A couple of years ago, I taught an undergraduate class introducing various aspects of classical geometry, learning the (beautiful!) subject as I went. I found one thing that really bothered me: the definition of lines in the upper half-plane model of the hyperbolic plane felt rather *ad hoc*. Although there is a nice explanation in terms of differential geometry, the algebraist in me was hoping that they could be alternatively explained through Galois theory. I gave this a shot and was unable to get it to come together, and (as an obvious novice in the subject) I was hoping someone could point me to where this has already been considered.

$ \newcommand{\Re}{\mathrm{Re}} \newcommand{\Im}{\mathrm{Im}} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\RP}{\R\mathrm P} \newcommand{\CP}{\C \mathrm P} \newcommand{\Q}{\mathbb Q} \DeclareMathOperator{\Gal}{Gal}$ First, a few relevant facts about the upper half-plane model:

- Lines take one of two forms: they are the (nonempty) complex solution sets to $ \Re(z) = b$ or $ z \bar z – b(z + \bar z) + c = 0$ for $ b$ and $ c$ real.
- Lines in the upper half-plane are determined by their pair of incidences on the horizon line $ \RP^1$ .
- Isometries of the upper half-plane fix the horizon $ \RP^1$ and are also determined by their behavior on it.

A common feature of the above facts is the governance of the reals: everything in sight is Galois-invariant for $ \Gal(\C/\R)$ . Accordingly, rather than privilege the upper half-plane, I would like to interpret this model as the quotient of $ \CP^1$ by the Galois action of $ \Gal(\C/\R)$ . In particular, isometries of this object are then inherited from those of $ \CP^1$ which fix $ \RP^1$ : these are the expected real linear fractional transformations.

In an effort to recover the formulas for the lines, let’s pick incidence points $ p$ and $ q$ in $ \RP^1$ and go looking for a quadratic Galois-invariant expression that contains these points in its zero locus. (Forgive me for working away from infinity in what follows.) We might first write down $ (z – p)(z – q)$ —and this obviously does not have the expected solution set in $ \CP^1$ , as it is a polynomial and hence has finitely many zeroes. The next thing to try, knowing that $ \bar z$ appears in the expected equation, is averaging this polynomial with its Galois conjugate: $ $ \frac{(z – p)(z – q) + (\bar z – p)(\bar z – q)}{2} = \Re(z^2) – (p+q)\Re(z) + pq.$ $ While this does have an infinite solution set, it again is not the expected result. Instead, though, if we tweak the original expression by shuffling the Galois action through, we arrive at $ $ 0 = \frac{(z-p)(\bar z – q) + (\bar z – p)(z – q)}{2} = z \bar z – c(z + \bar z) + (c^2 – r^2)$ $ for $ c = (p+q)/2$ and $ |r| = |p – q|/2$ .

At this point, I’m excited to have guessed the right answer, but I’m unable to conceptualize why I should have jumped right here from the start. Accordingly, my question is a bit nebulous:

Question:Is there a conceptual reason why it’s reasonable to have shuffled the Galois conjugation action through the expression for the real quadratic with roots $ p$ and $ q$ ? Is there a purely Galois perspective on why this equation deserves to be called anything like a “line” in $ \CP^1\;//\;\Gal(\C/\R)$ ?

Question:What happens if $ \C/\R$ is replaced with something like $ E/\Q$ or $ E_p / \Q_p$ a quadratic extension? Is there a fruitful theory of plane geometry here? (What about larger extensions?)

I previously asked a version of this question on math.stackexchange, and while I learned something, I don’t think it got properly answered.