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\(p\)-adic \(L\)-functions for unitary Shimura varieties. I: Construction of the Eisenstein measure. (English) Zbl 1143.11019

This paper is the first of an interesting project devoted to studying the relations between \(p\)-adic \(L\)-functions for \(\text{GL}(n)\) (and unitary groups), congruences between stable and endoscopic automorphic forms on unitary groups and Selmer groups for \(p\)-adic representations.
They first recall the theory of modular forms on unitary Shimura varieties in setting of M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. I. [Invent. Math. 82, 151–189 (1985; Zbl 0598.14019); II. Compos. Math. 60, 323–378 (1986; Zbl 0612.14019)]. Then, following Hida’s generalization of the construction of Deligne and Katz for \(\text{GL}(2)\), they present the theory of \(p\)-adic modular forms on unitary Shimura varieties. They have highlighted some special features adapted to the embedding of Igusa towers.
The last part of the paper is devoted to the calculation of the local coefficients of Eisenstein series and the relation of local to global coefficients. They construct the Eisentein measures in several variables on a quasi-split unitary group, as a first step toward the construction of the \(p\)-adic \(L\)-function of families of ordinary holomorphic modular forms on unitary groups.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11R23 Iwasawa theory
14G35 Modular and Shimura varieties
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