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  "body": "\n## Summary\n\nArithmetic geometry applies the methods of algebraic geometry to problems of number theory.\nIts central objects are varieties and [[schemes|5A35TYZD02]] over [[number fields|MWBDFS2KJ5]], finite fields, local fields and rings of integers, and their [[rational points|SWJYG87QQH]].\nQuestions about [[Diophantine equations|08B1RTW7VZ]] become questions about the geometry of the variety that the equations define.\nLandmark results include [[Faltings's theorem|H0DB07WM20]], the [[Mordell–Weil theorem|MXRCPAH09H]] and the [[Weil conjectures|81TXP32VDM]].\n\n## Explanations\n\n### L1 human\n\nThe equation $x^2 + y^2 = 1$ describes a circle, and its solutions in rational numbers are rational points on that circle.\nArithmetic geometry studies such solutions through the shape of the curve or higher-dimensional object that the equations define.\nA striking lesson is that the shape controls the arithmetic.\nOver $\\Q$, a smooth curve of genus $0$ with one rational point, like the circle, has infinitely many rational points.\nA smooth curve of genus at least $2$ has only finitely many.\nThe field combines algebra, geometry and number theory, and it is the setting of Wiles's proof of Fermat's Last Theorem.\n",
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