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Context: I've been reading Tate's thesis, and in it, we defined the character group for $k^{*}$ and $k^{+}$ for a local field $k$. Here we take the range of the characters to be $S_{1}$ for $k^{+}$ and $\mathbb{C}$ for $k^{*}$, where $S_{1}$ is the circle group. It's explained brilliantly in an answer to a previous question I asked about why $S_{1}$ is sufficient for the character group of $k^{+}$ and not for $k^{*}$. This got me thinking, why we don't study characters going to $\mathbb{Q}_{p}$ or $\mathbb{R}$? Studying characters is basically helping us study the group well, right? So doesn't studying these types of continuous homomorphisms $\chi: k^{*} \to \mathbb{Q}_{p}$ give us anything? (I'm asking this in a general setting and not exclusive to Tate's thesis, but some sort of motivation connected to Tate's thesis would be great too).

What do I think: The case about characters to $\frac{\mathbb{R}}{\mathbb{Z}}$ is the same as characters to $S_{1}$, so that question is partially dealt for the case of $\mathbb{R}$. For $\mathbb{Q}_{p}$, we have the totally disconnected topology, so does that make things uninteresting? Or is it because $\mathbb{C}$ is algebraically closed but $\mathbb{R}$ and $\mathbb{Q}_{p}$ are not?

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    $\begingroup$ One reason may be that the structure of multiplicative groups of local fields is very explicit (it’s on wikipedia). Just looking at it , in most cases I don’t think there are many interesting continuous homomorphisms from the multiplicative group of a local field $K$ to $\mathbb{Q}_p$, especially when $p$ differs from the residue characteristic of $K$. Also, morphisms from $K^\times$ to $\mathbb{C}$ include all morphisms to $\mathbb{R}$ already. $\endgroup$
    – Vik78
    Commented Jul 23, 2023 at 15:14
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    $\begingroup$ I'm not familiar enough with Tate's thesis (shame on me), but i guess he uses some kind of (Fourier) inversion formula, and a Poisson summation formula consequence thereof, which do not hold for smaller "dualizing" groups. Characters of the absolute Galois group have abelian image so factor through the Galois group of the maximal abelian extension, which Kronecker-Weber says is generated by roots of unity. But i think that people also use $GL(n,\mathbb Q_p)$ and $GL(n,\mathbb F_p)$ (or with $\ell\neq p$) as targets to extract data from Galois groups, and such representations arise in geometry. $\endgroup$
    – plm
    Commented Jul 23, 2023 at 16:52
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    $\begingroup$ I expect that an important reason the circle group is used is because of Pontrjagin duality. $\endgroup$
    – Daniel Asimov
    Commented Jul 23, 2023 at 19:10
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    $\begingroup$ As @Vik78 says$\newcommand\m{\mathbb}\newcommand\t{^\times}$ (although I think both you and they meant to pass to multiplicative groups), there are interesting homomorphisms $\m Q_p\t\to\m C\t$, even $\m Q_p\t\to\m R\t$; but, because $\m Q_p\t$ is totally disconnected, only one homomorphism $\m C\t\to\m Q_p\t$, and only two $\m R\t \to \m Q_p\t$. That is, if you want to pick one target that's interesting for all local fields, it shouldn't be $\m Q_p\t$. (Also, there is one infinite place, but infinitely many finite places; which one?) $\endgroup$
    – LSpice
    Commented Jul 23, 2023 at 21:06

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