An explicit construction of non-tempered cusp forms on $O(1,8n+1)$

TL;DR

This paper explicitly constructs non-tempered cusp forms on orthogonal groups of signature (1,8n+1) via theta lifts from Maass forms, analyzing their automorphic properties and L-functions.

Contribution

It introduces a new explicit construction method for non-tempered cusp forms on orthogonal groups using Borcherds' Fourier expansion, replacing the previous reliance on Maass' converse theorem.

Findings

Constructed cusp forms are irreducible and non-tempered at non-archimedean places.

Archimedean components of the cusp forms are tempered if originating Maass forms are Hecke eigenforms.

Standard L-functions factor into symmetric square L-functions and shifted Riemann zeta functions.

Abstract

We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. The standard $L$-functions of the cusp forms are proved to be products of symmetric square $L$-functions of the Hecke-eigen Maass cusp forms with shifted Riemann zeta functions.