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I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of viewing all this stuff geometrically and how?

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Dear Yoshinobu, Could you please give a little more context? Do you mean, e.g. Riemann--Roch as stated in Tate's thesis (so some theorem about adeles satisfying certain properties), or do you mean the Riemann--Roch theorem for curves (maybe over a finite field) being applied in an arithmetic situation? Best wishes, –  Matt E Jun 14 '11 at 11:47
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My guess is that the phrase "learn number theory recently" means a first course in algebraic number theory which probably means Riemann-Roch for curves say as given in Neukirch. –  Matt Jun 14 '11 at 14:08
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I really mean riemann roch for curves and corresponding divisors as mentioned in Neukrich's book,there he make a few comments which I do not understand such as the anology with function fields of genus ,euler characteristic etc. –  Yoshinobu Osawa Jun 14 '11 at 14:11

2 Answers 2

Have a look at Tate's thesis "Fourier analysis in number theory and Hecke's zeta function" (theorem 4.2.1), where he describes it as an equivariant Poisson summation formula.

In theorem 4.4.1 he proves the functional equation for $L$ functions with it.

It works equally well for function fields. I think this is elaborated in Bump "Automorphic representations" in chapter 3(?).

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I don't know a lot about Riemann-Roch, but the most important theorem I have seen being proved using R-R is a theorem on the dimension of spaces of modular/cusp forms. Here the Riemann surface in question is a (compactified) modular curve, and Riemann-Roch is applied in this context. From my point of view, this is already a huge application of R-R, but I am sure there are even more important applications.

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