---
tag: YXWGV80Z2V
kind: concept
name: Arakelov theory
msc: [14G40, 11G50]
sources:
  - url: https://en.wikipedia.org/wiki/Arakelov_theory
same_as:
  wikidata: Q4783822
  wikipedia: Arakelov theory
---

## Summary

Arakelov theory is an intersection theory for arithmetic varieties.
These are schemes that are flat and of finite type over the ring of integers of a [[number field|MWBDFS2KJ5]].
Such a scheme behaves like a family over a curve that is missing its points at infinity, which correspond to the archimedean places.
Arakelov added these fibers at infinity by equipping the associated complex manifolds with Hermitian metrics and Green functions, so that intersection numbers become well defined.
The theory gives a geometric meaning to [[heights|F1FF505PWH]] and is a main tool of Diophantine geometry.

## Explanations

### L1 human

The ring of integers $\Z$ is in many ways like a curve, with one point for each prime.
But this curve is not complete: there is no point for the usual absolute value, the "infinite prime".
Arakelov theory fills in this missing point with analysis on complex manifolds.
After 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.
Heights of rational points appear as arithmetic degrees in this theory.