COMPOSITION OF BINARY QUADRATIC FORMS: UNDERSTANDING THE APPROACHES OF GAUSS, DIRICHLET AND BHARGAVA FRANÇOIS SÉGUIN Abstract. In 2004, Bhargava introduced a new way to understand the com- position law of integral binary quadratic forms through what he calls cubes of integers. The goal of this article is to introduce the reader to Bhargava’s cubes and this new composition law, as well as to relate it to the composition law as it is known classically. We will present a historical exposition of the subject, from Gauss to Bhargava, and see how the different formulations of the composition laws are equivalent. 1. Introduction Binary quadratic forms were already studied in the seventeenth and eighteenth centuries. Originally the main questions formulated in terms of binary qua- dratic forms were about representation of numbers. Given a binary quadratic form f(x, y) = ax2 + bxy + cy2, we can ask which integers n can be written as n = f(x0, y0) for some x0, y0, in which case f is said to represent n. Given an in- teger n, we can also ask which binary quadratic forms could represent n. Finally, given such a representation, how many other representations can we find. It is interesting to note that these very questions will eventually come to play a role in subjects such as Diophantine equations, quadratic reciprocity and even class field theory. Mathematicians like Fermat and Euler tried to classify all the possible binary quadratic forms, but the biggest results came from Carl Friedrich Gauss in 1801 in his Disquisitiones Arithmeticae. Gauss spells out the underlying structure that those binary quadratic forms have, structure that will later came to be known as a group. What is now known as Gauss composition law for binary quadratic forms has been modernized using algebraic number theory through the works of Dirichlet amongst others. This construction plays a critical role in number theory. In particular, it is one of the primary tools to understand class groups of quadratic number fields. 2010 Mathematics Subject Classification. 11E16. Key words and phrases. Binary quadratic forms, Composition laws. Research of the author partially supported by a FRQNT B2 Research Scholarship. 1 Recently, Manjul Bhargava formulated a brand new way to approach the com- position law for binary quadratic forms. In his thesis and then a series of articles [2, 3, 4, 5], Bhargava uses a geometric approach to describe the composition law, giving insight into a new possible way to understand it. Bhargava also gener- alizes the composition law to other objects, establishing a correspondence with structures in higher degree number fields. In this article, we will explore how the composition law came to be understood by Gauss, Dirichlet and finally Bhargava, as well as see how these different formu- lations are really equivalent. The proofs for most of the propositions we will be presenting are easy exercises, accessible to undergraduate students. We encourage the reader to take them as exercises when possible. 2. Preliminaries We define an (integral) binary quadratic form as a homogenous degree 2 poly- nomial in two variables, i.e. of the form F (x, y) = ax2 + bxy + cy2, where a, b, c ∈ Z. Since any binary quadratic form is completely determined by its three coefficients a, b and c, we sometimes denote the above polynomial by (a, b, c). The discriminant of a binary quadratic form (a, b, c) is defined as Disc(a, b, c) = b2 − 4ac. Also, we say that a binary quadratic form is primitive if the three coefficients are coprime, i.e. gcd(a, b, c) = 1, and positive definite if the binary quadratic form takes only positive values, i.e. ax2 + bxy + cy2 > 0 for all x, y 6= 0. Proposition 2.1. The binary quadratic form (a, b, c) is positive definite if and only if Disc(a, b, c) < 0 and a > 0. Proof. If f(x, y) = ax2 + bxy + cy2 is of discriminant D, then 4af(x, y) = (2ax + by)2 − Dy2. From now on, we will use the term binary quadratic forms to mean integral primitive positive definite binary quadratic forms. Recall that the group SL2(Z) is defined as ( ! ) a b SL2(Z) := : a, b, c, d ∈ Z, ad − bc = 1 . c d 2 We define the action of SL2(Z) on a binary quadratic form as ! α β (ax2 + bxy + cy2) ∗ = a(αx + βy)2 + b(αx + βy)(γx + δy) + c(γx + δy)2. γ δ Note that this is simply a change of variable. An alternative way of expressing this action is, for Q(x, y) a binary quadratic form and M ∈ SL2(Z), Q(x, y) ∗ M = Q(x0, y0) where ! ! x0 x = M . y0 y ! 0 −1 Example 2.1. The result of the action of the matrix on the binary 1 0 quadratic form (a, b, c) is ! 0 −1 (a, b, c) ∗ = ay2 − bxy + cx2 1 0 = (c, −b, a). Proposition 2.2. For any matrix M, Disc ((a, b, c) ∗ M) = det(M)2 Disc(a, b, c). Proof. Note that for f(x, y) = ax2 + bxy + cy2 of discriminant D, ! ! a b/2 x f(x, y) = x y b/2 c y and a b/2 Disc(f(x, y)) = −4 . b/2 c Therefore, ! ! a b/2 x f(x, y) ∗ M = x y M T M b/2 c y and computing the determinant we are done. From this Proposition, we can conclude that the action of SL2(Z) preserves the discriminant. We now fix a certain integer D and consider all the possible binary quadratic forms of discriminant D. On this set, we define the following equivalence relation. If Q1 and Q2 are binary quadratic forms of discriminant D, then Q1 ∼ Q2 if and only if Q1 ∗ M = Q2 3 for some M ∈ SL2(Z). We call the set of equivalence classes under this relation GD, i.e. GD = {(a, b, c): a, b, c ∈ Z, Disc(a, b, c) = D} / ∼ . We denote [a, b, c] the equivalence class containing (a, b, c) in GD. Proposition 2.3. Any binary quadratic form that is equivalent to a primitive binary quadratic form is primitive. Proof. Consider the set of integers that can be written as Q(x, y) for a binary quadratic form Q and x, y ∈ Z. Q is non-primitive if and only if those integers are all multiple of some N (consider (x, y) = (1, 0), (0, 1) and (1, 1) for the reverse implication). However, since equivalence of binary quadratic forms is given by a simple in- vertible change of variables, any equivalent binary quadratic forms represent the same integers. We say that a binary quadratic form (a, b, c) is reduced if |b| ≤ a ≤ c. Proposition 2.4. There are finitely many reduced binary quadratic forms of a fixed discriminant D. Furthermore, every binary quadratic form is equivalent to a unique reduced bi- nary quadratic form, with the exception of (a, b, a) ∼ (a, −b, a) and (a, a, c) ∼ (a, −a, c) where the reduced form is not unique. Finally, we conclude that |GD| is finite for any discriminant D. Proof. Recall that two generators for SL2(Z) are ! ! 1 1 0 −1 (2.1) T = and S = . 0 1 1 0 Using T n, we can show that (2.2) (a, b, c) ∼ (a, b + 2an, c0) Note that c0 is uniquely determined from the first two entries by the fact that the discriminant is fixed. Here, c0 = an2 + bn + c. Using S, we have (2.3) (a, b, c) ∼ (c, −b, a). Using (2.2) we can find a representative with |b| ≤ |a|, and using (2.3) we can insure that |a| ≤ |c|. Uniqueness follows from the fact that T and S are generators for SL2(Z), and any transformation can be expressed in terms of these two. The last statement is an obvious consequence of the first two. 4 3. Gauss’s composition law We recall the following identity attributed to 7th century Indian mathematician Brahmagupta (see [12]). Proposition 3.1. For any integers x1, y1, x2, y2,D, 2 2 2 2 2 2 x1 + Dy1 x2 + Dy2 = (x1x2 − Dy1y2) + D (x1y2 + x2y1) . √ Proof. Note that N(x + y −D) = x2 + Dy2. Therefore, √ √ 2 2 2 2 x1 + Dy1 x2 + Dy2 = N x1 + y1 −D N x2 + y2 −D √ = N x1x2 − Dy1y2 + (x1y2 + x2y1) −D by the multiplicativity of the norm, and so 2 2 = (x1x2 − Dy1y2) + D (x1y2 + x2y1) . We can summarize this previous identity by saying that the numbers of the form x2+Dy2 are closed under multiplication. In 1801, Gauss asked in his Disquisitiones Arithmeticae whether it was possible to generalize this to numbers of a more general form, namely ax2 + bxy + cy2. He comes up with the answer: yes! 2 2 2 2 Theorem 3.2 (Gauss). Let a1x1 +b1x1y1 +c1y1 and a2x2 +b2x2y2 +c2y2 be binary quadratic forms of discriminant D. Then, there exists an (explicit) transformation (change of variables) x1 x2 ! ! X p p p p x y 0 1 2 3 1 2 = Y q0 q1 q2 q3 y1 x2 y1 y2 and integers A, B and C, such that 2 2 2 2 2 2 a1x1 + b1x1y1 + c1y1 a2x2 + b2x2y2 + c2y2 = AX + BXY + CY . Moreover, B2 − 4AC = D. From this, it is easy to conclude the following. Proposition 3.3. The set GD forms a finite abelian group. Proof. We need to show that the composition law defined this way is well defined 0 ! ! 0 ! x1 x1 x2 on GD. Indeed, the two changes of variables 0 = M1 and 0 = y1 y1 y2 5 ! x2 M2 correspond to the change of variable y2 0 0 x1x2 x1 x2 x0 y0 x y 1 2 = (M ⊗ M ) 1 2 0 0 1 2 y1x2 y1 x2 0 0 y1y2 y1 y2 so the matrix 0 0 0 0 ! ! p0 p1 p2 p3 p0 p1 p2 p3 0 0 0 0 = (M1 ⊗ M2) q0 q1 q2 q3 q0 q1 q2 q3 0 0 will yield the correct change of variable for the multiplication of Q1(x1, y1) and 0 0 Q2(x2, y2).