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Proper equivalence of binary quadratic forms
Definition
Let and be integral binary quadratic forms. We say that and are properly equivalent when there exists a matrix
such that
for all integers .
When this holds we also write
Remarks
- The determinant condition is , so proper equivalence uses only orientation-preserving unimodular substitutions.
- On the companion page, an example shows that allowing determinant can merge two distinct proper-equivalence classes.
Depends on
Used by
- Improper equivalence can merge two distinct proper classes Example
- A non-reduced positive-definite form admits an equivalent positive-definite form with smaller reduction measure Lemma
- Integral substitution defines a right action of SL₂(ℤ) on integral binary quadratic forms Lemma
- Properly equivalent reduced forms with the same leading coefficient are equal Lemma
- The leading coefficient of a reduced positive-definite form is minimal in its proper-equivalence class Lemma
- A positive integer n is primitively represented by some discriminant Δ form exactly when Δ is a square modulo 4n Proposition
- Proper equivalence preserves discriminant and primitivity of the form Proposition
- Properly equivalent binary quadratic forms represent the same integers, with primitive representations in bijection Theorem
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Sources
- William Stein, Elementary Number Theory and Elliptic Curves, Definition 9.2.2 (standard reference, not scraped)
- Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)