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  "body": "\n## Summary\n\nA height function measures the arithmetic complexity of a rational or algebraic point, roughly the number of digits needed to write down its coordinates.\nThe basic example is the Weil height on projective space; composing it with a morphism gives heights on any projective variety.\nThe key property, due to Northcott, is that there are only finitely many points of bounded height and bounded degree.\nHeights drive finiteness proofs such as [[Mordell's theorem|BZHGKN6Q3K]] and appear in the [[Gross–Zagier theorem|BX558DTV2D]] and the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].\n\n## Explanations\n\n### L1 human\n\nThe rational numbers $1/2$ and $1000001/2000001$ are close in size, but the second is much more complicated.\nA height captures this: the height of a fraction $a/b$ in lowest terms is $\\max(\\abs{a}, \\abs{b})$, so the first has height $2$ and the second has height $2000001$.\nThere are only finitely many fractions of height at most a given bound, so heights let one turn \"infinitely many points\" into \"points of ever larger complexity\".\nFor points with several coordinates, one takes the largest coordinate after clearing denominators.\nOn an elliptic curve, a refined height behaves like a quadratic form under the group law, which is the engine of the descent proofs.\n",
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