Binary quadratic form

A quadratic form in two variables, i.e. having the form

$$ \tag{* } f = f (x, y) = \ ax ^ {2} + bxy + cy ^ {2} . $$

If $ a, b $ and $ c $ are integers, the binary quadratic form is said to be integral. The expression $ d = ac - {b ^ {2} } /4 $ is called the discriminant or determinant of the binary quadratic form. The expression $ b ^ {2} - 4ac $ is also sometimes referred to as the discriminant. The arithmetic theory of binary quadratic forms originated with P. Fermat, who proved that any prime number of the form $ 4k + 1 $ can be represented as the sum of two squares of integers. The theory of quadratic forms was completed by J.L. Lagrange and by C.F. Gauss. This theory is a special case of the theory of quadratic forms in $ n $ variables; the arithmetic theory of binary quadratic forms is equivalent to the theory of ideals in quadratic fields, and is one of the origins of algebraic number theory (cf. Quadratic form; Quadratic field).

The number of genera of binary quadratic forms with discriminant $ d $ equals $ 2 ^ {s-1 } $, where $ s $ is the number of different prime divisors of $ d $, except for $ d \equiv 1 $( $ \mathop{\rm mod} 4 $), $ d \equiv 0 $( $ \mathop{\rm mod} 8 $), when $ s $ is increased by one; if $ -d $ is a square, the number of different binary quadratic forms is doubled. The number $ r(d, m) $ of essentially different primitive representations of a number $ m $ by a complete system of binary quadratic forms with discriminant $ d $ is equal to the number of solutions of the equation

$$ x ^ {2} \equiv -d ( \mathop{\rm mod} m). $$

As in the general case, there exists an algorithm which reduces the problem of solving a given second-degree Diophantine equation in two unknowns (in particular, an equation $ f(x, y) = m $) to the problem on the arithmetic equivalence of two binary quadratic forms.

All integral automorphisms of a primitive form $ f $ with $ a \neq 0 $ can be represented in the form

$$ \left \| \begin{array}{cc} t - bu/2 &-cu \\ au &t + bu/2 \\ \end{array} \ \right \| , $$

where $ t ^ {2} + du ^ {2} = 1 $, and $ 2t $ and $ u $ are integers (cf. Pell equation). Therefore, the problem on the equivalence of two forms is solved by the reduction theory of binary quadratic forms. The reduction theory of positive-definite binary quadratic forms is a special case of the reduction theory of positive-definite quadratic forms according to H. Minkowski. The reduction theory of integral indefinite binary quadratic forms can be reduced to the reduction theory of quadratic irrationalities ([2], [3]).

An important role in the theory of numbers is played by the arithmetic function $ h(d) $— the number of classes of primitive integral binary quadratic forms with determinant $ d $. It is known that $ h(d) < + \infty $. Some idea of the rate of increase of the function $ h(d) $ can be obtained from Siegel's theorem: Let $ d > 0 $, then for any $ \epsilon > 0 $ there exist constants $ c _ \epsilon $ and $ c _ \epsilon ^ \prime > 0 $ which satisfy the condition

$$ c _ \epsilon ^ \prime d ^ {1/2 - \epsilon } < \ h (d) < c _ \epsilon d ^ {1/2 + \epsilon } $$

(a similar formula is also valid for $ d < 0 $).

Let $ \Delta $ be an integer, $ \Delta \equiv 1 $ or $ 0 $( $ \mathop{\rm mod} 4 $), suppose that if $ s ^ {2} \mid \Delta $ then $ s = 1 $ or $ s = 2 $, and let $ F = \mathbf Q ( \sqrt \Delta ) $ be the quadratic field which is obtained by adjoining $ \sqrt \Delta $ to the field of rational numbers. A correspondence has been established between the integral ideals $ [ \alpha _ {1} , \alpha _ {2} ] $ of the field $ F $ and the integral quadratic forms

$$ f (x, y) = \ \frac{N ( \alpha _ {1} x + \alpha _ {2} y) }{N [ \alpha _ {1} , \alpha _ {2} ] } $$

with $ - \Delta /4 $ as determinant. This results in a one-to-one correspondence (up to a transition to conjugate classes of ideals) between the ideal classes of the field $ F $ and the classes of binary quadratic forms. In this correspondence, multiplication of ideal classes defines a composition of classes of binary quadratic forms.

As in the case of forms in $ n $ variables, the theory of binary quadratic forms can be generalized to include forms (*) with coefficients $ a, b $ and $ c $ in a given algebraic number field.

There are various variants in the definitions of an integral form, the discriminant of the form, equivalence of forms, and the classes and the genera of forms. The definition of integral forms given above is due to L. Kronecker. Gauss [1] stipulated that $ b $ be even. In determining the equivalence (and the class of forms), only the substitutions with discriminant $ + 1 $ may be considered; in other cases discriminants $ \pm 1 $ are considered. The definition of a genus given in [6] is wider than that given by Gauss.

References

[1]	C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin) MR0197380 Zbl 0136.32301
[2]	B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) MR0265267 Zbl 0204.37101
[3]	B.W. Jones, "The arithmetic theory of quadratic forms" , Math. Assoc. Amer. (1950) MR0037321 Zbl 0041.17505
[4]	A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian) MR201368 Zbl 0142.01403
[5]	E. Landau, "Vorlesungen über Zahlentheorie" , 3 , Hirzel (1927) MR0250844 Zbl 53.0123.17
[6]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[7]	O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018