An explicit lifting construction of CAP forms on O(1, 5) (Automorphic form, automorphic $L$-functions and related topics)

Abstract

This article is the write-up of what the fist named author presented on January 25th in 2022 during the RJMS workshop. We explicitly construct non-tempered cusp forms on the orthogonal group O(1, 5) of signature (1+, 5-). Given a definite quaternion algebra B over ℚ, the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of B. As well as the explicit construction we study the cuspidal representations generated by our cusp forms in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms. Our construction can be viewed as a generalization of [8]to the case of any definite quaternion algebras, for which we note that [8] takes up the case where the discriminant of B is two. Unlike [8] the method of the construction is to consider the theta lifting from Maass cusp forms to O(1, 5), following the formulation by Borcherds.