I am looking for a problem in (algebraic) number theory in which a 1st/2nd year graduate student in number theory can make some nontrivial progress. I am not looking for something that can conceivably be answered or has deep implications, but rather something that is more of a project (aka relatively accessible).
Background Knowledge: I have taken graduate algebraic NT(ramification, valuations, little bit of CFT) and am currently taking a course in elliptic curves. I know a fair amount of commutative algebra, modular forms and undergraduate algebraic geometry. I haven't read through Hartshorne yet, was planning to do that next semester.