On the residue method for period integrals

Abstract

By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic Lfunctions. In some cases, we also study the local multiplicity of the spherical varieties.

The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods

“…Theorem 1.5 of [PWZ19] proves one direction of this conjecture under the assumption that π is discrete series at a split place of F . In [Les19a], we outlined a comparison of relative trace formulas conjectured by Wei Zhang (first suggested in a less precise form in [GW14]) which would prove this conjecture if established.…”

“…Recently, these periods have appeared in the literature in several ways (for example, [IP18], [PWZ19], and indirectly in [LZ19]) with several interesting applications to arithmetic and relative functoriality. As a simple example, we have the following conjecture, which is a special case of conjectures of Getz and Wambach [GW14].…”

Periods of automorphic forms over reductive subgroups

“…Periods of Eisenstein series induced from maximal parabolic subgroups. Since we will use the results of this paper in [37] in the particular case of Eisenstein series induced from maximal parabolic subgroup of G we include this case here. We hope this Paragraph will also demonstrate that the constructions of this article are very explicit in practice.…”

“…The above mentioned work of Ichino-Yamana [15] also follows this path using a mixed truncation. In a future collaboration [37] with A. Pollack and C. Wan, we will apply the mixed truncation and regularization developed here to several new cases that emerged from considering the conjectures of Sakellaridis-Venkatesh [40]. The explicit formula of Corollary 4.3 is particularly useful for such questions.…”

On the p-adic interpolation of unitary Friedberg--Jacquet periods

“…• The "relative trace formula approach" in forthcoming work of Jingwei Xiao and Wei Zhang, and the work of Spencer Leslie [Les19a], [Les19b]. • Applications of the residue method in the work of Pollack-Wan-Zydor [PWZ19].…”