Applications of integral p-adic Hodge theory

Question

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.

Answer 1

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.

E.g. have a look at this famous paper:

Breuil, Conrad, Diamond, Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises.

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").

This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.

Answer 2

Let me highlight some quite unexpected (to me) geometric applications of the recent work on integral $p$-adic Hodge theory. Namely, prismatic cohomology, along with a $p$-adic form of the Riemann--Hilbert correspondence, was used critically in Bhatt's proof of Cohen-Macaulayness of absolute integral closures. The main theorem is the following:

Let $R$ be an excellent noetherian domain with an absolute integral closure $R^+$. Then the $R/p^n$-module $R^+/p^n$ is Cohen-Macaulay for all $n\geq 1$.

This is a version in mixed characteristic of a theorem of Hochster--Huneke in positive characteristic. (Interestingly, the theorem fails completely in equal characteristic $0$, and the result proved by Bhatt was largely considered too optimistic, so it was never conjectured in print.) The theorem immediately implies the direct summand conjecture and various strengthenings, including the existence of weakly functorial big Cohen-Macaulay algebras (proved previously by André). In fact, one can now simply use (the $p$-adic completion of) $R^+$ as a big Cohen-Macaulay algebra.

There is also a geometric version, giving a mixed-characteristic variant of Kodaira vanishing, see Theorem 1.2 in Bhatt's paper. Briefly, while Kodaira vanishing may fail, it can be corrected by passing to a finite cover.

These results are in turn used to establish the minimal model program for arithmetic threefolds, see here and here.