I saw the following claim on Godement-Jacquet's classical book "Zeta functions of simple algebras": (on page 153) Let $F$ be a global function field, $\Phi$ a Schwartz function on $\mathbb{A}_F$. (i.e. linear combinations of those $\otimes_v\Phi_v$, where $v$ runs over all the places of $F$ and for almost all $v$, $\Phi_v$ is the characteristic function of the ring of integers of $F_v$)
Claim: There exists a constant $M>0$, such that for all $a\in\mathbb{A}_F$ with its idelic norm $|a|>M$, we have $$|\sum_{\xi\in F^*}\Phi(a\xi)|=0.$$
I didn't understand this. Because in the number field case this classical lemma looks like the following:
Number field version: For any $p\in \mathbb{Z}_{\geq 1}$, there exists a constant $c_p>0$ (depending on $p$), such that for all $a\in\mathbb{A}_F$ with its idelic norm $|a|\geq 1$, we have $$|\sum_{\xi\in F^*}\Phi(a\xi)|\leq c_p|a|^{-p}.$$ The proof in the number field case is very classical. One way is to reduce this to $F=\mathbb{Q}$ and then using some explicit computations to reduce it to summation of Schwartz functions on $\mathbb{R}$. But for the function field case, even in the most typical example: $F=\mathbb{F}_q(T)$ and $\Phi=\otimes_v\Phi_v$, where every $\Phi_v$ is the characteristic function of the ring of integers of $F_v$, I don't understand this phenomenon.
I guess this must be some well-known fact among experts. But I have seldom dealt with global or local fields of characteristic $p$ before, so forgive me if this question looks very naive. Any suggestion or hint would be welcome. Thanks a lot in advance!