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

Given an automorphic line bundle of weight k on the Drinfeld 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_dR¹( X, Sym_K^k(St)) with coefficients in the k-th symmetric power of the standard representation of SL₂(K) (where k0) of projective curves X uniformized by X: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing H_dR¹( X, Sym_K^k(St)), we re-prove the Hodge decomposition of H_dR¹( X, Sym_K^k(St)) and show that the monodromy operator on H_dR¹( 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 -quotient.Mathematics Subject Classification (2000): 11F33, 11F12, 11G09, 11G18I wish to thank Peter Schneider and Jeremy Teitelbaum for generously providing me with some helpful private notes on their own work, and for their interest. I am also grateful to Matthias Strauch for useful discussions on odd weight modular forms. I thank Christophe Breuil for his interest and his insisting on lattices for the entire G-action. Finally I thank the referee for his suggestions concerning the presentation of several technical constructions.