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.
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A Structure Theorem on Intersections of General Doubling Measures and Its Applications
Abstract In this paper, we construct an explicit family of measures that are $p$adic doubling for any given pair of primes, yet not doubling. This generalizes the construction by Boylan, Mills, and Ward on a structure theorem on the intersection of dyadic doubling measures and triadic doubling measures. As some byproducts, we apply these results to show analogous statements about the reverse Hölder and Muckenhoupt $A_p$ classes of weights.
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 Award ID(s):
 1954407
 NSFPAR ID:
 10406226
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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