Arithmetic differential operators over the p-adic integers by Claire C. Ralph

By Claire C. Ralph

The examine of mathematics differential operators is a singular and promising quarter of arithmetic. this whole advent to the topic begins with the fundamentals: a dialogue of p-adic numbers and a few of the classical differential research at the box of p-adic numbers resulting in the definition of mathematics differential operators in this box. Buium's thought of mathematics jet areas is then constructed succinctly in an effort to outline mathematics operators as a rule. positive factors of the booklet contain a comparability of the behaviour of those operators over the p-adic integers and their behaviour over the unramified of completion, and a dialogue of the connection among attribute capabilities of p-adic discs and mathematics differential operators that disappears once a unmarried root of solidarity is adjoined to the p-adic integers. This booklet is vital examining for researchers and graduate scholars who need a first advent to mathematics differential operators over the p-adic integers.

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Our use of this result will be in the form of an easy consequence: if two power series of the indicated form agree over infinitely many p-adic integers, then they must be identical. 12 to introduce the notion of arithmetic differential operators over the ring Zp . This is the essential idea in our work here. In the following chapter, we shall present the theory of these operators in general, discussing the jet spaces whose global sections give rise to them. We shall even expand our explanations to outline the general theory in the case of multiple primes, something that we start also in a simplified manner here.

12) where NK/Qp (α) is the value at x = 0 of the unique monic irreducible polynomial iα (x) of α, and [K : Qp ] is the degree of K over Qp . 12 is clearly multiplicative in α, it is only zero when α = 0, and it yields the p-adic norm of α in the case when α ∈ Qp . Thus, this function is an extension of the p-adic norm function on Qp . 2), fact whose proof we leave to the interested reader. 12 defines a non-Archimedean norm on Qp that is an extension of the p-adic norm on Qp . For the same reasons as before, the set {x ∈ Qp : ||x||p ≤ 1} forms a ring, the ring of integers of Qp .

In what follows, we shall say that a sequence {an }∞ n=0 ⊂ Qp is restricted if an → 0 p-adically. ) Let S = {{an }∞ n=0 ⊂ Qp , an → 0} be the set of restricted sequences. 6) x n an is a bijection. 5). We show first that this mapping is well defined. The elementary combinatorial identity n x n = k=0 x−y k y n−k holds for p-adic integers x, y also. 1 to determine the coefficient of tn in the series expansion of the identity (1 + t)x = (1 + t)x−y (1 + t)y . Given an arbitrary integer x in Zp , let y be a nonnegative rational integer such that x−y n!

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