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

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

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.

Forms with integer coefficients

Representations

Diophantus considered whether, for an odd integer ${\displaystyle n}$ , it is possible to find integers ${\displaystyle x}$  and ${\displaystyle y}$  for which ${\displaystyle n=x^{2}+y^{2}}$ .^[1] When ${\displaystyle n=65}$ , we have

${\displaystyle {\begin{aligned}65&=1^{2}+8^{2},\\65&=4^{2}+7^{2},\end{aligned}}}$ 

so we find pairs ${\displaystyle (x,y)=(1,8){\text{ and }}(4,7)}$  that do the trick. We obtain more pairs that work by switching the values of ${\displaystyle x}$  and ${\displaystyle y}$  and/or by changing the sign of one or both of ${\displaystyle x}$  and ${\displaystyle y}$ . In all, there are sixteen different solution pairs. On the other hand, when ${\displaystyle n=3}$ , the equation

${\displaystyle 3=x^{2}+y^{2}}$ 

does not have integer solutions. To see why, we note that ${\displaystyle x^{2}\geq 4}$  unless ${\displaystyle x=-1,0}$  or ${\displaystyle 1}$ . Thus, ${\displaystyle x^{2}+y^{2}}$  will exceed 3 unless ${\displaystyle (x,y)}$  is one of the nine pairs with ${\displaystyle x}$  and ${\displaystyle y}$  each equal to ${\displaystyle -1,0}$  or 1. We can check these nine pairs directly to see that none of them satisfies ${\displaystyle 3=x^{2}+y^{2}}$ , so the equation does not have integer solutions.

We say that a binary quadratic form ${\displaystyle q(x,y)}$  represents an integer ${\displaystyle n}$  if it is possible to find integers ${\displaystyle x}$  and ${\displaystyle y}$  satisfying the equation ${\displaystyle n=q(x,y)}$ . The pair ${\displaystyle (x,y)}$  is then called a representing pair for ${\displaystyle n}$ . The oldest problem in the theory of binary quadratic forms is the representation problem: describe the set of integers represented by a given quadratic form q. The above paragraph discusses the representation problem for the binary quadratic form ${\displaystyle x^{2}+y^{2}}$ . We see that 65 is represented by ${\displaystyle x^{2}+y^{2}}$  in sixteen different ways, while 3 is not represented by ${\displaystyle x^{2}+y^{2}}$  at all.

For each ${\displaystyle n}$ , the equation ${\displaystyle n=x^{2}+y^{2}}$  can have only a finite number of solutions since ${\displaystyle x^{2}+y^{2}}$  will exceed ${\displaystyle n}$  unless the absolute values ${\displaystyle |x|}$  and ${\displaystyle |y|}$  are both less than ${\displaystyle {\sqrt {n}}}$ . There are only a finite number of pairs of integers with absolute values satisfying this constraint. In contrast, the form ${\displaystyle x^{2}-2y^{2}}$  represents 1 with infinitely many different representing pairs. One such pair is pair ${\displaystyle (x,y)=(3,2)}$ , that is, there is an equality ${\displaystyle 1=3^{2}-2\cdot 2^{2}}$ . If ${\displaystyle (x,y)}$  is any pair that represents 1, then ${\displaystyle (3x+4y,2x+3y)}$  is another such pair. For instance, from the pair ${\displaystyle (3,2)}$ , we compute

${\displaystyle (3\cdot 3+4\cdot 2,2\cdot 3+3\cdot 2)=(17,12)}$ ,

and we can check that this satisfies ${\displaystyle 1=17^{2}-2\cdot 12^{2}}$ . Iterating this process, we find further pairs that represent 1:

${\displaystyle {\begin{aligned}(3\cdot 17+4\cdot 12,2\cdot 17+3\cdot 12)&=(99,70),\\(3\cdot 99+4\cdot 70,2\cdot 99+3\cdot 70)&=(477,408),\\&\vdots \end{aligned}}}$ 

These values will keep growing in size, so we see there are infinitely many representing pairs.

If there are a finite number of representations of an integer ${\displaystyle n}$  by a form ${\displaystyle q}$ , then further questions ask for an algorithm to enumerate all of the representing pairs and for a closed formula for the number of representing pairs.

Equivalence and class invariants

Two forms $f$ and $g$ are called equivalent if there exist integers ${\displaystyle \alpha ,\beta ,\gamma ,{\text{ and }}\delta }$  such that the following conditions hold:

${\displaystyle {\begin{aligned}f(\alpha x+\beta y,\gamma x+\delta y)&=g(x,y)\\\alpha \delta -\beta \gamma &=1\end{aligned}}}$ 

(Lagrange and Legendre allowed ${\displaystyle \alpha \delta -\beta \gamma =\pm 1}$ , but since Gauss it has been recognized that the above definition has more interesting properties. If there is a need to distinguish, sometimes forms are called properly equivalent using the definition above and improperly equivalent if they are equivalent in Lagrange's sense.)

This defines an equivalence relation on the set of integral quadratic forms. From the theory of equivalence relations, it follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms. A class invariant is any property shared by all forms in the same class.

Equivalent forms necessarily have the same discriminant

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

Lagrange 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.^[2]

From a modern perspective, the class group of a fundamental discriminant D is isomorphic to the narrow class group of the quadratic field ${\displaystyle \mathbf {Q} ({\sqrt {D}})}$  of discriminant D.^[3] 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.

History

There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms.^[4] The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares. Pell's equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE. Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the chakravala method, attributed to either of the Indian mathematicians Jayadeva or Bhāskara II.^[5] The problem of representing integers by sums of two squares was considered in the 6th century by Diophantus.^[6] In the 17th century, inspired while reading Diophantus's Arithmetica, Fermat made several observations about representations by specific quadratic forms, including that which is now know as Fermat's theorem on sums of two squares.^[7] Euler provided the first proofs of Fermat's observations and added some new conjectures about representations by specific forms, without proof.^[8]

The general theory of quadratic forms was initiated by Lagrange in 1775 in his Recherches d'Arithmétique. Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms."^[9] 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".^[10] 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 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 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 with integer coefficients continues to the present. This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including Shank's infrastructure, Zagier's reduction algorithm, Conway's topographs, and Bhargava's reinterpretation of composition through Bhargava cubes.

See also

• Bhargava cube
• Legendre symbol

Notes

1. Weil, 2001 & p.30 harvnb error: no target: CITEREFWeil2001p.30 (help)
2. Cohen 1993, §5.2
3. Fröhlich & Taylor 1993, Theorem 58
4. Weil 2001, Ch.I §§VI, VIII
5. Weil 2001, Ch.I §IX
6. Weil 2001, Ch.I §IX
7. Weil 2001, Ch.II §§VIII-XI
8. Weil 2001, Ch.III §§VII-IX
9. Weil 2001, p.318
10. Weil 2001, p.317

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
• Weil, André (2001), Number Theory: An approach through history from Hammurapi to Legendre, Birkhäuser Boston

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