Intuition for Little Picard's theorem

Solution 1:

$\mathbb{C} - \{ 0, 1 \}$ can be thought of as the modular curve $Y(2) \cong \mathbb{H}/\Gamma(2)$ parameterizing elliptic curves together with a basis for their $2$-torsion. This parameterization takes a point $\lambda \in \mathbb{C} - \{ 0, 1 \}$ to the elliptic curve $y^2 = x(x - 1)(x - \lambda)$ together with the ordered basis $(0, 0), (1, 0)$ (say) for the $2$-torsion. Explicit formulas for the covering map can be found, for example, in Dolgachev's Lectures on Modular Forms, Lecture 9.

As for intuition, I suppose one could say the following: generically we should expect a hyperbolic structure. $\mathbb{C}$ is not generic as it has a group structure, and $\mathbb{C} - \{ 0 \}$ is also not generic as it also has a group structure, but $\mathbb{C} - \{ 0, 1 \}$ has no obvious group structure so we can expect generic behavior.