Binary Quadratic Forms Overview

Updated 18 August 2025

Binary quadratic forms are homogeneous degree-2 polynomials in two variables with integer coefficients, fundamental to classical and modern number theory.
They are classified via discriminants, equivalence relations, and reduction theory, providing a finite group structure for a fixed discriminant.
Applications include representation theory and modular forms connections, with implications spanning analytic, arithmetic, and geometric studies.

A binary quadratic form is a homogeneous degree-2 polynomial in two variables with integer (or more generally, ring) coefficients. The paper of such forms—traditionally denoted $f(x, y) = ax^2 + bxy + cy^2$ —lies at the heart of classical and modern number theory, touching on algebraic, analytic, geometric, and arithmetic themes. Central topics include the classification of forms up to equivalence, the structure of their value sets, composition laws, reduction theory, connections with class field theory, and deep links to modular forms, arithmetic geometry, and combinatorics. Recent developments have extended the reach of quadratic form techniques, providing refined invariants, geometric interpretations, and new tools for applications ranging from the representation of numbers to the analysis of diophantine problems and factorization.

1. Fundamental Definitions and Classification

A binary quadratic form over a ring $R$ is

$Q(x, y) = ax^2 + bxy + cy^2$

with $a, b, c \in R$ . Over $\mathbb{Z}$ , key notions include:

Discriminant: $\Delta_Q = b^2 - 4ac$ . The discriminant governs the arithmetic and geometry of the form, distinguishing positive definite ( $\Delta_Q < 0, a > 0$ ), indefinite ( $\Delta_Q > 0$ ), and degenerate cases.
Primitive forms: $\gcd(a, b, c) = 1$ .
Equivalence: Two forms $Q$ and $Q'$ are (properly) equivalent if $Q'(x, y) = Q(ax + by, cx + dy)$ for some $\begin{pmatrix} a & b \ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})$ .
Reduced forms: For $\Delta < 0$ , reduction selects a unique representative per class with canonical inequalities (e.g., $|b| \leq a \leq c$ ), as per Gauss reduction.

The set of equivalence classes of primitive forms with fixed discriminant $\Delta$ (up to proper equivalence) is finite, forming the form class group. For negative fundamental $\Delta$ , its order is the class number $h(\Delta)$ of the corresponding imaginary quadratic order.

2. Composition, Genus Theory, and Class Groups

Gauss's composition law provides a group structure on the set of equivalence classes of primitive forms of discriminant $\Delta$ :

Given two classes, their product corresponds to a new class whose representative can be explicitly constructed (for coprime $a$ , symmetric matching, and specific congruence conditions on $b$ ).
The group is abelian and isomorphic, for fundamental discriminants, to the ideal class group of $\mathbb{Q}(\sqrt{\Delta})$ (Abdesselam et al., 2010, Lemmermeyer, 2011, Zeytin, 15 Jan 2025).
Genus theory partitions the class group by local invariants ("genus characters"), and connects representation of squares by forms to extracting roots in the class group (Lemmermeyer, 2011).
Ambiguous and reciprocal classes play a structural role, often detectable via symmetries in geometric or graph-theoretic realizations (Uludağ et al., 2015).

Over number fields: The composition law generalizes via orientation data to relate (properly oriented) ideal classes in a quadratic extension to classes of quadratic forms over the ring of integers of the base field (Zemková, 2017). For totally negative discriminants, total positive definiteness requires the leading coefficient to be totally positive.

3. Representation Theory and Value Sets

A central question is which integers are represented by a binary quadratic form, or by all forms in a given collection, and under what conditions representations are unique:

Prime representation: Each positive definite form represents, up to automorphisms, a certain set of positive integers; for primitive Pythagorean triples, various associated sides and sums can be expressed as values of different binary quadratic forms, uniquely representing distinct congruence classes of odd primes (Perez, 2011).
Simultaneous representation: For a finite set of forms, the intersection of their value sets can be characterized using the parity of class numbers and the structure of composite fields (Donnay et al., 2017).
Value set equivalence: Two integer-coefficient forms have exactly the same value set on $\mathbb{Z}^2$ if and only if they are either $\mathrm{GL}(2, \mathbb{Z})$ -equivalent, or form a specific pair of "extraordinary" forms with discriminants related by $d$ and $4d$, subject to congruence and class number coincidences (Fouvry et al., 15 Apr 2024).
Essential representation: For irreducible forms, any pair of representations of the same integer are related via a rational automorphism of the form, rendering all representations essentially equivalent up to automorphism (Xiao, 2017).

Table: Value Set Equivalence (Fouvry et al., 15 Apr 2024)

Case	Conditions	Structure
Ordinary	$D = d$	Unique under $\mathrm{GL}(2,\mathbb{Z})$
Extraordinary	$D = 4d$ , $d \equiv 5 \pmod{8}$ , $h(4d) = h(d)$	Two distinct classes

4. Analytic, Modular, and Combinatorial Aspects

Binary quadratic forms attain further significance through their connections to analytic number theory and modular forms:

Theta functions: The generating function $\theta_Q(z) = \sum_{m,n \in \mathbb{Z}} e^{2\pi i Q(m,n)z}$ is a modular form of weight 1; differences of such theta functions yield cusp forms whose Fourier coefficients encode subtle invariants of forms, especially when the class number is odd (Akbary et al., 31 Jul 2024).
Eta quotients and Rogers–Ramanujan identities: Explicit eta product expressions for differences of theta series are rare; in the odd class number (notably D = 23) case, unique eta quotient expansions can occur. These reflect deep modular relations and extend van der Blij's results (Akbary et al., 31 Jul 2024, Berkovich et al., 2012).
Fourier averages: For quaternary forms, the average number of representations of binary forms relates closely to the average of Fourier coefficients of corresponding Siegel modular forms, allowing asymptotic analysis and bounds on the proportion of represented classes (Schulze-Pillot, 2012).
Arithmetic progressions: Sharp bounds exist for the maximal length of arithmetic progressions contained in the value set of a given binary quadratic form, depending logarithmically and polynomially on coefficients governed by the discriminant (Elsholtz et al., 2018).

5. Structural and Geometric Models

Contemporary developments include geometric, combinatorial, and group-theoretic models for understanding forms:

Extended modular group and reduction: The action of the extended modular group ( $\bar{\Pi}$ ) on the upper half-plane and on base points encodes equivalence, reduction, and canonical form transformations (Malik et al., 2012).
Çarks (modular graphs): For indefinite forms, equivalence classes correspond to infinite graphs called çarks, whose spines encode periodicity under Gauss reduction, with symmetry properties tracking ambiguous and reciprocal classes (Uludağ et al., 2015, Zeytin, 15 Jan 2025).
Bhargava cubes: The reinterpretation of composition using $2 \times 2 \times 2$ cubes (octuples) gives an elegant, highly symmetric formulation of the triple product relation among forms and relates to higher composition laws and arithmetic invariant theory (Zeytin, 15 Jan 2025).
Generalizations: Relations between binary forms and higher-dimensional phenomena (e.g., norm forms, quaternary lattices, and composition over number fields) are accessible via ideal-theoretic correspondences and explicit reduction algorithms (Zemková, 2017, Schulze-Pillot, 2012).

6. Local-Global Principles, Counterexamples, and Isolation Phenomena

Binary quadratic forms provide a testing ground for the local-global principle and understanding failures thereof:

Counterexamples to Hasse principle: Explicit construction of counterexamples for certain quartic curves (e.g., $px^4 - my^4 = z^2$ ) exploits quadratic forms whose class is a square but not a fourth power in the form class group, leading to the existence of local but not global solutions (Lemmermeyer, 2011).
Isolation and irrecoverability: The question whether every representation of proper subforms of a given form implies a representation of the form itself ("isolation") relates to the fine structure of the form class group. Notably, no binary isolation exists for any unary quadratic form, and nonexistence results for ternary isolations of binary forms and discriminant constraints for quaternary isolations are settled using composition laws and genus-theoretic arguments (Ju et al., 11 Aug 2025).

7. Extensions and Generalizations

Ongoing work extends binary quadratic form theory in several directions:

Generalization to number fields, with careful treatment of units and orientation, connects quadratic forms intimately to relative class fields and modularity (Zemková, 2017).
Congruence and genus refinements: Refined equivalence relations ( $\Gamma$ -equivalence, $N$ -representation, and $N$ -genus) provide a framework for studying representation with congruence constraints, isomorphisms with ideal class groups with level structure, and subdivided genera (Cho, 2017).
Completely p-primitive forms: Necessary and sufficient conditions for a form to be "completely $p$ -primitive"—that is, able to represent all admissible values with solutions whose coordinates are coprime to $p$ —involve explicit conditions on class group elements and local behaviors (Oh et al., 2017).

The arithmetic, combinatorial, geometric, and analytic aspects of binary quadratic forms form a dense web of interconnections underpinning diverse areas of mathematics. Advances in explicit reduction theory, modular and automorphic structures, and representation phenomena continue to expand the landscape both in classical and modern settings, unifying approaches from algebraic number theory, arithmetic geometry, and the theory of modular forms.