I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what I've been concentrating on recently which has been learning algebraic geometry from a scheme viewpoint (mainly just for my own interest but also for my masters thesis next year, in which I will develop the basic theory and then apply it to a few as-yet undecided problems).
I'd really like to incorporate elliptic curves into this project by looking at their geometric and possibly arithmetic properties using this modern machinery. However with my current very basic knowledge of elliptic curves it seems like the full power (and elegance!) of schemes, categories etc might be unnecessary unless we look at either a) situations such as elliptic curves over general rings; b) moduli spaces of curves; or c) abelian varieties. Whilst all of these are nice topics I worry they're a bit too far away from actually studying elliptic curves!
So my question is - is there any "interesting" geometric/arithmetic information about elliptic curves over fields with some number-theoretic relation which can be studied most effectively using modern algebraic geometry? Or would it be better to study elliptic curves separately at first and find another application with which to better demonstrate the use of the theory that I develop in the first part of the thesis? Thanks very much in advance, and I'd welcome any reading recommendations.