-ADIC -FUNCTIONS FOR UNITARY GROUPS

This paper completes the construction of$p$-adic$L$-functions for unitary groups. More precisely, in Harris, Li and Skinner [‘$p$-adic$L$-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’,Doc. Math.Extra Vol.(2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such$p$-adic$L$-functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and$p$-adic differential operators [Eischen, ‘A$p$-adic Eisenstein measure for unitary groups’,J. Reine Angew. Math.699(2015), 111–142; ‘$p$-adic differential operators on automorphic forms on unitary groups’,Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local$\unicode[STIX]{x1D701}$-integrals occurring in the Euler product (including at$p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.