Functorial products for GL2×GL3 and functorial symmetric cube for GL2

Infinite Dimensional Groups and Automorphic L-Functions

“…The recent progress made towards the Ramanujan and Selberg conjectures relies on establishing certain new cases of Langlands functoriality conjecture [1,28,33,34,38]. In this section we will explain the case of symmetric powers for GL 2 .…”

“…Here is a table for pairs (G, M D ) which are used in [34] to give the analytic properties of L(s, (π 1 ⊠ π 2 ) × σ) for each n, n = 1, 2, 3, 4,…”

“…Applying the techniques of the method [19,27,48 (5.8) Theorem [34]. Conditions 5.6a), b) and c) are valid and therefore there exists an automorphic representation…”

“…In particular, one needs to use base change, both normal and non-normal cubic [3,37] , as well as a result from theory of K-types [7] which is an appendix to [34]. As a bonus one gets the equality of triple product L-functions L(s, π 1v × π 2v × σ v ) and root numbers ε(s, π 1v × π 2v × σ v , ψ v ) defined by Langlands-Shahidi method [52] with Artin factors for all v. Note that L(s, π 1v × π 2v × σ v ) and ε(s, π 1v × π 2v × σ v , ψ v ) are defined completely by methods of harmonic analysis (cf.…”

“…We note that to obtain the fifth symmetric power transfer of a cusp form π on GL 2 (A F ), it would be enough to prove that π × Sym 4 π can be transferred to an automorphic representation π ⊠ Sym 4 π of GL 10 (A F ), from which the existence of Sym 5 π follows by an appeal to the classification of automorphic forms on GL N (A F ). This was successfully accomplished for the transfer of forms on GL 2 (A F )×GL 3 (A F ) to GL 6 (A F ) in [34]. No higher transfers of forms on products of general linear groups is available at present.…”