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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.

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    $\begingroup$ Just to be clear -- are you interested in a problem which has already been solved, or a problem which is not yet solved? And are you intending to work on this problem under the guidance of a PhD supervisor, or are you going it alone for some reason? Finally just to remark -- if I run into such problems (and I do occasionally) then I tend to give them to my own PhD students rather than posting them on the internet, because they can be quite hard to find. $\endgroup$
    – Kevin Buzzard
    Commented Feb 8, 2017 at 8:26
  • $\begingroup$ Yes, I am intending to work on this problem with my adviser (we are having trouble finding one). Problems that have been solved, but can be approached from another viewpoint (ex: through more elementary means) are still good. I understand the scarcity of these problems and why wouldn't post them on the internet, but was hoping someone could push us in the right direction. $\endgroup$
    – CL1337
    Commented Feb 8, 2017 at 18:56

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