I recently learned why the Tate's thesis over a function field $F = \mathbb{F}_q(X)$ of a smooth projective curve $X_{/ \mathbb{F}_q}$ is Riemann-Roch theorem. This essentially follows from (a version of) Poisson summation formula
$$ \sum_{a \in F} f(ax) = \frac{1}{|x|} \sum_{a \in F} \widehat{f}\left(\frac{a}{x}\right) $$
and apply this to $f = \mathbf{1}_{\mathcal{O}_{\mathbb{A}_F}}$ & interpret both sides as cardinalities of linear systems of certain divisors (I followed Ramakrishnan-Valenza, Chapter 7.2). Here we choose $x = x(D)$ an idele with $v(x(D)_v) = n_v$ for $D = \sum_v n_v v$. After reading this, I have two question arose in my mind:
We also have a similar Poisson summation formula for $\mathrm{GL}_n$ that is used by Godement-Jacquet for their theory of automorphic $L$-functions of $\mathrm{GL}_n$. If we read the formula for a function field again, does it give any Riemann-Roch-like theorem for vector bundles over $X$? (I'm not sure which one is the right direction for generalization - increasing the dimension of $X$ or the rank of bundle on $X$). Especially, I wonder if this gives a proof of Riemann-Roch for vector bundles of rank $n$.
As Tate did in his thesis, we can choose a different test function $f$ other than the simplest one $\mathbf{1}_{\mathcal{O}_{\mathbb{A}_F}}$. Especially, we can choose $f = \otimes_v f_v$ with $f_v = \psi_v$ an additive character with conductor strictly larger than $0$, for all but finitely many $v$. In this case, both sides might count something else other than just $\#\mathcal{L}(D)$ or $\#\mathcal{L}(K - D)$. Can we explicitly describe what those object would be? (I think it could be just some translation $D' = D + D_0$ for some explicit $D_0$ given in terms of $\psi_v$'s, so may not give anything new..)
If 1 and 2 are both nontrivial, then we may mix these two.