Functorial Products for GL 2 × GL 3 and the Symmetric Cube for GL 2

Ω-Result on Coefficients of Automorphic L-Functions Over Sparse Sequences

“…Lao and Sankaranarayanan [10] established the asymptotic formula of the sum n≤x λ 2 f (n j ), where j = 2, 3, 4. By using the properties of symmetric power L-functions and their Rankin-Selberg L-function, which have been established in the references [3,7,8,13,22], they showed that…”

“…However, for the sum n x λ 2 f (n 3 ), the condition (C) in Lemma 2.1 is not satisfied, and for the sum n x λ 2 f (n 4 ), the corresponding generating function has a factor L(sym 6 f ×sym 6 f, s), whose analytic properties are not clear (since the automorphy of the jth symmetric power lift of an automorphic cuspidal representation over GL 2 (A Q ) is only known for j ≤ 4, see e.g. [3,7,8,22]). …”

Special values of symmetric power L-functions and Hecke eigenvalues

“…It is has been proved in the cases m ∈ {1, 2, 3, 4} (see [GJ78], [KS02b], [KS02a] and [Kim03]). The second hypothesis is concerned with the eventual Landau-Siegel zero of the mth symmetric power Lfunctions, it is denoted by LSZ m (N ) and has been proved for m ∈ {1, 2, 4} (see [HL94], [GHL94], [HR95] and [RW03]).…”

Strong spectral gaps for compact quotients of products of PSL(2,ℝ)

“…Serre conjectures that the congruence subgroup property holds for such groups, this being the most elementary and fundamental case for which the congruence subgroup problem is open (see [34,Chapter 7]). If true, this coupled with the Jacquet-Langlands correspon-dence [21] implies that E( \G) is empty if the Ramanujan-Selberg conjecture [44] is true, and that E( \G) ⊆ (2, 18/7] by [24].…”