A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing

TY - JOUR

T1 - A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties

T2 - Splicing and dicing

AU - Diamond, Fred

AU - L Kassaei, Payman

AU - Sasaki, Shu

PY - 2023/4/15

Y1 - 2023/4/15

N2 - 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 GL_2 in characteristic p and generalizes a result of Serre for classical modular forms. Finally we study the fibres 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.

AB - 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 GL_2 in characteristic p and generalizes a result of Serre for classical modular forms. Finally we study the fibres 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.

UR - https://arxiv.org/abs/2001.00530

U2 - 10.24033/ast.1198

DO - 10.24033/ast.1198

M3 - Article

SN - 0303-1179

VL - 439

JO - Astérisque

JF - Astérisque

ER -