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arXiv · 2003.02557

Twisting moduli for GL(2)

Benjamin Bedert·George Cooper·Thomas Oliver·Pengcheng Zhang
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Abstract

We prove various converse theorems for automorphic forms on Γ_0(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 is 18, 20, or 24 - these integers are the smallest multiples of 4 or 9 not covered by earlier work of Conrey-Farmer. This development is a consequence of finding generating sets for Γ_0(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 N=1,...12,16,18 (resp. N=1,...,12,14,15,16,17,18,20,23,24).

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Benjamin Bedert, George Cooper, Thomas Oliver, Pengcheng Zhang. 2020-03-05. Twisting moduli for GL(2). https://doi.org/10.1016/j.jnt.2020.04.008

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