The triply punctured sphere

I’ll develop this post later, but the main purpose for now is to consolidate and share some thoughts/notes of mine that have been floating around for a few semesters. Let denote the doubly punctured plane, or equivalently, the triply punctured Riemann sphere. We will show that the universal cover of is the upper half-plane.

We have a natural covering map . To define this, associate to each a complex torus given by , where is generated by . Moreover, fix a basis for the points of order on this quotient; let .

The Weierstrass -function gives a natural degree-two map and note that is uniquely determined by its evenness (up to composition with a M"{o}bius transformation). Define by the cross-ratio and note that this modular function is invariant under the congruence subgroup . Thus, it descends to an isomorphism . I’ll edit this post later to justify some of these facts in greater detail, and I’ll also prove Picard’s Little Theorem with it.