Abstract
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups U_{n}times U_{n+1} in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called ∗-regular, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods introduced by Jacquet-Lapid-Rogawski. The other, built upon Zeta integrals, is expressed in terms of functionals on the Whittaker model. A direct proof of the equality between the two expressions is also given. Finally several useful auxiliary results about the spectral expansion of the Jacquet-Rallis trace formula are provided.
Highlights
One of the main motivation of the paper is the obtention of the remaining cases, the so-called “endoscopic cases”, of the Gan-Gross-Prasad and the Ichino-Ikeda conjectures for unitary groups
We fix most notation to be used in the paper, we explain our convention on normalization of measures, we introduce the various spaces of functions we need and we discuss several properties of Langlands decomposition along cuspidal data as well as kernel functions that are important for us
The goal of this section is to get the spectral expansion of the Flicker-Rallis integral of the automorphic kernel attached to a linear group and a specific cuspidal datum
Summary
The formation of ZFR(s, Wφ, φ) commutes with the spectral expansion (1.3.3.6) when (s) 1 and, as follows from the local theory, the Zeta integrals ZFR(s, Wφλ, φ) for λ ∈ ia∗M are essentially Asai L-functions whose meromorphic continuations, poles and growths in vertical strips are known Combining this with an application of the Phragmen-Lindelöf principle, we are able to deduce Theorem 1.3.3.1. As φ is not necessarily a cusp form, we get extra terms in the course of the unfolding but, thanks to the special nature of the cuspidal datum χ , we are able to show that they all vanish At this point, we use the spectral decomposition (1.3.3.6) to express ZRS(s, Wφ) as the integral of ZRS(s, Wφλ) when (s) 1. We can conclude that it is equal to Iχ (f ) by applying Theorem 1.2.4.1 to the expression (1.2.4.5)
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