17 research outputs found
Analytic continuation on Shimura varieties with -ordinary locus
We study the geometry of unitary Shimura varieties without assuming the
existence of an ordinary locus. We prove, by a simple argument, the existence
of canonical subgroups on a strict neighborhood of the -ordinary locus
(with an explicit bound). We then define the overconvergent modular forms (of
classical weight), as well as the relevant Hecke operators. Finally, we show
how an analytic continuation argument can be adapted to this case to prove a
classicality theorem, namely that an overconvergent modular form which is an
eigenform for the Hecke operators is classical under certain assumptions.Comment: Redaction and references improve
The compatibility with the duality for partial Hasse invariants
We give a simple and natural proof for the compatibility of the Hasse
invariant with duality. We then study a -divisible group with an action of
the ring of integers of a finite ramified extension of . We
suppose that it satisfies the Pappas-Rapoport condition ; in that case the
Hasse invariant is a product of partial Hasse invariants, each of which can be
expressed in terms of primitive Hasse invariants. We then show that the dual of
the -divisible group naturally satisfies a Pappas-Rapoport condition, and
prove the compatibility with the duality for the partial and primitive Hasse
invariants.Comment: 12 page
Partial Hasse invariants, partial degrees and the canonical subgroup
If the Hasse invariant of a -divisible group is small enough, then one can
construct a canonical subgroup inside its -torsion. We remark that, assuming
the existence of a subgroup of adequate height in the -torsion whose dual
has small degree, the expected properties of the canonical subgroup can be
easily proven. A fundamental relation is the equality between the Hasse
invariant and the degree of the dual of the canonical subgroup. When one
considers a -divisible group with an action of the ring of integers of a
(possibly ramified) finite extension of , then much more can be
said. One can define partial Hasse invariants ; they are natural in the
unramified case, and generalize a construction of Reduzzi and Xiao in the
general case. One can also define partial degrees for finite flat subgroups of
. We prove some properties for these partial Hasse invariants and partial
degrees, and compute the partial degrees of the canonical subgroup.Comment: 28 pages, 2 tables. To appear in Canad. J. Mat
Formes modulaires surconvergentes, ramification et classicit\'e
We prove in this paper a classicality result for overconvergent modular forms
on PEL Shimura varieties of type (A) or (C), without any ramification
hypothesis. We use an analytic continuation method, which generalizes previous
results in the unramified setting. We work with the rational model of the
Shimura variety, and use an embedding into the Siegel variety to define the
integral structures on the rigid space.Comment: 42 pages. In french. To appear in Ann. Inst. Fourie
On the geometry of the Pappas-Rapoport models in the (AR) case
We study some integral model of P.E.L. Shimura varieties of type A for
ramified primes. Precisely, we look at the Pappas-Rapoport model (or splitting
model) of some unitary Shimura varieties for which there is ramification in the
degree 2 CM extension. We show that the model isn't smooth, but that it is
normal and a local complete intersection. We moreover study its special fiber
by introducing a combinatorial stratification for which we can compute the
closure relations. Even if there are "extra" components in special fiber, we
prove that those do not contribute to mod p modular forms in regular degree. We
also study the interaction of the stratification with the natural
stratification given by the vanishing of some partial Hasse invariants, in the
case of signature (1,n-1)
Classicité de formes modulaires surconvergentes sur une variété de Shimura
We deal with overconvergent modular forms défined on some Shimura varieties, andprove classicality results in the case of big weight. First we study the case of varieties with good reduction, associated to unramified groups in p. We deal with Shimura varieties of PEL type (A) and (C), which are associated respectively to unitary and symplectic groups. To prove a classicality theorem, we use the analytic continuation method, which has been developed by Buzzard and Kassaei in the case of the modular curve. We then generalize this classicality result for varieties without assuming that the associated group is unramified in p. In the case of Hilbert modular forms, we construct integral models of compactifications of the variety, and prove a Koecher principle. For more general Shimura varieties, we work with the rationnal model of the variety, and use an embedding to a Siegel variety to define the integral structures.Nous nous intéressons aux formes modulaires surconvergentes définies sur certaines variétés de Shimura, et prouvons des théorèmes de classicité en grand poids. Dans un premier temps, nous étudions les variétés ayant bonne réduction, associées à des groupes non ramifiés en p. Nous nous intéressons aux variétés de Shimura PEL de type (A) et (C), qui sont associées respectivement à des groupes unitaires et symplectiques. Pour démontrer un théorème de classicité, nous utilisons la méthode du prolongement analytique, qui a été développée par Buzzard et Kassaei dans le cas de la courbe modulaire. Nous généralisons ensuite ce résultat de classicité à des variétés en ne supposant plus que le groupe associé est non ramifié en p. Dans le cas des formes modulaires de Hilbert, nous construisons des modèles entiers des compactifications de la variété, et démontrons un principe de Koecher. Pour des variétés de Shimura plus générales, nous travaillons avec le modèle rationnel de la variété, et utilisons un plongement vers une variété de Siegel pour définir les structures entières
A quotient of the Lubin-Tate tower II
In this article we construct the quotient M_1/P(K) of the infinite-level
Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form
(n-1,1) as a perfectoid space, generalizing results of one of the authors (JL)
to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results
for certain Harris-Taylor Shimura varieties at infinite level. As an
application of the quotient construction we show a vanishing theorem for
Scholze's candidate for the mod p Jacquet-Langlands and the mod p local
Langlands correspondence. An appendix by David Hansen gives a local proof of
perfectoidness of M_1/P(K) when n = 2, and shows that M_1/Q(K) is not
perfectoid for maximal parabolics Q not conjugate to P.Comment: with an appendix by David Hanse