---
tag: D3HEJTVY7G
kind: concept
name: Complex multiplication
msc: [11G15, 14H52, 14K22]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_multiplication
same_as:
  wikidata: Q2654923
  wikipedia: Complex multiplication
---

## Summary

An [[elliptic curve|427N6S6QQW]] has complex multiplication when it has more [[endomorphisms|0J7H0YBFDJ]] than the multiplication-by-$n$ maps.
In characteristic zero its endomorphism ring is then an order in an imaginary [[quadratic field|7SV4WA92SM]].
The theory of complex multiplication describes the [[j-invariants|FEC2XH822R]] and torsion points of such curves.
These values generate abelian extensions of imaginary quadratic fields, such as the [[Hilbert class field|WWCJ7CR2R3]].

## Explanations

### L1 human

On every elliptic curve one can add a point to itself $n$ times, which gives an endomorphism "multiplication by $n$".
Some special curves have extra symmetries.
For example, on $y^2 = x^3 - x$ the map $(x, y) \mapsto (-x, iy)$ sends the curve to itself and respects addition, and doing it twice gives $P \mapsto -P$.
So this map behaves like multiplication by the complex number $i$, hence the name.
These special curves are rare, but they have a particularly rich arithmetic.