Comparison theorem between Fourier transform and Fourier transform with compact support
This paper is of interest to those studying number theory. It compares the Fourier transform and the Fourier transform with compact support in the context of arithmetic \(\mathcal{D}\)-modules. Recall that for a function \(f: \mathbb{R} \rightarrow \mathbb{C}\) the classical Fourier transform is defined by \[ \hat{f}(\omega) = \int_{\mathbb{R}}f(t)e^{-\omega t} dt \] where \(\omega\) is on the imaginary axis. The Fourier transform satisfies the identities \(\widehat{f'}=w\hat{f}\) and \((\hat{f})'=\widehat{-tf}\). Considering multiplication by \(\omega\) and \(t\) as endomorphisms of the sheaf of smooth functions, the Fourier transform interchanges \(t \leftrightarrow -\frac{d}{d\omega}\) and \(\frac{d}{dt} \leftrightarrow \omega\). Identifying the local coordinates \(t\) and \(\omega\) on the real and imaginary axis as a single variable \(x\) and generalizing to dimension \(n\), the Fourier transform induces an algebraic automorphism of the Weyl algebra \(\mathcal{D}_{\mathbb{R}^n}=\mathbb{R}\langle x_1,\dots,x_n, \partial_1,\dots,\partial_n\rangle\) by \(x_i \mapsto -\partial_i\) and \(\partial_i \mapsto x_i\). This can be thought of as a noncommutative change of coordinates on the cotangent space \(T^*\mathbb{R}^n\). The Fourier transform of a \(\mathcal{D}\)-module \(\mathcal{M}\), \(\widehat{\mathcal{M}}\), is defined as the pushforward along this endomorphism. This is a categorification of the process of solving differential equations by replacing differential operators with polynomials. The Fourier transform functor can be reformulated in terms of pullback functor and a pushforward functors on a certain diagram. This permits definitions of the Fourier transform in a variety of algebraic, analytic, and arithmetic settings. It also helps to show that variations such as Fourier transform for constructible sheaves and Fourier transform with compact supports are reasonably defined. In the setting of constructible \(\ell\)-adic sheaves, \textit{N. M. Katz} and \textit{G. Laumon} [Publ. Math., Inst. Hautes Étud. Sci. 62, 145--202 (1985; Zbl 0603.14015)] proved a comparison theorem between Deligne's Fourier transform functors. Their comparison theorem proves that the Fourier transform and the Fourier transform with compact support are naturally isomorphic functors. The paper under review extends this result to the setting of arithmetic \(\mathcal{D}\)-modules.
scientific article; zbMATH DE number 6173999