On the finite order of Whittaker functions, Eisenstein series, and automorphic L-functions

The theory of automorphic forms has seen an important breakthrough in the past few years. Shahidi and Kim [15] [16] [14], have been able to prove a few new cases of functoriality. One case of particular interest, is that they established the functoriality of Sym3 : GL₂ → GL₄. Further, these new results lead to the best estimates towards Selberg's conjecture for GL₂ Maass forms. Of course, central to these new developments is the Langlands-Shahidi method, coming from the theory of cuspidally induced Eisenstein series.; The proof of functoriality in these cases requires the use of a converse theorem. For this, one must verify analytic information about L-functions related to the candidate functorial representation. One condition, is boundedness in vertical strips of the complex plane. The paper of Gelbart and Shahidi [7] proves this. They employ the theory of Eisenstein series. However, the proof of meromorphic continuation of Eisenstein series due to Langlands [20] requires the use of a functional equation in a somewhat analytically delicate way. Thus, so does the result in [7]. Further, they need to quote Müller's estimates [23].; There is an ingenious Fredholm theory proof of meromorphic continuation due to Selberg where one does not require a functional equation. This thesis is a first attempt to make Selberg's proof effective for the purpose of obtaining a self-contained result of the following form. For our group G, the Eisenstein series can be written as a ratio F( s, g)/D(s), where D and F are entire of finite order, with the bounds for F depending on g in a compact set of G. We establish this result, but with the additional restriction that the cusp form have trivial K-type (so as to use spherical inversion). At the present time, we have also to rely on Müller's estimates.; We also prove a finite order result on Whittaker functions, the remaining local issue. In a handful of examples, we show how the main result combined with this local result gives the result of [7], without the delicate appeal to a functional equation.