Canonical height pairing ($⟨P,Q⟩=\frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))$)

The canonical height pairing is $⟨P,Q⟩=(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))/2$. The height parallelogram identity makes it bilinear. It descends through torsion and is positive definite on the free part of the Mordell-Weil group. The Cauchy-Schwarz inequality gives $\hat{h}(P-Q)\le 2\hat{h}(P)+2\hat{h}(Q)$, useful for height descent.