Appears in collection : New Structures and Techniques in $p$-adic Geometry

Let $p > 5$ be a prime, and let $F$ be a totally real field in which $p$ is unramified. We study mod $p$ Hilbert modular forms for $F$ of level prime to $p$ and weight $(k, l)$, where $k$ and $l$ are tuples of integers. To a mod $p$ Hilbert modular Hecke eigenform of weight $(k, l)$, Diamond and Sasaki associate a two-dimensional mod $p$ Galois representation of ${\rm Gal}(Fp/F)$. The local–global compatibility (LGC) conjecture predicts that, at each place above $p$, the restriction of this representation admits crystalline lifts with Hodge–Tate weights determined explicitly by $(k, l)$. In this talk, we will discuss a proof showing that LGC for regular $p$-bounded weights (each entry of $k$ between 2 and $p+1$) implies LGC in the partial weight one $p$-bounded case (each entry of $k$ between 1 and $p+1$). Our approach combines computations of scheme-theoretic intersections on the Emerton–Gee stack with weight-changing arguments on quaternionic Shimura varieties, using restriction to Goren–Oort strata. This is joint work in progress with Brandon Levin and David Savitt.