One way to study (mixed signature) quadratic forms in two variables is to study the topograph. Looks like the signatrue doesn’t matter: here is (1,1) and (1-,1).
Literally derived from the word “topography”, I read, that if you’re really good at reading these things you can extract information like the genus, class number, solve Pell eq etc.
There are two resources I found for this type of thing:
- The Sensual Quadratic Form John H Conway
- Topology of Numbers Allen Hatcher
They are basically drawing the dual tree of the Farey Tesselation, which is a tiling of $ mathbb{H}$ or $ mathbb{D}$ by hyperbolic triangles. Is there a more serious name for putting trees on $ mathbb{H}$ ?
This question emerges, for example, trying to draw these things with a computer and I needed to decide a natural place to put the interior vertices, and I couldn’t think of one. The “outer” vertices are indexed by $ text{P}mathbb{Q}^1$ and the interior vertices could be in any reasonable place.
There could be a serious name for this structure, like the Bruhat-Tits building or maybe it’s in Serre’s book on Trees. Any guidance?
