TY - JOUR
T1 - Twisting moduli for GL(2)
AU - Bedert, Bejnamin
AU - Cooper, George
AU - Oliver, Thomas
AU - Zhang, Pengcheng
PY - 2020
Y1 - 2020
N2 - We prove various converse theorems for automorphic forms on Γ0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N∈{18, 20, 24}– these integers are the smallest multiples of 4or 9not covered by earlier work of Conrey–Farmer. This development is a consequence of finding generating sets for Γ0(N)such that each generator can be written as a product of special matrices. As for real-analytic Maass forms of even (resp. odd) weight we prove the analogous statement for 1 ≤N≤12and N∈{16, 18}(resp. 1 ≤N≤12, 14 ≤N≤18 and N∈{20, 23, 24}).
AB - We prove various converse theorems for automorphic forms on Γ0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N∈{18, 20, 24}– these integers are the smallest multiples of 4or 9not covered by earlier work of Conrey–Farmer. This development is a consequence of finding generating sets for Γ0(N)such that each generator can be written as a product of special matrices. As for real-analytic Maass forms of even (resp. odd) weight we prove the analogous statement for 1 ≤N≤12and N∈{16, 18}(resp. 1 ≤N≤12, 14 ≤N≤18 and N∈{20, 23, 24}).
U2 - https://doi.org/10.1016/j.jnt.2020.04.008
DO - https://doi.org/10.1016/j.jnt.2020.04.008
M3 - Article
SN - 0022-314X
VL - 217
SP - 142
EP - 162
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -