Converse theorems assuming a partial Euler product

2004/08/17 by David W. Farmer, Farmer, David W., Kevin Wilson +1
Mathematics · #11F66 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F66

paper · pdf · doi:10.48550/arxiv.math/0408221

12 pages, LaTeX. Final version. To appear in The Ramanujan Journal

arxiv created 2005/04/06 · arxiv updated 2009/12/01

Abstract

Associated to a newform f(z) is a Dirichlet series L_f(s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F(s) has a functional equation of a particular form, then F(s)=L_f(s) for some holomorphic newform f(z) on Γ(1). Weil extended this result to Γ₀(N) under an assumption on the twists of F(s) by Dirichlet characters. Conrey and Farmer extended Hecke's result for certain small N, assuming that the local factors in the Euler product of F(s) were of a special form. We make the same assumption on the Euler product and describe an approach to the converse theorem using certain additional assumptions. Some of the assumptions may be related to second order modular forms.

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