Sommes de Kloosterman généralisées : l’équation fonctionnelle

Résumé

Soit EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa % aaleaaceWG4bGbaebaaeqaaaaa!3811!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${V_{\bar x}}$$ la variété définie sur EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVraaa!41D8!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\mathbb{F}$$ _q par l’équation EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara % Waa0baaSqaaiaaigdaaeaacaWGNbWaaSbaaWqaaiaaigdaaeqaaaaa % kiabgEna0kabgwSixlabgwSixlabgwSixlabgEna0kqadshagaqeam % aaDaaaleaacaWGUbaabaGaam4zamaaBaaameaacaWGUbaabeaaaaGc % cqGH9aqpceWG4bGbaebaaaa!4B3A!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\bar t_1^{{g_1}} \times \cdot \cdot \cdot \times \bar t_n^{{g_n}} = \bar x$$, où g ₁,..., g _n sont des entiers naturels premiers entre eux dans leur ensemble. Dans un travail précédent [2], nous avions construit la cohomologie p-adique associée aux sommes exponentielles:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaacI % cacaWGwbWaaSbaaSqaaiqadIhagaqeaaqabaGccaGGSaGaeuiMdeLa % aiilaiabeE8aJjaacMcacqGH9aqpdaaeqbqaaiabeE8aJnaaBaaale % aacaaIXaaabeaakiaacIcaceWG0bGbaebadaWgaaWcbaGaaGymaaqa % baGccaGGPaaaleaacaGGOaGabmiDayaaraGaaiilaiaac6cacaGGUa % GaaiOlaiaacYcaceWG0bGbaebadaWgaaadbaGaamOBaaqabaWccaGG % PaGaeyicI4SaeqyVd42aaSbaaWqaaiqadIhagaqeaaqabaWccaGGOa % Wefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVr % daWgaaadbaGaamyCaaqabaWccaGGPaaabeqdcqGHris5aOGaey41aq % RaeyyXICTaeyyXICTaeyyXICTaey41aqRaeq4Xdm2aaSbaaSqaaiaa % d6gaaeqaaOGaaiikaiqadshagaqeamaaBaaaleaacaWGUbaabeaaki % aacMcacqqHyoqucaGGOaGabm4yayaaraWaaSbaaSqaaiaaigdaaeqa % aOGabmiDayaaraWaa0baaSqaaiaaigdaaeaacaWGRbWaaSbaaWqaai % aaigdaaeqaaaaakiabgUcaRiabgwSixlabgwSixlabgwSixlabgUca % RiqadogagaqeamaaBaaaleaacaWGUbaabeaakiqadshagaqeamaaDa % aaleaacaWGUbaabaGaam4AamaaBaaameaacaWGUbaabeaaaaGccaGG % Paaaaa!8C50!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$S({V_{\bar x}},\Theta ,\chi ) = \sum\limits_{(\bar t,...,{{\bar t}_n}) \in {\nu _{\bar x}}({\mathbb{F}_q})} {{\chi _1}({{\bar t}_1})} \times \cdot \cdot \cdot \times {\chi _n}({\bar t_n})\Theta ({\bar c_1}\bar t_1^{{k_1}} + \cdot \cdot \cdot + {\bar c_n}\bar t_n^{{k_n}})$$

où χ ₁, ..., χ _n sont des caractères multiplicatifs et Θ un caractère additif non trivial de EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVraaa!41D8!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\mathbb{F}$$ _q.