In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form.
This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fields, but advances specific to binary quadratic forms still occur on occasion.
Pierre Fermat stated that if p is an odd prime then the equation has a solution iff , and he made similar statements about the equations , , and . In addition, and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems.
Another instance of quadratic forms is Pell's equation .
Binary quadratic forms are closely related to ideals in quadratic fields. This 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 , if is a positive definite quadratic form then is a theta function.
Equivalence
Two forms f and g are called equivalent if there exist integers such that the following conditions hold:
For example, with and , , , and , we find that f is equivalent to , which simplifies to .
The above equivalence conditions define an equivalence relation on the set of integral quadratic forms. 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 . 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.
In matrix terminology, which is used occasionally below, when
has integer entries and determinant 1, the map is a (right) group action of on the set of binary quadratic forms. The equivalence relation above then arises from the general theory of group actions.
If , then important invariants include
The discriminant . The content, equal to the greatest common divisor of a, b, and c.
Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant is definite if , degenerate if 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
Automorphisms
If f is a quadratic form, a matrix
in is an automorphism of f if . For example, the matrix
is an automorphism of the form . The automorphisms of a form are a subgroup of . When f is definite, the group is finite, and when f is indefinite, it is infinite and cyclic.
Representation
A binary quadratic form represents an integer if it is possible to find integers and satisfying the equation Such an equation is a representation of by .
Examples
Diophantus considered whether, for an odd integer , it is possible to find integers and for which . When , we have
so we find pairs that do the trick. We obtain more pairs that work by switching the values of and and/or by changing the sign of one or both of and . In all, there are sixteen different solution pairs. On the other hand, when , the equation
does not have integer solutions. To see why, we note that unless or . Thus, will exceed 3 unless is one of the nine pairs with and each equal to or . We can check these nine pairs directly to see that none of them satisfies , so the equation does not have integer solutions.
A similar argument shows that for each , the equation can have only a finite number of solutions since will exceed unless the absolute values and are both less than . There are only a finite number of pairs satisfying this constraint.
Another ancient problem involving quadratic forms asks us to solve Pell's equation. For instance, we may seek integers x and y so that . Changing signs of x and y in a solution gives another solution, so it is enough to seek just solutions in positive integers. One solution is , that is, there is an equality . If is any solution to , then is another such pair. For instance, from the pair , we compute
,
and we can check that this satisfies . Iterating this process, we find further pairs with :
These values will keep growing in size, so we see there are infinitely many ways to represent 1 by the form . This recursive description was discussed in Theon of Smyrna's commentary on Euclid's Elements.
The representation problem
The oldest problem in the theory of binary quadratic forms is the representation problem: describe the representations of a given number by a given quadratic form f. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any representations exist.
The examples above discuss the representation problem for the numbers 3 and 65 by the form and for the number 1 by the form . We see that 65 is represented by in sixteen different ways, while 1 is represented by in infinitely many ways and 3 is not represented by 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 is always finite. The sum of squares function gives the number of representations of n by as a function of n. There is a closed formula
where is the number of divisors of n that are congruent to 1 modulo 4 and is the number of divisors of n that are congruent to 3 modulo 4.
There are several class invariants relevant to the representation problem:
The set of integers represented by a class. If an integer n is represented by a form in a class, then it is represented by all other forms in a class. The minimum absolute value represented by a class. This is the smallest nonnegative value in the set of integers represented by a class. The congruence classes modulo the discriminant of a class represented by the class.
The minimum absolute value represented by a class is zero for degenerate classes and positive for definite and indefinite classes. All numbers represented by a definite form have the same sign: positive if and negative if . For this reason, the former are called positive definite forms and the latter are negative definite.
The number of representations of an integer n by a form f is finite if f is definite and infinite if f is indefinite. We saw instances of this in the examples above: is positive definite and is indefinite.
Equivalent representations
The notion of equivalence of forms can be extended to equivalent representations. Representations and are equivalent if there exists a matrix
with integer entries and determinant 1 so that and
The above conditions give a (right) action of the group on the set of representations of integers by binary quadratic forms. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.
As an example, let and consider a representation . Such a representation is a solution to the Pell equation described in the examples above. The matrix
has determinant 1 and is an automorphism of f. Acting on the representation by this matrix yields the equivalent representation . This is the recursion step in the process described above for generating infinitely many solutions to . 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 . A complete set of representatives for these classes can be given in terms of reduced forms defined in the section below. When , 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 . When , Zagier proved that every representation of a positive integer n by a form of discriminant is equivalent to a unique representation in which f is reduced in Zagier's sense and , . The set of all such representations constitutes a complete set of representatives for equivalence classes of representations.
Reduction and class numbers
Joseph-Louis Lagrange proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the 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 ever since has been the reduction algorithm most commonly given in textbooks. In 1981, Don Zagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss's.
Composition
Composition most commonly refers to a binary operation on primitive equivalence classes of forms of the same discriminant, one of the deepest discoveries of Gauss, which makes this set into a finite abelian group called the form class group (or simply class group) of discriminant . Class groups have since become one of the central ideas in algebraic number theory. From a modern perspective, the class group of a fundamental discriminant is isomorphic to the narrow class group of the quadratic field of discriminant . For negative , the narrow class group is the same as the ideal class group, but for positive it may be twice as big.
"Composition" also sometimes refers to, roughly, a binary operation on binary quadratic forms. The word "roughly" indicates two caveats: only certain pairs of binary quadratic forms can be composed, and the resulting form is not well-defined (although its equivalence class is). The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes.
"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated 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, as follows from Brahmagupta's identity.
Composing forms and classes
A variety of definitions of composition of forms has been given, often in an attempt to simplify the extremely technical and general definition of Gauss. We present here Arndt's method, because it remains rather general while being simple enough to be amenable to computations by hand. An alternative definition is described at Bhargava cubes.
Suppose we wish to compose forms and , each primitive and of the same discriminant . We perform the following steps:
Compute and , and