---
tag: 824JAXQ5M6
kind: concept
name: Arithmetic geometry
msc: [14G25, 11G35, 14G05]
sources:
  - url: https://en.wikipedia.org/wiki/Arithmetic_geometry
same_as:
  wikidata: Q2179749
  wikipedia: Arithmetic geometry
---

## Summary

Arithmetic geometry applies the methods of algebraic geometry to problems of number theory.
Its central objects are varieties and [[schemes|5A35TYZD02]] over [[number fields|MWBDFS2KJ5]], finite fields, local fields and rings of integers, and their [[rational points|SWJYG87QQH]].
Questions about [[Diophantine equations|08B1RTW7VZ]] become questions about the geometry of the variety that the equations define.
Landmark results include [[Faltings's theorem|H0DB07WM20]], the [[Mordell–Weil theorem|MXRCPAH09H]] and the [[Weil conjectures|81TXP32VDM]].

## Explanations

### L1 human

The equation $x^2 + y^2 = 1$ describes a circle, and its solutions in rational numbers are rational points on that circle.
Arithmetic geometry studies such solutions through the shape of the curve or higher-dimensional object that the equations define.
A striking lesson is that the shape controls the arithmetic.
Over $\Q$, a smooth curve of genus $0$ with one rational point, like the circle, has infinitely many rational points.
A smooth curve of genus at least $2$ has only finitely many.
The field combines algebra, geometry and number theory, and it is the setting of Wiles's proof of Fermat's Last Theorem.