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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

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14G20 Local ground fields in algebraic geometry 43A32 Other transforms and operators of Fourier type
##### Keywords:
$$\ell$$-adic Fourier transformation
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##### References:
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