Geodesic
Intersection theory of divisors on an arithmetic surface
Izvestiya. Mathematics , Tome 8 (1974) no. 6, pp. 1167-1180.

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In this article it is explained how to construct for a nonsingular model of a curve defined over a number field a theory analogous to the theory of divisors, and the intersection numbers of divisors, on a compact algebraic surface. 
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     title = {Intersection theory of divisors on an arithmetic surface},
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S. Yu. Arakelov. Intersection theory of divisors on an arithmetic surface. Izvestiya. Mathematics , Tome 8 (1974) no. 6, pp. 1167-1180. http://geodesic.mathdoc.fr/item/IM2_1974_8_6_a0/

[1] Shiffer M., Spenser D. K., Funktsionaly na konechnykh rimanovykh poverkhnostyakh, IL, M., 1957