CONGRUENCE PROPERTIES OF A DRINFELD MODULAR FUNCTION μ

The Drinfeld modular function µ is a generator of the func-tion field of the Drinfeld modular curve X 0 ( T ) and has an t -expansionwith the integral coefficients at infinity. In this paper, we show that thecoefficients of µ has congruence properties modulo powers of T . 1. IntroductionVincent Bosser [1] showed that the coefficients of the Drinfeld modular in-variant j has congruence properties modulo powers of polynomials of degree 1in F q [ T ]. It can be applied for a generator µ of the function field of the Drinfeldmodular curve X 0 ( T ). The generator µ plays an important role in the study of X 0 ( T ) and the construction of class fields over function fields. Jeon and Kim[2] show that µ gives a plane model for X 0 ( T ) and the singular values of µ generate class fields over imaginary quadratic function fields.In this paper, by using tools of Bosser we show that the coefficients of µ hascongruence properties modulo powers of T .2. PreliminariesLet K be the rational function field F q