Motivic Poisson summation.

*(English)*Zbl 1184.03027The 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.

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.

Reviewer: Immanuel Halupczok (Münster)