On the fundamental automorphic L-functions of SO(2n + 1)

Abstract

The fundamental automorphic L-functions of SO2n+1 are by definition the Langlands automorphic L-functions attached to irreducible cuspidal automorphic representations σ of SO2n+1(A) and the fundamental complex representations ρ1, ρ2, . . . , ρn of the complex dual group Sp2n(C) of SO2n+1,whereA is the ring of adeles of the number field k. These Lfunctions are denoted by L(s, σ, ρj)which is given by a Euler product of all local L-factors. The precise definition will be given in Section 2. It is known by a theorem of Langlands that the L-functions L(s, σ, ρj) converge absolutely for the real part of s large [6]. The Langlands conjecture asserts that L(s, σ, ρj) should have meromorphic continuation to the whole complex plane C, satisfy a functional equation relating the value at s to the value at 1−s, and have finitely many poles on the real line. When j = 1, L(s, σ, ρ1) is the standard L-function. In this case, the Langlands conjecture has been verified through the doubling method of Gelbart, Piatetski-Shapiro, and Rallis [11] and through the Langlands-Shahidi method [12]. For j ≥ 2, it is clear that the Langlands conjecture for L(s, σ, ρj) is beyond reach via the Langlands-Shahidi method or via any currently known integral representation of the Rankin-Selberg-type (except for certain cases of n ≤ 4 and j = 2 [7]).