# Analogues of the Riemann-Roch Theorem

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

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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