Why are elliptic points called elliptic?

Question

Points on the upper half plane $\mathbb H := \{ z \in \mathbb C : \Im(z)>0 \}$ are called elliptic with respect to some $\gamma \in \operatorname{SL}_2(\mathbb Z)$ if they are fixpoints of the Möbius transformation induced by $\gamma$.

Why elliptic?

Answer 1

This is because of the condition on elements $\gamma \in \mathrm{SL}_2(\mathbb{Z})$ you get by imposing that it has exactly one fixed point in the interior of the upper half plane. If you solve the equation $\gamma \tau = \tau$ for some $\tau \in \mathbb{H}$, and $\gamma$ doesn't fix any other point, it implies that the absolute value of the trace of $\gamma$ is less than 2. Elements of the modular group with absolute value of trace equal to or greater than 2 are called parabolic and hyperbolic respectively. So the monikers are with respect to some inequality being satisfied.

Answer 2

We consider Möbius transformations $z\to\frac{az+b}{cz+d}$ with corresponding matrices in $SL(2,\mathbb{R})$. These Möbius transformations can be characterized by the trace of the matrices as well as by the number of fixed points.

The connection with fixed points can be found in the paper Isometries of the hyperbolic plane by A. Chang. We find in section $4$: Characterization of Isometries:

Now, we will see how Möbius transformations can be characterized by the trace of their matrices. ... Since the determinants of these matrices must equal $1$, the absolute value of the traces of the matrices will be respectively less than $2$, called elliptic, greater then $2$, called hyperbolic, and equal to $2$, called parabolic if it is not the identity transformation.

However, another way to reach this same conclusion is through the use of fixed points.

We find the following statements:

• Any Möbius transformation $A\in PSL(2,\mathbb{R})$ has at least one fixed point in $\mathbb{C}\cup\{\infty\}$.

• If a Möbius transformation fixes three or more points, then it is the identity transformation.

• If the Möbius transformation $A\in PSL(2,\mathbb{R})$ has one fixed point in $\mathbb{R}\cup\{\infty\}$, then the absolue value of the trace of its matrix is $2$.

• If the Möbius transformation $A\in PSL(2,\mathbb{R})$ has two fixed point in $\mathbb{R}\cup\{\infty\}$, then $\left|\mathrm{tr(A)}\right|>2$.

• If the Möbius transformation $A\in PSL(2,\mathbb{R})$ has one fixed point in $\mathbb{H}^2=\{x+iy\big|y>0; x,y\in\mathbb{R}\}$, then $\left|\mathrm{tr(A)}\right|<2$.

Elliptic, hyperbolic and parabolic points are graphically shown here in section 1.4.

We conclude the naming convention of the fixed points is due to the classification in elliptic, hyperbolic and parabolic matrices of the corresponding Möbius transformations.