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I am not a specialist in automorphic forms, can someone explain to me typical elements of adelic Schwartz class, $\mathcal{S}(\mathbb{A})$. Over the real numbers there are obviously elements like: $$ p(x) \,e^{-x^2} $$ where $p(x)$ is polynomial. My notes have that Schwartz class over adeles is the tensor product over all places. $$ \mathcal{S}(X_\mathbb{A}) = \bigotimes_v \mathcal{S}(X_v) $$ where $X$ is just the line. So really I would just like to understand better, what are elements of the Schwartz space over $p$-adic numbers?

Abdelmalek Abdesselam
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john mangual
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    Read chapter 1 of Goldfeld and Hundley. Anyway, the $p$-adic component of a Bruhat-Schwartz function is a locally constant compactly supported function, and is the indicator function of $\mathcal{O}_v$ for all but finitely many nonarchimedean places $v$. – Peter Humphries Apr 07 '17 at 14:50
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    I found these lecture notes of Ngô, $p$-adic representation theory is a whole topic in itself, and here they write it in a few sentences. – john mangual Apr 07 '17 at 15:10
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    Those notes are not the place to learn automorphic representations for the first time. – Peter Humphries Apr 07 '17 at 15:11
  • @PeterHumphries that is what I need a whole book on $GL_2$ and all it's variants $GL_2(\mathbb{A})$ and $GL_2(\mathbb{Q}_p)$. – john mangual Apr 07 '17 at 15:14
  • I think about it some more if I have these monomial elements of Schwartz class, then I can put arbitrary linear combinations of these things and still remain in Schwartz class or add $\mathbb{Z}$-averages (or $SL(2, \mathbb{Z})$ averages?) In any case $\theta$ functions and other automorphic functions, hopefully remain in this safety class. – john mangual Apr 07 '17 at 16:31
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    Don't get ahead of yourself. A Bruhat-Schwartz function is a finite linear combination of these pure tensors. What is a $\mathrm{SL}_2(\mathbb{Z})$ average of a function on $\mathbb{A}$? Seriously, read Goldfeld and Hundley. It explains the dictionary between classical and adèlic automorphic forms. – Peter Humphries Apr 07 '17 at 16:35
  • A very literal answer to the request for a book on $\mathrm{GL}_2$ is http://www.ams.org/mathscinet-getitem?mr=401654 (SLN 114). If you're willing to go up to $\mathrm{GL}_3$, Bump may be more friendly http://www.ams.org/mathscinet-getitem?mr=765698 (SLN 1083). For $\mathrm{GL}_2$ again, I like the book of Bushnell and Henniart http://www.ams.org/mathscinet-getitem?mr=2234120 , but it's local. – LSpice Apr 07 '17 at 20:08

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For concreteness, let me just take the adeles of $\mathbb{Q}$. Let us call a function $f$ on $\mathbb{A}_{\mathbb{Q}}$ elementary or factorizable if it can be written as $$ f(x_{\infty},x_2,x_3,x_5,\ldots)=g_{\infty}(x_{\infty})\times \prod_{p\ {\rm prime}} g_{p}(x_p) $$ with

  1. $g_{\infty}$ in the usual Schwartz space of $\mathbb{R}$, i.e., smooth with fast decay at infinity and likewise for all its derivatives.
  2. for all $p$, $g_p$ is in the Schwartz-Bruhat space for $\mathbb{Q}_p$, i.e., locally constant and compactly supported.
  3. for all except at most finitely many $p$'s, $g_p$ is the indicator function of $\mathbb{Z}_p$.

The space $S(\mathbb{A}_{\mathbb{Q}})$ of Schwartz-Bruhat functions on the adele ring $\mathbb{A}_{\mathbb{Q}}$ is the space of all finite linear combinations of elementary functions as above.

Note that number theory references often do not tell what the topology is on this space. It turns out that as a topological vector space $S(\mathbb{A}_{\mathbb{Q}})$ is isomorphic to the space $\mathcal{D}(\mathbb{R})$ of smooth compactly supported test functions on $\mathbb{R}$.

Abdelmalek Abdesselam
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  • I didn't know that last statement (about the isomorphism of $\mathscr S(\mathbb A)$ with $\mathscr D(\mathbb R)$)! Is there an elementary construction of the isomorphism? – LSpice Apr 07 '17 at 20:08
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    @LSpice: I am not sure about elementary... It's not too hard to construct an isomorphism between $S(\mathbb{A}{\mathbb{Q}})$ and the intermediate sequence space $\oplus{\mathbb{N}} \mathfrak{s}$ where $\mathfrak{s}$ is the space of sequences with fast decay. The difficult part is the second isomorphism with $\mathcal{D}(\mathbb{R})$. See this MO question: https://mathoverflow.net/questions/187404/isomorphisms-between-spaces-of-test-functions-and-sequence-spaces – Abdelmalek Abdesselam Apr 07 '17 at 20:48