The height function is a way to measure the complexity or size of rational points on algebraic varieties, particularly in arithmetic geometry. It provides a quantitative tool to analyze the distribution of rational points, connecting deeply with concepts such as the Mordell-Weil theorem and the properties of elliptic curves represented by Weierstrass equations. This function plays a pivotal role in understanding not just rational points, but also preperiodic points and their dynamics within the framework of conjectures like the Dynamical Manin-Mumford conjecture.
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