Cartier operators on fields of positive characteristic p

TL;DR

This paper introduces Cartier operators acting on fields of positive characteristic, explores their relationship with Hasse derivatives, and demonstrates their role as orthonormal bases for continuous functions in function field arithmetic.

Contribution

It presents a new sequence of Cartier operators, establishes their binomial inversion with Hasse derivatives, and shows they form orthonormal bases for continuous functions over finite fields and p-adic integers.

Findings

Cartier operators form an orthonormal basis for continuous functions on $qn[[T]]$.

A binomial inversion formula links Cartier operators and Hasse derivatives.

Cartier operators serve as substitutes for higher derivatives in function field arithmetic.

Abstract

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 suitable substitutes 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 $\Fq[[T]]$ in one variable $T$ over a finite field $\Fq$ of $q$ elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous $\Fq$-linear functions on $\Fq[[T]].$ According to the digit principle, every continuous function on $\Fq[[T]]$ is uniquely written in terms of a $q$-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the $p$-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on $\Zp.$