Henry H. Kim

Cuspidality of symmetric powers with applications

The purpose of this paper is to prove that the symmetric fourth power of a cusp form on GL(2), whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the RamanujanPetersson and Sato-Tate conjectures. In particular, we establish the bound q 1/9 v for

Functorial products for GL~2 x GL~3 and the symmetric cube for GL~2

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on GL2 × GL3 as automorphic forms on GL6, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on GL2. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain L-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Sections 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for GL2. As noted below, many other applications follow. To be precise, let π1 and π2 be two automorphic cuspidal representations of GL2(AF ) and GL3(AF ), respectively, where AF is the ring of adèles of a number field F . Write π1 = ⊗vπ1v and π2 = ⊗vπ2v. For each v, finite or otherwise, let π1v ⊠ π2v be the irreducible admissible representation of GL6(Fv), attached to (π1v, π2v) through the local Langlands correspondence by Harris-Taylor [HT], Henniart [He], and Langlands [La4]. We point out that, if φiv, i = 1, 2, are the twoand the three-dimensional representations of Deligne-Weil group, parametrizing πiv, respectively, then π1v⊠π2v is attached to the six-dimensional representation φ1v ⊗ φ2v . Let π1 ⊠ π2 = ⊗v(π1v ⊠ π2v). Our first result (Theorem 5.1) is:

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

Abstract We prove that the functorial tensor product of cuspidal automorphic representations of GL2 and GL3 is an automorphic representation of GL6. As a consequence, we prove that the symmetric cube of a cuspidal automorphic representation of GL2 is an automorphic representation of GL4 which is in general cuspidal (conjectured by Langlands). We improve on present estimates for Ramanujan and Selberg conjectures on GL2, breaking the crucial 1/6 estimate.

Symmetric cube L-functions for GL~2 are entire

The purpose of this paper is to prove the long awaited holomorphy of the third symmetric power L-functions attached to nonmonomial cusp forms of GL_2 over an arbitrary number field on the whole complex plane.

Langlands-shahidi method and poles of automorphicL-functions II

We use Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations, to study the Rankin-SelbergL-functions of GLk × classical groups. Especially we prove thatL(s, σ ×τ) is holomorphic, except possibly ats=0, 1/2, 1, whereσ is a cuspidal representation of GLk which satisfies weak Ramanujan property in the sense of Cogdell and Piatetski-Shapiro andτ is any generic cuspidal representation of SO2l+1. Also we study the twisted symmetric cubeL-functions, twisted by cuspidal representations of GL2.

Quadratic unipotent Arthur parameters and residual spectrum of symplectic groups

The purpose of this paper is to study certain quadratic unipotent Arthur parameters in the sense of Moeglin and use them to parametrize a part of the residual spectrum of symplectic groups over number fields, coming from the conjugacy class of Borel subgroups. In particular, using certain identities satisfied by local intertwining operators, Arthurs multiplicity formula is established for them which remarkably enough appears by itself in the corresponding residue of the Eisenstein series.

Holomorphy of Rankin tripleL-functions; special values and root numbers for symmetric cubeL-functions

In this paper we prove the holomorphy of Rankin tripleL-functions for three cusp forms on GL(2) on the entire complex plane, if at least one of them is non-monomial. We conclude the paper by proving the equality of our root numbers for the third and the fourth symmetric powerL-functions with those of Artin through the local Langlands correspondence. We also revisit Deligne’s conjecture on special values of symmetric cubeL-functions.

Holomorphy of the 9th symmetric power L-functions for Re(s) > 1

Let F be a number field and denote by A its ring of adeles. Let π = ⊗vπv be a cuspidal automorphic representation of GL2(A). We normalize π so that ωπ | R+ ≡ 1, where ωπ is the central character of π and A∗ = A1 × R+. One of the consequences of the existence of Sym(π) and Sym(π) as automorphic representations of GL4(A) and GL5(A), respectively, recently proved in [2, 7], is a proof of certain analytic properties of L(s, π, Sym ρ2) for m ≤ 9 (cf. [6, 8]). While we succeeded in proving that when m ≤ 8, each of these symmetric power L-functions are invertible and even absolutely convergent for Re(s) > 1 (cf. [6, 13]), no such results were obtained for L(s, π, Sym ρ2). In fact, all that we knew (even though it is not stated explicitly in [6]) was that L(s, π, Sym ρ2) is holomorphic and nonvanishing for Re(s) ≥ 1, except possibly for finitely many poles and zeros on the real axis 1 ≤ s ≤ 2. On the other hand, Gelbart and Lapid [1] have recently established zero free regions for each of the L-functions obtained from Langlands-Shahidi method to the left