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12. Soient [Λ, M ] ∈ I et E ⊂ π −1 Λ/Λ de dimension i. Si p : π −1 Λ/Λ et si π −1 Λ K ⊂ (Id + πEnd(Λ)) ∩ Id + πEnd(p−1 (E)) il y a un isomorphisme ∼ ∂i,E D[Λ,M],K −−→ ∂n−i,(π−1 Λ/Λ)/E D[p−1 (E),Πi M],K induit par le quotient par η(E). 8. Mod`eles entiers des cellules 43 D´emonstration. 2, sur D[Λ,M],K le O-module formel universel H poss`ede une structure de niveau de Drinfeld de niveau 1 ´etendant la structure de niveau η sur la fibre g´en´erique. 4 qu’il existe un unique sous-groupe plat fini η(E) ⊂ H[π] induisant ponctuellement sur ∂i,E , en chaque point de la fibre g´en´erique, le sous-groupe des q i -points de plus grande valuation dans H[π].

Q i λn ). • de la forme f (α), o` u α est un z´ero de la s´erie formelle [π]Fx − β, avec β ∈ ker f \ {0}, et Fx d´esigne une loi de groupe formel associ´ee `a Hx . Le polygone de Newton de cette s´erie formelle est obtenu en prenant l’enveloppe convexe de (0, v(β)) et celui de [π]Fx 1 1− λ1 1 qi i n qn Fig. 4: Le polygone de Newton de [π]Fx − β, v(β) = λ1 Mais, sup β∈ker f \{0} v(x1 ) ≥ 1 − v(β) = λ1 . Or, λ1 < λn . En effet, qn 1 1 1 1 ⇒ λ1 ≤ et v(xn ) ≥ ⇒ λn ≥ n n n(q − 1) n n(q − q n−1 ) Donc, ∀β ∈ ker f \ {0}, Newt([π]Fx − β) est le segment joignant (0, v(β)) a` (q n , 0) avec v(β) ∈ {λ1 , .

Pour GL2 (Qp ), ce sch´ema formel u X∞ sera un sch´ema formel p-adique, muni d’une action de GL2 (Qp ) × D× , o` D|Qp est une alg`ebre de quaternions. On aura alors X∞ /π ai Z Z i∈I o` u I est un ensemble fini et ai ∈ N \ {0}. Bien sˆ ur, sur la construction pr´ec´edente, on ne voit pas l’action de D× , il faut utiliser le th´eor`eme de Serre-Tate pour la voir. 1 Hypoth`eses et notations Soit F |Qp une extension de degr´e fini, d’uniformisante π et de corps r´esiduel k = Fq = OF /πOF . On note F˘ = F nr le compl´et´e de l’extension maximale nonramifi´ee de F dans une clˆ oture alg´ebrique de celui-ci et F 0 l’extension maximale ˘ pour O ˘ .

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Complex Numbers and Complex Arithmetic [article] by Doe J.


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