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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)

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    $\begingroup$ mathoverflow.net/questions/1916/riemann-surfaces mentions several. $\endgroup$ – anon Jul 30 '10 at 11:59
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    $\begingroup$ The other answers are fine, but to mention two other approaches, you could use Rotman's homological algebra which requires very little background beside comm alg. Another obvious approach is the more conventional alg geom route for which I mention Ravi Vakil's notes on algebraic geometry. $\endgroup$ – BBischof Jul 30 '10 at 17:30
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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.

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Not being trained in Mathematics, I have the distinct impression that the British book by S. Donaldson (Oxford University Press) is the easiest to read.

I am not claiming it is the very best in terms of breadth and depth of knowledge in an attempt to acquire a fairly decent understanding of Riemann Surfaces, but I am impressed that at least a retired engineer (with graduate degrees in Electrical Engineering) like me, not being ingrained in topological concepts, can start reading his book here. Even the classic work by George Springer (Indiana University) can be challenging in that the writings after the third chapter can be daunting to those without the proper background and prerequisites, i.e., Real and Complex Analysis and Introductory Topology.

There may be other ones out there I am unaware of. That is just my two cents here.

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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.

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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.

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