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I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)..."

The lemma: for $\Phi\in \mathcal{S}(\mathbb{A}_F)$ (setting: $F$ is a global number field, $\mathcal{S}(\mathbb{A}_F)$ the adelic schwartz function space) and any $p\in \mathbb{Z}_{\geq 1}$, there exists a constant $c>0$ such that $$|\sum_{\xi\in F^\times}\Phi(a\xi)|\leq c|a|^{-p},~\forall a\in \mathbb{I}_F, with ~|a|_{\mathbb{I}}\geq 1.$$

O.K. So that's the lemma. I believe this fact is quite well known among the experts working in automorphic representation, otherwise Jacquet would not "take it for granted". I know this is a key convergence lemma for general definitions of theta functions (summation of an adelic Schwartz-Bruhat function over $F$). But I have not found a precise and complete proof on any paper or book. A professor told me Weil's famous paper 'Sur certains operateurs unitaire' (in which he dealt with adelic metaplectic groups and introduced the Weil representation) included some similar arguments, but they're very different.

I tried to prove it by myself. I tried to reduce this to the case $F=\mathbb{Q}$. In the case $F=\mathbb{Q}$ it's technically easy. Because when $F=\mathbb{Q}$ it suffices to consider $a$ in the argument with $a_{fin}\in \prod_p \mathbb{Z}_p^\times$ (finite part of the idele), since we can multiply some element in $F^\times$ to $a$ and adjust all the "orders" at finite places freely. (But this is not valid when $F$ isn't of class number one.) Now when $a_{fin}\in \prod_p \mathbb{Z}_p^\times$ the summation is just for a real Schwartz function over standard lattice $\mathbb{Z}$ in $\mathbb{R}$, and it's a handy analysis exercise. The difficulty for general cases is: I can't $\textbf{adjust and then fix}$ the orders at finite places to obtain a $\textbf{fixed}$ lattice embedded into the $\mathbb{A}_\infty=\prod_{v|\infty} F_v$, then I can't obtain an estimate while for different $a$ the lattice is changing.

Now I have no idea how to reduce this to $F=\mathbb{Q}$ case! Nor did I found any direct proof. So could anyone give me some idea or reference related to this convergence lemma? Thank you very much for your help in advance!

youknowwho
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  • We don't really care of the class number. Fix an idele representative of each $c\in C_F$ then for all $a\in \Bbb{A}F^\times$ there is $b\in F^\times$ and $c\in C_F$ such that $(abc){fin} \in \widehat{O_F}^\times$ – reuns May 04 '22 at 16:55
  • @reuns oh you’re right. It suffices to use the finiteness of class number to know that we can control them to finitely many lattices. Thanks! – youknowwho May 04 '22 at 17:19
  • I believe this fact is quite well known... otherwise Jacquet would not "take it for granted" - or he wasn't sure about it or he just didn't want to write details down – Kimball May 05 '22 at 02:00
  • @Kimball Ah ok, but many friends told me Jacquet is very careful while writing books or papers. Maybe that was misleading. – youknowwho May 05 '22 at 13:36
  • That was partly tongue in cheek. I believe Godement and/or Jacquet checked it in this case. Jacquet is generally pretty careful, but he's not infallible. – Kimball May 06 '22 at 00:33
  • @Kimball Ah, ok :) – youknowwho May 06 '22 at 07:28

1 Answers1

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Although many sources do argue by reducing to classical situations, thereby worry about class numbers, units, and so on, the more modern Iwasawa-Tate viewpoint does not truly require this, at all.

In brief, instead of insisting on writing things as a sum over a classical lattice (to prove convergence), we can estimate things by Euler products... and estimate those by the Euler products of zeta functions of number fields and/or powers of the zeta function of $\mathbb Z$.

Fairly careful estimates of this sort are carried out in the later sections of the second part of

https://www-users.cse.umn.edu/~garrett/m/number_theory/Notes_2011-12.pdf

paul garrett
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  • Thanks, professor Garrett! By the way, some professor told me the Godement-Jacquet’s theory is of fundamental importance due to its relations to inner forms and Jacquet-Langlands correspondence ( dealing with general simple algebras). So what’s the relations or difference of it and Iwasawa-Tate? – youknowwho May 04 '22 at 17:23
  • The Godement-Jacquet theory is a large extension of Iwasawa-Tate theory (which we could say is "$GL1$") to $GLn$ . Some aspects are similar, and Poisson summation is still a key analytical device. But, now, there are cuspforms, unlike $GL1$. And, yes, the Godement-Jacquet approach was the first to give analytic continuation of (standard) $L$-functions attached to cuspforms on $GLn$... – paul garrett May 04 '22 at 18:21
  • I see, thanks you very much! – youknowwho May 04 '22 at 18:45