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?
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asked Apr 7, 2017 at 14:27
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7$\begingroup$ 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 $v$. $\endgroup$– Peter HumphriesCommented Apr 7, 2017 at 14:50
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1$\begingroup$ 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. $\endgroup$– john mangualCommented Apr 7, 2017 at 15:10
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4$\begingroup$ Those notes are not the place to learn automorphic representations for the first time. $\endgroup$– Peter HumphriesCommented Apr 7, 2017 at 15:11
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$\begingroup$ @PeterHumphries that is what I need a whole book on $GL_2$ and all it's variants $GL_2(\mathbb{A})$ and $GL_2(\mathbb{Q}_p)$. $\endgroup$– john mangualCommented Apr 7, 2017 at 15:14
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4$\begingroup$ 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. $\endgroup$– Peter HumphriesCommented Apr 7, 2017 at 16:35
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1 Answer
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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
- $g_{\infty}$ in the usual Schwartz space of $\mathbb{R}$, i.e., smooth with fast decay at infinity and likewise for all its derivatives.
- for all $p$, $g_p$ is in the Schwartz-Bruhat space for $\mathbb{Q}_p$, i.e., locally constant and compactly supported.
- 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 $\mathbb{R}$.
answered Apr 7, 2017 at 14:54
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$\begingroup$ 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? $\endgroup$– LSpiceCommented Apr 7, 2017 at 20:08
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3$\begingroup$ @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: mathoverflow.net/questions/187404/… $\endgroup$ Commented Apr 7, 2017 at 20:48