My question concerns some statements in this essay:
https://www.maa.org/programs/maa-awards/writing-awards/number-theory-as-gadfly
Consider the map
$$\mathcal{U}:\mathbf{C}-\Lambda \to \mathrm{elliptic \, curve}-\mathrm{origin}$$
defined by
$$\mathcal{U}(z) = (X,Y) = (\mathcal{P}(z),\mathcal{P}'(z))$$
where $\mathcal{P}$ is Weierstrauss's $\mathcal{P}$-function and $\Lambda$ a lattice in the complex plane. Mazur says that $\mathcal{U}$ is a uniformization of an elliptic curve.
His places a lot of emphasis on this but I don't understand what he means by "uniformization." My understanding of the word is that to uniformize something means to take its constituents and give them all the same ("uni") properties ("form").
We have uniformly parameterized the complex points of any elliptic curve by the points of $\mathbf{C}$.
What does it mean to uniformly parameterize these points? What are the simplest examples of nonuniform parameterizations and of uniform parameterizations generally?
The uniformization $\mathcal{U}$ also uniformizes the conformal geometry of any particular elliptic curve, in the sense that it identifies the conformal geometry of the Riemann surface, locally, with Euclidean conformal geometry.
This is less opaque; the "same form" each open set is being given is that of the conformal structure of the plane. But I don't have a good picture of what a "local uniformizer" is. What are the simplest examples of local analytic functions on a Riemann surface which are (or aren't) local uniformizers?
Mazur's terminology is a bit nonstandard. However, it does fit within the confines of the following definition:
Definition. Let $X$ be a Riemann surface. A uniformization of $X$ is a holomorphic covering map $f: \Omega\to X$, where $\Omega$ is a domain in the Riemann sphere ${\mathbb C}\cup \{\infty\}$.
In practice, one usually insists on a simply-connected domain $\Omega$ (see this Wikipedia article), but there are some exceptions such as Schottky uniformization (due to Koebe) or the one used by Mazur. As for a good example which fails to be a uniformization, think of a locally biholomorphic surjective map $f: {\mathbb C}\to {\mathbb C}$ which is not 1-1. For instance: $$ z\mapsto \int_0^z e^{-t^2}dt. $$