p-adic Differential Operators on Automorphic Forms and Applications.

Abstract

We construct certain C ∞-differential operators and their p-adic analogues, which act on (vector- or scalar-valued) automorphic forms on the unitary groups U (n, n). We study properties of these operators, and we prove some arithmeticity theorems using them. These differential operators are a generalization to the p-adic case of the C ∞-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. They are a generalization to the vector-valued situation of the p-adic differential operators constructed in the one-dimensional setting by N. Katz. They should be useful in the construction of certain p-adic L-functions, in particular p-adic L-functions attached to p-adic families of automorphic forms on the unitary groups U (n) × U (n).