Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$.

Abstract. Let F be a totally real number field and let p be a prime unramified in F . We prove the existence of Galois pseudo-representations attached to mod p m Hecke eigenclasses of paritious weight which occur in the coherent cohomology of Hilbert modular Shimura varieties for F of level prime to p.

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We recall basic facts about p-divisible group; the main references are [Dem72], [Fon77], [Mes72] and [MR96b], 2.1-2.8. Basic definitionsFix a scheme S. By an S-group we mean a f.p.p.f. sheaf of commutative groups on the site (Sch/S) f.p.p.f. , whose underlying category is the category of S-schemes, endowed with the Grothendieck f.p.p.f. topology. An S-group that is representable is called an S-group-scheme. For any positive integer n and any S-group G we denote by GFix a positive prime integer p.An S-group G is a Barstotti-Tate group (or a p-divisible group) if the following three conditions are satisfied: (1) G = limBy the theory of finite group-schemes over a field, the rank of the fiber of G[p] at a point s ∈ S is of the form p h(s) , where h : S → Z is a locally constant function on S; the rank of the fiber of G[p n ] at s is p nh(s) for any n ≥ 1. If h is a constant function (e.g., when S = Spec(k) for a field k), its only value is called the height ht(G) of G.A morphism f : G → H of p-divisible groups over S is said to be an isogeny if it is an epimorphism of f.p.p.f. sheaves whose kernel is representable by a finite locally free S-group scheme. If S is a scheme over Spec(Z p ) in which p is locally nilpotent, then the kernel of an isogeny f : G → H is finite of rank p h ′ where h ′ : S → Z is locally constant; if h ′ is constant, its only value is called the height of f . The Z-module Hom S (G, H) of homomorphisms from G to H is a torsion-free Z p -module. A quasi-isogeny f from G to H is global section of the sheaf Hom S (G, H) ⊗ Z Q such that any point s of S has a Zariski open neighborhood on which p n f : X → Y is an isogeny for some positive integer n = n(s). We denote by Qisg S (G, H) the group of quasi-isogenies from G to H. We have the following rigidity property (cf. [MR96b], 2.8):Proposition 2.1 Let G and H be p-divisible groups over a scheme S in which p is locally nilpotent; let S ′ ⊂ S be a closed subscheme whose defining sheaf of ideals is locally nilpotent. Then the canonical homomorphismRecall that for any finite flat group scheme X over S, the Cartier dual D(X) (or X D ) of X is the finite locally free S-group-scheme defined by D(X)(T ) := Hom T (G T , G m × S T ) (T any S-scheme). The assignment D induces an additive anti-duality on the category of finite and locally free S-group schemes. Let G be a p-divisible group over S. The Serre dual of G, denoted by G (or G D , or D(G)), is the p-divisible group defined as G := lim) for any n. The assignment G → G extends to morphisms in an obvious way and gives rise to an anti-duality in the category of p-divisible groups over S (notice this category is not abelian) which is compatible with base changes. There is a canonical isomorphism of p-divisible groups G ∼ → G.

Let F be a totally real field of degree g, and let p be a prime number. We construct g partial Hasse invariants on the characteristic p fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for F with level prime to p, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when p ramifies in F , we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-representations attached to Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties mod p m , extending a previous result of M. Emerton and the authors which required p to be unramified in F .