# Applications of integral p-adic Hodge theory

What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $$\mathbb{Z}_p$$-lattices are OK but I am more interested in applications of comparison theorems (like the one recently established by Bhatt--Morrow--Scholze).

I am aware of certain applications to the question of nice reductions of varieties (e.g. this or this).

It appears that this question is not a duplicate since the paper of Berthelot et al mentioned there uses rational p-adic Hodge theory.

One major application of research in integral $$p$$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $$p$$-adic representations which have some reasonable local properties at $$p$$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $$p$$, which belongs squarely to integral $$p$$-adic Hodge theory.
This was the paper that completed the proof of modularity of elliptic curves over $$\mathbb{Q}$$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").