Hecke-Weil equivalances

Abstract

Any element f(z) of $D_k(Γ_{0}(N),χ) = {∈D_k(Γ_{0}(N)_{χ})|f|_{k}γ = χ(γ)f for any γ∈Γ_0(N)}$ has a Fourier expansion of the form $f(z) = \∑_{n=0}^{∞}a_{n}e^{2πinz}$. Never theless a holomorphic function $f(z)$ on $\BDbb H$ with a Fourier expansion is not necessarily a modular form. In Chpter 2, we shall provide a Hecke equivalence between the automorphy of f(z) and a certain functional equation of a Dirichlet series $L(s;f) = ∑_{n=0}^{∞} a_{n}n^{-s}$. In Chapter 3, we shall give a Weil equivalence between the automorphy of f(z) and a certain functional equation of a twisted Dirichlet series $L(s;f,φ) = ∑_{n=1}^{∞}φ(n)a_{n}n^{-s} with Dirichlet character χ$.