Abstract. We construct a generalization of the Hasse invariant for any Shimura variety of PEL type A over a prime of good reduction, whose non-vanishing locus is the open and dense µ-ordinary locus.
International audienceThe isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field of characteristic p is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D
Soient
F
F
un corps de fonctions sur
F
q
\mathbb {F}_q
,
A
A
l’anneau des fonctions régulières hors d’une place
∞
\infty
et
p
\mathfrak {p}
un idéal premier de
A
A
. En premier lieu, nous développons la théorie de Hida pour les formes modulaires de Drinfeld de rang
r
r
qui sont de pente nulle pour l’opérateur de Hecke
U
π
\mathrm {U}_{\pi }
convenablement défini. En second lieu, nous montrons en pente finie l’existence de familles de formes modulaires de Drinfeld variant continûment selon le poids. En troisième lieu, nous prouvons un résultat de classicité: une forme modulaire de Drinfeld surconvergente de pente suffisamment petite par rapport au poids est une forme modulaire de Drinfeld classique.
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