I had trouble parsing the statement of this conjecture. The modular curve I can think of is $\mathbb{H}/\text{SL}(2,\mathbb{Z})$ or I think it can also be written: $\text{PSL}(2,\mathbb{R})/\text{PSL}(2,\mathbb{Z})$. These parameterized complex structures on a single elliptic curve. The other modular curves $X_0(N)=\Gamma_0(N)\backslash\mathbb{H}$ could parameterize other objects, I don't know. Isn't the Fermat curve a modular curve?
It doesn't seem to matter either, because CM-points (otherwise known as Heegner points) are basically just the images of algebraic numbers in $\mathbb{H}$ and they have finite orbits under $\text{SL}(2,\mathbb{Z})$. So there are finitely many images on these various modular curves.
The conjecture seem to say that the Picard group (possibly $\text{SL}(2,\mathbb{Z})$ is mixing) on the images of the algebraic numbers in $\mathbb{H}$ in the limit of large norm.
What is the status of this conjecture? And why they say "packet" of CM points (I gave my best guess)? Is this related to the Quantum Mechanical wave packets?
- Joint Equidistribution of CM Points arXiv:1710.04557 Ilya Khayutin
The paper says more. If $G = PGL_2$ then $Y=G(\mathbb{Q})\backslash G(\mathbb{A})/K$ where $K \subset G(\mathbb{A})$ is compact and open. That having been said there is a working definition of packet.
I do not know the meaning of "class number 1" description. A "torus" I'm hoping is a collection of diagonal matrices in $G$ or else I am lost. The space $Y$ could be an analogue of the modular curve $\mathbb{H}/\text{SL}(2,\mathbb{Z})$ if I knew which adelic parameters to set.