The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the Newton groupoids. From the introduction:
The proof is based on the ideas that go back to the 17th century (namely, the implicit function theorem or equivalently, Newton's method for finding roots) and the 19th century (the Weierstrass division theorem).
The goal of the rest of the article is to clarify the geometric ideas behind the proof from §2. In particular, we introduce Newton groupoids. These are certain groupoids in the category of schemes, which are related to Newton's method for finding roots. They are associated to any generically étale morphism from a locally complete intersection to a smooth variety (both schemes are assumed separated).
The actual setting for the definition of Newton groupoids seems to me rather involved, so instead of writing down, I risk assuming many people read what Drinfeld writes.
Question. What is the geometric intuition behind Newton groupoids, and how are they related to the implicit function theorem/Newton's method of finding roots?