Integral structures in automorphic line bundles on the p-adic upper half plane

Abstract.

Given an automorphic line bundle of weight k on the Drinfel’d upper half plane X over a local field K, we construct a GL₂(K)-equivariant integral lattice in as a coherent sheaf on the formal model underlying Here is ramified of degree 2. This generalizes a construction of Teitelbaum from the case of even weight k to arbitrary integer weight k. We compute and obtain applications to the de Rham cohomology H _dR ¹(Γ∖ X, Sym _K ^k(St)) with coefficients in the k-th symmetric power of the standard representation of SL₂(K) (where k≥0) of projective curves Γ∖X uniformized by X: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing H _dR ¹(Γ∖ X, Sym _K ^k(St)), we re-prove the Hodge decomposition of H _dR ¹(Γ∖ X, Sym _K ^k(St)) and show that the monodromy operator on H _dR ¹(Γ∖ X, Sym _K ^k(St)) respects integral de Rham structures and is induced by a ‘‘universal’’ monodromy operator defined on , i.e. before passing to the Γ-quotient.