Compact periods of Eisenstein series of orthogonal groups of rank one at even primes

TL;DR

This paper computes local factors of Eisenstein series periods for orthogonal groups at even primes, providing explicit evaluations and examples that support the Gross-Prasad conjecture.

Contribution

It offers explicit calculations of local period factors at even primes for orthogonal groups, extending previous work to include these challenging cases.

Findings

Explicit evaluation of local factors at even primes

Verification of consistency with the Gross-Prasad conjecture

Concrete examples for the case k=Q

Abstract

Fix a number field k with its adele ring A. Let G=O(n+3) be an orthogonal group of k-rank 1 and H=O(n+2) a k-anisotropic subgroup. We have previously [arXiv:0908.3521] described how to factor the global period of a spherical Eisenstein series of G against a cuspform F of H into an Euler product. Here, we describe how to evaluate the factors at even primes. When the local field is unramified, we carry out the computation in all cases. We show also concrete examples of the complete period when k=Q. The results are consistent with the Gross-Prasad conjecture.