# What is Riemann-Roch in arithmetic all about?

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
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
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

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