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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 field|quadratic]] and more general [[number field]]s, but advances specific to binary quadratic forms still occur on occasion.
== Equivalence ==
Two forms ''f'' and ''g'' are called '''equivalent''' if there exist integers <math>\alpha, \beta, \gamma, \text{ and } \delta</math> such that the following conditions hold:▼
: <math>\begin{align} f(\alpha x + \beta y, \gamma x + \delta y) &= g(x,y),\\▼
\alpha \delta - \beta \gamma &= 1\end{align}.</math>▼
For example, with <math>f= x^2 + 4xy + 2y^2</math> and <math>\alpha = -3</math>, <math>\beta = 2</math>, <math>\gamma = 1</math>, and <math>\delta = -1</math>, we find that ''f'' is equivalent to <math>g = (-3x+2y)^2 + 4(-3x+2y)(x-y)+2(x-y)^2</math>, which simplifies to <math>-x^2+4xy-2y^2</math>.
Lagrange used a different notion of equivalence, in which the second condition is replaced by <math> \alpha \delta - \beta \gamma = \pm 1</math>. 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
: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
has integer entries and determinant 1, the map <math> f(x,y) \mapsto f(\alpha x + \beta y, \gamma x + \delta y)</math> is a [[group action]] of <math>\mathrm{SL}_2(\mathbb{Z})</math> on the set of binary quadratic forms. The equivalence relation above then arises from the general theory of group actions.
* The [[discriminant]] <math>\Delta=b^2-4ac. </math>.▼
* 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 <math>\Delta</math> is '''definite''' if <math>\Delta < 0</math>, '''degenerate''' if <math>\Delta</math> 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.<ref>{{harvnb|Cohen|1993|loc=§5.2}}</ref> Discriminants satisfy <math>\Delta\equiv 0,1 \pmod 4. </math>▼
=== Automorphisms ===
If ''f'' is a quadratic form, a matrix
: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
in <math>\mathrm{SL}_2(\mathbb{Z})</math> is an '''automorphism''' of ''f'' if <math>f(\alpha x + \beta y, \gamma x + \delta y) = f(x,y)</math>. For example, the matrix
: <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 form 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]].
==Representations==
We say that a 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 =
=== Examples ===
[[Diophantus]] considered whether, for an odd integer <math>n</math>, it is possible to find integers <math>x</math> and <math>y</math> for which <math>n = x^2 + y^2</math>.<ref>{{harvnb|Weil|2001|p.30}}</ref> When <math>n=65</math>, we have
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A similar argument shows that for each <math>n</math>, the equation <math>n =x^2+y^2</math> can have only a finite number of solutions since <math>x^2+y^2</math> will exceed <math>n</math> unless the absolute values <math>|x|</math> and <math>|y|</math> are both less than <math>\sqrt{n}</math>. There are only a finite number of pairs satisfying this constraint.
▲We say that a 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 = q(x,y).</math> 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 <math>n</math> 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 <math>x^2 + y^2</math>. We see that 65 is represented by <math>x^2 + y^2</math> in sixteen different ways, while 3 is not represented by <math>x^2+y^2</math> 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 <math>x^2+y^2</math> is always finite. The [[sum of squares function]] <math>r_2(n)</math> gives the number of representations of ''n'' by <math>x^2+y^2</math> as a function of ''n''. There is a closed formula<ref>{{harvnb|Hardy|Wright|2008||loc=Thm. 278}}</ref>▼
: <math> r_2(n) = 4(d_1(n) - d_3(n)), </math>▼
where <math>d_1(n)</math> is the number of [[divisor]]s of ''n'' that are [[Modular arithmetic|congruent]] to 1 modulo 4 and <math>d_3(n)</math> 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 <math>x^2 - 2y^2</math> represents 1 in infinitely many ways. One solution is <math>(x,y) = (3,2)</math>, that is, there is an equality <math>1 = 3^2 - 2 \cdot 2^2</math>. If <math>(x,y)</math> is any solution to <math>1 = x^2 - 2 y^2</math>, then <math>(3x+4y,2x+3y)</math> is another such pair. For instance, from the pair <math>(3,2)</math>, we compute
: <math>(3\cdot 3 + 4 \cdot 2, 2\cdot 3 + 3 \cdot 2) = (17,12)</math>,
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&\vdots \end{align}
</math>
These values will keep growing in size, so we see there are infinitely many ways to represent 1 by the form <math>x^2 - 2y^2</math>. This recursive description was discussed in Theon of Smyrna's commentary on [[Euclid's Elements]].
=== The representation problem ===
▲Two forms ''f'' and ''g'' are called '''equivalent''' if there exist integers <math>\alpha, \beta, \gamma, \text{ and } \delta</math> such that the following conditions hold:
The oldest problem in the theory of binary quadratic forms is the '''representation problem''': describe the representations of a given number <math>n</math> 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 respresentations, or even just determine whether any representations exist.
▲: <math>\begin{align} f(\alpha x + \beta y, \gamma x + \delta y) &= g(x,y)\\
▲ \alpha \delta - \beta \gamma &= 1\end{align}</math>
The examples above discuss the representation problem for the numbers 3 and 65 by the form <math>x^2 + y^2</math> and for the number 1 by the form <math>x^2 - 2y^2</math>. We see that 65 is represented by <math>x^2 + y^2</math> in sixteen different ways, while 1 is represented by <math>x^2 - 2y^2</math> in infinitely many ways and
▲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 [[partition]]ed 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.
▲
▲: <math> r_2(n) = 4(d_1(n) - d_3(n)), </math>
▲Lagrange used a different notion of equivalence, in which the second condition is replaced by <math> \alpha \delta - \beta \gamma = \pm 1</math>. 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.
▲where <math>d_1(n)</math> is the number of [[divisor]]s of ''n'' that are [[Modular arithmetic|congruent]] to 1 modulo 4 and <math>d_3(n)</math> is the number of divisors of ''n'' that are congruent to 3 modulo 4.
▲Some class invariants can be defined in terms of an arbitrarily chosen form in the class. If <math>f=ax^2+bxy+cy^2</math>, then important invariants include
There are several class invariants relevant to the representation problem:
▲* The [[discriminant]] <math>\Delta=b^2-4ac. </math>.
▲* The content, equal to the greatest common divisor of ''a'', ''b'', and ''c''.
* 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 congruence classes modulo the discriminant of a class represented by the 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 <math>f = ax^2 + bxy + cy^2</math> have the same sign: positive if <math>a>0</math> and negative if <math>a<0</math>. 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: <math>x^2+y^2</math> is positive definite and <math>x^2 - 2y^2</math> is indefinite.
=== Equivalent representations ===
The notion of equivalence of forms can be extended to '''equivalent representations'''. Representations <math>m = f(x_1,y_1)</math> and <math>n = g(x_2,y_2)</math> are equivalent if there exists a matrix
: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
with integer entries and determinant 1 so that <math>f(\alpha x + \beta y, \gamma x + \delta y) = g(x,y)</math> and
: <math>\begin{pmatrix} \delta& -\beta \\ -\gamma & \alpha\end{pmatrix} \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}</math>
The above conditions give a (right) action of the group <math>\mathrm{SL}_2(\mathbb{Z})</math> 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 <math>f = x^2 - 2y^2</math> and consider a representation <math>1 = f(x_1,y_1)</math>. Such a representation is a solution to the Pell equation described in the examples above. The matrix
: <math> \begin{pmatrix} 3 & -4 \\ -2 & 3 \end{pmatrix} </math>
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 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.
▲Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant <math>\Delta</math> is '''definite''' if <math>\Delta < 0</math>, '''degenerate''' if <math>\Delta</math> 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.<ref>{{harvnb|Cohen|1993|loc=§5.2}}</ref> Discriminants satisfy <math>\Delta\equiv 0,1 \pmod 4. </math>
== Reduction and class numbers ==
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