The Refined Gross-Prasad Conjecture for Unitary Groups

TL;DR

This paper refines the Gross-Prasad conjecture for unitary groups, establishing explicit links between period integrals and L-values, and proves initial cases using Waldspurger's theorem and Theta correspondence techniques.

Contribution

It extends the refined Gross-Prasad conjecture to unitary groups and proves the first cases using advanced automorphic methods.

Findings

Established the conjecture for the first case using Waldspurger's theorem.

Proved the second case employing Theta correspondence machinery.

Connected period integrals with special L-values for unitary groups.

Abstract

Let F be a number field, A_F its ring of adeles, and let {\pi}_n and {\pi}_{n+1} be irreducible, cuspidal, automorphic representations of SO_n(A_F) and SO_{n+1}(A_F), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to {\pi}_n and {\pi}_{n+1} is equivalent to the non-vanishing of L(1/2, {\pi}_n x {\pi}_{n+1}). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period integral. We make a similar conjecture for unitary groups, and prove the first few cases. The first case of the conjecture will be proved using a theorem of Waldspurger, while the second case will use the machinery of the {\Theta}-correspondence.