Here is one way to say Serre's (pen)(mage)(heorem). Consider the system of modular curve covers defined as the projective system of spaces , for an odd prime . This has a projective system of arithmetic and geometric monodromy groups which are known to be and , , respectively.
Then, the intersects the decomposition group of a projective system of points lying over an algebraic value of with the projective limit of the geometric monodromy groups of . The says there are just two types of groups depending on which you choose, called CM and GL.
Modular Towers gives another way to look at this same system, and thereby a road to generalizing it. We look at a mysterious aspect that arises early in Serre's result, the use of the Weil Pairing, and see how that interprets in a new system.
Comparing Serre's Open Image Theorem with a new system of Moduli spaces