---
tag: JDDXZ89N33
kind: concept
name: Abelian scheme
msc: [14K05, 14K10, 14L15]
specializes: [TJ04VQP4NB]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_variety
---

## Summary

An abelian scheme over a base scheme $S$ is a family of [[abelian varieties|R7W03TXRS9]] parametrized by $S$.
It is a smooth proper [[group scheme|TJ04VQP4NB]] over $S$ with geometrically connected fibres.
Abelian schemes describe abelian varieties with good reduction, and they are the objects classified by moduli spaces of abelian varieties.

## Explanations

### L1 human

An abelian variety is a single geometric object with an addition law.
An abelian scheme lets this object vary with a parameter, like a whole family of elliptic curves.
Over the integers, the parameter is a prime $p$, and the fibre at $p$ is the reduction of the variety modulo $p$.
For example, the curve $y^2 = x^3 - x$ gives an abelian scheme over $\Z[1/2]$, because it stays smooth modulo every odd prime.