Growth of Fourier coefficients of vector-valued automorphic forms

Abstract

In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated to even integer weight vector-valued automorphic forms of Fuchsian groups of the first kind. At the end, their L-functions and exponential sums have been discussed.

Lifting of vector-valued automorphic forms

“…In particular to have Fourier expansion of X(τ) at the cusps of G, we need to ensure that ρ(γ) has only unitary eigenvalues for every parabolic element γ of G ′ . This is indeed true, and proved by the authors in [2,Sect. 6].…”

“…R), we can not define j(γ, τ) in this way, because of one possible sign factor. In [1] and [2], this sign factor was redundant as the authors worked with even weights vector-valued automorphic forms and therefore it always comes with an even power.…”

“…This section reviews the basics of vector-valued automorphic forms that we need to understand and prove the main results of the article. In [1] and [2], the authors studied vvaf associated to Fuchsian groups of the first kind of even weight. The even weight is a necessary condition to be able to define j(•, •), as we mentioned in the previous section.…”

“…In [1] the first author showed that, the induction of an admissible representation is admissible and hence the induced function X(τ) is also a weakly holomorphic admissible vvaf associated to the induced representation. We are mainly interested in the case when ρ is not necessarily admissible and all eigenvalues of ρ(γ) are unitary for every parabolic element γ ∈ G. In [2], the authors explained the necessity behind assuming the eigenvalues to be unitary. In the literature, such a representation is called logarithmic representation, and any associated vvaf X(τ) is called as logarithmic vector-valued automorphic form.…”

“…Knopp and Mason [4,5] and Gannon [3] studied this case when G is the modular group. Recently, the authors studied them for any Fuchsian group of the first kind in [2]. In this article, we want to say what happens to logarithmic vector-valued automorphic forms after lifting them.…”