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Fix a prime number p and let E / F be a CM extension of number fields in which p splits relatively. Let π be an automorphic representation of a quasi-split unitary group of even rank with respect to E / F such that π is ordinary above p with respect to the Siegel parabolic subgroup. We construct the cyclotomic p -adic L -function of π , and a certain generating series of Selmer classes of special cycles on Shimura varieties. We show, under some conditions, that if the vanishing order of the p -adic L -function is 1, then our generating series is modular and yields explicit nonzero classes (called Selmer theta lifts) in the Selmer group of the Galois representation of E associated with π ; in particular, the rank of this Selmer group is at least 1. In fact, we prove a precise formula relating the p -adic heights of Selmer theta lifts to the derivative of the p -adic L -function. In parallel to Perrin-Riou's p -adic analogue of the Gross–Zagier formula, our formula is the p -adic analogue of the arithmetic inner product formula recently established by Chao Li and the second author. [ABSTRACT FROM AUTHOR]

In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number −1. [ABSTRACT FROM AUTHOR]

Abstract: We prove uniqueness of Fourier–Jacobi models for general linear groups, unitary groups, symplectic groups and metaplectic groups, over an Archimedean local field. [Copyright &y& Elsevier]

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of being the base change of a stable cuspidal representation of the quasi-split unitary group associated to the quadratic extension E/F, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of GLn,X′ is nonvanishing if and only if the corresponding local system E on X′ is conjugate self-dual with respect to an étale double cover X′/X of curves, which directly relates to the base change from the associated unitary group. In particular, the geometric setting makes sense over any base field. [ABSTRACT FROM AUTHOR]