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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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(?).
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.