Cartier operators on fields of positive characteristic

초록

From an analytic perspective, we introduce a sequence of Cartier operators which act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic. In this paper, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a great role in various objects of study as a suitable substitute for higher derivatives. For an applicable object, the Wronskain criteria associated with Cartier operators are introduced. These results result from a careful study of two types of Cartier operators on the power series ring $\text{Fq}[[T]]$ in one variable $T$ over a finite field $\text{Fq}$ of $q$ elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous $\text{Fq}$-linear functions on $\text{Fq}[[T]].$ The digit principle leads to that every continuous function on $\text{Fq}[[T]]$ is uniquely written in terms of a $q$-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of two cases. Moreover, the $p$-adic analogues of Cartier operators are discussed as orthonormal bases of the space of continuous functions on $\text{Zp}.$