Transformation de Fourier et majoration de sommes exponentielles. (Fourier transformation and majoration of exponential sums.). (French) Zbl 0603.14015

For a finite field \(k={\mathbb{F}}_ q\) of characteristic \(p\) one has the Lang torsor \((=\) principal fibre bundle) \({\mathcal L}\) on the affine line \({\mathbb{A}}^ 1_ k=Spec k[U]\), defined by the equation \(V^ q-V=U\) and for \(\ell \neq p\) the \({\bar {\mathbb{Q}}}^{\times}_{\ell}\)-valued additive characters \(\psi\) of \({\mathbb{F}}_ q\) give a set of \({\bar {\mathbb{Q}}}^{\times}_{\ell}\)-bundles \({\mathcal L}_{\psi}\) (Artin-Schreier sheaves) on the line. Let E be the \({\mathbb{F}}_ q\)-vector space \({\mathbb{F}}^ r_ q\) of rank r and \(E^{\vee}\) its dual. The category D(E,\({\bar {\mathbb{Q}}}_{\ell})\) is the derived category of \({\bar {\mathbb{Q}}}_{\ell}\)-sheaves on E (bounded complexes with negative degrees only and residual cohomology with constructible supports). One then has the \(\ell\)-adic Fourier transformation \({\mathcal F}: D(E,{\bar {\mathbb{Q}}}_{\ell})\to D(E^{\vee},{\bar {\mathbb{Q}}}_{\ell}):\) the definition formula for the elementary Fourier transform \({\mathcal F}(K)=<K,\psi >\) now reads like \({\mathcal F}(K)=Rpr_*^{\vee}(pr^*K^ L\otimes \mu^*{\mathcal L}_{\psi})[r],\) where \(\mu\) is the evaluation map \(E^{\vee}\times E\to {\mathbb{A}}^ 1\). Actually E may, more generally, be a vector bundle on a good \({\mathbb{F}}_ q\)-scheme. \({\mathcal F}\) commutes with duality. One proves a Riemann-Roch theorem for objects K adapted to a given stratification of E, relating the Euler-Poincaré characteristics \(\chi(F(K))\) and \(\chi(K)\) which are defined as functions on \(E^{\vee}\) and E. If in the definition formula for \(\chi\) (K) the signs \((-)^ i\) are all replaced by \(+\), one gets a function on E, denoted by \(\| K\|\). One of the main results is a uniformity theorem: \((\sup \| {\mathcal F}(K)\|)\leq \text{const. }(\sup \| K\|)\), where the suprema are taken along a fibre of \(E^{\vee}\) and E, with an integer constant determined by E (which is allowed to be given as a vector bundle over a \({\mathbb{Z}}\)-scheme of finite type). The uniformity theorem is applied in the following situation. Let R be a subring of \({\mathbb{C}}\) of finite type, X an affine R-scheme of relative dimension \(m\) and \(\phi: R\to k\) a ring homomorphism. Let \(f: X\to {\mathbb{A}}^ r_ R\) be an R- morphism and g an invertible rational function on X. Then, with \(f_{\phi}: X_{\phi}\to {\mathbb{A}}^ r_ R\) and \(g_{\phi}: X_{\phi}\to {\mathbb{G}}_ m={\mathbb{A}}^ 1_ k-\{0\},\) one takes \(K=(f_{\phi})_*{\mathcal L}_{\chi}(g_{\phi})[m]\), where this time \(\chi: {\mathbb{F}}^{\times}_ q\to {\mathbb{C}}\) is a multiplicative character, and thus obtains a majoration for trigonometric sums \(S=\sum_{x}\psi (a\cdot f_{\phi}(x))\chi (g_{\phi}(x))\) (with \(a\in {\mathbb{Z}}^ r\), sum taken over \(X({\mathbb{F}}_ q))\), namely \(| S| \leq const. (\sqrt{q})^ m\), with a constant determined by \(X_{{\mathbb{C}}}\), provided that a certain polynomial expression in a is not divisible by p. The paper also considers an analogous Fourier theory for \({\mathcal D}\)-modules instead of \(\ell\)-adic sheaves.
Reviewer: J.H.de Boer


14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
43A32 Other transforms and operators of Fourier type
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