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real case: In the very first course of Calculus, one learns that a real function $f \colon \mathbb{R} \to \mathbb{R}$ is called smooth, if it is differentiable as many times as one pleases. So the intuitive idea behind 'smoothness' is that the graph of $f$ is, well... smooth if you look at it.

p-adic case: Now the picture changes in the $p$-adics; one sets smooth := 'invariant w.r.t. an open subset', for example in representation theory (of let's say $\mathrm{GL}_n(K)$ of a local field $K$), and one can show that smooth = locally constant + compact support. I always took this as the definition. I also always assumed (and maybe I also heard or read) that one uses the word smooth in the $p$-adic setting, s.t. one can put things into a more general setting just by referring to smooth functions $K \to \mathbb{C}$ for whatever field $K \in \{\mathbb{Q}_p, \mathbb{R}, \mathbb{C}\}_{p \text{ prime}}$ is.

For example, one defines the space $\mathcal{S}(\mathbb{Q}_p)$ of (Bruhat)-Schwartz functions as the space of smooth functions $\mathbb{Q}_p \to \mathbb{C}$, and one has a (very) similar definition for $\mathcal{S}(\mathbb{R})$ (as smooth + rapidly decreasing derivatives). This might not be the best example but it surely reflects a little what I tried to tell.

But I am kind of not satisfied with the motivation for the definition. So my question is...

Question: Is there some deeper analogy with the real in the sense of maybe the $p$-adic differentiation? Or maybe generalizing the question.. I would really like to see some 'intuition' or rather 'motivation' behind the definition, so that I can say, "OK, this is really the right word for smoothness in the $p$-adic case".

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    $\begingroup$ Have you seen for instance nLab on Bruhat-Schwartz functions? $\endgroup$
    – Wojowu
    Commented Oct 10, 2022 at 21:14
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    $\begingroup$ Does it completely though? It gives a definition for general locally compact abelian group, by writing the group as an inverse limit of Lie groups and taking spaces of smooth functions on those. For (locally) profinite groups this happens to trivialize, since they are written as inverse limits of 0-dimensional manifolds, but it still fits into the same framework. $\endgroup$
    – Wojowu
    Commented Oct 10, 2022 at 22:42
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    $\begingroup$ Well, this is the definition of Bruhat-Schwartz functions. My question regards rather the motivation for $p$-adic 'smoothness' to define it the way it is defined. $\endgroup$
    – Maty Mangoo
    Commented Oct 10, 2022 at 23:10
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    $\begingroup$ Functions $\mathbf Q_p \to \mathbf C$, or $\mathbf Z_p \to \mathbf C$, are kind of strange things, don't you think? We're not going to be able to describe them using power series since $\mathbf Q_p$ and $\mathbf Z_p$ have no reasonable embedding into $\mathbf C$ that respects the algebra and topology (sure, there are embeddings algebraically by Zorn's lemma, but not in any natural way). So locally constant + compact support for complex-valued functions on the $p$-adics is about all we can think of as an analogue of the smooth + rapid decay definition for Schwartz functions. $\endgroup$
    – KConrad
    Commented Oct 30, 2022 at 20:18
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    $\begingroup$ Another thing to keep in mind is that there are many continuous characters $\mathbf Q_p \to S^1$ and $\mathbf Z_p \to S^1$ when you do Fourier analysis on these locally compact abelian groups. And such continuous characters have a property that reflects their $p$-adic domains: they must be locally constant. That's because $\mathbf Q_p$ and $\mathbf Z_p$ have arbitrarily small open subgroups around $0$, while $S^1$ has "no small subgroups" (the only subgroup in a small nbd. of 1 is $\{1\}$), and that forces each group hom $\mathbf Z_p \to S^1$ to be trivial on $p^N\mathbf Z_p$ for large $N$. $\endgroup$
    – KConrad
    Commented Oct 30, 2022 at 20:23

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