Arithmetic properties for the minus space of weakly holomorphic modular forms

Let M-k(!) (p) be the space of weakly holomorphic modular forms of weight k on Gamma(0) (p), and let M-k(!-) (p) be the minus space which is the subspace of M-k(!) (p) consisting of all eigenforms of the Fricke involution W-p with eigenvalue -1. We are interested in finding a canonical basis for the minus space M-k(!-) (p) for certain levels. Using this result, along with previous works of Choi and Kim [CK13], we find a canonical basis for the space M-k(!) (p), and investigate its arithmetic properties. We also give another generalization of [CK13] to the cases of square-free integer levels. (C) 2018 Elsevier Inc. All rights reserved.