Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3) - Choi SoYoung; Kim Chang Heon; Lee Kyung Seung (Open Mathematics)
For p∈{2,3}p\in \left\{2,3\right\} and an even integer kk, let Wk−2−(p)W_{k-2}^{-}\left(p) be the space of period polynomials of weight k−2k-2 on Γ0+(p){Γ }_{0}^{+}\left(p) with eigenvalue −1-1 under the Fricke involution. We determine the dimension formula for Wk−2−(p)W_{k-2}^{-}\left(p) and construct an explicit basis for it using period functions for weakly holomorphic modular forms. Furthermore, for a quadratic form QQ, we define the function F−(z,Q)F^{-}\left(z,Q) on the complex upper half-plane as a generating function of the cycle integrals of the canonical basis elements for the space of weakly holomorphic modular forms of weight kk and eigenvalue −1-1 under the Fricke involution on Γ0(p){Γ }_{0}\left(p). We also show that F−(z,Q)F^{-}\left(z,Q) is a modular integral on Γ0+(p){Γ }_{0}^{+}\left(p). Our approach focuses on calculating cycle integrals within Γ0(p){Γ }_{0}\left(p) rather than Γ0+(p){Γ }_{0}^{+}\left(p), which allows us to overcome certain technical challenges. This study extends earlier work by Choi and Kim (Rational period functions and cycle integrals in higher level cases, J. Math. Anal. Appl. 427 (2015), no. 2, 741–758) which focused on eigenvalue +1, providing new insights by examining eigenvalue −1-1 cases in the theory of rational period functions and cycle integrals in this setting.