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**1 - 6**of**6**### ARITHMETIC DIFFERENTIAL OPERATORS ON Zp

, 808

"... Abstract. Given a prime p, we let δx = (x−x p)/p be the the Fermat quotient operator over Zp. We prove that a function f: Zp → Zp is analytic if, and only if, there exists m such that f can be represented as f(x) = F(x, δx,..., δ m x), where F is a restricted power series with Zp-coefficients in m ..."

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Abstract. Given a prime p, we let δx = (x−x p)/p be the the Fermat quotient operator over Zp. We prove that a function f: Zp → Zp is analytic if, and only if, there exists m such that f can be represented as f(x) = F(x, δx,..., δ m x), where F is a restricted power series with Zp-coefficients in m + 1 variables. 1.

### RESEARCH DESCRIPTION

"... The aim of these notes is to explain 1) some of the ideas and results of my past research; 2) the main directions of my intended research. Roughly, the main purpose of my (past and intended) research is to develop an arithmetic analogue of differential calculus in which differentiable functions x(t) ..."

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The aim of these notes is to explain 1) some of the ideas and results of my past research; 2) the main directions of my intended research. Roughly, the main purpose of my (past and intended) research is to develop an arithmetic analogue of differential calculus in which differentiable functions x(t) is replaced by the Fermat quotient operator n ↦ → n−np p, where p is a prime integer. The Lie-Cartan geometric theory of differential equations (in which solutions are smooth maps) is then replaced by a theory of “arithmetic differential equations ” (in which solutions are integral points of algebraic varieties). In particular the differential invariants of groups in the Lie-Cartan theory are replaced by “arithmetic differential invariants ” of correspondences between algebraic varieties. A number of applications to diophantine geometry over number fields and to classical modular forms will be explained. This program was initiated in [9] and pursued, in particular, in

### DIFFERENTIAL CALCULUS WITH INTEGERS

"... Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory. ..."

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Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory.

### ARITHMETIC LAPLACIANS

, 2008

"... We develop an arithmetic analogue of elliptic partial differential equations. The rôle of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relation ..."

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We develop an arithmetic analogue of elliptic partial differential equations. The rôle of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, and analyze their spaces of solutions.