An explicit construction of non-tempered cusp forms on $O(1,8n+1)$ (Analytic and Arithmetic Theory of Automorphic Forms)

Abstract

This short note is a write-up of the results presented by the second named author at RIMS workshop "Analytic and arithmetic theory of automorphic forms" The main result is an explicit construction of the rcal analytic cusp forms on O ({imath}, 8n+1) by a lifting from Maass cusp forms of level one. The lifting is proved to be Hecke-equivariant. Our results include an explicit formula for Hecke eigenvalues of the lifts and explicit determination of the cusidal representations generated by them. This leads to showing the nontemperedness of the cuspidal representations at every finite place, namely our explicit construction provides "real analytic counterexamples to Ramanujan conjecture".