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  "body": "\n## Summary\n\nArakelov theory is an intersection theory for arithmetic varieties.\nThese are schemes that are flat and of finite type over the ring of integers of a [[number field|MWBDFS2KJ5]].\nSuch a scheme behaves like a family over a curve that is missing its points at infinity, which correspond to the archimedean places.\nArakelov added these fibers at infinity by equipping the associated complex manifolds with Hermitian metrics and Green functions, so that intersection numbers become well defined.\nThe theory gives a geometric meaning to [[heights|F1FF505PWH]] and is a main tool of Diophantine geometry.\n\n## Explanations\n\n### L1 human\n\nThe ring of integers $\\Z$ is in many ways like a curve, with one point for each prime.\nBut this curve is not complete: there is no point for the usual absolute value, the \"infinite prime\".\nArakelov theory fills in this missing point with analysis on complex manifolds.\nAfter that, one can intersect divisors on arithmetic surfaces as on ordinary projective surfaces, and prove analogues of the Riemann–Roch theorem and the Hodge index theorem.\nHeights of rational points appear as arithmetic degrees in this theory.\n",
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