Abstract
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.
Highlights
This paper completes the construction of p-adic L-functions for unitary groups
Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method
We study complex L-functions of automorphic representations of unitary groups of n-dimensional hermitian spaces, by applying the doubling method of Garrett and Piatetski—Shapiro–Rallis [Gar84, GPSR87] to the automorphic representations that contribute to the coherent cohomology of Shimura varieties in degree 0; in other words, to holomorphic modular forms
Summary
This paper completes the construction of p-adic L-functions for unitary groups. More precisely, in [HLS06], three of the authors proposed an approach to c The Author(s) 2020. One of the observations in the present project is that the integral information provided by these Betti periods can naturally be recovered in the setting of the doubling method, provided one works with Hida families that are free over their corresponding Hecke algebras, and one assumes that the Hecke algebras are Gorenstein These hypotheses are not indispensable, but they make the statements much more natural, and we have chosen to adopt them as a standard; some of the authors plan to indicate in a subsequent paper what happens when they are dropped. Our p-adic L-function, when specialized at a classical point corresponding to the automorphic representation π , gives the corresponding value of the classical complex L-function, divided by what appears to be the correctly normalized complex period invariant, and multiplied by a factor c(π ) measuring congruences between π and other automorphic representations This is a formal consequence of the Gorenstein hypothesis and is consistent with earlier work of Hida and others on p-adic L-functions of families.
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