Cuspidality of symmetric powers with applications

Abstract

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 Ramanujan-Petersson and Sato-Tate conjectures. In particular, we establish the bound q 1/9 v for unramified Hecke eigenvalues of cusp forms on GL(2). Over an arbitrary number field, this is the best bound available at present.

On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms

“…Proof This lemma follows from Gelbart and Jacquet [5] for k = 2 and from the recent works of Kim and Shahidi [11,12] and Kim [10] when k = 3, 4. The current explicit version of this lemma can be found in [17].…”

“…As a part of the far-reaching Langlands program, the analytic properties of symmetric power L-functions L(sym j f , s) are important topics in contemporary mathematics and have a significant impact on modern number theory. The analytic continuation of the symmetric power L-functions L(sym j f , s) with j = 2, 3, 4 over the whole complex plane and the predicted functional equations have been established by Gelbart and Jacquet [5], Kim and Shahidi [11,12], and Kim [10] respectively. Lemma 2.6 Let f (z) ∈ S k (Γ) be a Hecke eigencuspform of even integral weight k. The jth symmetric power L-function L(sym j f , s) is defined in (2.3).…”

“…From the famous works of Gelbart and Jacquet [5], Kim and Shahidi [11,12], and Kim [10], we learn that for 1 ≤ j ≤ 4 the j-th symmetric power L-function L(sym j f , s) agrees with the L-function associated with an automorphic cuspidal selfdual representation sym j π f of GL j+1 (A Q ). Then from the works of Jacquet and Shalika [8,9], Shahidi [20][21][22][23][24], and the reformulation of Rudnick and Sarnak [18], we know the analytic properties for the Rankin-Selberg L-functions L(sym i f ×sym j g, s) with i, j = 1, 2, 3, 4.…”

Cancellation of two classes of dirichlet coefficients over Beatty sequences

“…Although we lack the automorphy of π × π, it is known from the work of Kim and Shahidi [22] that sym 2 π is an automorphic irreducible cuspidal representation. Then by Landau's lemma (see [3]), we have…”

Standard zero-free regions for Rankin–Selberg L-functions via sieve theory

“…Moreover, cases (3) and (4) make full use of the advances in functoriality by Kim and Shahidi [15,16].…”