Converse theorems assuming a partial euler product

• Published in: The Ramanujan journal Vol. 15; no. 2; pp. 205 - 218
• Main Authors: Farmer, David W., Wilson, Kevin
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-02-2008
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; Hecke operator; Modular form; Converse theorem; function; 11F66
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-007-9073-1

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 Γ
0
(
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.