Context: I am a junior math major and am hoping to go to grad school after next year for a PhD. I have completed most of the standard undergraduate courses and have been consistently most interested by number theory. I've taken three courses in number theory; the first using Niven's text, the second using Ireland and Rosen, and the third using Montgomery and vaughan's Multiplicative Number Theory.
Next year, I have room for four independent study courses. One of them will likely be on (introductory) algebraic geometry but with the other three, I wish to gain some breadth within number theory to find out what I am most passionate about.
Questions: What are the popular branches of number theory being actively researched today and what are some good introductory texts in each of these?