---
date: 2022-07-30 13:50
modification date: Saturday 30th July 2022 13:50:44
title: "CM"
aliases: [CM]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #todo/untagged
- Refs:
	- Silverman's advanced topics book
	- Section by Serre in Cassels-Frohlich
	- Shimura's book
	- Milne's notes <https://www.jmilne.org/math/CourseNotes/CM.pdf>
- Links:
	- [Kronecker-Weber](Unsorted/Kronecker-Weber%20theorem.md)

---

# CM

![](attachments/2023-02-27hodgecm.png)

Example: the curve $y^2=x^3-x$ is defined over $\mathbf{Q}$, has no $\mathrm{CM}$ over $\mathrm{Q}$, but does have $\mathrm{CM}$ over $\mathrm{Q}(i)$ on account of the extra automorphism $(x, y) \mapsto(-x, i y)$

![](attachments/Pasted%20image%2020220730135103.png)
![](attachments/Pasted%20image%2020220730135124.png)

## Misc


- An elliptic curve $E/k$ for $k=\QQ(\sqrt{-d})$ has **complex multiplication** iff $\Endo(E) \supsetneq \ZZ$
![](attachments/Pasted%20image%2020220227152733.png)
![](attachments/Pasted%20image%2020220227153127.png)

![](attachments/Pasted%20image%2020220501131606.png)

![](attachments/Pasted%20image%2020220514171323.png)