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. 



Analogues of the Riemann-Roch Theorem
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. 
> 
> 
> 


