# Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

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.

• 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. – Vesselin Dimitrov Nov 6 '18 at 19:36
• 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 $\mathcal{M}(\mathbb{Z})$. – Robert Furber Nov 6 '18 at 20:49