Hecke structures of weakly holomorphic modular forms and their algebraic properties

Let Sk!(Γ1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ1(N) with an even integer k>2 and Mk!(Γ1(N)) be the space of weakly holomorphic modular forms of weight k on Γ1(N). Further, let z denote a complex variable and D:=12πi∂∂z. In this paper, we construct a basis of the space Sk!(Γ1(N))/Dk−1(M2−k!(Γ1(N))) consisting of Hecke eigenforms by using the Eichler-Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla-Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.