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Integral substitution defines a right action of ${S L}_{2} (Z)$ on integral binary quadratic forms

Statement

For an integral binary quadratic form $f$ and matrices $M, N \in {S L}_{2} (Z)$ , the substitution notation of Proper equivalence of binary quadratic forms satisfies

$f ∣ I = f, (f ∣ M) ∣ N = f ∣ (M N) .$

Thus integral substitution defines a right action of ${S L}_{2} (Z)$ on integral binary quadratic forms.

Facts & Assumptions

Given: An integral binary quadratic form $f = (a, b, c)$ and matrices $M, N \in {S L}_{2} (Z)$ .

[F1]

Proper equivalence is defined by the substitution $g (x, y) = f (p x + q y, r x + s y)$ for a determinant-one integer matrix $(\begin{array}{cc} p & q \\ r & s \end{array})$ (Proper equivalence of binary quadratic forms).

[L1]

Matrix multiplication is associative, and identity matrices act as units on either side whenever the shapes are compatible (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

Proof

technique · direct

1.1

If $I$ is the $2 \times 2$ identity matrix, then $(f ∣ I) (x, y) = f (1 \cdot x + 0 \cdot y, 0 \cdot x + 1 \cdot y) = f (x, y)$ for all $x, y$ , so $f ∣ I = f$ .

F1L1algebra

1.2

Write $M = (\begin{array}{cc} p & q \\ r & s \end{array})$ and $N = (\begin{array}{cc} u & v \\ w & z \end{array})$ . Then $((f ∣ M) ∣ N) (x, y) = (f ∣ M) (u x + v y, w x + z y) = f (p (u x + v y) + q (w x + z y), r (u x + v y) + s (w x + z y))$ , which is exactly $(f ∣ (M N)) (x, y)$ by the definition of matrix multiplication.

F1L1algebra

2.1

The coefficients of $f ∣ M$ are $a p^{2} + b p r + c r^{2}$ , $2 a p q + b (p s + q r) + 2 c r s$ , and $a q^{2} + b q s + c s^{2}$ , hence are integers. Therefore the substitutions stay inside the set of integral binary quadratic forms, and steps 1.1 and 1.2 are precisely the right-action axioms.

F1step 1.2algebra∎

Depends on

Used by

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Sources

William Stein, Elementary Number Theory and Elliptic Curves, Proposition 9.2.3 (standard reference, not scraped)
Andrew Granville, Binary Quadratic Forms, Exercise 4.1d (standard reference, not scraped)

Integral substitution defines a right action of SL2(Z)SL2​(Z) on integral binary quadratic forms