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## Fractional linear transformations and elliptic curves

The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even).

Lemma 1: Fix a field $F$. The group of fractional linear transformations $PGL_2(F)$ acts triple transitively on $\mathbb{P}^1(F)$ and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on $\mathbb{P}^1(\mathbb{C})$ preserving the upper half plane $\mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \}$ is $PSL_2(\mathbb{R})$.

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).