Arakelov Geometry


In 1974 Arakelov introduced an intersection pairing for divisors on an arithmetic surface in which one also takes into account contributions coming from complex analysis by using Green's functions on Riemann surfaces. His results were extended by Faltings in a 1983 paper entitled Calculus on Arithmetic Surfaces.

These two papers constitute the beginning of Arakelov Theory and were conceived with the aim of studying rational points on curves of genus at least two. Faltings's famous theorem (formely Mordell's conjecture) states that the set of rational points on a smooth projective curve of genus at least two over a number field is finite.

Parshin's proof of the function field analogue of Mordell's conjecture was achieved by using classical intersection theory on algebraic surfaces fibred over curves. The aim of Arakelov's and Faltings's work was to prove the Mordell conjecture (over number fields) using similar methods. This required introducing an intersection pairing on arithmetic surfaces taking into account the archimedean places which was achieved by Arakelov by considering metrized line bundles.

Thereafter, Arakelov theory was proven to be ubiquitous in the study of heights of points on varieties, forming a part in the proof of Bogomolov's conjecture on small points on abelian varieties given by Szpiro, Ullmo and Zhang. Also, quite notably, Vojta's proof of Faltings's theorem (i.e., Mordell's conjecture) relied on the extension of the theory of Arakelov and Faltings to higher-dimensional varieties, which was achieved by Gillet-Soulé in the early nineties.

Nowadays, Arakelov theory has found its way into several parts of mathematics such as tropical geometry, special cycles on Shimura varieties, computational aspects of Galois representations, and algorithmic aspects of computing heights on curves.