Generalized Hecke operators on weakly holomorphic modular forms of non-positive weight

Generalized Hecke operators, originating from the replication formula in Monstrous Moonshine, were extended in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] to apply to harmonic Maass functions on modular curves of higher genera, building on works in [M. Koike, On replication formula and Hecke operators, Nagoya University, preprint]. Their action was further applied to weakly holomorphic modular functions, deriving numerous arithmetic properties of Fourier coefficients. In this paper, we extend these operators to weakly holomorphic modular forms of arbitrary even non-positive weights. In the process, we show [Proposition 3.1 of L. Beneish and M. H. Mertens, On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras, Math. Z. 297(1–2) (2021) 59–80], which is used to obtain dimension formulas for certain vertex operator algebras, can be derived from our results. Additionally, we identify the conditions under which the action of the generalized Hecke operator preserves holomorphicity. Moreover, we show that this action can be expressed as a linear combination of same-level or lower-level forms, refining the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717]. Finally, we establish more general congruence relations on Fourier coefficients, with the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] emerging as a special case.