= Arithmetic cube identity for Weil heights
{title2=$h(P+Q+R)-h(P+Q)-h(P+R)-h(Q+R)+h(P)+h(Q)+h(R)-h(O)=O(1)$}
For a <line bundle> on an <elliptic curve> over a <number field>, the geometric cube identity remains exact as a line-bundle identity. The <Weil height machine> translates it into the displayed bounded-error identity for a chosen logarithmic height, uniformly on the rational triples. This distinction is essential: arbitrary naive heights do not obey an exact quadratic identity. Dividing the identity for $2^nP,2^nQ,2^nR$ by $4^n$ produces an exact identity for the <canonical height of an elliptic curve>.
Back to article page
