Quadratic forms and sums of squares

Abstract

Our goal in this chapter is to develop the theory of quadratic forms so we can give another proof of Theorem , especially in the three square case. Our exposition follows [31, Part 3, Chap IV] closely.

Appendices

Exercises

• 12.1 Verify Equation (12.3).

• 12.2 This exercise uses the notations of the proof of Lemma 12.2. Suppose $A, A' \in M_n(R)$ for some ring R, and suppose for all j we have $A(j) = A'(j)$. Show that $A = A'$.

• 12.3 Show that every positive definite binary quadratic form of discriminant d is equivalent to a quadratic form whose associated matrix $\begin{pmatrix} a &{} b \\ b &{} c \end{pmatrix}$ satisfies

  $$ 2 |b| \le a \le \frac{2}{\sqrt{3}} \sqrt{d}. $$

• 12.4 Show that a reduced binary quadratic form cannot be equivalent to a different reduced binary quadratic form.

• 12.5 Show that for every natural number d there are only finitely many equivalence classes of positive definite binary quadratic forms of discriminant d.

• 12.6 Find representatives for equivalence classes of positive definite binary quadratic forms of discriminant d when

  • a. $d =2$;

  • b. $d=3$;

  • c. $d=5$.

• 12.7 We say that a binary form f represents m properly if there are $a, b \in \mathbb Z$ with $\gcd (a, b) =1$ such that $f(a, b) = m$. Show that a binary quadratic form represents an integer m properly if and only if it is equivalent to a binary form $mx^2 + bxy + cy^2$ for some $b, c \in \mathbb Z$.

• 12.8 Find reduced forms that are equivalent to the following forms:

  • a. $4x^2 + y^2$;

  • b. $9x^2 + 2xy + y^2$;

  • c. $126x^2 + 74 xy + 13 y^2$.

• 12.9 ($\maltese$) List all reduced primitive positive definite binary quadratic forms of discriminant bounded by 100. For each d, find the number of forms with that discriminant.

• 12.10 Prove Lemma 12.15.

• 12.11 Suppose $a, b, c \in \mathbb Z$ are such that $\gcd (a, b, c) = 1$. Then prove that there are integers d, e, f, g, h, i such that the matrix

  $$ \begin{pmatrix} a &{} b &{} c \\ d &{} e &{} f \\ g&{} h &{} i \end{pmatrix} $$

  has determinant 1.

• 12.12 Prove that the Three Square Theorem implies the Four Square Theorem.

• 12.13 Finish the proof of the Three Square Theorem for $n \equiv 1, 5 \ \mathrm {mod}\ 8$.

• 12.14 Show that if $p>17$ is a prime number $p \equiv 5 \ \mathrm {mod}\ 1 2$ then p is a sum of three distinct positive squares. Hint: Use the identity,

  $$ 9(a^2 + b^2) = (2a-b)^2 + (2a+2b)^2 + (2b-a)^2. $$

Notes

1.1 Gauss Composition

The easy identity

$$\begin{aligned} (x^2+y^2)(z^2 +w^2) = (xz+yw)^2 + (xw-zy)^2 \end{aligned}$$

(12.6)

has been known for hundreds of years. As we noted in the Notes to Chapter 3, the master Indian mathematician Brahmagupta discovered the more general identity

$$\begin{aligned} (x^2 + dy^2)(z^2 + dw^2) = (xz+ dyw)^2 + d(xw-yz)^2 \end{aligned}$$

(12.7)

at some point in the seventh century CE. Over a thousand years later, Lagrange discovered the identities

$$\begin{aligned} (2x^2 + 2xy + 3y^2)(2z^2 + 2zw + 3w^2) = (2xz+xw+ yz+3yw)^2 + 5(xw-yz)^2, \end{aligned}$$

(12.8)

and

$$\begin{aligned} (3x^2 + 2xy+5y^2)(3z^2+2zw+5w^2)=(3x^2+xw+yz+5yw)^2 + 14(xw-yz)^2. \end{aligned}$$

(12.9)

All of these identities are of the form

$$\begin{aligned} f(x, y)f(z, w)=g(B_1(x, y, z, w), B_2(x, y, z, w)); \end{aligned}$$

(12.10)

with f and g positive definite binary quadratic forms of the same discriminant, and $B_1, B_2$ homogeneous quadratic forms in the four variables x, y, z, w. The binary quadratic forms in Equation (12.6) have discriminant 1, in Equation (12.7) they have discriminant d, in Equation (12.8) they have discriminant 5, and in Equation (12.9) they have discriminant 14. Gauss proved a truly impressive theorem that generalizes all such identities. In fact, he showed the following theorem: Let $f_1, f_2$ be positive definite binary quadratic forms of discriminant d. Then there are homogeneous polynomials $B_1, B_2$ of degree 2 in the variables x, y, z, t such that

$$ f_1(x, y)f_2(z, w)=g(B_1(x, y, z, w), B_2(x, y, z, w)); $$

for some positive definite binary quadratic form g of discriminant d. Gauss called the quadratic form g the composition of $f_1$ and $f_2$, and for that reason the theorem is called the composition law. The binary quadratic forms we studied in this chapter all had an even middle coefficient, i.e., they were of the form $ax^2 + 2bxy+ cy^2$ with b an integer. Gauss considered the more general quadratic forms $ax^2 + b xy+ cy^2$ with b integral. For such forms the discriminant as we defined it is not necessarily an integer, so the discriminant is generally defined to be $4ac - b^2 \in \mathbb Z$. Gauss illustrated his theory with the following example:

$$ (4x^2+3xy+5y^2)(3z^2+zw+6w^2) $$

$$ =(xz-3xw-2yz-3yw)^2+(xz-3xw-2yz-3yw)(xz+xw+yz-yw) $$

$$ + 9(xz+xw+yz-yw)^2. $$

Let us denote the composition of the forms $f_1$ and $f_2$ by $f_1 \circ f_2$. An important feature of Gauss’s composition is that if $f_1$ is equivalent to a form $f_1'$, then $f_1 \circ f_2 \sim f_1' \circ f_2$. This means that the composition provides a well-defined operation on the finite set of equivalence classes of binary quadratic forms of discriminant d, turning it into a finite abelian group, the class group of binary forms. It was Dirichlet who interpreted the composition of binary quadratic forms in terms of ideal multiplication, whereby connecting the class group of binary forms to the ideal class group of modern algebraic number theory. After about 200 years since the publication of [21], in a series of groundbreaking works, Manjul Bhargava generalized the Gauss composition laws and found numerous other composition laws. Gauss’s proof of his composition law is extremely complicated; see [21, Ch. V]. Cox [14, §3] contains a motivated introduction to Gauss’s theory of quadratic forms. We refer the reader to Andrew Granville’s lecture at a summer school in 2014 for a review of Gauss’s work and the works of other mathematicians that preceded it, as well as an introduction to Bhargava’s works:

http://www.crm.umontreal.ca/sms/2014/pdf/granville1.pdf