny^2=1</math>.

Binary quadratic forms are closely related to ideals in quadratic fields,. thisThis allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant.

The classical [[theta function]] of 2 variables is <math> \sum_{(m,n)\in \mathbb{Z}^2} q^{m^2 + n^2}</math>, if <math>f(x,y)</math> is a positive [[definite quadratic form]] then <math> \sum_{(m,n)\in \mathbb{Z}^2} q^{f(m,n)}</math> is a theta function.

== Equivalence ==

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: <math> \begin{pmatrix} 3 & -4 \\ -2 & 3 \end{pmatrix} </math>

is an automorphism of the form <math>f = x^2 - 2y^2</math>. The automorphisms of a form formare a [[subgroup]] of <math>\mathrm{SL}_2(\mathbb{Z})</math>. When ''f'' is definite, the group is finite, and when ''f'' is indefinite, it is infinite and [[cyclic group|cyclic]].

==RepresentationsRepresentation==

We say that aA binary quadratic form <math>q(x,y)</math> '''represents''' an integer <math>n</math> if it is possible to find integers <math>x</math> and <math>y</math> satisfying the equation <math>n = fq(x,y).</math> Such an equation is a '''representation''' of {{math|''n''}} by {{math|''fq''}}.

=== Examples ===

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: <math>\begin{align}

(3 \cdot 17 + 4 \cdot 12, 2 \cdot 17 + 3 \cdot 12) &= (99,70),\\

(3 \cdot 99 + 4 \cdot 70, 2 \cdot 99 + 3 \cdot 70) &= (477577,408),\\

&\vdots \end{align}

</math>

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has determinant 1 and is an automorphism of ''f''. Acting on the representation <math>1 = f(x_1,y_1)</math> by this matrix yields the equivalent representation <math>1 = f(3x_1 + 4y_1, 2x_1 + 3 y_1)</math>. This is the recursion step in the process described above for generating infinitely many solutions to <math>1 = x^2 - 2y^2</math>. Iterating this matrix action, we find that the infinite set of representations of 1 by ''f'' that were determined above are all equivalent.

There are generally finitely many equivalence classes of representations of an integer ''n'' by forms of given nonzero discriminant <math>\Delta</math>. A complete set of representatives[[representative (mathematics)|representative]]s for these classes can be given in terms of ''reduced forms'' defined in the section below. When <math>\Delta < 0</math>, every representation is equivalent to a unique representation by a reduced form, so a complete set of representatives is given by the finitely many representations of ''n'' by reduced forms of discriminant <math>\Delta</math>. When <math>\Delta > 0</math>, Zagier proved that every representation of a positive integer ''n'' by a form of discriminant <math>\Delta</math> is equivalent to a unique representation <math>n = f(x,y)</math> in which ''f'' is reduced in Zagier's sense and <math>x > 0</math>, <math>y \geq 0</math>.<ref>{{harvnb|Zagier|1981|loc=}}</ref> The set of all such representations constitutes a complete set of representatives for equivalence classes of representations.

== Reduction and class numbers<!--'Class number (binary quadratic forms)' redirects here--> ==

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"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated{{citation needed|date=March 2017}}<!--<ref>{{harvnb|Shanks|1989}}</ref> full citation not in article yet --> than composition of forms, but arose first historically. We will consider such operations in a separate section below.

Composition means taking 2 quadratic forms of the same discriminant and combining them to create a quadratic form of the same discriminant, itas isfollows afrom generalization of the 2-square[[Brahmagupta's identity <math>\left(a^2 + b^2\right)\left(c^2 + d^2\right) = \left(ac-bd\right)^2 + \left(ad+bc\right)^2</math>]].

=== Composing forms and classes ===

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The general theory of quadratic forms was initiated by [[Lagrange]] in 1775 in his ''[[List of important publications in mathematics#Recherches d'Arithmétique|Recherches d'Arithmétique]]''. Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms."<ref>{{harvnb|Weil|2001|loc=p.318}}</ref> 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".<ref>{{harvnb|Weil|2001|loc=p.317}}</ref> Lagrange showed that there are finitely many equivalence classes of given discriminant, thereby defining for the first time an arithmetic [[Ideal class group|class number]]. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of [[infrastructure (number theory)|infrastructure]]. In 1798, [[Adrien-Marie Legendre|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 [[Carl Friedrich Gauss|Gauss]] in Section V of ''[[List of important publications in mathematics#Disquisitiones Arithmeticae|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 (mathematics)|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 2''b'' in place of ''b''; the modern convention allowing the coefficient of ''xy'' to be odd is due to [[Gotthold Eisenstein|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 field]]s. 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 Section 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 ''[[List of important publications in mathematics#Vorlesungen über Zahlentheorie|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|Hilbert's]] ''[[List of important publications in mathematics#Zahlbericht|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]].

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*[[Fermat's theorem on sums of two squares]]

* [[Legendre symbol]]

* [[Brahmagupta's identity]]

==Notes==

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* 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

* David A. Cox, ''Primes of the form <math>x^2 + yny^2</math>, Fermat, class field theory, and complex multiplication''

* {{Citation

| last=Cohen