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

Abstract

Given an automorphic line bundle OX(k) of weight k on the Drinfel'd upper half plane X over a local field K, we construct a GL2(K)-equivariant integral lattice O X(k) in OX(k) ⊗K K, as a coherent sheaf on the formal model X underlying X ⊗K K. Here K/K 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 H ∗ ( X, O X (k)) and obtain applications to the de Rham cohomology H 1 dR (� \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 � \X uniformized by X: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing H 1 dR (� \X, Sym k K (St)), we re-prove the Hodge decomposition of H 1 dR (� \X, Sym k K (St)) and show that the monodromy operator on H 1 dR (� \X, Sym k K (St)) respects integral de Rham struc- tures and is induced by a "universal" monodromy operator defined on X, i.e. before passing to the � -quotient.