By A.D. Thomas
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Extra resources for Zeta functions, introduction to algebraic geometry
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On en d´eduit (cf. Lie E(H)rig → (Lie E(H)rig ) est une filtration localement facteur direct de codimension 1. D’o` u une filtration V (H)rig ⊂ DO (H) ⊗ OXrig localement facteur direct de codimension 1. Cette filtration d´efinit l’application des p´eriodes Xrig −→ P(DO (H))rig Bien sˆ ur, cette application s’´etend sur tout l’espace de Rapoport-Zink en un morphisme ´etale D× -´equivariant M −→ P(DO (H))rig d´efini de la mˆeme fa¸con que pr´ec´edemment en rempla¸cant X par M. L’espace des p´eriodes P(DO (H))rig est en quelque sorte l’espace M quotient´e par la relation d’isog´enie, c’est-`a-dire le quotient de la tour de Lubin-Tate par le groupe GLn (F ).
2. Si G est un groupe de Lubin-Tate et K ⊂ Id + π k End(Λ), alors sur XK le groupe G est muni d’une structure de niveau de Drinfeld, de niveau k. C’est une cons´equence du fait que l’espace classifiant des structures de Drinfeld de niveau k est fini au-dessus de X a mˆeme fibre g´en´erique que XK (en fibre g´en´erique toutes les d´efinitions des structures de niveau co¨ıncident) et est donc en-dessous du normalis´e. 3. , la donn´ee pour i entre 1 et n − 1 de nombres rationnels αi , 0 < αi < 1, tels que le polygone commen¸cant en (0, 1), passant par les (q i , αi ) et finissant en (q n , 0) soit convexe.
12, il y a un isomorphisme naturel ∼ Da→a ,K −−→ Da →a,K d’o` u deux immersions ouvertes V ppp p p B ppp Da→a ,Kt xxx xxx xx8 Da,K Da ,K Ces applications de face sont ´equivariantes pour l’action de GLn (F ) × D× . 14. Plus g´en´eralement, soit a0 → a2 → · · · → ar 44 Chapitre I. D´ecomposition cellulaire de la tour de Lubin-Tate un simplexe orient´e de I. Mettons-le sous la forme ai = [Λi , Mi ], o` u Λ0 ⊂ Λ1 ⊂ · · · ⊂ Λr ⊂ π −1 Λ et Mi = Π[Λ0 :Λi ] M0 . Cela d´efinit un type i, avec ia = dimFq Λi /Λ0 , et un drapeau E • de type i dans π −1 Λ0 /Λ0 .