Given a lattice $\wedge = \{\omega_1, \omega_2 \}$ in $\mathbb{C}$, $\omega_1 / \omega_2 \not\in \mathbb{R}$, we know that $\wedge' = \{\omega_1', \omega_2' \}$ defines the same lattice precisely when $$\begin{pmatrix} \omega_1' \\ \omega_2' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d\end{pmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}$$ for the acting matrix an element in $SL_2(\mathbb{Z})$.
I am currently reading about extensions of $SL_2(\mathbb{Z})$ to larger groups, specifically $SL_2(\mathcal{O}_d)$, $\mathcal{O}_d$ the ring of quadratic integers in a quadratic field $K$, for some $d < 0$ (namely Bianchi groups).
In most of the literature I have read, I have not found an intuitive reason for why we might want to study such an extension. We care about $SL_2(\mathbb{Z})$ because lattices in $\mathbb{C}$ are invariant under an action by a fractional linear transformation described by elements of $SL_2(\mathbb{Z})$, and this in turn relates to studying elliptic curves and modular forms and so on.
In a nutshell, here are some of my questions:
- Is the space that we are acting on with $SL_2(\mathcal{O}_d)$ still the upper half plane, or do we extend that as well?
- Is there such an analogue reason for why we might want to consider the action of $SL_2(\mathcal{O}_d)$? For example, are lattices, or specific sets of lattices, now invariant under this transformation? If so, what kinds of consequences result from that?
- How does the class number of the field $K$ that $\mathcal{O}_d$ is derived from affect our interpretation here? I ask because not all rings for arbitrary $d$ have characteristic 1, and this should affect whether or not the ring is a P.I.D., and then a unique factorization domain. In turn, I believe that this would affect the resulting structure of whatever analogue of a Hecke Algebra (if we can form one) we could form.
- Is there a general way to understand the action of $SL_2(\mathcal{O}_d)$ for arbitrary $d$? Currently I mostly care about $d < 0$ (generating an imaginary quadratic field), but if there is a general framework / intuition that motivates all $d$, I would be interested to hear it. I am aware that for $d > 0$, we generate a real quadratic field, and such actions described lead to the study of Hilbert Modular forms, but I am unsure if this is too far removed or too large of topic to tackle all at once.
- If you have any good sources to read about this from, I would appreciate references! Currently I am reading through John Cremona's thesis, Modular Symbols, although I find that he has omitted many details, and that is making it hard for me to work through his text without the proper motivation or background.
I apologize for the long winded questions, but I would really appreciate answers to them!