# Centralizers of Cartan subgroups

Let $$E$$ be an elliptic curve with CM by an order $$\mathcal O$$ in an imaginary quadratic field $$K$$. Choose a basis for $$E[N]$$ to get an isomorphism $$\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\mathbb Z/N\mathbb Z)$$. Complex multiplication on $$E$$ induces a homomorphism $$(\mathcal O/ N\mathcal O )^\times\rightarrow \operatorname{GL}_2(\mathbb Z/N\mathbb Z)$$. Let $$C_N$$ be the image.

Is $$C_N$$ equal to its centralizer in $$\operatorname{GL}_2(\mathbb Z/N\mathbb Z)$$?

This should occur in some cases, and should give rise to a rational point on certain modular curves. See Burcu Baran: Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem, especially Proposition 4.1.

## 1 Answer

I believe the answer is yes. It is really a group theory question, so I am not the best person to answer this.

If $$N$$ is prime: The centraliser $$Z_G(C)$$ is included in $$N_G(C)$$. Since the action by conjugation of $$N_G(C)$$ is not trivial on $$C$$ (both in the split and non-split case), $$Z_G(C)\neq N_G(C)$$. As the index of $$C$$ in $$N_G(C)$$ is 2, we must have $$Z_G(C)=C$$.

For square-free $$N$$ use the Chinese remainder theorem to reduce to the prime case.

I have not thought about the case of a prime power, but my guess would be that it holds there too.

I don't know how this question relates to the rational point question. But yes, for CM curves, depending on the splitting behaviour of primes $$p\mid N$$ in the field $$\operatorname{End}(E)\otimes\mathbb{Q}$$, we get rational points on the (split and non-split) Cartan modular curves. The big question however is if there are any other rational points on these curves.