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Curieux·euse de découvrir l’histoire derrière notre nouveau style ?
We consider Hilbert modular varieties in characteristic p with Iwahori level at p and construct a geometric Jacquet- Langlands relation showing that the irreducible components are isomorphic to products of projective bundles over quaternionic Shimura varieties of level prime to p. We use this to establish a relation between mod p Hilbert and quaternionic modular forms that reflects the representation theory of GL2 in characteristic p and generalizes a result of Serre for classical modular forms. Finally we study the fibers of the degeneracy map to level prime to p and prove a cohomological vanishing result that is used to associate Galois representations to mod p Hilbert modular forms.
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