There's also the topology on the integers in Fürstenberg's proof of the infinitude of primes.

I was doing math research for some years after the PhD and enjoying it. I'm not doing research now, but that's due to family matters.

Some will read them later as they need facts, and will realize how beautiful they are. That is so true. I needed some facts concerning differential operators for my PhD thesis. After asking several experts in related fields and obtaining no clear answers I gave up asking and decided to lift through EGA (or SGA, I've forgot). And there it all was: very clearly and thoroughly stated. I was ashamed I let the immensity of the work scare me away before. If I had had the courage to go to it instead of learning from Hartshorne's alone, I would have been much more happy.

The generalization from $\exp(2\pi iz)$ to elliptic curves isn't that difficult to imagine (although it certainly is beautiful!). The values of $\exp(2\pi iz)$ at rational numbers are roots of unity which are the torsion elements of the algebraic group $\Q^{x}$. The points of finite order on an elliptic curve are the torsion elements of the algebraic group that is the elliptic curve. So one gets abelian extensions by adjoining torsion elements of algebraic groups.

There is a discussion of this fact in Milne's "Modular Functions And Modular Forms" notes, page 95-96.