×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] A. A. Beilinson, I. N. Bernstein etP. Deligne, Faisceaux pervers, Conférence de Luminy, juillet 1981, Analyse et Topologie sur les expaces singuliers, I,Astérisque 100 (1982).
[2] I. N. Bernstein, Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients,Funct. Anal. 5 (1971), 1–16. · Zbl 0246.17008 · doi:10.1007/BF01075841
[3] I. N. Bernstein, Lectures onD-Modules, Conférence de Luminy, juillet 1983, “ Systèmes différentiels et singularités ”, preprint.
[4] J. E. Björk,Rings of Differential Operators, North-Holland (1979).
[5] J. L. Brylinski,Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, prepint de l’Ecole polytechnique (1983).
[6] J. L. Brylinski, (Co)-homologie d’intersection et faisceaux pervers, Séminaire Bourbaki 1981–1982, no 585,Astérisque 92–93 (1982), p. 129–157.
[7] J. L. Brylinski, A. S. Dubson etM. Kashiwara, Formule de l’indice pour les modules holonomes et obstruction d’Euler locale,C.R.A.S. 293 (30 novembre 1981), 573–576. · Zbl 0492.58021
[8] P. Deligne, La conjecture de Weil II,Publ. Math. IHES 52 (1980), 313–428.
[9] P. Deligne, Equations différentielles à points singuliers réguliers,Lecture Notes in Math.163, Springer Verlag (1970).
[10] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zéro, I et II,Annals of Math. 79 (1964), 109–326. · Zbl 0122.38603 · doi:10.2307/1970486
[11] N. M. Katz, Sommes exponentielles, Cours à Orsay, automne 1979,Astérisque 79 (1980).
[12] G. Laumon, Semi-continuité du conducteur de Swan (d’après Deligne), dans Caractéristique d’Euler-Poincaré, Séminaire E.N.S. 1978–1979,Astérisque 82–83 (1981), 173–219.
[13] G. Laumon, Comparaison de caractéristiques d’Euler-Poincaré en cohomologie -adique,C.R.A.S. 292 (19 janvier 1981), 209–212. · Zbl 0468.14005
[14] G. Laumon, Majoration de sommes exponentielles attachées aux hypersurfaces diagonales,Ann. scient. Ec. Norm. Sup., 4e série,16 (1983), 1–58.
[15] M. Raynaud, Caractéristique d’Euler-Poincaré d’un faisceau et cohomologie des variétés abéliennes,Séminaire Bourbaki 1964–1965, exposé no 286, W. A. Benjamin (1966).
[16] J.-P. Serre, Cohomologie galoisienne,Lecture Notes in Math. 5, Springer Verlag (1964). · Zbl 0143.05901
[17] J.-P. Serre, Majoration de sommes exponentielles, Journées Arithmétiques de Caen,Astérisque 41–42 (1977), 111–126.
[18] T. Ekedahl, On the adic formalism, to appear. · Zbl 0821.14010
[19] E. L. Ince,Ordinary Differential Equations, Dover (1956). · Zbl 0063.02971
[20] Z. Mebkhout, The Riemann-Hilbert problem in higher dimension,Proc. Conf. Generalized Functions Appl. in Math. Phys. (Moscow, Nov. 1980), Steklov Inst. 1981, 334–341.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.