On the Ramanujan Conjecture and Finiteness of Poles for Certain L- Functions

Infinite Dimensional Groups and Automorphic L-Functions

“…Estimate (3.6), proved in [33], and its archimedean counterpart, proved in [29], are obtained by applying a general result of [51,52], Lemma 5.8, to the case E 8 − 2 of [51], with the representation of M(A F ), tailored from Sym 3 π and Sym 4 π as explained in [33]. Observe that the derived group of M is isomorphic to SL 4 × SL 5 .…”

“…We shall assume π is not monomial. It then follows that either Theorem 5.1 is proved by applying a version of converse theorems of Cogdell and Piatetski-Shapiro [10,11] to certain triple product L-functions L(s, (π 1 ⊠ π 2 ) × σ) whose analytic properties are obtained from the Langlands-Shahidi method [19,27,48,49,50,51,52].…”

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“…When m ¼ 1 and p 0 is the trivial representation 1, this L-function agrees with the standard L-function. There are two distinct methods for defining these L-functions, the first using the gcds of integral representations, due to Jacquet, Piaietski-Shapiro and Shalika ( [JPSS83]), and the second via the constant terms of Eisenstein series on larger groups, due to Langlands and Shahidi ([Sh88,90]); see also [MW89]. The fact that they give the same L-functions is nontrivial but true.…”

Irreducibility and Cuspidality

“…Let π, π ′ be isobaric automorphic representations in A(n, F ), A(n ′ , F ) respectively. Then there exists an associated Euler product L(s, π × π ′ ) ( [JPSS], [JS1], [Sh1,2], [MW], [CoPS2]), which converges in {ℜ(s) > 1}, and admits a meromorphic continuation to the whole s−plane and satisfies the functional equation, which is given in the unitary case by…”

“…π). The Lfunction on the right is the Rankin-Selberg L-function, whose mirific properties have been established in the independent and complementary works of Jacquet, Piatetski-Shapiro and Shalika ( [JPSS], and of Shahidi ([Sh1,2]); see also [MW]. A theorem of Jacquet and Shalika ([JS1]) asserts that the order of pole at s = 1 of L S (s, π × π ∨ ) is 1 iff π is cuspidal.…”