Inclusions between p-bounded crystalline loci in dimension two

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1.

Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K.

We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion.

We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations.

We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations).

GPT-5.5 Pro was used extensively in the course of this work.