TITLE: Analogues of the Riemann-Roch Theorem
QUESTION [15 upvotes]: 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.

REPLY [6 votes]: 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.