I have had a course in number theory where I studied Marcus and also a course in differential geometry. I have read Koblitz's introductory book on $p$-adic numbers. I am roughly interested in both $p$-adic numbers and elliptic curves. I would like to know what I can read so that I get a bird's eye view and a feel for things that happen in these two areas and also should go deep enough to know at least a few things in good depth.
I don't know any of these: algebraic geometry, representation theory.
What can I do with this modest background? If you have had any student in a similar situation, I would like to know what suggestion you gave him/her. Thank you very much!