Abstract
We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.
Highlights
Let f ∈ Mk( 0(N ), ξ ) be a classical holomorphic modular form of weight k, level N and nebentypus character ξ, and defineS
In this paper we propose a replacement for the Euler product that, we conjecture, characterizes the modular forms of any level N, yet retains the linearity of (1.3): Conjecture 1.1 Let ξ be a Dirichlet character modulo N, k a positive integer satisfying fn, gξn(−=1)O=(n(σ−) 1fo)kr,saonmde{σfn>}∞ n=01. ,F{ogrnq}∞ n∈=1Ns,elqeutences of complex numbers satisfying
In Lemma 4.10, we show that if we start from a pair of modular forms f, g satisfying
Summary
Let f ∈ Mk( 0(N ), ξ ) be a classical holomorphic modular form of weight k, level N and nebentypus character ξ , and define. Given sequences { fn}∞ n=1, {gn}∞ n=1 of at most polynomial growth, if the functions f (s) and g(s) defined by (1.2) continue to entire functions of finite order and satisfy (1.3) fn and gn are the Fourier coefficients of modular forms of level N and weight k, related by (1.1). It has been conjectured (see [5, Conjecture 1.2]) that if f (s) and g(s) have Euler product expansions of the shape satisfied by primitive Hecke eigenforms the single functional equation (1.3) should suffice to imply modularity, without the need for character twists. In this paper we propose a replacement for the Euler product that, we conjecture, characterizes the modular forms of any level N , yet retains the linearity of (1.3): Conjecture 1.1 Let ξ be a Dirichlet character modulo N , k a positive integer satisfying fn, gξn(−=1)O=(n(σ−) 1fo)kr,saonmde{σfn>}∞ n=01. In this paper we propose a replacement for the Euler product that, we conjecture, characterizes the modular forms of any level N , yet retains the linearity of (1.3): Conjecture 1.1 Let ξ be a Dirichlet character modulo N , k a positive integer satisfying fn, gξn(−=1)O=(n(σ−) 1fo)kr,saonmde{σfn>}∞ n=01. ,F{ogrnq}∞ n∈=1Ns,elqeutences of complex numbers satisfying
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