Binary Quadratic Form - Main Questions

Main Questions

A classical question in the theory of integral quadratic forms (those with integer coefficients) is the representation problem: describe the set of numbers represented by a given quadratic form q. If the number of representations is finite then a further question is to give a closed formula for this number. The notion of equivalence of quadratic forms and the related reduction theory are the principal tools in addressing these questions.

Two integral forms are called equivalent if there exists an invertible integral linear change of variables that transforms the first form into the second. This defines an equivalence relation on the set of integral quadratic forms, whose elements are called classes of quadratic forms. Equivalent forms necessarily have the same discriminant

Gauss 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.

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