# zbMATH — the first resource for mathematics

Motivic Poisson summation. (English) Zbl 1184.03027
The first result of this article is a motivic version of the Poisson summation formula for function fields. Here “motivic” is to be understood as in motivic integration, i.e., numbers are replaced by elements of a certain Grothendieck ring of varieties.
More precisely: Let $$\mathfrak{f}$$ be a field and let $$C$$ be a smooth, projective, absolutely irreducible curve over $$\mathfrak{f}$$. Let me first recall the “classical” Poisson summation formula for function fields, so suppose that $$\mathfrak{f}$$ is finite. Write $$\mathbb{A}_{\mathfrak{f}(C)}$$ for the adèles of the function field $$\mathfrak{f}(C)$$. For suitable functions $$\varphi$$ on $$\mathbb{A}_{\mathfrak{f}(C)}$$, one can define a Fourier transform $$\mathcal{F}\varphi$$ of $$\varphi$$; after some choices, we may suppose that $$\mathcal{F}\varphi$$ is again a function on $$\mathbb{A}_{\mathfrak{f}(C)}$$. Using the embedding $$\mathfrak{f}(C) \subset \mathbb{A}_{\mathfrak{f}(C)}$$, define the sum over the rational points $$\delta^K (\varphi) := \sum_{y \in \mathfrak{f}(C)} \varphi(y)$$. The Poisson summation formula states that (under suitable assumptions) $$\delta^K (\varphi) = \delta^K (\mathcal{F}\varphi)$$.
In the motivic version, first of all we allow $$\mathfrak{f}$$ to be an arbitrary field. Let $$\mathfrak{K}_{\mathfrak{f}}$$ be a suitable Grothendieck ring, built out of the Grothendieck ring of varieties over $$\mathfrak{f}$$. Now instead of working with functions $$\varphi$$ with values in $$\mathbb{C}$$, we use functions with values in $$\mathfrak{K}_{\mathfrak{f}}$$. (To be more precise, $$\varphi$$ is not just a map $$\mathbb{A}_{\mathfrak{f}(C)} \to \mathfrak{K}_{\mathfrak{f}}$$, but a geometric object which yields a map $$\mathbb{A}_{\mathfrak{f}'(C)} \to\mathfrak{K}_{\mathfrak{f'}}$$ for any field extension $$\mathfrak{f}' \supset \mathfrak{f}$$.) In this setting, it is possible to make sense of the Fourier transform $$\mathcal{F}$$ and of the sum over the rational points $$\delta^K$$ (which is also an element of $$\mathfrak{K}_{\mathfrak{f}}$$). The result (Theorem 5.10) is that the Poisson summation formula still holds.
The connection between the motivic setting and the classical one is the following.
Suppose that $$\mathfrak{f}$$ is finite. Then one can define a “specialization” ring homomorphism $$\mu_{\mathfrak{f}}: \mathfrak{K}_{\mathfrak{f}} \to \mathbb{Z}$$ sending the class of a variety $$X$$ (defined over $$\mathfrak{f}$$) to its number of $$\mathfrak{f}$$-valued points $$|X(\mathfrak{f})|$$. This specialization commutes with summation over rational points (Lemma 5.7): $$\delta^K (\mu_{\mathfrak{f}} \circ \varphi) = \mu_{\mathfrak{f}}(\delta^K(\varphi))$$.
In the remainder of the article, this is applied to obtain some results on division algebras. Namely, let $$L$$ be a cyclic Galois extension of $$\mathfrak{f}$$ of prime order $$n$$, let $$g$$ be a generator of the Galois group $$\text{Gal}(L/\mathfrak{f})$$, and let $$D$$ be the division algebra over $$\mathfrak{f}(t)$$ generated by $$L(t)$$ and an element $$s$$, subject to the relations $$sa = g(a)s$$ (for $$a \in L$$) and $$s^n = t$$.
For each place $$v$$ of $$\mathfrak{f}(t)$$, a subring $$R_v$$ of $$D$$ is defined (for most $$v$$, $$R_v$$ consists of those elements of $$D$$ which “have no pole at $$v$$”). Two elements of $$D$$ are called integrally conjugate if, for each $$v$$, they are conjugate by an element of $$R^*_v$$ (possibly after base change). In the classical setting, one can ask for the number of rational points of the integral conjugacy class $$O_c$$ of an element $$c \in D$$. In the motivic setting, this number becomes an element of (a variant of) the Grothendieck ring $$\mathfrak{K}_{\mathfrak{f}}$$; the authors explicitly compute this element in Section 8.
Finally, the authors compare such integral conjugacy classes in different forms of $$D$$. Suppose that $$\dot{D}$$ is another form, obtained in the same way as $$D$$ but using a different generator $$\dot{g} \in \text{Gal}(L/\mathfrak{f})$$. Then there is a canonical bijection between the integral conjugacy classes of $$D$$ and $$\dot{D}$$, obtained using an isomorphism between $$D$$ and $$\dot{D}$$ over $$\mathfrak{f}^{\mathrm{alg}}(t)$$ (where $$\mathfrak{f}^{\mathrm{alg}}$$ denotes the algebraic closure of $$\mathfrak{f}$$). Now suppose that $$\varphi$$ and $$\dot{\varphi}$$ are (suitable) functions on $$D$$ and $$\dot{D}$$ respectively (with values in $$\mathfrak{K}_{\mathfrak{f}}$$). Let us say that $$\varphi$$ and $$\dot{\varphi}$$ match essentially if both are invariant under integral conjugacy if they take the same value on corresponding integral conjugacy classes. Suppose that $$\varphi$$ and $$\dot{\varphi}$$ do match. The authors show that then motivically we have $$\sum_{y \in D} \varphi(y) = \sum_{y \in \dot{D}} \dot{\varphi}(y)$$ (Proposition 8.17); in particular, the motivic number of points is the same for corresponding integral conjugacy classes. Moreover, this is used to deduce that the Fourier transforms $$\mathcal{F}\phi$$ and $$\mathcal{F}\dot{\varphi}$$ also match (Theorem 1.1).
This article uses model-theoretic methods. One problem that arises is to find a suitable way to treat global questions model-theoretically (the problem being that global fields are undecidable). The idea is to essentially work with the theory of algebraically closed fields and to treat global fields as ind-definable: if $$k$$ is algebraically closed and $$C$$ is a curve, then $$k(C)$$ can be seen as an infinite union of sets definable over $$k$$.
The motivic Fourier transform needs some version of motivic integration. The idea is the one from Cluckers-Loeser [R. Cluckers and F. Loeser, “Constructible exponential functions, motivic Fourier transform and transfer principle” (preprint); see also C. R., Math., Acad. Sci. Paris 341, No. 12, 741–746 (2005; Zbl 1081.14032)]; however, due to the restricted class of functions that are considered in the present article, only a very small fragment of that theory is needed.

##### MSC:
 03C60 Model-theoretic algebra 11R56 Adèle rings and groups 16K20 Finite-dimensional division rings
Full Text: