Congruences for the Coefficients of Weakly Holomorphic Modular Forms

[摘要] Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. In each case, the values of the given arithmetic function appear as the Fourier coefficients of a weakly holomorphic modular form. We show that similar congruences exist for the coefficients of any weakly holomorphic modular form on any congruence subgroup Gamma0(N). In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. Finally, we make a more general statement about simultaneous congruences for the coefficients of weakly holomorphic modular forms on Gamma1(N).

[关键词] Mathematics