Constructible exponential functions, motivic Fourier transform and transfer principle. (English) Zbl 1246.14025

Cluckers, Raf; Loeser, François

Ann. Math. (2) 171, No. 2, 1011-1065 (2010).

In [“Constructible motivic functions and motivic integration”, Invent. Math. 173, No. 1, 23–121 (2008; Zbl 1179.14011)], the authors laid the ground stone for a new version of motivic integration which in particular allows to treat parametrized integrals. The main result of the present article is to add additive characters to the class of functions which can be motivically integrated. This in particular allows to compute integrals involving additive characters in a uniform way over all non-archimedean local fields of sufficiently big residue characteristic. As an application, one obtains a transfer principle for equalities of such integrals between local fields of characteristic \(0\) and local fields of positive characteristic. Moreover, a motivic Bruhat-Schwartz space is defined and a formalism of Fourier transforms is developed in that setting.
Recall the formalism from [Zbl 1179.14011]. A “definable subassignment” is essentially a first order formula in the language of valued fields defining a subset of \(K((t))^m \times K^n \times \mathbb Z^r\) for each field \(K\) of characteristic \(0\) (where \(\mathbb Z\) is appears as the value group of \(K((t))\)). For any such definable subassignment \(X\), one has an abstract ring \(\mathcal{C}(X)\) of “constructible motivic functions on \(X\)”. Motivic integration is essentially a map \(\mathcal{C}(X) \to \mathcal{C}(\{\mathrm{point}\})\); more generally, for any morphism \(f: X \to Y\) of definable subassignments, one has a map \(f_! :\mathcal{C}(X) \to \mathcal{C}(Y)\) which “integrates over the fibers of \(f\)”. (I am omitting some technicalities here, like integrability issues.)
Motivic integration specializes to usual integration over non-archimedean local fields \(F\) of sufficiently big residue characteristic. More precisely, if \(F\) is such a field, with residue field \(\mathbb F_q\), then a definable subassignment \(X\) specializes to a set \(X_F \subseteq F^m \times \mathbb F_q ^n \times \mathbb Z ^r\), a constructible motivic function specializes to a function \(X_F \to \mathbb C\), and motivic integration specializes to actual integration with respect to the Haar measure on \(F^m\) (and to the counting measure on \(\mathbb F_q ^n \times \mathbb Z ^r\)).
In the present paper, the above formalism is extended to larger rings \(\mathcal{C}(X)^{\mathrm{exp}} \supseteq \mathcal{C}(X)\) of “constructible exponential functions” which in particular contain a new abstract symbol \(E\) which stands for an additive character of the valued field. Theorem 4.1.1 gives an abstract, axiomatic description of this formalism. A typical example of a constructible exponential function on \(K((t))^m\) is \(\text{ord}(f(x))\cdot \mathbb L^{\text{ord}(g(x))}\cdot E(h(x))\), where \(x\) is a variable running over \(K((t))^m\), \(f\), \(g\) and \(h\) are definable function from \(K((t))^m\) to \(K((t))\), and \(\mathbb L\) is a formal symbol which stands for the number of elements of the residue field.
To be able to specialize constructible exponential functions to a local field \(F\), one has to additionally choose an additive character \(\psi: F \to \mathbb C\) to which \(E\) should specialize (where \(\psi\) should satisfy some conditions). If \(F\) has residue field \(\mathbb F_q\), then the above example specializes to the function \(F^m \to \mathbb C\), \(x \mapsto \text{ord}(f(x))\cdot q^{\text{ord}(g(x))} \cdot\psi(h(x))\). (Note that \(f\), \(g\), and \(h\) induce well-defined functions \(F^m \to F\) as soon as the residue characteristic of \(F\) is big enough.)
In Section 7, a formalism of Fourier transforms is developed for constructible exponential functions. More precisely, a space of constructible Schwartz-Bruhat functions is defined (in local fields, these functions specialize to locally constant and compactly supported functions). Theorem 7.5.1 states that on this space, the motivic Fourier transform behaves as one would expect: it induces an isomorphism from the Schwartz-Bruhat space to itself and applied twice, it is the identity up to a factor and a change of sign.
In Section 8, the abstract formalism of constructible exponential functions is redeveloped in a fixed finite field extension \(F\) of \(\mathbb Q_p\) (with motivic integration replaced by \(p\)-adic integration). This yields a class of functions which contains all additive characters and which is closed under partial integration. For sufficiently big \(p\), this would follow directly by specializing the abstract formalism, but now, any \(p\) is allowed and moreover, this is also done in a context allowing sub-analytic sets.
Finally, some new transfer principles between \(\mathbb F_p((t))\) and \(\mathbb Q_p\) are deduced: for any equation of integrals of constructible exponential functions, there exists a bound \(M\) such that for \(p > M\), the equation holds in \(\mathbb F_p((t))\) holds iff it holds in \(\mathbb Q_p\). Many different versions of the fundamental lemma of the Langlands program are examples of equations of this type, so the theory can be applied to it; see [R. Cluckers, T. Hales and F. Loeser, “Transfer principle for the fundamental lemma.”, in: On the stabilization of the trace formula, (Int. Press, Somerville, MA), 309–347 (2011); arXiv:0712.0708].

Reviewer: Immanuel Halupczok (Münster)

MSC:

14E18	Arcs and motivic integration
14G20	Local ground fields in algebraic geometry
03C98	Applications of model theory
22E50	Representations of Lie and linear algebraic groups over local fields

Keywords:

motivic integration; Fourier transform; additive character; Bruhat-Schwartz space; \(p\)-adic integration; definable subassignment

Citations:

Zbl 1179.14011

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