CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Ternary isotropy via the Hilbert symbol

Statement

Let $a, b, c \in Q_{v}^{\times}$ . The ternary diagonal form

a X^{2} + b Y^{2} + c Z^{2}

is isotropic over $Q_{v}$ if and only if

(- a c, - b c)_{v} = 1.

Facts & Assumptions

Given: A place $v$ of $Q$ and nonzero elements $a, b, c \in Q_{v}$ .

[L1]

The binary form $a X^{2} + b Y^{2}$ represents $t$ exactly when $(a t, b t)_{v} = 1$ (Binary quadratic representation via the Hilbert symbol).

Proof

technique · direct

1.1

If the binary form $a X^{2} + b Y^{2}$ represents $- c$ , then some $(u, v)$ satisfies $a u^{2} + b v^{2} = - c$ , and therefore $(u, v, 1)$ is a nontrivial isotropic vector of $a X^{2} + b Y^{2} + c Z^{2}$ . Conversely, let $(x, y, z)$ be a nontrivial isotropic vector. If $z \neq 0$ , then dividing by $z^{2}$ shows that $a X^{2} + b Y^{2}$ represents $- c$ . If $z = 0$ , then $y \neq 0$ and $r : = x / y$ satisfies $b = - a r^{2}$ , so for any target $t$ the explicit choice $U : = (t + a) / (2 a)$ and $V : = (t - a) / (2 a r)$ gives $a U^{2} + b V^{2} = t$ . In particular, the binary form represents $- c$ . Thus ternary isotropy is equivalent to representation of $- c$ by $a X^{2} + b Y^{2}$ .

givenalgebra

2.1

Apply [L1] with $t = - c$ . Then the representation condition of step 1.1 is exactly $(- a c, - b c)_{v} = 1$ .

L1step 1.1algebra∎

Depends on

Binary quadratic representation via the Hilbert symbol

Used by

One local place is determined by the others for ternary forms Corollary
A local obstruction to a rational conic Example
Finite bad places for a ternary form Example
Hasse-Minkowski for ternary forms over Q Theorem
Hasse-Minkowski theorem over Q Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew V. Sutherland, 18.782 Lecture 11, Theorem 11.10 (standard reference, not scraped)
Sam Raskin, Introduction to the Arithmetic Theory of Quadratic Forms, section 4.5 (standard reference, not scraped)