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Cartier operators on compact discrete valuation rings and applications. (English) Zbl 1429.11211

Summary: 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 \(\mathbb{F}q[[T]]\) in one variable \(T\) over a finite field \(\mathbb{F}q\) of \(q\) elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous \(\mathbb{F}q\)-linear functions on \(\mathbb{F}q[[T]].\) According to the digit principle, every continuous function on \(\mathbb{F}q[[T]]\) 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 \(\mathbb{Z}p.\)

MSC:

11S85 Other nonanalytic theory
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