Remarks on the Symmetric Powers of Cusp Forms on Gl(2)

where the unordered pair {αv , βv} defines the diagonal conjugacy class in GL2(C) attached to πv. Even at a ramified (resp. archimedean) place v, one has by the local Langlands correspondence a 2-dimensional representation σv of the extended Weil groupWFv ×SL(2,C) (resp. of the Weil group WFv), and the v-factor of the symmetric m-th power L-function is associated to sym(σv). A basic conjecture of Langlands asserts that there is, for each m, an (isobaric) automorphic representation symm(π) of GL(m + 1,A) whose standard (degree m + 1) L-function L(s, symm(π)) agrees, at least at the primes not dividing N , with L(s, π; symm). It is well known that such a result will have very strong consequences, such as the Ramanujan conjecture and the Sato-Tate conjecture for π. The modularity, also called automorphy, has long been known for m = 2 by the pioneering work of Gelbart and Jacquet ([GJ]); we will write Ad(π) for the selfdual representation sym2(π)⊗ ω−1, ω being the central character of π. A major breakthrough, due to Kim and Shahidi ([KS2, KS1, Kim])), has established the modularity of symm(π) for m = 3, 4, along with a useful cuspidality criterion (for m ≤ 4). Furthermore, when F = Q and π is defined by a holomorphic newform f of weight 2, Q-coefficients and level N , such that at some prime p, the component πp is Steinberg, a recent dramatic theorem of Taylor ([Tay3]), which depends on earlier works of his with Clozel, Harris and ShepherdBaron, furnishes the potential modularity of sym2m(π) (for every m ≥ 1), i.e., its modularity over a number field K, thereby finessing the Sato-Tate