Explicit constructions of non-tempered cusp forms on orthogonal groups of low split ranks

The aim of this talk is to report a recent research on explicit constructions of cusp forms on orthogonal groups of split rank one or two by some lifts from cusp forms on the complex upper half plane. We also discuss cuspidal representations generated by them in terms of the explicit determination of their local components. As for the representation theoretic treatment, the point is to use Sugano’s non-archimedean local theory of ``Jacobi form formulation” of Oda-Rallis-Schiffman lifting to orthogonal groups of rank two. Sugano's local theory turns out to be useful also for the case of rank one. Such argument leads to non-temperedness of the non-archimedean local components and the explicit determination of the standard $L$-functions. It should be remarked that the cusp forms taken up in this talk are counterexamples to the Ramanujan conjecture and those in rank one case are real analytic but non-holomorphic. This talk includes a recent joint work with Ameya Pitale for the case of the rank one.