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The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ where $p \in \mathcal M(\mathbb Z)$ is the trivial norm, $j: U \to \mathcal M(\mathbb Z)$ is the inclusion of its complement, and $\mathcal O_\mathbb Z$ is the structure sheaf of analytic functions on $\mathcal M(\mathbb Z)$.

(Of course, rigid analytic geometry seems to mostly be done locally at a prime $p$, but I think there is some work that proceeds globally with spaces like $\mathcal M(\mathbb Z)$.)

Now, this is my ignorance speaking, but the coolest thing I know about the adèles is that taking them seriously is the starting point for proving the functional equation for $\zeta(s)$ à la Tate's thesis. So if the adèles have a natural rigid-analytic interpretation, it seems natural to want to interpret the rest of Tate's thesis in this framework.

Questions:

  1. Do the main concepts of Tate's thesis have natural interpretations in rigid-analytic geometry? I have in mind the following:

    • Haar measure on the additive and unit groups of a Banach algebra.

    • The Fourier transform on the additive group of a Banach algebra.

    • Eigenfunctions for this Fourier transform.

    • Schwartz functions on a Banach algebra.

    • The Fourier inversion formula on the additive group of a Banach algebra.

    • Characters and quasi-characters on the group of units of a Banach algebra.

    • The discreteness of $\mathbb Q \subset \mathbb A$ (more generally, if $\mathcal F$ is a coherent sheaf on a rigid-analytic space $X$ and $p \in X$ with inclusion $j: X \setminus p \to X$, then is $\mathcal F_p \to (j_\ast j^\ast \mathcal F)_p$ a topological embedding?).

    • The Poisson summation formula.

  2. If all the above components have good rigid-analytic interpretations, then by putting them together as Tate does it seems one should arrive at a good rigid-analytic interpretation of the function equation of $\zeta(s)$. Even if this does not work out, is there some other rigid-analytic interpretation of the functional equation?

EDIT: Here are some more appearances of the adeles $\mathbb A$ and the ideles $\mathbb I$:

  • An element of $\mathbb I$ is precisely the data needed to extend a trivial line bundle on $U \subset \mathcal M(\mathbb Z)$ (the complement of the trivial norm) to a line bundle on $\mathcal M(\mathbb Z)$, so I'm tempted to identify $\mathbb I$ with the divisor group of $\mathcal M(\mathbb Z)$.

  • Modding out global sections of $\mathcal O_\mathbb Z^\times$, $\mathbb I / \mathbb Q^\times$ is a sort of moduli space of line bundles on $\mathcal M(\mathbb Z)$. I don't know if this refines to a moduli Berkovich space "$Pic(\mathcal M(\mathbb Z))$" of line bundles.

  • Similarly, $\mathbb A / \mathbb Q$ is a moduli space of $\mathcal O_\mathbb Z$-principal bundles.

  • On a scheme $X$, the structure sheaf $\mathcal O_X$ locally has the property that every restriction map is a localization. But on $\mathcal M(\mathbb Z)$, the structure sheaf $\mathcal O_\mathbb Z$ fails to have this property (although restrictions are completions of localizations). A resolution by sheaves which do have this property is given by $j_\ast j^{-1} \mathcal O_\mathbb Z \to i_\ast \mathbb A / \mathbb Q$ where $i$ is the inclusion of the trivial norm into $\mathcal M(\mathbb Z)$ and $j$ as before is the inclusion of the complement. I'm not sure if this is significant.

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  • $\begingroup$ This paper of Bost ( arxiv.org/pdf/1512.08946.pdf ) somehow comes to mind, and while it is not (yet) what you want, I just thought I would include its link for the case (unlikely) you haven't yet come across it. $\endgroup$ – Vesselin Dimitrov Nov 6 '18 at 19:36
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    $\begingroup$ The Haar measure on a complete metric group only exists if it is locally compact (see for example 443K of Fremlin's Measure Theory). It is well known that Banach spaces over $\mathbb{R}$ are locally compact iff they are finite-dimensional, and by looking at the nuclear spaces section of Schneider's Nonarchimedean Functional Analysis it appears that compactness of a neighbourhood implies finite-dimensionality over non-archimedean fields as well. Unfortunately, as this is not my area, I don't know what this implies for $\mathcal{M}(\mathbb{Z})$. $\endgroup$ – Robert Furber Nov 6 '18 at 20:49

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