---
tag: QT4CF7ZKK5
kind: concept
name: Shimura variety
msc: [14G35, 11G18]
sources:
  - url: https://en.wikipedia.org/wiki/Shimura_variety
same_as:
  wikidata: Q7497063
  wikipedia: Shimura variety
---

## Summary

A Shimura variety is a quotient of a Hermitian [[symmetric space|PAA93R4WYD]] by a congruence subgroup of a [[reductive group|2XS84PWE3S]] over $\Q$.
It is viewed as an algebraic variety.
The family includes [[modular curves|T5W0CMS0DM]] and the Siegel moduli spaces of abelian varieties.
Shimura varieties are defined over number fields, and their cohomology carries both Galois actions and automorphic forms.
This makes them a main bridge in the Langlands program.

## Explanations

### L1 human

The upper half-plane, divided out by a group such as $\mathrm{SL}_2(\Z)$, gives a modular curve, which parametrizes elliptic curves.
Shimura varieties are the higher-dimensional versions of this picture.
They start from a symmetric space with a complex structure and divide it by an arithmetic group.
An example is the Siegel upper half-space of symmetric complex matrices with positive definite imaginary part.
The result is an algebraic variety that often parametrizes abelian varieties with extra structure.
Because it has a natural model over a number field, it links geometry, automorphic forms and Galois representations.