Binary quadratic form

Representations

Diophantus considered whether, for an odd integer n , it is possible to find integers x and y for which n = x 2 + y 2 . When n = 65 , we have

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

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

A similar argument shows that for each n , the equation n = x 2 + y 2 can have only a finite number of solutions since x 2 + y 2 will exceed n unless the absolute values | x | and | y | are both less than n . There are only a finite number of pairs satisfying this constraint.

We say that a binary quadratic form q ( x , y ) represents an integer n if it is possible to find integers x and y satisfying the equation n = q ( x , y ) . Such an equation is a representation of n by q. The oldest problem in the theory of binary quadratic forms is the representation problem: describe the representations of a given number n by a given quadratic form q. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of respresentations, or even just determine whether any representations exist.

The examples above discuss the representation problem for the numbers 3 and 65 by the form x 2 + y 2 . We see that 65 is represented by x 2 + y 2 in sixteen different ways, while 3 is not represented by x 2 + y 2 at all. In the first case, the sixteen representations were explicitly described. It was also shown that the number of representations of an integer by x 2 + y 2 is always finite. The sum of squares function r 2 ( n ) gives the number of representations of n by x 2 + y 2 as a function of n. There is a closed formula

where d 1 ( n ) is the number of divisors of n that are congruent to 1 modulo 4 and d 3 ( n ) is the number of divisors of n that are congruent to 3 modulo 4.

It is possible for there to be an infinite number of solutions to the representation problem: the form x 2 − 2 y 2 represents 1 in infinitely many ways. One solution is ( x , y ) = ( 3 , 2 ) , that is, there is an equality 1 = 3 2 − 2 ⋅ 2 2 . If ( x , y ) is any solution to 1 = x 2 − 2 y 2 , then ( 3 x + 4 y , 2 x + 3 y ) is another such pair. For instance, from the pair ( 3 , 2 ) , we compute

and we can check that this satisfies 1 = 17 2 − 2 ⋅ 12 2 . Iterating this process, we find further pairs ( x , y ) with 1 = x 2 − 2 y 2 :

These values will keep growing in size, so we see there are infinitely many ways to represent 1 by the form x 2 − 2 y 2 .

Equivalence classes and class invariants

Two forms f and g are called equivalent if there exist integers α , β , γ ,  and  δ such that the following conditions hold:

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 can mean either a function defined on equivalence classes of forms or a property shared by all forms in the same class.

Lagrange used a different notion of equivalence, in which the second condition is replaced by α δ − β γ = ± 1 . Since Gauss it has been recognized that this definition is inferior to that given above. 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.

Some class invariants can be defined in terms of an arbitrarily chosen form in the class. If f = a x 2 + b x y + c y 2 , then important invariants include

Other class invariants arise naturally as functions on classes. Examples include

Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant Δ is definite if Δ < 0 , degenerate if Δ is a perfect square, and indefinite otherwise. A form is primitive if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a fundamental discriminant, then the form is primitive. Discriminants satisfy Δ ≡ 0 , 1 ( mod 4 ) .

Reduction and class numbers

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.

Gauss gave a superior reduction algorithm in Disquisitiones Arithmeticae, which has ever since the reduction algorithm most commonly given in textbooks. In 1981, Zagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss's.

Composition

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.

From a modern perspective, the class group of a fundamental discriminant D is isomorphic to the narrow class group of the quadratic field Q ( D ) of discriminant D. 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.

Genera of forms

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.

History

There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. 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. The problem of representing integers by sums of two squares was considered in the 6th century by Diophantus. 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. 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 "a coherent general theory required the simulatenous consideration of all forms." 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 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.