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In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that certain sums converge uniformly, then for all ideles $a$, we have

$\frac{1}{|a|}\displaystyle\sum_{\xi\in k}\hat{f}(\xi/a)=\displaystyle\sum_{\xi\in k}f(a\xi)$.

Tate further refers to this theorem as the "number theoretic analogue of Riemann-Roch". My question is how this relates to the geometric Riemann-Roch theorems and why this deserves to be called an analogue of these theorems.

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Larry, Section 7.2 of Ramakrishnan and Valenza's "Fourier Analysis on Number Fields" answers precisely this question (as Theorem 7.12). Basically, when $k$ is a function field, applying Poisson summation to a certain $f$ (depending on a divisor $D$) gives the Riemann-Roch formula, in that the RHS is $q^{l(D)}$ and the LHS is $q^{l({\cal K}-D)+{\rm deg}(D)-g+1}$. –  B R Apr 16 '12 at 1:14
Minor correction: the point $a$ depends on $D$, the function does not. –  B R Apr 16 '12 at 1:57
May be you ca also give a look at Chapter 3 in Neukirch's book "Algebraic Number Theory" –  Filippo Alberto Edoardo Apr 16 '12 at 3:00

2 Answers 2

up vote 2 down vote accepted

Perhaps you might want to look in the book project of Frankenhuijsen http://research.uvu.edu/Math/machiel/RH.pdf and search for Riemann Roch theorem.

Quote from page 3: The function $ζ_C$ (zeta function of a curve) satisfies the functional equation $ζ_C (1 − s) = ζ_C (s)$. This functional equation can be proved using the Riemann–Roch theorem $$l(D) = deg D + 1 − g + l(K − D),$$ which is the analogue of (1) above.

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I think this one is really for the historians. I would guess that the thinking goes back to Emil Artin, and would have been developed in his lectures. Some of that can be seen in the polished edition "Algebraic Numbers and Algebraic Functions", called second edition (1967), but we can assume the ideas are from 20 years before. Artin's ideas can be seen in Lang, "Algebraic Number Theory" also, and (on different topics such as the Newton polygon) in Guido Weiss.

In other words, this is one "school of thought". The connection with the sheaf-theoretical view was probably worked out in greatest detail by Iwasawa (1950s). The Chevalley-Weil school of thought, which after all initiated the techniques, had different ideas on exposition, and the material surfaced in Weil, "Basic Number Theory".

Exposition and motivation is where this all belongs. The "global field" concept is a powerful set of analogies for those coming from number theory. For those coming from geometry, the fact that there are multiple ways of dealing with one dimensional Riemann-Roch is less suggestive, given that the issue for about 60 years has been about all dimensions.

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