@Ian You need to know about completeness for Cauchy sequences... because such series need not converge if your (wherever you're doing math) isn't complete!
But, it's kind of obnoxiously advanced, if not neat. For, if you change that definition into the context of ellpitic curves you get the Tate-Shafarevich group.
@PedroTamaroff Sure. Whenever $\mathfrak{m}$ is the maximal ideal of a valuation ring, you can just understand it as the valuation theoretic completion
@FernandoMartin It depends on how you define algebraic number theory. ABC is one, whether or not that is conjectural anymore is to be seen. Mordell's conjecture was a big one. The sections conjecture could loosely be defined to be number theory.
@FernandoMartin Once again, I guess that depends on how loosely you define analytic number theory. If you mean things that involve analysis, how about BSD?
Let $V(\mathfrak{a})$ be all elements of ${\rm Spec}(A)$ containing $\mathfrak{a}$. How does an arbitrary intersection of $V(\mathfrak{a}_i)$'s equal another $V(\mathfrak{a}')$ ?
@KarlKronenfeld It motivates me in the sense that I get feedback from those more knowledgeable than me, I get to talk to peers at length including doing study groups with them, and I have time to do it, in the sense that I am not also working another job.
@KarlKronenfeld It's not that there is no enjoyment in the idea of studying math not in school, I would also probably do it, it's just a much more difficult endeavor. Both in terms of motivational factors and logistically.
@AlexYoucis I am actually going to school in the fall for precisely the first two reasons you gave, so I am not ardently against school. That said, I highly suspect my appreciation for math and my motivation to do math will remain unchanged even in that environment.
@KarlKronenfeld Yeah, it's not about local concern, it's easy to do math if you know you'll eventually be around others to chat with, it's just hard globally. Math is the best, of course, but it really is, in my humble opinion, a team sport.
