An explicit lifting construction of CAP forms on O(1,5)

TL;DR

This paper explicitly constructs non-tempered CAP cusp forms on the orthogonal group O(1,5) using theta lifting from Maass forms, generalizing previous work to arbitrary definite quaternion algebras.

Contribution

It introduces a new explicit theta lifting method to construct CAP forms on O(1,5) for any definite quaternion algebra, extending prior results.

Findings

Constructed explicit non-tempered CAP cusp forms on O(1,5)

Determined all local components of the associated cuspidal representations

Showed that the constructed cusp forms are CAP forms

Abstract

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 $\mathbb{Q}$, the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of B. Our construction can be viewed as a generalization of [22] to the case of any definite quaternion algebras, for which we note that [22] takes up the case where the discriminant of B is two. Unlike [22] the method of the construction is to consider the theta lifting from Maass cusp forms to O(1,5), following the formulation by Borcherds. The cuspidal representations generated by our cusp forms are studied in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms.