On the rapid decay of cuspidal automorphic forms

TL;DR

This paper proves generalized rapid decay properties of automorphic cusp forms, including smooth forms, and establishes convergence results for Rankin-Selberg integrals, providing a foundation for new methods to analyze Langlands L-functions.

Contribution

It extends decay results to larger sets and smooth cusp forms, and proves convergence of integrals crucial for L-function analysis, advancing the analytic theory of automorphic forms.

Findings

Decay of cusp forms on larger sets established

Decay for smooth, non-K-finite cusp forms proven

Convergence of Rankin-Selberg integrals shown

Abstract

Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets, and furthermore state and prove the decay for smooth but not necessarily K-finite cusp forms. We also state a general theorem about the convergence of Rankin-Selberg integrals involving unipotent periods, closing a gap in the literature on L-functions. These properties serve as the analytic basis of a new method to establish holomorphic continuations of Langlands L-functions, in particular the exterior square L-functions on GL(n). Keywords: Automorphic forms, rapid decay, cusp forms, L-functions, Rankin-Selberg, integral representations, uniform moderate growth.