$p$-adic $L$-functions for unitary Shimura varieties. I: Construction of the Eisenstein measure

Abstract

We construct the Eisenstein measure in several variables on a quasi-split unitary group, as a first step towards the construction of $p$-adic $L$-functions of families of ordinary holomorphic modular forms on unitary groups. The construction is a direct generalization of Katz' construction of$p$-adic $L$-functions for CM fields, and is based on the theory of p-adic modular forms on unitary Shimura varieties developed by Hida, and on the explicit calculation of non-degenerate Fourier coefficients of Eisenstein series.