>>12576656I'll sketch something for you.
Pre-reqs 1:
One variable real analysis and an introductory course in algebra, linear algebra doesn't hurt either.
Introductory books:
Ireland's already cited; Ivan Niven's, Cox's "Primes of the form x^2+ny^2" is a great intro for the more algebraic (and modular) side of NT. There are others. Pic one, you don't want to spend too much time of your life on this stage. You can start simutanously with pre-req 1 courses.
Pre-req 2 (here I assume you learned most of elementary number theory from the previous stage).
For analytic number theory:
-Measure and Lebesgue integration;
-Complex Analysis
-(Optional) Something on Fourier analysis, harmonic analysis, complex geometry etc.
For Algebraic Number Theory:
-Groups, rings, modules, Galois theory... other undergrad stuff like topology
-Commutative Algebra helps a lot, though it's not so much necessary to start
-Notions of measure theory and integration, not that necessary but it helps in a few parts
For Arithmetic Geometry:
-All the undergrad algebra (groups, rings, etc), and a few others like topology
-Commutative Algebra
-A course in algebraic curves can help, see Fulton's for instace (not so necessary though)
For Modular Forms and Elliptic Curves:
-Complex Analysis
-Having a notion of basic algebraic curves helps
-undergrad courses of algebra, topology too
There's also additive number theory that I don't know much about, I think basic undergrad courses in combinatorics and probability would be enough. Knowledge on asymptotics would help a lot.
