Books/Notes recommendation request: Multivalued functions/Riemann surfaces

Question

I'm trying to read a document that applies Riemann-Roch left, right and center. I don't know this theorem or the theory it comes from so I need to build up a bit more background before I can tackle this.

Can you please recommend good books or (online) lecture notes which cover "multi-valued functions", Riemann Surfaces and similar up to (at least) Riemann-Roch? (I also want to pick up a bit about modular forms and the relation between lattices and elliptic curves).

(I've done a bit of complex calculus but it was all with analogy to real analysis so I am not sure that it will really give me any head start here. Also apologies for being so vauge with this but I don't know enough about this subject to be any more precise)

Answer 1

The book by Otto Forster on Riemann Surfaces is pretty good. I never finished reading it myself, but it covers things like Riemann-Roch and Abel's theorem from a sheafish viewpoint. In particular, the proof of Riemann-Roch is analogous to the one in Hartshorne; it follows from the Serre duality theorem and an inductive argument. Learning about sheaves is definitely a plus.

Also, there's a book by Springer, though the level is a bit more elementary.

Answer 2

You might like Raghavan Narasimhan's "Compact Riemann Surfaces". It does things like the construction of the Jacobian. It seems to suit your interests best.

Answer 3

I recommend Serge Lang's book 'introduction to algebraic and abelian functions'. It is a bit expensive but worth the price. Of course you may find a library copy somewhere.