The aim of Arakelov geometry is to extend intersection theory to the case of curves over $Spec(\mathbb{Z})$.
Arakelov complemented the algebraic geometry at finite primes with a holomorphic piece at a point at infinity. Then using complex analytic geometry and Green functions he defined the intersections numbers using the complementary piece at infinity.
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Recently, found (arxiv)
S. J. Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR Izv. 8 (6): 1167–1180, 1974, doi; Theory of intersections on an arithmetic surface, Proc. ICM Vancouver 1975, vol. 1, 405–408, Amer. Math. Soc. 1975, djvu, pdf After Arakelov there were main improvements by Faltings and Gillet and Soulé.
Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424, MR86e:14009, doi; Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), no. 3, 337–347, MR85m:14061, doi
H. Gillet, C. Soulé, An arithmetic Riemann–Roch Theorem, Invent. Math. 110: 473–543, 1992, doi
Shou-Wu Zhang, Small points and Arakelov theory, Proc. ICM 1998, vol. 2, djvu, pdf
wikipedia: Arakelov theory
In a recent Bonn thesis under Faltings’ supervision,
a completely algebraic replacement (using generalized schemes whose local models are spectra of commutative algebraic monads) for the original mixed approach is proposed; it is not known if that approach can be closely and precisely compared with the traditional.