The arithmetic study of Taylor coefficients of modular forms at CM points has its roots in the classical theory of complex multiplication, beginning with the algebraicity the values of the j-function at CM points. The study of Taylor expansions of modular forms at CM points has since developed through work of Shimura on nearly holomorphic modular forms and became more explicit in later work of e.g. Rodriguez-Villegas and Zagier. Recently, Bogo, Li, and Schwagenscheidt studied Laurent expansions of meromorphic modular forms with poles at CM points and observed arithmetic patterns in examples. In this talk, I will discuss ongoing joint work with Rolen aimed at explaining and characterizing the arithmetic behavior of these Laurent coefficients.