Geometric weight-shifting operators on Hilbert modular forms in characteristic p

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic $p$, generalizing the author's prior work with Sasaki to the case where $p$ is ramified in the totally real field $F$. In particular we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial $\Theta$-operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore we describe the kernels of partial $\Theta$-operators in terms of images of geometrically constructed partial Frobenius operators. Finally we apply our results to prove a partial positivity result for minimal weights of mod $p$ Hilbert modular forms.


Introduction
The study of weight-shifting operations on modular forms has a rich and fruitful history. Besides those naively obtained from the graded algebra structure on the space of classical modular forms of all weights, there is a deeper construction due to Ramanujan [30] which shifts the weight by two using differentiation, leading to a more general theory of Maass-Shimura operators. Analogous weight-shifting operations in characteristic p, first studied by Swinnerton-Dyer and Serre [34], take on special significance in the context of congruences between modular forms and the implications for associated Galois representations. In particular one has the following linear maps from the space of mod p modular forms of weight k and some fixed level N prime to p: • multiplication by a Hasse invariant H, to forms of weight k + p − 1; • a differential operator Θ, to forms of weight k + p + 1; • a linearized p-power map V , to forms of weight pk. These maps all have simple descriptions in terms of associated q-expansions: if f has q-expansion a n q n , then that of Hf (resp. Θf , V f ) is a n q n (resp. na n q n , a n q pn ). Following the work of Swinnerton-Dyer and Serre, there were further significant developments to the theory due to Katz [23,24] (interpreting the constructions more geometrically), Jochnowitz [21,22] (on the weight filtration and Tate's Θ-cycles) and Gross [20] (in the study of companion forms), providing crucial ingredients for Edixhoven's proof of the weight part of Serre's Conjecture in [14].
Suppose now that F is a totally real field of degree d = [F : Q] and consider spaces of Hilbert modular forms of weight k ∈ Z Σ and fixed level prime to p, where Σ denotes the set of embeddings {σ : F ֒→ Q p }. For such spaces of p-adic modular forms, Katz [25] constructed a family of commuting differential operators Θ σ , indexed by the d-embeddings σ ∈ Σ. The theory was further developed by Andreatta and Goren [1] who, building on Katz's work and Goren's definition of partial Hasse invariants in [18,19] (if p is unramified in F ), defined partial Θoperators on spaces of mod p Hilbert modular forms.
Under the assumption that p is unramified in F , some aspects of the construction of partial Θ-operators in [1] were simplified in [12], which also went on to define partial Frobenius operators (generalizing V ) geometrically and use their image to describe kernels of partial Θ-operators. When p is ramified in F , the effectiveness of the approach in [1] was limited by the singularities of the available (Deligne-Pappas) model for the Hilbert modular variety. Since then however, a smooth integral model was constructed by Pappas and Rapoport [29], and the theory of partial Hasse invariants was further developed in this context by Reduzzi and Xiao in [32]. The theory of partial Θ-operators was revisited in that light by Deo, Dimitrov and Wiese in [8], where they closely follow [1]. Here we instead exploit the observations and techniques introduced in [12], applying them directly to the special fibre of the Pappas-Rapoport model to construct and relate partial Θ and Frobenius operators. In particular this eliminates extraneous multiples of partial Hasse invariants that appear in [8], and yields results whose implications for minimal weights are motivated by the forthcoming generalization to the ramified case of the geometric Serre weight conjectures of [12]. The main contributions of this paper may be summarized as follows: • a construction of operators Θ τ with optimal effect on weight (Theorem 5.2.1); • a geometric construction of partial Frobenius operators V p (see §6.2); • a description of the kernel of Θ τ in terms of the image of V p (see §9.1); • an application to positivity of minimal weights (Theorem 9.2.1). We should emphasize that the focus of this paper is entirely on Hilbert modular forms in characteristic p. There is also a rich theory of Θ-operators on p-adic automorphic forms which has seen major progress recently in the work of de Shalit and Goren [7], and Eischen, Fintzen, Mantovan and Varma [15], which in turn has implications in the characteristic p setting [6,7,16,17]. Another advance in characteristic p has been Yamauchi's construction [36] of Θ-operators for mod p Siegel modular forms of degree two. We remark however that all of the work just mentioned only considers automorphic forms on reductive groups which are unramified at p; the novelty of this paper is largely in the treatment of ramification at p.
We now describe the contents in more detail. We first set out some basic notation and constructions in §2.1. In particular we fix a prime p, a totally real field F of degree 1 d = [F : Q] > 1, and let O F denote the ring of integers of F and S p the set of prime ideals of O F dividing p. For each p ∈ S p , let Σ p,0 denote the set of f p embeddings O F /p → F p and Σ p the set of e p f p embeddings F p → Q p , where f p (resp. e p ) is the residual (resp. ramification) degree of p. We let where each τ p,0 ∈ Σ p,0 is chosen arbitrarily, τ p,i = τ p i p,0 and θ p,i,1 , . . . , θ p,i,ep is any ordering of the lifts of τ p,i to Σ p . We also define a "right-shift" permutation σ of 1 Including the case F = Q would introduce different complications in the treatment of cusps and provide no new results. Σ by σ(θ p,i,j ) = θ p,i,j−1 , if j > 1; θ p,i−1,ep , if j = 1. In §2.2 we recall the definition of the Pappas-Rapoport model Y U for the Hilbert modular variety of level U , where U is any sufficiently small open compact subgroup of GL 2 (A F,f ) of level prime to p. This may be viewed as a coarse moduli space for Hilbert-Blumenthal abelian varieties with additional structure, where this additional structure includes a suitable collection of filtrations on direct summands of its sheaf of invariant differentials. The scheme Y U is then smooth of relative dimension d over O, where O is the ring of integers of a finite extension of Q p in Q p . Since the main results of the paper concern Hilbert modular forms in characteristic p, we will restrict our attention to this setting for the remainder of the Introduction, and let Y U = Y U,F where F is the residue field of O.
In § §3.1-3.2 we construct the automorphic line bundles A k,l on Y U for all k, l ∈ Z Σ and sufficiently small U (of level prime to p), and define the space of Hilbert modular forms of weight (k, l) and level U over F to be a smooth admissible representation of GL 2 (A F,f ) over F. A key point, as already observed in [12] in the unramified case, is that the parity condition on k imposed in the definition of Hilbert modular forms in characteristic zero (for the group Res F/Q GL 2 ) disappears in characteristic p. We remark also that the effect of the weight parameter l (in characteristic p) is to introduce twists by torsion bundles that make various constructions, in particular that of partial Θ-operators, compatible with the natural Hecke action. In § §3.3-4.2 we recall results of Reduzzi and Xiao [32] that will underpin our construction of partial Θ-operators. Firstly there is a natural Kodaira-Spencer filtration on direct summands of Ω 1 Y U /F whose graded pieces are isomorphic to the automorphic line bundles A 2e θ ,−e θ (where e θ denotes the basis element of Z Σ indexed by θ). Secondly for each θ = θ p,i,j ∈ Σ, there is a partial Hasse invariant where θ = θ p,i,ep . Note in particular that if p is ramified, then the shift 2 in the weight parameter k is by e σ −1 θ + e θ . The idea of the construction, inspired by the one in [1, §12], is to divide by fundamental Hasse invariants to get a rational function on the Igusa cover of Y U , differentiate, project to the top graded piece of the τcomponent of the Kodaira-Spencer filtration, and finally multiply by fundamental Hasse invariants to descend to Y U and eliminate poles. The argument also gives a direct (albeit local) definition of the Θ-operator without reference to the Igusa cover in (12), and establishes the following result (Theorem 5.2.1) generalizing [12,Thm. 8

.2.2]:
Theorem A. Let τ = τ p,i and θ = θ p,i,ep . Then Θ τ (f ) is divisible by H θ if and only if either f is divisible by H θ or p|k θ .
We turn to the construction of partial Frobenius operators V p in §6. This essentially generalizes a definition in [12, §9.8], but requires significantly more work to actualize if p is ramified. We do this using Dieudonné theory to define a partial Frobenius endomorphism Φ p of Y U and an isomorphism Φ * p A k,l ∼ = A k ′′ ,l ′′ , where k ′′ = k + θ∈Σp k θ h θ and l ′′ = l + θ∈Σp l θ h θ , in order to obtain, for p ∈ S p , commuting Hecke-equivariant operators We will use q-expansions to relate the kernel of Θ τ to the image of V p for τ ∈ Σ p,0 , so we recall the theory in §7. This is a straightforward adaptation to our setting of results and methods developed in [31,4,13,5]. In §8 we compute the (constant) qexpansions of the invariants H θ and G θ at each cusp of Y U , and we obtain formulas generalizing the classical ones for the effect of the operators Θ τ and V p on all qexpansions. In particular this shows that the operators Θ τ for varying τ commute.
In §9.1 we turn our attention to the description of the kernel of Θ τ . The qexpansion formulas also show that Θ τ • V p = 0 if τ ∈ Σ p,0 , and that ker(Θ τ ) is the same for all τ ∈ Σ p,0 . Theorem A then reduces the study of the kernel to the case of weights of the form (k ′′ , l ′′ ) where k ′′ , l ′′ are as in the definition of V p , for which the argument proving [12,Thm. 9.8.2] gives the following: 3 Theorem B. If k, l ∈ Z Σ and τ = τ p,i and θ = θ p,i,ep , then the sequence Before discussing the application to positivity of minimal weights, we remark that a less precise relation among the weight-shifting operations can be neatly encapsulated in terms of the algebra of modular forms of all weights M tot (U ; F) := k,l∈Z Σ M k,l (U ; F), or even its direct limit M tot (F) := lim − →U M tot (U ; F) (over all sufficiently small levels prime to p). It follows from its definition that the operator V p (resp. Θ τ ) on the direct sum is an F-algebra endomorphism (resp. F-derivation) of M tot (F). One also finds that V p maps the ideal to itself. 4 Furthermore Θ τ (H ′ θ ) = Θ τ (G ′ θ ) = 0 for all θ ∈ Σ, so V p (resp. Θ τ ) induces an F-algebra endomorphism (resp. F-derivation) of the quotient M tot (F)/I. We then have the following consequence of Theorem B (see Theorem 9.3.4): In §9.2 we apply our results to refine the main result of [10], which we recall states that minimal weights of non-zero forms always lie in Ξ min . The geometric Serre weight conjectures of [12] (and its forthcoming generalization to the ramified case) predict that if f is a mod p Hecke eigenform which is non-Eisenstein (in the sense that the associated Galois representation is irreducible), then k min (f ) should be totally positive. We use Theorems A and B to prove a partial result in this direction (Theorem 9.2.1): Theorem D. Suppose that p ∈ S p is such that F p = Q p and p fp > 3. Suppose that f ∈ M k,l (U ; F) is non-zero and k = k min (f ). If k θ = 0 for some θ ∈ Σ p , then k = 0.
Since the Hecke action on forms of weight (0, l) is Eisenstein (see Proposition 3.2.2), the theorem implies the total positivity of minimal weights of non-Eisenstein eigenforms in many situations, for example if p > 3 and there are no primes p ∈ S p such that F p = Q p . We remark that the hypothesis F p = Q p cannot be removed from Theorem D: if Σ p = {θ}, then there are non-zero forms whose minimal weight k satisfies k θ = 0 and k θ ′ > 0 for some θ ′ = θ. However forthcoming work with Kassaei will show that the Hecke action on such forms is Eisenstein; like in [9,10], the case of split primes seems to require a completely different method. Unfortunately the case of p ≤ 3, f p = 1, e p > 1 slips through the crack between the two methods. We do not know whether Theorem D should hold in this case, but we still at least conjecture the failure is Eisenstein.

Preliminaries
2.1. Embeddings and decompositions. We first set out notation and conventions for various constructions associated to the set of embeddings of a totally real field F , which together with a prime p, will be fixed throughout the paper.
We assume that F has degree d = [F : Q] > 1, let O F denote its ring of integers, d its different, and Σ the set of embeddings F → Q, where Q is the algebraic closure of Q in C.
We also fix an embedding Q → Q p . We let S p denote the set of primes of O F dividing p, and identify Σ with p∈Sp Σ p under the natural bijection, where Σ p denotes the set of embeddings F p → Q p .
For each p ∈ S p , we let F p,0 denote the maximal unramified subextension of F p , which we identify with the field of fractions of W (O F /p). We also let f p denote the residue degree [F p,0 : Q p ], e p the ramification index [F p : F p,0 ], and Σ p,0 the set of embeddings F p,0 → Q p , which we may identify with the set of embeddings . For each p ∈ S p , we fix a choice of embedding τ p,0 ∈ Σ p,0 , and for i ∈ Z/f p Z, we let τ p,i = φ i • τ p,0 where φ is the Frobenius automorphism of F p (or W (F p ) or its field of fractions), so that Σ p,0 = {τ p,1 , τ p,2 , . . . , τ p,fp }. We also let Σ 0 = p∈Sp Σ p,0 . Letting q = p∈Sp p denote the radical of p in O F , note that Σ 0 may also be identified with the set of For each τ = τ p,i ∈ Σ 0 , we let Σ τ ⊂ Σ p denote the set of embeddings restricting to τ , for which we choose an ordering θ p,i,1 , θ p,i,2 , . . . , θ p,i,ep , so that We also define a permutation σ of Σ whose restriction to each Σ p is the e p f p -cycle corresponding to the right shift of indices with respect to the lexicographic ordering, i.e., . . .
Let E ⊂ Q be a number field containing the image of θ for all θ ∈ Σ, let O be the completion of O E at the prime determined by the choice of Q → Q p , and let F be its residue field. For any

In particular, for any
We also fix a choice of uniformizer ̟ p for each p ∈ S p . We let f p (u) denote the minimal polynomial of Furthermore we have f τ (u) = θ∈Στ (u − θ(̟ p )), and we define elements . , e p (with the obvious convention that s τ,0 = t τ,ep = 1). Note that each of the ideals (s τ,j ) and (t τ,j ) is the other's annihilator; furthermore the quotients of O[u]/(f τ (u)) by these ideals are free over O, and the corresponding ideals in O F,p ⊗ W (OF /p),τ O may be described as kernels of projection maps to products of copies of O, hence depend only on j and the ordering of embeddings, and not on the choice of uniformizer ̟ p .
For an invertible O F -module L and an embedding θ = θ p,i,j ∈ Σ τ , we define L θ to be the free rank one O-module Note that L θ is not to be identified with L ⊗ OF ,θ O; rather there is a canonical map L θ → L ⊗ OF ,θ O which is an isomorphism if and only if j = e p . If L and L ′ are invertible O F -modules, we will write LL ′ for L⊗ OF L ′ and L −1 for Hom OF (L, O F ). Note that there are natural maps , but again these are isomorphisms if and only if j = e p .

2.2.
Pappas-Rapoport models. In this section we recall the description of the Hilbert modular variety as a coarse moduli space for abelian varieties with additional structure, along with the construction by Pappas and Rapoport of a smooth integral model (see [29] and [33]).
is sufficiently small, in a sense to be specified below.
We consider the functor which associates, to a locally Noetherian O-scheme S, the set of isomorphism classes of data (A, ι, λ, η, F • ), where: • s : A → S is an abelian scheme of relative dimension d; • η is a level U p structure on A, i.e., for a choice of geometric point s i on each connected component S i of S, the data of a π 1 (S i , (p) denotes the product over ℓ = p of the ℓ-adic Tate modules, and g ∈ U p acts on η i by pre-composing with right multiplication by g −1 ; • F • is a collection of Pappas-Rapoport filtrations, i.e., for each The proof of [12, Lemma 2.4.1] does not assume p is unramified in F , and shows that if U p is sufficiently small and α is an automorphism of a triple (A, ι, η) over a connected scheme S, then α = ι(µ) for some µ ∈ U ∩ O × F . If we assume further that −1 ∈ U ∩ O × F , then it follows from standard arguments that the functor above is representable by an infinite disjoint union of quasi-projective schemes over O, which we denote by Y U , and the argument in the proof of [33,Prop. 6] shows that Y U is smooth of relative dimension d over O. Furthermore defining an action of , we see that the resulting action of O × F,(p),+ /(U ∩O × F ) 2 is free and the quotient is representable by a smooth quasi-projective scheme over O, which we denote by Y U .
We also have a natural right action of GL 2 (A (p) F,f ) on the inverse system of schemes Y U induced by pre-composing the level structure η with right-multiplication by g −1 .
More precisely suppose that U 1 and U 2 are as above (with U p 1 and U p 2 sufficiently small) and g ∈ GL 2 (A . Together with the other data inherited from A, we obtain an object (A ′ , ι ′ , λ ′ , η ′ , F ′• ) corresponding to a morphism ρ g : Y U1 → Y U2 and descending to a morphism Y U1 → Y U2 which we denote ρ g . These morphisms satisfy the evident compatibility ρ g2 • ρ g1 = ρ g1g2 whenever g −1 Finally we remark that the schemes Y U define smooth integral models over O for the Hilbert modular varieties associated to the group G (with the usual choice of Shimura datum), and their generic fibres and resulting GL 2 (A (p) F,f )-action may be 5 Note the conventions in place with respect to the different, which are motivated by the point of view that we wish to systematically trivialize modules defined by cohomological constructions. identified with those obtained from a system of canonical models. In particular for any O → C, we have isomorphisms compatible with the right action of GL 2 (A (p) F,f ) on the inverse system, and inducing a bijection between the set of geometric components of Y U and (p),+ det(U p ) These isomorphisms arise in turn from ones of the form under which the set of geometric components of Y U is described by (A (p)

Automorphic bundles
3.1. Pairings and duality. Before introducing the line bundles whose sections define the automorphic forms of interest in the paper, we present a plethora of perfect pairings provided by Poincaré duality.
We fix a sufficiently small U as in §2.2 and consider the de Rham cohomology sheaves H 1 dR (A/S) = R 1 s * Ω • A/S on the universal abelian scheme A over S = Y U . Recall that these sheaves are locally free of rank two over O F ⊗ O S . Furthermore Poincaré duality and the polarization λ induce an O F ⊗ O S -linear isomorphism where c depends on the connected component of S and disappears from the last expression since it is prime to p). This in turn induces O F,p ⊗ W (OF /p),τ O S -linear isomorphisms we view as defining a perfect O S -bilinear pairing ·, · 0 τ between H τ := H 1 dR (A/S) τ and d −1 p ⊗ OF,p H τ for τ ∈ Σ p,0 . Furthermore the pairing is alternating in the sense that x, c ⊗ y 0 τ = − y, c ⊗ x 0 τ on sections. Alternatively, we may apply the canonical O F ⊗ R-linear isomorphism Note that H τ is locally free of rank two over O F,p ⊗ W (OF /p),τ O S , hence a vector bundle of rank 2e p over O S . Furthermore (s * Ω 1 A/S ) τ is a subbundle of H τ of rank e p , but is not locally free over O F,p ⊗ W (OF /p),τ O S if e p > 1 (in which case the failure is on a closed subscheme of codimension one), and more generally F (j) τ is a subbundle of rank j for j = 0, 1, . . . , e p .
Recall that for j = 0, 1, . . . , e p , we defined (see (1)) elements s τ,j and t τ,j of the ring O F,p ⊗ W (OF /p),τ O ∼ = O[u]/(f τ (u)), where f τ is the image under τ of the Eisenstein polynomial associated to our choice of uniformizer ̟ p . In the following, we shall fix τ and omit the subscripts τ , p and i to disencumber the notation; we also write simply W for W (O F /p). Note that since H is locally free over O F,p ⊗ W O S and F (j) is annihilated by s j , we have that F (j) is in fact a subbundle of t j H.
For a subsheaf E ⊂ H of O F,p ⊗ W O S -submodules, we define E ⊥ to be its orthogonal complement under the pairing ·, · , i.e., the kernel of the morphism: or equivalently the orthogonal complement of d −1 p ⊗ OF,p E under the pairing ·, · 0 . Note from the latter description that if E is an O S -subbundle of H, then so is E ⊥ .
Proof. We prove the lemma by induction on j, the case of j = 0 being obvious. Suppose then that 1 ≤ j ≤ e and that (F (j−1) ) ⊥ = t −1 j−1 F (j−1) . Note that (F (j) ) ⊥ and t −1 j F (j) are both kernels of surjective morphisms from H to vector bundles of rank j on S, so each is a subbundle of rank 2e − j, and hence it suffices to prove the inclusion t −1 j F (j) ⊂ (F (j) ) ⊥ . To do so, we may work locally on S, and assume that w, x j = w, t j y j = t j w, y j since t j w ∈ F (j−1) and y j ∈ t −1 j−1 F (j−1) (V ) = (F (j−1) (V )) ⊥ . Finally since the pairing is alternating, we have We have now shown that t −1 We now define G (j) = (u − θ j (̟)) −1 F (j−1) for j = 1, . . . , e. Thus G (j) is a rank j + 1 subbundle of H, and we have inclusions of subbundles F (j−1) ⊂ F (j) ⊂ G (j) , so that L j := F (j) /F (j−1) is a rank one subbundle of the rank two vector bundle P j := G (j) /F (j−1) . Furthermore all the inclusions are morphisms of O F,p ⊗ W O Smodules, and O F acts on P j via θ j .
Note that G (j) is annihilated by s j , so that G (j) ⊂ t j H, and we have t −1 j G (j) = t −1 j−1 F (j−1) = (F j−1 ) ⊥ by Lemma 3.1.1, from which it follows also that (The last equality can be seen by arguing locally on sections, or by noting that the diagram commutes, where the left horizontal morphisms are defined by the pairing, the right by restriction, and all the vertical morphisms by t j . The kernel of the composite along the top is (G (j) ) ⊥ , whereas t −1 j (t −1 j G (j) ) ⊥ is the kernel of the composite along the left and bottom. Since t j : t −1 j G (j) → G (j) is a surjective morphism of vector bundles, the leftmost vertical arrow is injective, so these kernels coincide.) Therefore multiplication by t j defines an isomorphism (F (j−1) ) ⊥ /(G (j) ) ⊥ ∼ −→ G (j) /F (j−1) , and composing its inverse with the isomorphism induced by the pairing on H, we obtain an alternating (O F,p ⊗ W O S )-valued pairing ·, · j on P j := G (j) /F (j−1) , whose description on sections is given in terms of (4) by t j x, t j y j = x, t j y = t j x, y .
Note that since O F,p acts via τ j on P j , we in fact have the identification where I j is the sheaf of ideals, and trivial rank one O S -subbundle, of O F,p ⊗ W O S generated by the global section j ′ =j (u − τ j ′ (̟ p )). We thus obtain a trivialization of ∧ 2 OS P j corresponding to a perfect O S -valued pairing ·, · 0 j , which an unravelling of definitions shows is given in terms of the original pairing ·, · 0 of (3) by the formula t j x, y 0 j = x, f ′ (̟ πp ) −1 ⊗ y 0 on sections (where f ′ is the derivative of the Eisenstein polynomial f ).

Automorphic line bundles.
Recall that the Pappas-Rapoport model Y U is equipped with line bundles 6 L j , which we described in §3.1 as sub-bundles of the rank two vector bundles P j . It is natural and convenient to consider also the twists of L j by powers of the determinant bundle of P j : As we will now consider these bundles for varying τ , we resume writing the indicative subscripts; thus for τ = τ p,i , we will denote G (j) by G (j) τ , P j by P p,i,j , and similarly for M j and N j . We also freely replace the subscript "p, i, j" by θ, where θ = θ p,i,j , so that for each θ ∈ Σ, we have now defined a rank two vector bundle P θ and line bundles L θ , M θ , N θ on S = Y U , along with exact sequences F , and one finds that the automorphism of P θ obtained from α ν is multiplication by θ(µ), so the natural action of O × F, (p),+ on the bundles fails to define descent data with respect to the cover Y U → Y U . We do however obtain descent data after taking suitable tensor products or base-changes of these bundles, which we now consider.
For any O-algebra R, we will use · R to denote the base-change to R of an O-scheme X, as well as the pull-back to X R of a quasi-coherent sheaf on X. Let { e θ | θ ∈ Σ } denote the standard basis of Z Σ . For k = k θ e θ and l = l θ e θ ∈ Z Σ , we define the line bundle where all tensor products are over O S . For n = n θ e θ ∈ Z Σ , we let χ n : O × F → O × denote the character defined by χ n (µ) = θ θ(µ) n θ , and we let χ n,R denote the associated R × -valued character. If k, l, R and U are such that χ k+2l,R is trivial on O × F ∩ U , then the action of O × F,(p),+ on A k,l,R (over its action on Y U,R ) factors through O × F,(p),+ /(U ∩ O × F ) 2 and hence defines descent data, in which case we denote the resulting line bundle on Y U,R by A k,l,R .
Def inition 3.2.1. For k, l, U and R as above, we call A k,l,R the automorphic line bundle of weight (k, l) on Y U,R , and we define the space of Hilbert modular forms of weight (k, l) and level U with coefficients in R to be We note some general situations in which this space is defined: • The paritious setting: We also have a natural left action of GL 2 (A (p) F,f ) on the direct limit over U of the spaces M k,l (U ; R). More precisely suppose that U 1 and U 2 are as above and g ∈ GL 2 (A (p) F,f ) is such that g −1 U 1 g ⊂ U 2 , in which case recall that in §2.2 we defined a morphism ρ g : Y U1 → Y U2 descending to a morphism ρ g : Y U1 → Y U2 . Furthermore the morphism ρ g is obtained from a prime-to-p quasi-isogeny A → A ′ where A is the universal abelian scheme over Y U1 and A ′ is the pull-back of the universal abelian scheme. We thus obtain isomorphisms ρ * g P θ,2 → P θ,1 compatible with (5) and the action of O × F, (p),+ (augmenting the notation for the automorphic bundles on Y Ui and Y Ui,R with the subscript i). Note that l,R,2 descends to Y U2,R , then A k,l,R,1 descends to Y U1,R , and we obtain an isomorphism ρ * g A k,l,R,2 ∼ = A k,l,R,1 . We then define These maps satisfy the obvious compatibility, namely that Finally we remark that the action of ν ∈ O × F, (p),+ on the trivialization of N θ is given by multiplication by θ(ν), so their products do not descend to trivializations of line bundles on Y U,R . However since the stabilizer of each geometric connected component of Y U is O × F,+ ∩ det(U ), we can obtain a (non-canonical) trivialization of A 0,l,R as in [12,Prop. 3.6.1], provided χ l,R is trivial on O × F,+ ∩ det(U ) and the geometric connected components of Y U are defined over R. Furthermore the same argument as in the proof of [12, Lemma 4.5.1] shows the following: 3.3. The Kodaira-Spencer filtration. In this section we define a filtration on Ω 1 YU /O whose pieces are described by automorphic bundles with weight components k θ = 2, l θ = −1. The construction of the filtration is due to Reduzzi and Xiao (see [32, §2.8]), but their presentation is complicated by the fact they wish to prove smoothness simultaneously, and it obscures the fact that the bundles we denoted G (j) automatically satisfy the orthogonality condition appearing in the definition of their counterparts in [32]. We will show below that, with smoothness already established, one can give a more direct conceptual description of the filtration and its properties. 7 Furthermore in the case p = 2, the argument in [32] appeals to a very general flatness assertion for divided power envelopes for which we could not find a proof or reference, so it is not used here.
together with an increasing filtration Proof. As usual, we first prove the analogous result for S := Y U and then descend to Y U .
We let δ 0 : S ∼ −→ ∆ ֒→ Z 0 denote the first infinitesimal thickening of the diagonal embedding, and we view Ω 1 S/O as δ * 0 I, where I denotes the sheaf of ideals defining ∆ in Z 0 . Letting s : A → S denote the universal abelian scheme, the transition maps for the crystal R 1 s cris, * O A/Zp and canonical isomorphisms with de Rham We then define the sheaf of ideals I τ,1 on Z 0 to be the image of β τ,1 , and we let Z τ,1 denote the subscheme of Z 0 defined by I τ,1 , and p τ,1 and q τ,1 the resulting projection maps Z τ,1 → S. By construction the pull-back of β τ,1 to Z τ,1 is trivial, and hence so is that of the morphism The simultaneous treatment in [32] seems natural in view of the inherent overlap in the analysis of deformations needed for both results. However the decision not to appeal to the results in [35] and [33] also makes reference to a perceived minor gap in the proof of [ (if e p > 1), and hence an isomorphism The same argument as above now yields a morphism , whose image is that of a sheaf of ideals on Z 0 we denote by I τ,2 .
Iterating the above construction thus yields, for each τ ∈ Σ p,0 , a chain of sheaves of ideals where Z τ,j denotes the closed subscheme of Z 0 defined by I τ,j and p τ,j , q τ,j are the projections Z τ,j → S.
Furthermore we claim that the map p∈Sp τ ∈Σp,0 I τ,ep → I is surjective. Indeed let J denote the image and let T denote the corresponding closed subscheme of Z 0 , so T is the scheme-theoretic intersection of the Z τ,ep , and let 8 p, q : T → S denote the projection maps. By construction α pulls back to for all τ and j. In particular t * Ω 1 (where t : p * A → T and u : q * A → T are the structure morphisms), which the Grothendieck-Messing Theorem implies is induced by an isomorphism p * A ∼ = q * A of abelian schemes lifting the identity over S. Since the isomorphism respects the filtrations F • , and the lifts of the universal auxiliary structures ι, λ and η over T are unique, it follows that p * A ∼ = q * A, which means that p = q ∈ S(T ), so T = ∆. Let us also note the interpretation of the Kodaira-Spencer filtration in terms of tangent spaces. For a closed point y of S corresponding to the data A 0 = is canonically identified with the set of isomorphism classes of data A 1 over k[ǫ] lifting A 0 , and the decomposition and filtrations of the theorem yield dual decompositions of T y (S) into components T y (S) τ with decreasing filtrations Fil j (T y (S) τ ). From the proof of the theorem one sees immediately that . We note also that the theorem yields a canonical (Kodaira-Spencer) isomorphism . Furthermore the decomposition, filtrations and isomorphisms of the theorem (and hence also the Kodaira-Spencer isomorphism) are Hecke-equivariant in the obvious sense. More precisely the same argument as for the compatibility with the O × F,(p),+ -action, but using the quasi-isogeny in the construction of ρ g , shows that if U 1 , U 2 and g ∈ GL 2 (A commute (where the top arrow is defined in the discussion preceding (6)).

Partial Hasse invariants
4.1. Construction of H θ and G θ . We now recall the definition, due to Reduzzi and Xiao [32], of generalized partial Hasse invariants on Pappas-Rapoport models. These will be, for each θ = θ p,i,j ∈ Σ, a Hilbert modular form We also define below a(n in)variant G θ of weight (0, h θ ).
We will now be working in the mod p setting, so until further notice S will denote Y U,F , and s : A → S the universal abelian scheme over it. Thus H 1 dR (A/S) is a locally free sheaf of rank two over where u acts via ι(̟ p ) * on the p-component of We will also now be working with a fixed p and omit the subscript from the notation, so that with each H i locally free of rank two over O S [u]/(u e ) (where we have also abbreviated the subscript τ p,i by i). Furthermore for each i ∈ Z/f Z, we have a filtration Firstly note that if j > 1, then u : We have now defined a morphism L θ −→ L ⊗n θ σ −1 θ for all θ, and hence a section of A h θ ,0,F over S. Furthermore it is straightforward to check that the section is invariant under the action of O × F,(p),+ and therefore descends to an element which we call the partial Hasse invariant (indexed by θ). Furthermore the partial Hasse invariants are stable under the Hecke action, in the sense that if U 1 , U 2 and g ∈ GL 2 (A (Note also that the partial Hasse invariants are dependent on the choice of uniformizer ̟ = ̟ p only up to a scalar in F × : if ̟ is replaced by a̟ for some a ∈ O × F,p , then H θ is replaced by τ (a)H θ if j > 1 and by τ (a) 1−e H θ if j = 1.) We remark also that the line bundles A 0,h θ ,F have canonical trivializations. Indeed for each i ∈ Z/f Z and j = 2, . . . , e, we have the exact sequence . Furthermore these isomorphisms are Hecke-equivariant in the usual sense, but note that they depend via u on the choice of ̟ p . For each θ, we let G θ ∈ M 0,h θ (U ; F) denote the canonical trivializing section.

Stratification.
We also recall how the partial Hasse invariants define a stratification of the Hilbert modular variety in characteristic p. For any θ ∈ Θ, we define Z θ (resp. Z θ ) to be the closed subscheme of S = Y U,F (resp. Y U,F ) defined by the vanishing of H θ , and for any subset T ⊂ Σ, we let Note that the schemes Z T are stable under the Hecke action, in the strong sense We then have the following consequence ([32, Thm. 3.10]) of the description of the Kodaira-Spencer filtration on tangent spaces at closed points; we give a proof here as some of the details are relevant to the construction of Θ-operators in §5.2.
Proof. We prove the result for Z T , from which the result for Z T is immediate.
Let y be a closed point of S with local ring R = O S,y , maximal ideal m and residue field k = R/m. For each θ ∈ Σ, choose a basis b θ for L θ,y over R and write Identifying m/m 2 with the fibre of Ω 1 YU /O at y and writing Fil j (m/m 2 ) τ for the subspaces obtained from the Kodaira-Spencer filtration, we claim that if y ∈ Z θ , then where τ = τ p,i and θ = θ p,i,j . Comparing dimensions, we see it suffices to prove the inclusion of the left-hand side in the right, or equivalently that if and v is orthogonal to x θ , then in fact v τ ∈ Fil j (T y (S) τ ) (using the notation of the discussion following the proof of Theorem 3.3.1).
). With τ = τ p,i fixed for now, we will suppress p from the notation and replace the subscript For v i to be orthogonal to x θ means that the morphism induced by H θ vanishes, and we need to show this implies that F Suppose first that j > 1. Then (9) is simply and is compatible with u, it follows that it also sends F On the other hand if j = 1, then the vanishing of (9) means that u 1−e F (1) , and φ 0 will denote the absolute Frobenius on k). Since the diagram commutes, where the vertical maps are induced by Verschiebung, the top arrow is (8) and the bottom one is given by the identification of φ * . This completes the proof of the claim. Now note that if y ∈ Z T , then (7) implies that the elements x θ for θ ∈ T can be extended to a basis for m/m 2 over k, hence are linearly independent. Since R is regular of dimension Finally we recall the definition of the minimal weight of a non-zero mod p Hilbert By the main result of [10], the minimal weight of f always lies in the cone: Note that the result stated in [10] applies to forms on a finiteétale cover of Y U,F , from which the analogous result for forms on Y U,F is immediate.

Fundamental Hasse invariants.
In order to define the partial Θ-operators (in §5.2 below), we first define a canonical factorization of the partial Hasse invariants over a finite flat (Igusa) cover of the Hilbert modular variety over F. We fix a sufficiently small U that the line bundles L θ , M θ , N θ (and hence A k,l,F ) on Y U,F descend to Y U,F for all θ ∈ Σ (and all k, l ∈ Z Σ ), and we write simply Y for Y U,F , and L τ,j , M τ,j and N τ,j for the line bundles on Y . For each p ∈ S p and τ ∈ Σ p,0 , we let (where all tensor products are over O Y ). We then define an action of (  (1) and (2) , the sections h τ,ep are injective, and hence so are the h τ,j for all τ and j. We write simply h θ for the section h τ,j of π * L θ = π * L τ,j , and we call the h θ the fundamental Hasse invariant (indexed by θ).

5.2.
Construction of Θ τ . We now explain how the construction of Θ-operators in [12] directly generalizes to the case where p is ramified in F , yielding an operator that shifts the weight k by (1, 1) in the final two components corresponding to embeddings with the same reduction, i.e., θ p,i,ep−1 , θ p,i,ep (and hence, by composing with multiplication by partial Hasse invariants, one can shift weights by +1 for any pair of embeddings with the same reduction).
Indeed for each τ ∈ Σ 0 , we define the operator Θ τ exactly as in [12, §8], but using the morphism provided by Theorem 3.3.1 via projection to the top graded piece of the filtration of the τ -component of Ω 1 Y /F . More precisely, fix p 0 ∈ S p and τ 0 = τ p0,i , let θ 0 = θ p0,i,ep 0 , and consider the morphism Suppose now that f ∈ M k,l (U ; F), and write h k = θ∈Σ h k θ θ and g l = θ∈Σ g l θ θ for any choice of trivializations g θ of the line bundles N θ on Y . We then define the section Furthermore, the section is independent of the choices of g θ and invariant under the Proof. We see exactly as in [12] that Θ τ0 (f ) is regular on the ordinary locus of Y , i.e., the complement of the divisor ∪ θ∈Σ Z θ (where Z θ was defined in §4.2), so the theorem reduces to proving that if z is the generic point of an irreducible component of Z θ1 for some θ 1 ∈ Σ, then Let R denote the discrete valuation ring O Y ,z , and for each τ ∈ Σ p,0 and θ ∈ Σ τ , let y θ = y τ,j be a basis for the stalk L θ,z = L τ,j,z over R (for j = 1, . . . , e p ). For each θ ∈ Σ, we may then write for some r θ = r τ,j ∈ R, and we let r τ = ep j=1 r τ,j . By construction, we have where each x τ is the dual basis of y τ,ep . We then have h τ,ep = x τ y τ,ep (in (π * π * L τ,ep ) z ), from which it follows that h τ,j = r τ,j+1 r τ,j+2 · · · r τ,ep x τ y τ,j for j = 1, . . . , e p − 1, and hence that h k = ϕ k y k , where y k = θ∈Σ y k θ θ and (writing k τp,i,j for k θp,i,j as usual, and working over the field of fractions of T ).
Writing f = ϕ f y k g l , we see that We are therefore reduced to showing that ord z KS τ0 (dr θ1 ) > 0 if and only if θ 1 = θ 0 . However the proof of Proposition 4.2.1 shows that if y is a closed point of Z θ1 , then KS τ0 (dr θ1 ) vanishes at y if and only if θ 1 = θ 0 . Remark 5.2.2. The Kodaira-Spencer isomorphism is defined in [12] using the Gauss-Manin connection. Much of the work in [12, §8.2] amounts to an explicit translation of this to the context of deformation theory. Here however we defined the morphism KS τ0 more directly using deformation theory, so the analogue of [12,Lem. 8.2.3] was not needed here.
Remark 5.2.3. It is straightforward to check directly that the right-hand side of (12) is independent of the choice of local trivializations y τ and g τ , and can therefore be used to define the partial Θ-operator without reference to the Igusa cover Y Ig .
We call Θ τ0 the partial Θ-operator (indexed by τ 0 ). It is immediate from its definition that the resulting map on F-algebras given by the direct sum over all weights of the operators Θ τ0 , is an F-linear derivation, i.e. that It is also clear that Θ τ0 (H θ ) = 0 for all θ ∈ Σ, and hence that Θ τ0 commutes with multiplication by partial Hasse invariants.
6. Partial Frobenius operators 6.1. Partial Frobenius endomorphisms. In order to define partial Frobenius operators on Hilbert modular forms (in §6.2 below), we first need to define partial Frobenius endomorphisms of Hilbert modular varieties over F.
Fix a prime p dividing p, and a level U , assumed as usual to be sufficiently small and prime to p. We will draw on ideas from [11, §7.1] to construct an isogeny on the universal abelian variety s : We begin by associating Raynaud data to the line bundles L p,i,ep over S, which we write simply as L i for i ∈ Z/f Z = Z/f p Z (omitting the subscripts for the fixed p and j = e = e p ). We define f i : L ⊗p i → L i+1 to be zero, and we define v i : L i+1 → L ⊗p i to be the morphism induced by the restriction of Recall that the Dieudonné crystal of ker(Frob A ) is canonically isomorphic to Φ * (s * Ω 1 A/S ), with F = 0 and V induced by Φ * (Ver * A ) (in the notation of [2, §4.4.3]).

On the other hand the Dieudonné crystal of
and there is a unique isogeny β : We now equip A ′ with auxiliary data corresponding to an element of Y U,F (S).
for all geometric points s of S, we immediately have a level U p structure η ′ on A ′ inherited from A.
Next we claim that the quasi-polarization λ on A induces an isomorphism 10 which amounts to the claim that H corresponds to the kernel of under the isomorphism induced by λ. Denoting this kernel by I, we have that H and I are finite flat group schemes over S of the same rank, so it suffices to prove that the composite is trivial. Taking Dieudonné modules, this in turn amounts to the vanishing of the composite We have already noted that the image of the first map has p-component ⊕ i I i ; on the other hand the kernel of the last map is the image of the map corresponding to the adjoint of We are therefore reduced to proving that I i is orthogonal to the kernel of β * i for each i ∈ Z/f Z under the pairing ·, · i defined by (4). Note however that the kernel of β * i is u e−1 I i , as can be seen for example from the commutative diagram -modules for s ∈ S(F p ). Finally the orthogonality of I i and u e−1 I i is immediate from that of F (e−1) i−1 and u −1 F (e−1) i−1 provided by Lemma 3.1.1, completing the proof of the claim. We may then define the quasi-polarization on A ′ by α * (λ ′ ) = δλ for any totally positive generator δ = δ p of pO F, (p) , so that λ ′ induces an , and hence equality holds. We thus obtain an exact sequence We may thus define a Pappas-Rapoport filtration on F ′ i by setting Note that Φ p depends on the choice of δ in the definition of λ ′ ; however it is straightforward to check that Φ p is compatible with the O × F,(p),+ -action on S and descends to an endomorphism Φ p of Y U which is independent of this choice. We call Φ p (resp. Φ p ) the partial Frobenius endomorphism (indexed by p) of Y U,F (resp. Y U ); the terminology is justified by the next proposition.
For the statement of the proposition, we also define the endomorphism Φ of Note that Φ is not the absolute Frobenius φ S on S (unless F = F p ), but we may write φ S = Φ • ǫ where ǫ is the isomorphism defined by the commutative diagram where the square is Cartesian and ε is the inverse of the isomorphism associated to φ * A = A × F,φ F with the evident auxiliary data. We thus have an isomorphism ǫ * A ∼ = A compatible with ι, λ and η, and inducing ǫ * F φ•τ for all τ and j.
(Note also that Φ may be viewed as the base-change of the absolute Frobenius on the descent of S to F p defined by the diagram.) The endomorphism Φ is compatible with the O F,(p),+ -action on S = Y U,F , and we let Φ denote the resulting endomorphism of Y U . Similarly ǫ descends to a φ-linear automorphism ǫ of Y U such that the absolute Frobenius on Y U is Φ • ǫ. Proof. We first prove the commutativity and analogous formula for the maps Φ p on S = Y U,F , from which the corresponding assertions for Φ p follow. To that end it suffices to consider the maps on geometric closed points s ∈ S(F p ), which we will do in order to facilitate computations on Dieudonné modules.
In particular taking r = e p gives which in turn implies the desired formula. Since Φ is finite (and Y U is separated), it follows that Φ p is finite, and therefore also flat since Y U regular. Note furthermore that Φ p is therefore bijective on closed points and induces isomorphisms on their residue fields, so the degree of Φ p in a neighborhood of any closed point x of Y U is that of the extension of completed regular local rings Φ p. Note also that we may replace Y U by S = Y U,F , x by any point in its pre-image in S and Φ p by Φ p .
Suppose then that x corresponds to the data (A 0 , ι 0 , λ 0 , η 0 , F • 0 ) over the residue field k, and its image y = Φ p (x) corresponds to the data (A ′ 0 , ι ′ 0 , λ ′ 0 , η ′ 0 , F ′• 0 ). Recall that the Kodaira-Spencer filtration on the fibre of Ω 1 S/Fp at x is dual to one on T x (S) which was described using Grothendieck-Messing deformation theory (see the discussion following the proof of Theorem 3.3.1). In particular, we have a decomposition T x (S) = ⊕ τ ∈Σ0 T x (S) τ and a decreasing filtration of length e p ′ on for all τ and j ≤ j τ under the canonical isomorphism We claim that the f p -dimensional subspace ⊕ τ ∈Σp,0 Fil ep−1 T x (S) τ is contained in the kernel of T x (S) → T y (S). Indeed if A 1 is a lift corresponding to an element of this subspace and A ′ 1 is its image in T y (S) and α i : A i → A ′ i are the specializations of the universal isogeny α : A → A ′ , then the commutativity of the diagram and the definition of Φ p imply that F for all τ and j. (Note in particular that F 1,τ = ker(α * 1,τ ) for all τ ∈ Σ p,0 , and that for all τ ∈ Σ 0 .) It follows that A ′ 1 is the trivial deformation of A ′ 0 , so the kernel of T x (S) → T y (S) has dimension n ≥ f p as required.
. Furthermore a similar argument shows that the map preserves the Kodaira-Spencer decomposition and filtration, in the obvious sense, and induces isomorphisms 6.2. Construction of V p . In this section we generalize the construction 11 of [12, §9.8] to define partial Frobenius operators, similar to the V p -operator on classical modular forms. We maintain the notation of §6.1, so that Φ p is an endomorphism of S = Y U,F corresponding to the data (A ′ , ι ′ , λ ′ , η ′ , (F ′ ) • ), where A ′ = A/H for a certain finite flat subgroup scheme H ⊂ A [p], and α is the projection A → A ′ .
It is immediate from the definition of F ′(j) τ that α * τ induces an isomorphism L ′ τ,j ∼ −→ L τ,j for all j if τ ∈ Σ p,0 , as well as We thus obtain a surjection , as can be seen on closed points, so we obtain a surjection, hence isomorphism, of line bundles The operators defined here differ slightly from the ones defined in [12] in the unramified case. The construction there is tailored to be compatible with the classical case and to be simply interpreted on q-expansions at cusps at ∞. Doing this in the general ramified case would introduce complications that make it seem not worthwhile.
Remark 6.2.1. One can check that the resulting isomorphisms Φ * p A 2e θ ,−e θ ,F ∼ = A 2e θ ,−e θ ,F (for θ ∈ Σ p ), and Φ * p A 2e θ ,−eτ ,F ∼ = A 2e σ −1 θ ,−e σ −1 θ ,F (for θ = θ p,i,j , j = 2, . . . , e p ) are compatible via the Kodaira-Spencer isomorphisms of Theorem 3.3.1 with the corresponding isomorphisms We are now ready to define the partial Frobenius operator (indexed by p) where the second map is the isomorphism (13). It is immediate from the definition that V p is injective, and that taking the direct sum over all weights yields an Falgebra homomorphism for all sufficiently small U containing GL 2 (O F,p ). It is also straightforward to check that V p is compatible with the Hecke action in the usual sense, and hence defines a GL 2 (A where the spaces are defined in (6) as direct limits over sufficiently small U containing GL 2 (O F,p ).
It will also be convenient at times to consider instead the operator where G θ is the trivialization of A 0,h θ ,F defined at the end of §4.1. Thus V 0 p is also Hecke-equivariant, but depends on the choice of uniformizer ̟ p .
We also record the relation between the partial Frobenius operators and the ppower map. First note that the identification Φ * (F (j) for all τ and j. We similarly have Φ * N τ,j ∼ = N ⊗p φ −1 •τ,j , and taking tensor products and descending to Y yields isomorphisms Φ * A k,l,F ∼ = A pk φ ,pl φ ,F for all k, l, where k φ θp,i,j = k θp,i+1,j , and hence an operator Furthermore the above isomorphisms of line bundles on S are compatible in the sense that the resulting diagram commutes, as does its analogue for the N θ , from which it follows that the composite is the p-power map.
Returning to the partial Frobenius operators, the isomorphisms between Φ * p L θ and L ⊗n θ σ −1 θ (resp. L θ ) for θ ∈ Σ p (resp. θ ∈ Σ p ) for different p ∈ S p are compatible with each other in the obvious sense, and taken together with the formula p∈Sp Φ ep p = ν · Φ and the canonical isomorphism ν * L θ ∼ = L θ yield the isomorphisms Φ * L τ,j ∼ = L ⊗p φ −1 •τ,j defined above. A similar assertion holds for the line bundles N θ , and it follows that the operators V p for p ∈ S p commute with each other and that p V

7.
Compactifications and q-expansions 7.1. Toroidal compactifications. We next recall how q-expansions of Hilbert modular forms are obtained using compactifications of Hilbert modular varieties. In this section we review properties of the toroidal compactification constructed by Rapoport [31] (see also [4] and [13]). We will consider toroidal compactifications only in the case U = U (N ), but we first describe the set of cusps adelically for any U of level prime to p.
For an arbitrary open compact subgroup U of GL 2 (A F,f ) containing GL 2 (O F,p ), we define the set of cusps of Y U to be denotes the subgroup of GL 2 consisting of upper-triangular matrices. Similarly we define the set of cusps of Y U to be We also have a natural bijection between Y ∞ U and the set of isomorphism classes of data (H, I, (p) .
F ⊗ OF H. The bijection is defined by associating the data (H g , I g , [λ g ], [η g ]) to the coset B(O F, (p) ) + gU p , where H g = O 2 F g −1 ∩ F 2 , I g is its intersection with the subspace {0} × F , λ g is induced by the determinant, and η g is induced by right premultiplication by g −1 .
Note that to give a prime-to-p orientation of ∧ 2 OF H is equivalent to giving an such that the set of (geometrically) connected components of its (reduced) closed subscheme Z tor We let ξ : S → Y U denote the natural morphism of formal schemes, and we write F S for the field of fractions of Γ( S, O S ) and µ ν for the automorphism of S defined by ν ∈ O × F,+ . The construction of the toroidal compactification also extends the universal abelian scheme A to a semi-abelian scheme A tor whose pull-back to S is identified with that of the Tate semi-abelian scheme 14 associated to a quotient of the form (19) T I,J : I is the homomorphism corresponding to the tautological element under the canonical isomorphism . Similarly its dual A ∨ extends to a Tate semi-abelian scheme (A ∨ ) tor whose pullback via ξ is associated to T dJ −1 ,dI −1 , with the isomorphism cd ⊗ OF A tor → (A ∨ ) tor defined by the quasi-polarization pulling back to the composite the first morphism is the isomorphism induced by λ, and the second is the canonical one.
The subschemes Z θ of Y U (defined in §4.2 by the vanishing of the partial Hasse invariants H θ ) are closed in Y tor U , and we let Y ord U (resp. Y tord U ) denote the complement of their union, i.e., the ordinary locus, in Y U (resp. Y tor U ), and we use similar 12 Compactified in the sense that its (infinitely many) connected components are proper over O. 13 The formal scheme depends on the chosen cone decomposition {σ C α } and is denoted [4, 3.4.2]. 14 More precisely, the formal scheme S has an open cover by affine formal subschemes Spf Rσ (indexed by cones σ) such that Spec Rσ × Y tor U A tor is identified with the semi-abelian scheme T I,J over Spec Rσ. The compatibilities in the discussion that follows are then systematically checked by verifying them over the open subschemes Spec R 0 notation for the restrictions of A tor and (A ∨ ) tor . Since the sheaf Furthermore its pull-back to S is identified (in the notation of (1)) with under the canonical isomorphism We thus obtain extensions L tor θ of the line bundles L θ = L p,i,j to Y tor U whose pullback to S is identified with ( It follows that each M θ extends to a line bundle M tor We can thus identify the pull-back ξ * N tor θ of the line bundle N tor which the polarization in turn identifies with (cd) θ ⊗ O O S in the notation of (2). Finally it follows that the automorphic bundles A k,l extend to line bundles A tor k,l on Y tor U such that (the tensor products being over O). We refer to this isomorphism as the canonical trivialization of ξ * A tor k,l . Next we consider the completion of Y tor U along the component corresponding to the cusp C represented by (H, I, λ, [η]), which we denote ( Y U tor ) ∧ C . We now describe the global sections of the completions of the line bundles A tor k,l using the identification ( Y tor U ) ∧ C = S/V 2 N and taking invariants under the action of V 2 N on their trivializations over S. Note firstly that Γ(( Y tor under the isomorphism of (18). One then finds that the descent data for ξ * A tor is provided by the isomorphisms T I,J ∼ −→ µ * α 2 T I,J induced by α⊗1 on d −1 I⊗G m , from which it follows that the descent data for ξ * L tor θ is provided on the trivialization by the isomorphisms On the other hand the descent data for ξ * M tor θ is similarly induced on the canonical trivialization by θ(α) −1 , so that the resulting trivialization of ξ * N tor θ descends to ( Y U tor ) ∧ C (in fact extending the one already defined over Y U via the choice of generator t τ,  (20) identify where b is any choice of basis for D k,l .

Minimal compactifications.
We now recall the construction due to Chai [4] of minimal compactifications of Hilbert modular varieties. The presentation in [4] is very concise with numerous typos, but a more detailed treatment of the construction can be found in [13] in the case of U 1 (n) (with different conventions than ours), and of the descriptions of q-expansions in that case in [5]. We continue to assume for the moment that U = U (N ) for some sufficiently large N prime to p. The minimal compactification Y U ֒→ Y min U is then constructed as in [4, §4] or [13, §8]. More precisely, letting t = e θ and taking the global sections of ⊕ k≥0 A tor kt,0 over each component of Y tor ) is the ring described by (21), where we have written C for the corresponding point of Y min U . Furthermore the argument of [31,Prop. 4.9] shows that ι * A k,l = π * A tor k,l (see the discussion following [31, Def. 6.10], or view Y U as a disjoint union of PEL Shimura varieties and apply [28,Thm. 2.5]), so the Theorem on Formal Functions gives that (ι * A k,l ) ∧ , C -module described in Proposition 7.1.1. (Note that ι * A k,l is coherent, but not necessarily invertible.) Similarly for any O-algebra R, we may identify ( ι R, * A k,l,R ) ∧ C with where ι R : Y U,R → Y min U,R is the base-change of ι to R, the completions are at the fibre over C, and b is any basis for D k,l .
The compatibility of the choices of polyhedral cone decompositions ensures that the natural action of O F,(p),+ on Y U extends (uniquely) to one on Y tor U . Furthermore the stabilizer of each component of Z tor U is V N,+ , and the action of V N,+ on each completion ( Y U tor ) ∧ C = S/V 2 N is induced by an action of V N,+ on S such that the effect of ν ∈ V N,+ on global sections of O S is induced by multiplication by ν −1 on M . We see also that the canonical isomorphism A → ν * A extends to an isomorphism A tor → ν * A tor whose pull-back via ξ is induced by the identity on d −1 I ⊗ G m , from which it follows that the action of ν is compatible with the canonical trivialization of the line bundle L tor  (18) on a choice of splitting of the exact sequence 0 → I → H → J → 0). Now suppose that R is an O-algebra such that χ k+2l,R is trivial on V N , so that the line bundle A k,l,R descends to one over Y U,R which we denote by A k,l,R . We then see that ι * A k,l,R is a coherent sheaf on Y min U,R whose completion at the (base-change to R of the) cusp C is identified under (22) with In particular ι * A k,l,R is a line bundle if χ l,R is trivial on V N,+ . Suppose now that U ′ is any sufficiently small open compact subgroup of GL 2 ( O F ) containing GL 2 (O F,p ). One can then carry out a construction similar to the one above to obtain the minimal compactification, or choose an N prime to p such that U (N ) ⊂ U ′ , extend the natural (right) action of U ′ /U (N ) on Y U(N ) to Y min U(N ) and take the quotient; we do the latter (see [13] for the former in the case of U ′ = U 1 (n)).
Firstly our choice of polyhedral cone decompositions ensures that the natural right action of U ′ /U on Y U extends to Y tor U , where U = U (N ) for some choice of N as above. Denoting the resulting automorphism of Y tor U by ρ g for g ∈ U ′ , the canonical identification of the universal A over Y U with its pull-back extends to an identification A tor = ρ * g A tor , giving rise to canonical isomorphisms ρ * g A tor kt,0 = A tor kt,0 , and hence to an action of U ′ /U on Y min U extending its action on Y U . Moreover the action commutes with the natural action of O F,(p),+ , so it descends to an action on Y min U extending the action on Y U . We denote the resulting automorphisms of Y min U by ρ g , and define Y min U ′ to be the quotient of Y min U by the action of U ′ /U (which we will shortly see is independent of the choice of N in its definition).
Identifying the set of components of Y min U − Y U with Y ∞ U , the resulting action of g ∈ U ′ is given by pre-composing η with right multiplication by g −1 , so the set of which we view as acting on J × I by right multiplication. The stabilizer of C is then the set of classes U g = gU ∈ U ′ /U such that and we let Γ C,U ′ = Γ C ∩ σηU ′ η −1 σ −1 . Thus the stabilizer of C is the image of the homomorphism s : Γ C,U ′ −→ U ′ /U defined by γ → η −1 σ −1 γσηU . We claim that if g = s(γ), then ρ * g on O ∧ Y min U ,C = H 0 ( S, O S ) VN,+ is induced by an automorphism ψ γ of S whose effect on global sections corresponds to the map defined by under (18) and the identification M * = J −1 I of (17). Indeed letting ν denote αδ (as well as the automorphism of Y tor U defined by its effect on the universal polarization), we see that δ ⊗ 1 on T I,J defines an isomorphism ξ * ρ * g ν * A tor ∼ −→ ψ * γ ξ * A tor compatible with all auxiliary data, from which one deduces that ν •ρ g •ξ = ξ•ψ γ . Note also that (25) defines an action of Γ C on Γ( S, O S ) which factors through the surjection and the latter group acts on Γ( S, O S ) VN,+ via its quotient where we recall that the isomorphism may depend on the choice of the splitting σ and that we view β as an element of M * . We note in particular if U ′ = U (N ′ ) for some N ′ |N , then the resulting description of O ∧ Y min U ′ ,C ′ coincides with the one previously obtained, from which it follows that the same holds for the scheme Y min U ′ , and hence that Y min U ′ is independent of the choice of N in its definition (for any sufficiently small U ′ containing GL 2 (O F,p ).) Suppose now that k, l ∈ Z Σ and R is an O-algebra such that χ k+2l,R is trivial on Using that the isomorphism ξ * A tor ∼ −→ ψ * γ ξ * A tor is induced by δ ⊗ 1 on the Tate semi-abelian scheme, we find that the resulting automorphism multiplies the canonical trivialization (20) of ξ * A tor k,l,R by χ l,R (α)χ k+l,R (δ). We therefore conclude: U ′ ,R whose completion at (the fibre over) C ′ is identified by the Koecher Principle and Proposition 7.
) ∧ C ′ may be viewed as a special case (with k = l = 0), as can the prior formula for U = U (N ). Furthermore the identifications are compatible in the obvious senses with base changes R → R ′ , inclusions U ′′ ⊂ U ′ (provided the splittings σ are chosen compatibly), and the natural algebra structure on k,l A ′ k,l,R (taking the direct sum over k, l as in the statement).
Recall that the q-expansion Principle allows one to characterize Hilbert modular forms in terms of their q-expansions: U ′ is any set of cusps containing at least one on each component of Y U ′ , then the natural map Note also that we may replace D k,l ⊗ O · with D k,l,R ⊗ R · in the description of q-expansions over R. In particular if R is an F-algebra, the identification The analogous formula holds for the factors (d(IJ) −1 ) θ appearing in the definition of D k,l .
The condition on the q-expansion coefficients in the description of the completions in Proposition 7.2.1 simplifies for certain standard level structures and cusps, as in [12,Prop. 9.1.2]. Suppose that n is an ideal of O F such that χ k+2l,R is trivial for every cusp C of Y U . Note that m ∈ n −1 M if and only if β(N m) ∈ N Z for all β ∈ nM * , and that α, δ ∈ V n implies that χ l,R (α)χ k+2l (δ) = χ l,R (αδ −1 ), so we see that Keep the same assumption on n, but now let U ′ = U 1 (n) and suppose that C ′ is a cusp of Y U ′ "at ∞" in the sense that η(0, 1) ∈ I + n H (p) . We then find that and we similarly conclude that We remark that every component of Y U ′ contains such cusps (in the obvious sense), and that in this case the isomorphism is independent of the choice of splitting σ.

7.3.
Kodaira-Spencer filtration. We next explain how the Kodaira-Spencer filtration on differentials extends to compactifications. We maintain the notation from the preceding section. In particular, we first assume U = U (N ) for some N prime to p before deducing results for more general level structures. The construction of Y tor U via torus embeddings then yields a canonical isomorphism (27) ξ for each cusp C of Y U under which the descent data relative to the quotient map S = S C → ( Y tor U ) ∧ C corresponds to that induced by the obvious action of V 2 N on N −1 M , and the completion of the canonical derivation pulls back to a derivation O S → N −1 M ⊗ O S whose effect on global sections corresponds under (18) to the map defined by Recall also that and are locally free sheaves of O F ⊗ O Y ord U -modules over Y ord U , and therefore so is Decomposing this sheaf over embeddings τ ∈ Σ 0 and equipping it with the filtration defined by the images of the endomorphisms t τ,j defined by (1), we see that the successive quotients (where τ = τ i , t j = t τ,j and θ = θ p,i,j ) are canonically identified with the automorphic bundles A 2e θ ,−e θ over Y ord U . Furthermore the proof of Theorem 3.3.1 shows that the natural map (28) s arising from Grothendieck-Messing theory, or equivalently the Gauss-Manin connection on H 1 dR (A ord / Y ord U ) (see [27, §2.1.7]), is an isomorphism. In particular the Kodaira-Spencer filtration on Ω 1 Y ord U /O corresponds under (28) to the one defined by the images of the endomorphisms t τ,j . Furthermore (28) extends over Y tord U to an isomorphism whose pull-back via ξ = ξ C for each cusp C of Y U is compatible with the canonical isomorphisms of the pull-back of each with M ⊗ O S = N −1 M ⊗ O S (the latter via (27)). Indeed the existence of the extension and the claimed compatibility follow from the analogous well-known result after base-change to C. We therefore conclude that the Kodaira-Spencer filtration on Ω 1 YU /O extends over Y tor U in the form of a decomposition together with an increasing filtration of length e p on each component (Ω 1 Furthermore for each cusp C of Y U and embeddings τ = τ p,i and θ = θ p,i,j , the pull- (27), and the resulting isomorphism coincides with the canonical trivialization of (20).
We now interpret this in the context of minimal compactifications. First we note that the argument of [31,Prop. 4.9] yields a Koecher Principle for Ω 1 We see also from the description of the extension of the Kodaira-Spencer filtration to Ω 1 with completion at C given by in terms of which the canonical derivation is r m q m → m⊗r m q m . Furthermore the completion at C of Fil j (ι * (Ω 1 VN,+ , and the natural maps are isomorphisms whose completions at the cusps are induced by the surjections Suppose now that U ′ is an arbitrary sufficiently small open compact subgroup of GL 2 (A F,f ) of level prime to p, and choose N so that U (N ) ⊂ U ′ . The constructions above are then also compatible with the natural actions of U ′ , so we arrive at similar conclusions with minor modifications to the descriptions of completions that result from taking invariants under Γ C,U ′ . We omit the details, but we remark that letting L (resp. V ) denote the kernel (resp. image) of the homomorphism . Since V acts freely on L * (twisting q-expansion coefficients by a possibly non-trivial cocycle valued in L ⊗ µ N (O)), we still obtain the isomorphism of (29) with U replaced by U ′ , identifying the graded pieces of the Kodaira-Spencer filtration on ι * Ω 1 The description of the extension of the Kodaira-Spencer filtration over compactifications also applies after base-change to an arbitrary O-algebra R, with one significant difference. If R is not flat over O, then the modules M ⊗ R (and their subquotients) may have invariants under the action of the unit groups V N,+ (or more generally the isotropy groups Γ C,U ′ ), so that q-expansions of meromorphic differentials on Y U,R (and forms of weight (2e θ , −e θ )) may have non-zero constant terms, and the morphism analogous to (29) may fail to be an isomorphism. (Note that in this case the relevant base-change morphisms (ι * F ) R → ι R, * (F R ) fail to be surjective at the cusps.) We can however simplify matters by placing ourselves in the situation when this fails in the extreme. Suppose then that p n R = 0 for some n > 0, and that N is sufficiently large that ν ≡ 1 mod p n O F for all ν ∈ V N,+ . Arguing exactly as above, we find that ι R, * (Ω 1 YU,R/R ) is now a vector bundle over Y min U,R whose completion at with the canonical derivation given by r m q m → m ⊗ r m q m . Furthermore each is a sub-bundle whose completion at C is identified with and the natural maps are isomorphisms of line bundles over Y min U,R whose completions at the cusps are induced by the surjections t j (M ⊗ O) τ → M θ . We remark also that this carries over with U replaced by arbitrary U ′ , provided U ′ is sufficiently small that (in additional to the usual hypotheses) α ≡ δ mod p n O F for all α β 0 δ ∈ Γ C,U ′ and cusps C ′ of Y U ′ (the condition being independent of the choice of N and C in the definition of Γ C,U ′ ).

Operators on q-expansions
8.1. Partial Hasse invariants. We next describe the effect of the various weightshifting operators on q-expansions, beginning with the simplest case of (multiplication by) partial Hasse invariants. We will now only be working in the setting of R = F, and we will use · to denote base-changes from O to F. Since the formation of q-expansions is compatible in the obvious sense with pull-back under the projections Y U → Y U ′ , it will suffice to consider the case U = U (N ).
Recall that in §4.1 we defined the partial Hasse invariants as certain elements where h θ := n θ e σ −1 θ − e θ , with n θ = p if j = 1 and n θ = 1 if j > 1. In particular if j > 1, then H θ is defined by the morphism u : L τ,j → L τ,j−1 induced by ̟ p on the universal abelian variety over Y U,F , which evidently extends to the endomorphism ̟ p of A tor over S := Y tor U,F . Since its pull-back via ξ is defined by ̟ p on T I,J , the resulting morphism of line bundles is compatible with their canonical trivializations, and more precisely with the morphism where the constant corresponds to the basis element ̟ p under the identification provided by (26).
For j = 1 we use also that the morphism of line bundles So in this case we again find that H θ has constant q-expansion, the constant now corresponding to the basis element ̟ given by (26).
The q-expansions of the canonical sections G θ ∈ M 0,h θ (U ; F) may be described similarly. Indeed for j > 1, the composites U,F are isomorphisms whose tensor product defines N θ ∼ = N σ −1 θ . Its unique extension to Y tord U,F , and hence to Y tor U,F , is therefore the isomorphism whose pull-back via ξ is the tensor product of the isomorphisms defined on canonical trivializations by (I −1 ) θ ⊗ O F u → (I −1 ) σ −1 θ ⊗ O F and by the identity on (dJ −1 ⊗F) τ /u(dJ −1 ⊗F) τ , and hence corresponds to Therefore G θ has constant q-expansion, with constant corresponding to the basis element ̟ p under the identification provided by the analogue of (26) for (d( induced by Frob A . The extensions to Y tord U,F are again compatible with the canonical trivializations, now corresponding to maps whose tensor product is the inverse of the isomorphism given by (26) for (d(IJ) −1 ) θ .

8.2.
Partial Θ-operators. We now compute the effect of Θ-operators on q-expansions exactly as in [12]. Recall from §5.2 that for each τ 0 = τ p,i ∈ Σ 0 , the associated partial Θ-operator is a map where k ′ = k + n θ0 e σ −1 θ0 + e θ0 , l ′ = l + e θ0 and θ 0 = θ p,i,ep . It is defined for all sufficiently small U of level prime to p, and is Hecke-equivariant. In particular it is compatible with restriction for U ⊂ U ′ , so we may assume U = U (N ) for some N sufficiently large that ν ≡ 1 mod pO F for all ν ∈ V N,+ . Recall from the proof of Theorem 5.2.1 that Θ τ0 is defined by a morphism A k,l,F → A k ′ ,l ′ ,F given locally on sections by formula (12). Our assumptions on U imply that is free of rank one over O ∧ Y min U,F ,C for all weights k, l and cusps C, so the completion at C of ι F, * Θ τ0 is the map (12), where y θ is any basis for (ι F, * L θ ) ∧ C and y k = θ y k θ θ . In particular we may choose y θ = b θ ⊗ 1 where b θ is a basis for D e θ ,0,F = (I −1 ) θ ⊗ O F. The fact that H θ has (non-zero) constant q-expansion at C then means the same holds for the element r θ ∈ O ∧ Recall also that g l = θ g l θ θ in (12), where each g θ is a trivialization of N θ over Y U,F . Therefore g θ trivializes ι F, * N θ over Y min U,F , from which it follows that g θ = c θ ⊗1 for some basis c θ of D 0,e θ ,F = (d(IJ) −1 ) θ ⊗ O F. The formula (12) therefore takes the form Finally the descriptions in §7.3 of the canonical derivation, the Kodaira-Spencer filtration and the isomorphism (29) in terms of q-expansions yield the formula As noted above, it follows that (30) holds with U = U (N ) replaced by any sufficiently small open compact U ′ of level prime to p and C replaced by any cusp of Y U ′ . In this case the q-expansions are necessarily invariant under the natural action of Γ C,U ′ (whose compatibility with (30) is a consequence of the construction, but is straightforward to check directly).
We see immediately from (30) that the operators Θ τ for varying τ commute. We see also that where τ 1 = τ 0 • φ = τ p,i+1 and θ 1 = σ ep θ 0 = θ p,i+1,ep . Indeed this follows from (30) together with the fact that the q-expansions of are constants given by the canonical isomorphisms (( where the downward arrow is multiplication by the (constant) q-expansion of H 2 τ1 G −1 τ1 .

Partial Frobenius operators.
Finally we compute the effect on q-expansions of the partial Frobenius operators V p defined in §6.2. We must first extend the partial Frobenius endomorphisms Φ p (and Φ p ) defined in §6.1 to compactifications. To that end let h p denote the matrix 1 0 0 δ , viewed as an element of GL 2 (A (p) )h −1 p gU p , and similarly let Φ ∞ p denote the induced permutation of Y ∞ U . Then Φ ∞ p translates to the map on corresponding data sending (H, I, λ, Suppose now that U = U (N ) for some sufficiently large N . One then checks that the morphism Φ p : , and the resulting map on completions pulls back to a morphism S ′ C,F → S C ′ ,F whose effect on global sections corresponds under the isomorphisms of (18) to the homomorphism induced by the canonical inclusion Furthermore the pull-back of A ′tor = ( Φ tor p ) * A tor to S ′ C,F is identified with the Tate semi-abelian variety T p −1 I,J , and the isomorphisms defined in §6.2 relating the line bundles Φ * p L θ to L θ or L σ −1 θ extend to isomorphisms ( Φ tor p ) * L tor θ ∼ = L tor ′ θ for θ ∈ Σ p , and ( Φ tor p ) * L tor θ ∼ = (L tor ′ σ −1 θ ) ⊗n θ for θ ∈ Σ p over Y tor ′ U,F whose pull-backs are compatible via their canonical trivializations with isomorphisms induced by the canonical (O F ⊗ F) τ -equivariant maps More precisely if τ ∈ Σ p,0 , then the second map is an isomorphism identifying (pI −1 ) θ ⊗ O F with (I −1 ) θ ⊗ O F, and if τ = τ p,i and θ = θ p,i,j , then this map also induces the desired isomorphisms On the other hand if τ = τ p,i and θ = θ p,i,1 , then the first map induces an isomorphism whose composite with the ones induced by The relations between the line bundles Φ * p N θ and N θ or N σ −1 θ extend similarly over Y tor ′ U,F , so for k ′′ , l ′′ as in (14) we obtain isomorphisms whose pull-backs to S ′ C,F are compatible via their canonical trivializations with the isomorphisms obtained as the tensor products of the ones just defined (where D ′ k,l is associated to the data for the cusp C ′ , and D k ′′ ,l ′′ to the data for C).
It follows from the above description of Φ tor p that Φ p extends to the morphism Φ min p : Y min U,F → Y min U,F restricting to Φ ∞ p on the set of cusps, with the induced maps on completed local rings 15 being the restriction to V 2 N -invariants of the canonical inclusion where M = d −1 I −1 J and M ′ = pM . Furthermore the commutativity of the diagram (where the top vertical arrows are defined by pulling back via Φ min p , Φ tor p and the map S ′ C,F → S C ′ ,F ) shows that the resulting map on q-expansions is the restriction , obtained as the tensor product of (31) and (32). (Note that the isomorphism (31) is V 2 N -equivariant, but we can also choose N sufficiently large that the action is trivial.) The constructions above are all compatible with the natural action of O × F,(p),+ , so the morphism Φ min p induces a morphism Φ min p on cusps, and its effect on completed local rings is given by the V N,+ -invariants of (32). Furthermore the map V p : M k,l (U ; F) −→ M k ′′ ,l ′′ (U, F) is described on q-expansions by taking the V N,+ -invariants of the tensor product of (31) and (32). (Note that Φ min p is proper and quasi-finite, hence finite, but not necessarily flat at the cusps.) Similarly for any sufficiently small U ′ , we may choose N so U = U (N ) ⊂ U ′ and take invariants under the natural action of U ′ /U , with which the above constructions are also easily seen to be compatible. We thus obtain the description of V p on q-expansions (under the identifications of Proposition 7.2.1) as the resulting map (33) (Note that the maps (31) and (32) are in fact Γ C -equivariant, where Γ C is defined in (24) and its action on the target is via the natural inclusion in Γ C ′ .) Finally we note that the effect of the operator V 0 p : M k,l (U ; F) −→ M k ′′ ,l (U, F) on q-expansions has the same description, but with (31) replaced by its composite with the isomorphism given by choosing the basis element of D 0,l−l ′′ to be the (constant) q-expansion of θ∈Σ G −l θ θ .
9. The kernel of Θ 9.1. Determination of the kernel. In this section we analyze the kernel of the partial Θ-operator Θ τ : M k,l (U ; F) → M k ′ ,l ′ (U ; F) for τ ∈ Σ p,0 and relate it to the image of a partial Frobenius operator. We allow U to be any sufficiently small open compact subgroup of GL 2 (A F,f ) of level prime to p, and (k, l) any weight such that χ k+2l,F is trivial on U ∩ O × F . First note that by (30)  ) and in particular is independent of N prime to p such that U (N ) ⊂ U .) Note that the condition is the same for all τ ∈ Σ p,0 , so that ker(Θ τ ) = ker(Θ τ ′ ) for all τ, τ ′ ∈ Σ p,0 .
Note also that the condition is invariant under multiplication by the Hasse invariants H θ (and of course the forms G θ ) for all θ, so that Θ τ (f ) = 0 if and only if ≥0 . (Alternatively note that this follows from the fact the partial Θ-operators commute with multiplication by the G θ and H θ , as can be seen directly from their definition.) Suppose now that k = k min (f ), so that f is not divisible by any partial Hasse invariants (see §4.2). Then if f ∈ ker(Θ τ ), and hence f ∈ ker(Θ τp,i ) for all i ∈ Z/f p Z, then Theorem 5.2.1 implies that p|k θp,i,e p for all i ∈ Z/f p Z. Therefore k is of the form k ′′ 0 for some k 0 , where k ′′ 0 is as in the definition of V p in §6.2, or equivalently V 0 p . Furthermore it is immediate from the description of the effect of V 0 p on q-expansions in (33) that its image is contained in the kernel of Θ τ . We now use the method of [12,Thm. 9.8.2] to prove the kernel is precisely the image of V 0 p . Theorem 9.1.1. Suppose that f ∈ M k ′′ 0 ,l (U ; F) and τ ∈ Σ p,0 . If Θ τ (f ) = 0, then f = V 0 p (g) for some g ∈ M k0,l (U ; F).
Proof. Let ι denote the embedding Y U ֒→ Y min U , and choose a set of cusps S ⊂ Y ∞ U consisting of precisely one on each connected component of Y U . Note that since Φ min p (defined in §8.3) is bijective on cusps as well as connected components, the set S ′ := Φ ∞ p (S) also includes exactly one cusp on each connected component. Recall from Proposition 7.2.1 that the sheaves ι * A k0,l,F and ι * A k ′′ 0 ,l,F are coherent, as is ι * Φ p, * A k ′′ 0 ,l ′′ ,F = Φ min p, * ι * A k ′′ 0 ,l,F since Φ min p is finite. For each C ∈ S, let C ′ = Φ ∞ p (C), so that Φ min, * p defines a finite extension O Y min U ,C ′ ֒→ O Y min U ,C of local rings. We let N C ′ = (ι * A k0,l,F ) C ′ denote the stalk at C ′ of ι * A k0,l,F , and similarly let N ′′ C = (ι * A k0,l,F ) C = (ι * Φ p, * A k ′′ 0 ,l,F ) C ′ . The stalk at C ′ of ι * of the adjoint of Φ * p A k0,l,F ∼ −→ A k ′′ 0 ,l,F then defines an injective homomorphism N C ′ → N ′′ C of finitely generated O Y min U ,C ′ -modules, extending V 0 p to a map Similarly localizing at the generic points of Y U (or equivalently Y min , so we obtain a commutative diagram of injective maps (34) M k0,l (U ; F) / / . (Note that the horizontal maps, defined by localization, are injective since S and S ′ each contain a unique cusp on each component of Y U .) Let N ∧ C ′ denote the completion of N C ′ with respect to the maximal ideal of O Y min U ,C ′ , and similarly let N ′′∧ C denote the completion of N ′′ C with respect to the maximal ideal of O Y min U ,C , or equivalently O Y min U ,C ′ . Note that the map N ∧ C ′ → N ′′∧ C is the one described by (33), or more precisely its variant for V 0 p . Recall from (30) that if f ∈ ker Θ τ , then for each C ∈ S, the q-expansion of f : satisfies r m = 0 for all m ∈ pN −1 M = N −1 M ′ + , where M ′ = pM , and is therefore in the image of N ∧ C ′ . Since the completion O ∧ Y min U ,C ′ is faithfully flat over O Y min U ,C ′ , it follows that the image of f in C∈S N ′′ C is of the form V 0 p (g) for some g ∈ C ′ ∈S ′ N C ′ , and hence that its image in H 0 (Y U , A k ′′ 0 ,l,F ⊗ Y U F U ) is of the form V 0 p (g) for some g ∈ H 0 (Y U , A k ′′ 0 ,l,F ⊗ Y U F U ). It just remains to prove that g ∈ M k0,l (U ; F), or equivalently that ord z (g) ≥ 0 for all prime divisors z on Y U . To that end, note that the operators V p ′ for all p ′ ∈ S p , and ǫ k,l for all k, l ∈ Z Σ (see §6.2)) similarly extend to maps on stalks at generic points satisfying (16), so that V p (g) = f θ∈Σ G l θ θ and Therefore p · ord z (g) ≥ 0, and hence ord z (g) ≥ 0.
For the following corollary, recall that Ξ min is defined by (10) and that the main result of [10] states that if f is a non-zero form in M k,l (U ; F), then k min (f ) ∈ Ξ min . Corollary 9.1.2. Suppose that f ∈ M k,l (U ; F) and τ ∈ Σ p,0 . Then Θ τ (f ) = 0 if and only if there exist k 0 ∈ Ξ min , n ∈ Z Σ ≥0 and g ∈ M k0,l (U ; F) such that k = k ′′ 0 + θ n θ h θ and f = V 0 p (g) θ H n θ θ .
Proof. We have already seen that if f = V 0 p (g) θ H n θ θ , then Θ τ (f ) = 0. For the converse, note that we may assume k = k min (f ), so that k ∈ Ξ min and k = k ′′ 0 for some k 0 ∈ Z Σ . Therefore the theorem implies that f = V 0 p (g) for some g ∈ M k0,l (U ; F). Finally it is immediate from the definitions of Ξ min and k ′′ 0 that k 0 ∈ Ξ min if (and only if) k ′′ 0 ∈ Ξ min . 9.2. Forms of partial weight 0. We now apply our results on partial Θ-operators to prove a partial positivity result for minimal weights of Hilbert modular forms. Recall the main result of [10] proves that minimal weights k = k θ e θ of Hilbert modular forms necessarily lie in the cone Ξ min , and hence satisfy k θ ≥ 0 for all θ, and that forms with k = 0 are easily described by Proposition 3.2.2. We prove the following restriction on possible minimal weights k with k θ = 0 for some (but not all) θ ∈ Σ. Theorem 9.2.1. Suppose that p ∈ S p is such that f p > 1, or e p > 1 and p > 3. Suppose that f ∈ M k,l (U ; F) is non-zero and k = k min (f ). If k θ = 0 for some θ ∈ Σ p , then k = 0.
Proof. Writing simply f = f p and e = e p , note that the hypotheses mean that ef > 1 and p f > 3. Choose any τ = τ p,i ∈ Σ p,0 and let θ 0 = θ p,i,e . We will first prove that Θ τ (f ) = 0.
(with the obvious collapsing here and in subsequent inequalities if e or f = 1). In particular all the expressions in (35) are non-negative, so we have m 1 ≥ m 2 ≥ · · · m e ≥ pm e+1 ≥ pm e+2 ≥ · · · ≥ p f −1 m ef −1 ≥ p f −1 m ef and 2 + m ef − pm 1 ≥ 0, which implies that (p f − 1)m ef ≤ 2. Since p f > 3, it follows that m ef = 0, so pm 1 ≤ 2, which implies that either m 1 = 0, or m 1 = 1 and p = 2. If m 1 = 0, then m r = 0 for all r, which contradicts the final inequality in (35). On the other hand if m 1 = 1 and p = 2, then all the expressions in (35) are zero, which in turn implies that m 1 = p f −1 m ef , which again yields a contradiction.
We have now shown that Θ τ (f ) = 0. Note that k ′′ = k since k θ = 0 for all θ ∈ Σ p , so Theorem 5.2.1 implies that f = V 0 p (f 1 ) for some f 1 ∈ M k,l (U ; F). We may therefore iterate the above argument to conclude that f 1 = V 0 p (f 2 ) for some f 2 ∈ M k,l (U ; F), and by induction that for all n ≥ 1, we have f = (V 0 p ) n (f n ) for some f n ∈ M k,l (U ; F). It follows that for all n ≥ 1, the q-expansion of f at every cusp of Y U satisfies r m = 0 for all m ∈ p n M , so in fact the q-expansion of f at every cusp is constant.
To prove that k = 0, recall that Ξ min is contained in the cone spanned by the partial Hasse invariants, so k = θ∈Σ s θ h θ for some s θ ∈ Q ≥0 . Furthermore the denominators are divisors of M = lcm{ p fq − 1 | q ∈ S p }, so that M k = m θ h θ for some m θ ∈ Z ≥0 . Similarly M l = n θ h θ for some n θ ∈ Z. Since f has constant q-expansions, so does f M , and therefore For each θ ∈ Σ, the assumption that k = k min (f ) means that f is not divisible by H θ , so ord z (f ) = 0 for some irreducible component z of Z θ . On the other hand we have M ord z (f ) = ord z (f M ) ≥ m θ , so m θ = 0. As this holds for all θ ∈ Σ, we conclude that k = 0. 9.3. The kernel revisited. Finally we present a cleaner, but less explicit, variant of Corollary 9.1.2 describing the kernels of partial Θ-operators.
We first record the effect of V p on the partial Hasse invariants H θ . For each prime p ∈ S p , we let β p = p −1 ̟ ep p ∈ O × F,p . It is straightforward to check, directly from the definition of V p or from the description (33) of its effect on q-expansions (and those of the H θ in §8.1), that if θ ∈ Σ p then V p (H θ ) = H θ , but if θ = θ p,i,j then if e p > 1 and j = 2; H σ −1 θ , otherwise.
Therefore we define the modified partial Hasse invariant to be H ′ θ = θ(β p ′ )H θ if θ = θ p ′ ,i,1 for some p ′ ∈ S p and i ∈ Z/f p ′ Z, and H ′ θ = H θ otherwise, so that Similarly letting G ′ θ = θ(β p ′ )G θ if θ = θ p ′ ,i,1 and G ′ θ = G θ otherwise, we have V p (G ′ θ ) = G ′n θ σ −1 θ if θ ∈ Σ p and V p (G ′ θ ) = G θ if θ ∈ Σ p . Now for any sufficiently small U of level prime to p, consider the F-algebra of Hilbert modular forms of all weights and level U (where we let M k,l (U ; F) = 0 if χ k+2l,F is non-trivial on U ∩ O × F ). We may then consider V p (resp. Θ τ ) as an Falgebra homomorphism (resp. F-linear derivation) M tot (U ; F) → M tot (U ; F) for any p ∈ S p and τ ∈ Σ p,0 . Furthermore letting I denote the ideal H ′ θ − 1, G ′ θ − 1 θ∈Σ in M tot (U ; F) and R U = M tot (U ; F)/I, we see that V p (I) ⊂ I and Θ τ (I) ⊂ I, so we obtain an F-algebra homomorphism V p and derivation Θ τ such that the composite is zero for any p ∈ S p , τ ∈ Σ p,0 .
Let Λ denote the subgroup θ∈Σ Zh θ of Z Σ = θ∈Σ Ze θ , so Λ is the image of the image of the endomorphism of Z Σ defined by θ m θ e θ → θ m θ h θ . Writing h θ = one gets a description of c θ by multiplying by θ(b p ) if θ = θ p,i,1 .) Similarly let d θ ∈ D C 0,h θ denote the (constant) q-expansion of G ′ θ at C, and define I C to be the ideal c θ − 1, d θ − 1 θ∈Σ of D C tot . We may then view the quotient D Proof. The inclusion I ⊂ ker(q) is clear from the definitions. Suppose then that q(f ) = 0 and write f = k,l∈W f k,l for some finite subset W of Z Σ . For each χ ∈ Ψ, choose k χ ∈ ̺ −1 (χ) sufficiently large that k ≤ Ha k χ for all k ∈ ̺ −1 (χ) ∩ W . Thus for each k ∈ W , there is a unique m k = θ m k,θ e θ ∈ Z Σ ≥0 such that k ̺(k) = k + θ m k,θ h θ . Now note that g := k,l∈W H ′m k G ′m l f k,l ∈ χ,ψ∈Ψ M kχ,k ψ (U ; F) and that f − g ∈ I (where H ′m k = θ H ′m k,θ θ and G ′m l = θ G ′m l,θ θ ). Since I ⊂ ker(q), it follows that q(g) = 0. However q restricts to the q-expansion map which is injective by Proposition 7.2.2, so g = 0, and hence f = f − g ∈ I.
We also extract the following observation from the proof of the lemma: We are now ready to interpret the description of the kernel of the partial Θoperator in terms of the algebra R U .