Converse Theorems, Functoriality, and Applications

Abstract

Converse Theorems traditionally have provided a way to characterize Dirichlet series associated to modular forms in terms of their analytic properties. The prototypical Converse Theorem was due to Hamburger who characterized the Riemann zeta function in terms of its analytic properties [23]. More familiar may be the Converse Theorems of Hecke and Weil. Hecke first proved that Dirichlet series associated to modular forms enjoyed “nice” analytic properties and then proved “Conversely” that these analytic properties in fact characterized modular Dirichlet series [26]. Weil extended this Converse Theorem to Dirichlet series associated to modular forms with level [62]. In their modern formulation, Converse Theorems are stated in terms of automorphic representations instead of modular forms. This was first done by Jacquet and Langlands for GL2 [31]. For GLn, Jacquet, Piatetski-Shapiro, and Shalika have proved that the L-functions associated to automorphic representations have nice analytic properties similar to those of Hecke [32] (see also [15, 9]). The relevant “nice” properties are: analytic continuation, boundedness in vertical strips, and functional equation. Converse Theorems in this context invert this process and give a criterion for automorphy, or modularity, in terms of these L-functions being “nice” [31, 32, 13, 9]. The first application of a Converse Theorem that might come to mind is to the question of modularity of arithmetic or geometric objects. To use the Converse Theorem in this context one must first be able to prove that the Lfunctions of these arithmetic/geometric objects are “nice”. However, essentially the only way to show that an L-function is nice is to have it associated to an automorphic form already! So, on second thought, as a direct tool for establishing modularity the Converse Theorem seems a bit lacking. Instead, the most natural direct application of the Converse Theorem is to Functoriality, in this case the