Weierstrass preparation theorem and singularities in the space of non-degenerate arcs

It has been long expected that there exists a deep connection between singularities of certain arc spaces and harmonic analysis over nonarchimedean fields. For instance, certain functions appearing naturally in harmonic analysis can be interpreted as the function attached to the trace of the Frobenius operator on what should be the stalks of the intersection complex of certain arc spaces, see [1]. For the time being however, a proper foundation of a theory of perverse sheaves on arc spaces is still missing even though a recent work of Bouthier and Kazhdan [2] outlines a strategy for setting it up. As the theory of perverse sheaves is originally built for schemes of finite type, the basic difficulty in extending it to arc spaces is that those spaces are almost always infinite dimensional. The first inroad into this new territory is made by Grinberg and Kazhdan who prove that the formal completion of the arc spaces at a point representing a non-degenerate arc is isomorphic to the formal completion of a scheme of finite type, augmented by infinitely many free formal variables, under the assumption that the base field is the field of complex numbers. This result is later improved by Drinfeld who prove it over an arbitrary base field. Let us fix the notations in order to state Grinberg-Kazhdan-Drinfeld’s theorem. Let k be a field. Let X be an affine k-scheme of finite type. For every n ∈ N, we consider the space of nth jets on X representing the functor R 7→ LnX (R) = X (R[t]/tn+1) on the categories of k-algebras. The arc space of X is the limit of LnX as n→∞: