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As a follow-up to a comment on this answer, I'm wondering if there are expected to be applications of the new point of view on integral $p$-adic Hodge theory, à la Bhatt-Morrow-Scholze and others, to modularity lifting and the Fontaine-Mazur conjecture. Most previous work on those topics uses Fontaine-Lafaille theory (e.g., p.42 of these notes, this more recent result, among many others), which restricts greatly the Hodge-Tate weights one can consider.

While Bhatt-Morrow-Scholze does not refer to Fontaine-Lafaille theory, it does represent a great step forward in integral $p$-adic Hodge theory (along with the more recent prismatic cohomology). Therefore, is there any hope that this new machinery would be able to generalize Fontaine-Lafaille theory to something that allows more general modularity lifting theorems?

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  • $\begingroup$ From the arithmetic perspective, (integral) $p$-adic Hodge theory is the study of $p$-adic Galois representations. That’s why it plays a role in the proof of automorphy lifting theorems. However, Bhatt-Morrow-Scholze‘s work is geometric i.e. to study the geometry of (proper smooth) schemes over a $p$-adic field. So they are not directly related but we can construct some functors to link them and study $p$-adic Hodge theory via the interplay between the arithmetic and geometric perspectives. $\endgroup$
    – coLaideronnette
    Commented Jun 25 at 21:36
  • $\begingroup$ On the other hand, the modern results on automorphy lifting theorems are heavily dependent on the study of the geometry of Shimura varieties, where the $p$-adic geometry plays an essential role. $\endgroup$
    – coLaideronnette
    Commented Jun 25 at 21:50
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    $\begingroup$ I disagree with the premise of this question. You seem to suggest that integral p-adic HT did not stand still in the multiple decades between Fontaine--Laffaille and Bhatt--Scholze, and many of those advances (e.g. Breuil-Kisin modules) were motivated by & immediately applied to modularity-lifting problems. $\endgroup$
    – David Loeffler
    Commented Jun 27 at 6:03
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    $\begingroup$ Your two example citations are (a) a set of introductory lecture notes which deliberately uses non-cutting-edge methods for simplicity, and (b) a paper somewhat outside the mainstream of research in this area. For instance, Kisin's work on potentially semistable deformation rings – which relies on Breuil-Kisin modules to go well beyond the Fontaine--Laffaille range – has been fundamental to virtually every major advance in modularity lifting since it was written (in 2007). $\endgroup$
    – David Loeffler
    Commented Jun 27 at 6:10
  • $\begingroup$ (typo in first comment: "did not stand still" should be "stood still".) $\endgroup$
    – David Loeffler
    Commented Jun 27 at 6:11

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