CARTIER OPERATORS ON COMPACT DISCRETE VALUATION RINGS AND APPLICATIONS

초록

From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic p. In this work, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a prominent role in various objects of study in function field arithmetic, as a suitable substitute for higher derivatives. For an applicable object, the Wronskian criteria associated with Cartier operators are introduced. These results stem from a careful study of two types of Cartier operators on the power series ring F-q[[T]] in one variable T over a finite field F-q of q elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous F-q-linear functions on F-q[[T]]. According to the digit principle, every continuous function on F-q[[2]1 is uniquely written in terms of a q-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the p-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on Z(p).

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