Let p be a prime number, k a perfect field of characteristic p > 0, X a smooth k-scheme and D (m) X the algebra of (arithmetic) differential operators of level m ≥ 0. We study the Azumaya nature of this algebra and show how to construct, using an additional data, a splitting of a completion (along some ideal contained in its center) of it.

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.

In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We introduce here a notion of twisted divided powers (relative to q) and show that one can recover the twisted Weyl algebra and obtain a twisted p-curvature map that describes the center of the twisted Weyl algebra. We also build a divided p-Frobenius that will give, by duality, a formal Azumaya splitting of the twisted Weyl algebra as well as a twisted Simpson correspondence.

Let f (X) ∈ Z[X] be an irreducible polynomial of degree D ≥ 2 and let N be a sufficiently large positive integer. We estimate the number of positive integers n ≤ N such that the productis a perfect square. We also consider more general questions and give a lower bound on the number of distinct quadratic fields of the form Q( F (n)), n = 1, . . . , N .

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