Compactifications of Iwahori-level Hilbert modular varieties

We study minimal and toroidal compactifications of $p$-integral models of Hilbert modular varieties. We review the theory in the setting of Iwahori level at primes over $p$, and extend it to certain finer level structures. We also prove extensions to compactifications of recent results on Iwahori-level Kodaira--Spencer isomorphisms and cohomological vanishing for degeneracy maps. Finally we apply the theory to study $q$-expansions of Hilbert modular forms, especially the effect of Hecke operators at primes over $p$ over general base rings.


Introduction
In this paper we study compactifications of Hilbert modular varieties, augmenting and extending the existing theory in several ways we expect to be useful.
We start by recalling, in § §2-3, some of the main results on toroidal and minimal compactifications of integral (at p) models of Hilbert modular varieties with (at most) Iwahori level structure at primes p over p.The models are defined using moduli problems introduced in [Pap95] and [PR05], and the results on compactifications can be obtained by adapting the methods of Rapoport ( [Rap78]; see also [Cha90,Dim04,Dia21a]) or by specializing much more general results of Lan (especially [Lan18]).
The new results are presented in § §4-6.Firstly in §4 we consider level U 1 (p)structure, using models based on the moduli problem in [Pap95].These are finite and flat over those with Iwahori (i.e., U 0 (p)) level structure, and we construct and study compactifications which preserve this property.These were already introduced in [DS22] under the assumption that p is unramified in the totally real field F , where they are used in establishing existence and properties of Galois representations associated to mod p Hilbert modular eigenforms.The results here will similarly be used in the sequel [DS], where the assumption on ramification is removed.
In §5 we extend the main results of [Dia21b] to toroidal compactifications.Recall that in [Dia21b] we construct a Kodaira-Spencer isomorphism describing the dualizing sheaf of the integral Iwahori-level model, and prove relative cohomological vanishing results for degeneracy maps, the latter generalizing results in [DKS], where p is assumed unramified in F .Our main motivation for this is to extend the "saving trace" to compactifications.Recall that the saving trace is introduced in [Dia21b] in order to conceptualize and generalize the construction of Hecke operators at primes over p.Its extension here to toroidal compactifications is used in §6 to prove the operators T p have the desired effect on q-expansions.(See [DW20,DM20,DS22] for similar formulas, based on different, less general, constructions of T p .)This also implies the commutativity of the operators T p (for varying p over p), tying up a loose end from [Dia21b].
Finally in §7 we take the opportunity to list a few minor corrections to [Dia21a].
Notation.We adopt much of the set-up and notation from [Dia21a] and [Dia21b].
In particular, we fix a prime p and a totally real field F = Q.We let O F denote the ring of integers of F and d = d F/Q the different.We write O F,(p) for the localization of F at the prime p of Z, and O F,p for the p-adic completion of O F .We let Σ p denote the set of prime ideals of O F over p, so that O F,(p) is also the localization of O F at the complement of p∈Σp p and O F,p = p∈Σp O F,p .For each p ∈ Σ p , we let |O F /p| = p fp and e p = v p (p), so [F p : Q p ] = e p f p .We also fix a finite extension K of Q p , and let O denote its ring of integers and k its residue field.We assume that K is sufficiently large to contain the image of all embeddings of F in the algebraic closure of K, and we let Θ denote the set of such embeddings.For each p ∈ Σ p we let Θ p denote the set of embeddings F p → K, and we identify Θ with p∈Σp Θ p via the canonical bijection.Similarly for each p ∈ Σ p we may write Θ p = τ Θ τ , where τ runs over the set of f p embeddings O F /p → k, and for each such τ , we arbitrarily choose an ordering of Θ τ = { τ 1 , . . ., τ ep } of the e p elements of Θ p restricting to τ : W (O F /p) → O.
We let A F,f = O F ⊗ Q denote the finite adeles of F , and F ⊗ Q the prime-to-p finite adeles of F (where • (p) denotes prime-to-p completion).We let U be a sufficiently small1 open compact subgroup of GL 2 (A F,f ) containing GL 2 (O F,p ), so that U = U p GL 2 (O F,p ) for a (sufficiently small) open compact subgroup U p of GL 2 (A (p) F,f ).
Acknowledgements.This work originated as an appendix to the forthcoming paper with Shu Sasaki [DS], but took on a life of its own.We are grateful to him, and also to Payman Kassaei, for numerous helpful conversations in relation to the topic.The author also thanks Mladen Dimitrov for encouraging him to pursue some of the questions considered here, Najmuddin Fakhruddin for calling his attention to the notion of rational singularities (and hence the reference [Dan78]), and Kai-Wen Lan for quickly rectifying a minor oversight in [Lan18] that came to light.

Level prime to p
In this section, we recall the construction and properties of toroidal and minimal compactifications of p-integral models of Hilbert modular varieties of level prime to p.Our main focus will be on the "splitting" models constructed by Pappas and Rapoport as in [PR05], but we first consider the"naive" models of Deligne and Pappas [DP94].Since the ordinary loci of these models coincide, we may view the compactifications of the former as obtained from the latter.
2.1.The Deligne-Pappas model.We let Y − denote the (infinite disjoint union of Deligne-Pappas PEL) fine moduli scheme(s) of level U (defined as in [Dia21a, §2.1], but without filtrations), and Y − := O × F,(p),+ \ Y − the resulting model for the Hilbert modular variety of level U .More precisely, for a locally Noetherian Oscheme S, Y − (S) is identified with the set of isomorphism classes of data (A, ι, λ, η), where: • s : A → S is an abelian scheme of relative dimension d; • ι : O F → End S (A) is an embedding such that (s * Ω 1 A/S ) p is, locally on S, free of rank e p over W (O F /p) ⊗ Zp O S for each p ∈ S p ; • λ is an O F -linear quasi-polarization of A such that for each connected component S i of S, λ induces an isomorphism c i d ⊗ OF A Si → A ∨ Si for some fractional ideal c i of F prime to p; • η is a level U p structure on A, and the O × F,(p),+ -action is defined by ν • (A, ι, λ, η, F • ) = (A, ι, νλ, η, F • ).We thus have a universal abelian scheme over Y − , and the determinant of its cotangent bundle along the zero section defines an ample line bundle ω, descending to one on Y − which we denote by ω.

2.2.
Cusps of level U .We define the set of cusps of level U as in [Dia21a, §7.1] (called there the cusps of Y U and denoted Y ∞ U ) to be F,f )/U p , where B ⊂ GL 2 is the subgroup of upper-triangular matrices, and the subscript + denotes those with totally positive determinant.Similarly we define the set of polarized cusps of level U to be ) for the associated isomorphism class.

2.3.
Toroidal compactification of Y − .For simplicity, we assume U = U (N ) for some integer N ≥ 3 (prime to p) in the consideration of toroidal compactifications, which are obtained as in [DS22] or [Dia21a] by applying the method of [Rap78, §5] (see also [Cha90], [Dim04] or [Lan13]) to the connected components of Y − .More precisely, choosing a polyhedral cone decomposition (as in [Rap78, §4]) of We remark that the cone decompositions can be chosen so that the action of O × F,(p),+ on Y − extends to one on Y tor − , yielding a toroidal compactification Y tor − of Y − as the quotient.One can furthermore remove the restriction that U = U (N ), but we will make no use of this here, our ultimate focus (except in §5) being on the minimal compactifications for more general U .

2.4.
Relation with Lan's definition.We note that our Y − is an infinite disjoint union of schemes of the form M naive H p ,O in the notation of [Lan18].More precisely for each ǫ ∈ (A − denote the open and closed subscheme of Y − over which the diagram commutes (in the notation of [Dia21a, §2.2], cf.[DKS, §2.1.2]).Letting [ǫ] denote the fractional ideal of F defined by ǫ and L = δd −1 ⊕ [ǫ]d −1 for any totally positive δ ∈ [ǫ] −1 such that δO F,p = dO F,p , the scheme Y ǫ − is then isomorphic to the one defined in [Lan18] as M naive H p ,O using the standard alternating pairing on L (composed with the trace and twisted by any choice of Z ∼ −→ Z(1)).The isomorphism (and choice of H p ) are given by modifying the level structure (resp.quasi-polarization) of [Dia21a] by multiplication by diag(δ −1 , ǫ −1 ) (resp.δ).
By Corollary 2.4.8 of [Lan18], we also have an isomorphism 2 a priori algebraic spaces, but in fact projective schemes by (the method of) [Lan13, §7.3], or alternatively by the relation described in §2.4 with the compactifications defined in [Lan17] where a is an invertible O F,p -submodule of F p ). Furthermore the condition in Theorem 6.1(6) of [Lan17] is satisfied 3 by the connected components of Y tor − , yielding morphisms to the toroidal compactificatons M tor H,Σ,O satisfying the conclusion of [Rap78, Thm.3.5], so that the identification above extends to one between Y tor − and an infinite disjoint union of toroidal compactifications M tor H,Σ,O as in [Lan17].
2.5.Minimal compactification of Y − .We continue to assume for the moment that U is of the form U (N ).The universal abelian scheme over Y − extends to a semiabelian scheme over Y tor − , yielding also an extension of the line bundle ω.Moreover the line bundle ω M tor H,Σ ,J on M tor H,Σ,O (with J a singleton) is identified with the pullback of our ω ⊗a for some integer a > 0 by [Lan17, Thm.6.1(2)].The scheme there denoted M min H,O therefore coincides with the projective scheme associated to the global sections of the symmetric algebra on ω over the corresponding connected components, and taking their disjoint union yields the minimal compactification Y − ֒→ Y min − .The scheme Y min − is normal, independent of the choice of cone decompositions in the definition of Y tor − , and its connected components are flat and projective over O.We thus have a commutative diagram of morphisms 3 To make the translation between our set-up and that of [Lan17] at the polarized cusp such that for j = 1, . . ., e p , the quotient (again with H = H p U p (L )).Just as for Y − , it follows that the isomorphism extends to one between Y tor and an infinite disjoint of the schemes M spl,tor H,Σ,O defined in [Lan18], where the universal property is now the one in [Lan18,Thm. 3.4 over the corresponding ones for Y − ; again the diagonal morphisms are open immersions and the vertical morphism is projective.
We claim also that Y min is isomorphic to an infinite disjoint of the schemes denoted M spl,min H,O in [Lan18].Indeed by the Koecher Principle for the vertical morphisms in the diagram ([Lan18, Thm.4.4.10],[Lan17, Thm.8.7]), the fact that the top arrow is an isomorphism on the ordinary locus implies that so is the bottom arrow.

Iwahori level at p
We now recall how the theory reviewed in §2 applies to yield toroidal and minimal compactifications of Hilbert modular varieties with Iwahori level at primes over p.

3.1.
The Iwahori-level model.Let P be a divisor of the radical of pO F , and let We let Y 0 (P) denote the corresponding fine moduli scheme parametrizing pairs of objects A 1 , A 2 of Y , equipped with a P-isogeny ψ : A 1 → A 2 respecting the additional structures (as defined in [Dia21b, §2.4], thus depending on a choice of ̟ P in the notation there).Similarly we let Y 0 (P) denote its quotient by the action of O × F,(p),+ , so that Y 0 (P) is a model for the Hilbert modular variety of level U 0 (P) (and is independent of ̟ P ).We thus have a pair of forgetful morphisms defined by the natural projection and multiplication by α P := p|P 1 0 0 ̟p p for any choice of uniformizers ̟ p at p.

3.2.
Cusps of level U 0 (P).We define the set of cusps of level U 0 (P), denoted C 0 (P), exactly as we did for level U (see §2.2), but with U now replaced by U 0 (P).We thus have the natural projections Furthermore we have a bijection between C 0 (P) and the set of isomorphism classes of triples (H 1 , H 2 , α), where Under this bijection the maps π ∞ i correspond to the obvious forgetful maps, and we have a pair of bijections We also let C 0 (P) denote the set of isomorphism classes of triples ( H 1 , H 2 , α), where now the H i (resp.α) are as in the description of C (resp.C 0 (P)), with the additional condition that λ 2 • ∧ 2 α = ̟ P λ 1 .We then have a pair of bijections defined in the same way as for C 0 (P), and we write [ H 1 , H 2 , α] for the isomorphism class of ( H 1 , H 2 , α).

3.3.
Toroidal compactification of Y 0 (P).Once again we assume that U = U (N ) for some N ≥ 3, with the N th roots of unity contained in O, and define the toroidal compactification Y 0 (P) ֒→ Y 0 (P) tor as in [Rap78].More precisely for each cusp in C 0 (P), we choose an admissible cone decomposition of (J −1 2 I 1 ⊗R) ≥0 and construct the toroidal compactification using the morphism of semi-abelian schemes defined by over the resulting formal scheme.The universal isogeny ψ : of semi-abelian schemes over Y 0 (P) tor whose completion along the complement of Y 0 (P) is described by (3.1).
Our Y 0 (P) can again be identified with an infinite disjoint union of schemes of the form M spl H p ,O considered in [Lan18], where L is now the set of lattices in F 2 , where now H = H p U p (L ) for this choice of L , and that the identification extends to one between Y 0 (P) tor and an infinite disjoint of the schemes M spl,tor H,Σ,O defined in [Lan18].Furthermore if the admissible cone decomposition for each cusp [ H 1 , H 2 , α] in C 0 (P) is chosen to refine those for the cusps [ H 1 ] and [ H 2 ] in C, then the same universal property implies that each π i extends to a morphism π tor i : Y 0 (P) tor → Y tor under which the semi-abelian scheme A tor i is the pull-back of the extension A tor of the universal abelian scheme A over Y .
We summarize the main results as follows: Theorem 3.3.1.There is a normal scheme Y 0 (P) tor over O and an open immersion Y 0 (P) ֒→ Y 0 (P) tor with the following properties: (1) Y 0 (P) tor is flat and Cohen-Macaulay over O with projective connected components, and there is a canonical bijection c ←→ Z c between C 0 (P) and the set of connected components of the reduced complement of , where S 0 (P) is the completion of the complement of Spec (O[N −1 M ]) in the torus embedding defined by the chosen cone decomposition of (3) the cone decompositions may be chosen so that the degeneracy maps π 1 and π 2 extend to morphisms Y 0 (P) tor → Y tor , and the universal isogeny ψ : A 1 → A 2 over Y 0 (P) extends to an isogeny of semi-abelian schemes over Y 0 (P) tor whose completion along Z c has the form (3.1).
Just as for Y − (see §2.3), one can obtain a toroidal compactification Y 0 (P) tor of Y 0 (P) as a quotient of Y 0 (P) tor .This applies in particular to Y = Y 0 (O F ); furthermore one can relax the restriction that U = U (N ).
3.4.Minimal compactification of Y 0 (P).Let Y 0 (P) ord (resp.Y 0 (P) tord ) denote the ordinary locus in Y 0 (P) (resp.Y 0 (P) tor ), and define where f : Y 0 (P) tord → Y minor is the restriction of the composite of the extension of π 1 with the projection Y tor → Y min .Since the restriction of π 1 to Y 0 (P) ord → Y ord is finite, we may identify Y 0 (P) ord with an open subscheme of Y 0 (P) minor and define Y 0 (P) min by gluing Y 0 (P) minor to Y 0 (P) along Y 0 (P) ord .Just as for Y (i.e., the case P = O F ), we see that Y 0 (P) min is normal and independent of choice of cone decompositions, its connected components are flat and projective over O, and we have a commutative diagram over the corresponding one for Y , the vertical (resp.diagonal) morphism(s) being projective (resp.open immersions).
We can again identify our minimal compactification Y 0 (P) min with an infinite disjoint union of the schemes denoted M spl,min (1) Y 0 (P) min is normal, flat and projective over O, and independent of the choice of cone decompositions in its construction, and there is a canonical isomorphism between the reduced complement of Y 0 (P) in Y 0 (P) min and c∈C0(P) Z c , where each Z c is isomorphic to Spec (O); (2) the completion of Y 0 (P) min along Z c is isomorphic to Spf (P c ), where (3) the morphisms π i extend to Y 0 (P) min → Y min , with the restriction to the complement of Y 0 (P) being π ∞ i and the completion at Z c being the inclusion of local rings obtained by replacing M by The assertions in the theorem all follow from analogous ones with Y 0 (P) min replaced by Y 0 (P) min .The first two parts can then be proved by minor modifications of the arguments in [Cha90, §8] (see also [Dim04,§4]) or seen as a particular case of [Lan18,Thm. 4 for each Φ H = 0).We note that, as in the discussion following [Dia21a,(18)], the isomorphism in (2) depends on a choice of splittings of the exact sequences 0 → I i → H i → J i → 0 for i = 1, 2, which we take to be compatible with α.Part (3) is then immediate from the construction of Y 0 (P) min and part (3) of Theorem 3.3.1.
More generally, for any sufficiently small level U and N prime to p such that U ′ = U (N ) ⊂ U , the action of U/U ′ on Y ′ 0 (P) extends to Y ′ 0 (P) min , and we define Y 0 (P) min to be the quotient.The resulting scheme is then independent of the choice of N (up to changing the base O), and Theorem 3.5.1 holds exactly as stated above, except that the general description of the completed local ring P c is slightly more complicated (see Proposition 3.7.1 below).More precisely, it is given by the expression in the discussion preceding [Dia21a, Prop.7.2.1], but with C ∈ C ′ 0 (P) lying over c ∈ C 0 (P) and J −1 I replaced by J −1 2 I 1 in the definition 6 of the group Γ C in [Dia21a,(24)].The resulting group Γ C,U appearing in the expression is thus isomorphic to Aut (H 1 , H 2 , α), with the isomorphism depending on the choice of (compatible) splittings, and hence to B(F ) + ∩gU 0 (P)g −1 if [H 1 , H 2 , α] corresponds to the double coset B(F ) + gU 0 (P).
3.6.Hecke action.Suppose that g ∈ GL 2 (A (p) F,f ), U and U ′ are any sufficiently small (prime-to-p) levels such that U ′ ⊂ gU g −1 .We then have the morphism ) is the universal triple over Y ′ 0 (P), then the abelian schemes A i (for i = 1, 2) are characterized by the existence of a prime-to-p quasi-isogeny ψ i : 6 Note that the roles of U and U ′ are reversed here with respect to [Dia21a], and that Γ C ⊂ Aut O F (J 2 ×I 1 ) may be identified with the intersection of the groups similarly defined for the cusps ) for all geometric points s of Y ′ 0 (P), with the rest of the data defining (A 1 , A 2 , α) determined by the obvious compatibilities between the pair (ψ 1 , ψ 2 ) and the triple (A ′ 1 , A ′ 2 , α ′ ).The resulting morphism ρ g thus descends to a morphism ρ g : Y ′ 0 (P) → Y 0 (P) giving rise to the map on complex points Similarly (but more simply), there is a map , where g denotes right-multiplication by g −1 and the rest of the data is determined by the obvious compatibilities.Again this descends to the map ρ ∞ g : C ′ 0 (P) → C 0 (P) induced by right multiplication by g on double cosets and satisfying the usual compatibility relation for varying g, U and U ′ .
We claim that ρ g extends via ρ ∞ g to a morphism Y ′ 0 (P) min → Y 0 (P) min whose completion at the cusps has a simple description.In order to make this precise, ) (with notation as in Theorem 3.5.1(3))and the matrix acting on (J i × I i ) (p) by right-multiplication), and the compatibilities with α and α ′ imply that ǫ 1 = ǫ • α and ǫ 2 = α • ǫ for some ǫ ∈ M * (p) = Hom OF (J 2 , I 1,(p) ).
Lemma 3.6.1.With notation as above, the morphism ρ g : Y ′ 0 (P) → Y 0 (P) extends uniquely to a morphism Y ′ 0 (P) min → Y 0 (P) min .Its restriction to the complement of Y ′ 0 (P) corresponds under Theorem 3.5.1(1) to the morphism defined by ρ ∞ g , and the resulting morphisms on completions correspond under Theorem 3.5.1(2) to the ones defined by Before discussing the proof, we note that the hypotheses on N and N ′ imply that End OF (J 2 × I 1 ) (as matrices acting via right multiplication), which implies that N ′ Hom OF (J ′ 2 , I ′ 1 ) ⊂ N Hom OF (J 2 , I 1 ) and N ′ ǫ ∈ N Hom OF (J 2 , I 1 ).It follows that N ′ M ⊂ N M ′ and N ′ ǫ ∈ N M * , so the formula in the statement does indeed define a map on the power series rings appearing in the statement of Theorem 3.5.1.Furthermore we have 1 −ǫ 0 1 ⊂ Γ C,U as subgroups of Aut OF (J 2 × I 1 ) acting via right multiplication, so the formula defines a map on the completions as described for arbitrary U and U ′ .(Note also that taking g = 1 in the proposition describes how the choice of splittings affects our uniformizations of the completions.)To prove the lemma, we may assume that U ′ = U (N ′ ) and U = U (N ).The assertions then follow from analogous ones for toroidal compactifications, namely that the cone decompositions in their construction may be chosen so that the morphism ρ g of Y 0 (P) ′ extends to Y 0 (P) ′tor → Y 0 (P) tor with the desired effect on (completions at) components of the complement.To that end, we may assume that if c = ρ ∞ g (c ′ ), then the cone decomposition of Hom (M ′ , R) ≥0 = Hom (M, R) ≥0 chosen for c ′ refines the one for c.The assertions then reduce (by [Lan18, Thm.3.4.1(4)]) to the claim that if V is a complete DVR with fraction field L and (with auxiliary data given by c ′ and the splittings σ ′ i ), then ρ g • s ′ = s where s is similarly defined by (3.1) with q : q ′m (and auxiliary data given by c and the σ i ).Finally the claim itself is straightforward to check in view of the commutativity of the diagram , where T ′ i is the prime-to-p adelic Tate module of (d −1 I ′ i ⊗ L × )/ q d −1 J ′ i , the top right horizontal isomorphism is defined by (x, y) → q ′x/N ′′ (ζ N ′′ ⊗ y) for (x, y) ∈ (J ′ i × I ′ i ) ⊗ Z/N ′′ Z and p ∤ N ′′ , T i and the bottom right isomorphism are defined similarly, and the right vertical isomorphism is induced by the canonical quasi-isogeny (d 3.7.The Koecher Principle.Consider now the line bundle A k, m,R on Y R , where denote its pull-back to Y 0 (P) R via π i (omitting the subscript R from the morphisms when clear from the context).We explain how the Koecher Principle applies to describe j * A tor is then formally canonical, so we may apply [Lan18, Thm.4.4.10] to identify j * A under the projection Y 0 (P) tor → Y 0 (P) min .In particular j * A is coherent and its completion along the cusp corresponding to (H i , q i ) is described as in [Dia21a,(22)], taking M = d −1 I −1 1 J 2 and using I i (resp.J i ) in place of I (resp.J) in the definition of the free rank one O-module D k, m in [Dia21a,(20)].
The analogous statements then follow for j * A and any sufficiently small U , in the sense that j * A (i) k, m,R is coherent and that its completion along each cusp is described as in [Dia21a, Prop.7.2.1], with the evident modifications.Furthermore, we may describe the effect on q-expansions of the composite morphisms in the setting of Proposition 3.6.1,where the latter map is induced by the quasiisogeny ψ i : (in the notation of [Dia21a, (2)]), we have the following: where b is a basis for ) is as in Theorem 3.5.1(2)(or more precisely its variant for more general U and R, so in particular Γ C,U ∼ = Aut (H 1 , H 2 , α)); (3) for U ′ as above and g ∈ GL 2 (A (p) F,f ) such that U ′ ⊂ gU g −1 , the morphism (3.2) corresponds under the isomorphisms in (2) to the map defined by , and N ′ and ǫ are as in the statement of Lemma 3.6.1.

Level U 1 (P)
We now consider a more refined level structure at primes over p; more precisely, we let We follow [DS22] to construct toroidal and minimal compactifications of p-integral models of the resulting Hilbert modular varieties, defined using a moduli problem based on [Pap95], thus diverging from those obtained by the methods of [Lan18].
We assume throughout this section that U is P-neat in the sense that Note that this holds for example if U = U 1 (n) for sufficiently small n.

4.1.
The model for Y 1 (P).Maintaining the notation from the preceding sections, we let ψ : A 1 → A 2 denote the universal isogeny over Y 0 (P), and write G = p|P G p where G = dP ⊗ OF ker(ψ) and each G p is a Raynaud (O F /p)-vector space scheme.Consider also the dual isogeny ψ ∨ : A ∨ 2 → A ∨ 1 over Y 0 (P), and identify its kernel with dP ⊗ OF G ∨ , where G ∨ = p|P G ∨ p is the Cartier dual of G.The scheme Y 1 (P) is then a certain closed subscheme of G, finite and flat over Y 0 (P) of degree #(O F /P) × (see [Dia21b, §5.1]).As usual, we let Y 1 (P) denote its quotient by the action of O × F,(p),+ , so that Y 1 (P) is a model for the Hilbert modular variety of level U 1 (P) and the morphism Y 1 (P) → Y 0 (P) corresponds to the usual projection on complex points.
The neatness hypothesis implies that Y 1 (P) → Y 0 (P) is again finite and flat, and hence that Y 1 (P) is Cohen-Macaulay over O. However our model for Y 1 (P) is not normal (unless P = O F ), and does not preserve the symmetry between the projections π 1 and π 2 .In particular the automorphism w P of Y 0 (P) does not lift to an automorphism of Y 1 (P), as follows for example from the fact that if the fibre over s ∈ Y 0 (P)(F p ) is étale, then that of w P (s) is connected.4.2.Cusps and clasps.Just as for level U 0 (P), we define the set of cusps of level U 1 (P) to be We thus have a bijection between C 1 (P) and the set of isomorphism classes of pairs (H, Ψ), where H corresponds to an element of C = C U and Ψ ∈ H/PH is such that Ann OF Ψ = P. (The bijection is induced by g → (H, Ψ), where H = O 2 F g −1 ∩ F 2 , I = H ∩ (0 × F ), λ = det, η is induced by right-multiplication by g −1 and Ψ is the image of (0, 1) • g −1 .) As before, the set C 1 (P) will parametrize the complement of Y 1 (P) K in its minimal compactification; now however collections of cusps coalesce in connected components of the complement on the integral model.With this in mind, we define a clasp of level U 1 (P) to be an isomorphism class of pairs (H, Ξ), where H is as in the description of C and Ξ ∈ J/PJ (with J = H/I and the obvious notion of isomorphism).We let C 1/2 (P) denote the set of clasps of level U 1 (P), so we have the natural degeneracy maps where Ξ is the image of Ψ, q = Ann OF Ξ, and as usual [•] denotes the associated isomorphism class.

Toroidal compactification.
Once again we initially assume U = U (N ) for some integer N (prime to p).Let Y 0 (P) tor be a toroidal compactification of Y 0 (P), and write ψ tor : A tor 1 → A tor 2 for the extension of the universal isogeny ψ.
We then have that G p extends uniquely over Y 0 (P) tor 1 to a finite flat subgroup scheme of dP ⊗ OF ker(ψ tor ), its completion along Z 2 being the subgroup corresponding to the image of Similarly dP⊗ OF G ∨ p extends uniquely over Y 0 (P) tor 2 to a finite flat subgroup scheme of the kernel of the extension of ψ ∨ , its completion along Z 1 being the subgroup corresponding to the image of (pJ 2 .Taking Cartier duals, it follows that G p extends to a finite flat group scheme over Y 0 (P) tor 2 as well, and gluing to its extension over Y 0 (P) tor 1 thus yields an (O F /p)-vector space scheme G tor p over Y 0 (P) tor .We then let G tor = p|P G tor p , and define Y 1 (P) tor as the extension of Y 1 (P) to a closed subscheme of G tor , finite and flat over Y 0 (P) tor .4.4.Minimal compactification.We define the minimal compactification of Y 1 (P) as where f is the composite Y 1 (P) tor → Y 0 (P) tor → Y 0 (p) min .As usual Y 1 (P) min is independent of the choice of cone decomposition in its definition, the action of O × F,(p),+ on Y 1 (P) extends to it uniquely, and we let Y 1 (P) min denote the quotient by this action.More generally, for any sufficiently small level U (P-neat and prime to p) we define Y 1 (P) min as the quotient of Y ′ 1 (P) min by the unique extension of the action of U/U ′ on Y ′ 1 (P), where Y ′ 1 (p) is defined using U ′ = U (N ) ⊂ U for suitable N (of which the resulting scheme Y 1 (P) min is independent).Lemma 4.4.1.For sufficiently small U , the morphism h : Y 1 (P) min → Y 0 (P) min is finite and flat.
Proof.Since the morphism is finite and its restriction to Y 1 (P) is flat, it suffices to prove that its completion at each cusp of Y 0 (P) min is flat.To that end, fix a cusp c ∈ C 0 (P), corresponding to the isomorphism class of (H 1 , H 2 , α), and write P = q 1 q 2 as above, so p|q 1 (resp.p|q 2 ) if and only if α(J 1 ) ⊂ pJ 2 (resp.α(I 1 ) ⊂ pI 2 ).Suppose for example that for some n (prime to p) such that if µ ∈ O × F and µ ≡ 1 mod n, then µ ≡ 1 mod P. Note in particular that U is P-neat, and hence so is U ′ := U (N ) for any N such that U (N ) ⊂ U .For i = 0, 1, let Y i denote the completion of Y ′ i (P) tor along the preimage of c, where c ∈ C ′ 0 (P) is any cusp lying over c.The construction of Y ′ 1 (P) tor and the condition on N then give an isomorphism (4.2) where T = p|P T p and T p is the finite flat O-algebra representing generators of the étale (resp.multiplicative) (O F /p)-vector space scheme if p|q 1 (resp.p|q 2 ).It then follows from the Theorem on Formal Functions that (4.2) holds with "tor" replaced by "min" in the definition of Y i for i = 0, 1. Furthermore the condition on U implies that if where the actions are by right multiplication), then α ≡ δ ≡ 1 mod P. The isomorphism (4.2) is therefore compatible with the action of the stabilizer of c in O × F,(p),+ × (U/U ′ ), so it holds with Y ′ i (P) min replaced by its quotient Y i (P) min for i = 0, 1. 4.5.Completions at clasps.The proof of 4.4.1 also yields a description of the completion of Y 1 (P) min along the complement of Y 1 (P).Maintaining the notation there, note that the hypothesis on U ensures that every automorphism of (H 1 , H 2 , α) fixes H 2 /α(H 1 ) pointwise.The proof of the lemma shows that the connected components of the fibre in Y 1 (P) min over c ∈ C 0 (P) correspond to generators of J 1 /q 1 J 1 , hence to isomorphism classes of pairs (H 1 , Ξ) such that Ann OF (Ξ) = q 1 , i.e., elements of C 1/2 (P) lying over c.We thus obtain a bijection between C 1/2 (P) and the set of connected components of the complement of Y 1 (P) in Y 1 (P) min . In

particular for each cusp [H] = [H, I, [λ], [η]
] ∈ Y min (for such U ), the connected components of its preimage in Y 1 (P) min under the composite ρ = π 1 • h : Y 1 (P) min −→ Y 0 (P) min −→Y min are in bijection with J/PJ.Furthermore the completion of Y 1 (P) min along the component corresponding to an element Ξ ∈ J/PJ has the same description as the completion of Y min at [H], but with M = d −1 I −1 J replaced by q −1 M and O replaced by the finite flat local O-algebra T r representing generators of µ p ⊗ (I/rI), where q = Ann OF Ξ and P = qr.For example if U = U (n) for some n as in the proof of Lemma 4.4.1, then the completion is isomorphic7 to Spf (P ) where We note also that the action of U 0 (P)/U 1 (P) ∼ = (O F /P) × on Y 1 (P) extends to Y 1 (P) min (with the obvious description on the completion at the fibre over a cusp of Y min ), and we may identify Y 0 (P) min with the quotient.For completeness, we consider more general U below, but first we record the following immediate consequence of Lemma 4.4.1:Suppose now only that U is P-neat and sufficiently small the usual sense (as in [Dia21a, §2.2]).Since Y 1 (P) min is the quotient of Y ′ 1 (P) min by the action of U/U ′ (where U ′ = U (N )), we see that the action of U 0 (P)/U 1 (P) ∼ = (O F /P) × on Y 1 (P) extends to Y 1 (P) min , with Y 0 (P) min as the quotient.Furthermore taking U -invariants yields the following description of the completion of Y 1 (P) min along the complement of Y 1 (P) (as in [Dia21a, §7.2]): (1) There is a bijection between C 1/2 (P) (resp.C 1 (P)) and the set of connected components of the complement of Y 1 (P) in Y 1 (P) min (resp.Y 1 (P) K in Y 1 (P) min K ), under which (4.1) is compatible (in the obvious sense) with the morphisms (2) Let [H, Ξ] ∈ C 1/2 (P), q = Ann OF (Ξ), r = q −1 P, and The extension is compatible in the obvious sense with the map C ′ 1/2 (P) → C 1/2 (P) induced by right multiplication by g, and the resulting map on completions is given by the same formula as in Lemma 3.6.1,where now t m ∈ T ′ r = T r via the identification I ′ ⊗µ p = I⊗µ p .We remark that the assumption that ζ p ∈ O is only made in order to incorporate the assertions about Y 1 (P) K and C 1 (P) into the statement.4.6.q-expansions.Suppose as usual that k, m ∈ Z Θ and R is a Noetherian Oalgebra such that χ k+2 m,R = 1 on O × F ∩ U , and consider the line bundle on Y 1 (P) R (writing ρ and h also for their restrictions to Y 1 (P) R and omitting the subscripts R).We claim that j ♭ * A ♭ k, m,R is coherent and describe its completion along the complement of the image of j ♭ : Y 1 (P) R ֒→ Y 1 (P) min R .Assume first that U is of the form U (N ) and let h tor denote the finite flat morphism Y 1 (P) tor → Y 0 (P) tor .The isomorphism (4.2) then shows that the vector bundle h tor * O Y1(P) tor is formally canonical, and hence so is . We may therefore apply the Koecher Principle (again in the form of [Lan18,Thm. 4.4.10]) to conclude that under Y 1 (P) tor R → Y 1 (P) min R , hence is coherent.Taking quotients by group actions then yields the following for any sufficiently small (in particular P-neat) U : where b is any basis for D k, m (in the notation of [Dia21a, (20)]) and the rest of the notation is as in Propositon 4.5.2,except that now t m ∈ T r ⊗ O R in the preceding expression and the definition of F,f ) and U ′ ⊂ gU g −1 , then the maps induced by the morphisms j ′♭ * A ′♭ k, m,R → j ♭ * A ♭ k, m,R on completions are given by the same formula as in Proposition 3.7.1(3).

Kodaira-Spencer and cohomological vanishing
We now proceed to explain how the main results of [Dia21b] extend to toroidal compactifications.We assume U = U (N ), and choose polyhedral cone decompositions for cusps in C so that the resulting toroidal compactification Y tor is smooth over O.We let Z denote the reduced complement of Y in Y tor .5.1.The Kodaira-Spencer isomorphism.Recall from [Dia21a, §7.3] that the Kodaira-Spencer filtration on Ω 1 Y /O extends to one on Ω 1 Y /O (log Z), with graded pieces canonically isomorphic to A 2e θ ,−e θ (omitting the subscript R when R = O).
It follows that , where as usual K Y tor /O denotes the dualizing sheaf of Y tor over O. Furthermore the extension is obtained from isomorphisms (5.1) where S is the formal scheme whose quotient by (O × F ∩ U ) 2 defines the completion of Y tor along the connected component of Z corresponding to a cusp (H, I, λ, [η]), ξ : S → Y tor is the composite of the quotient map with the completion, and the second isomorphism in (5.1) is given by the canonical trivializations Turning now to level U 0 (P), recall from [Dia21b, Thm.3.2.1]that the Kodaira-Spencer isomorphism on Y 0 (P) takes the form (5.2) To extend this to toroidal compactifications, let us choose the polyhedral cone decompositions for cusps in C 0 (P) so that the morphisms π i (for i = 1, 2) extend to Y 0 (p) tor → Y tor (which we still denote π i ), and furthermore Y 0 (P) tord is smooth, so that Y 0 (P) tor is a local complete intersection over O.
Proof.The same argument that establishes (5.1) gives an isomorphism where S 0 (P) is the formal scheme whose quotient defines the completion along a connected component corresponding to a cusp of the form (H 1 , H 2 , α), and ξ 0 : S 0 (P) → Y 0 (P) tor is the resulting morphism.Furthermore the isomorphisms are compatible with those of (5.1) under π 1 in the sense that the diagram commutes, where the top line is the pull-back of (5.1) via the morphism ζ : S 0 (P) → S over π 1 , the leftmost arrow is induced by the morphism π * 1 K Y tor /O → K Y0(P) tor /O , the middle by the morphism J 1 → J 2 defined by α, and the last by the morphism π * 1 ( δ −1 ω) → π * 2 ( δ −1 ω) induced by the extension of the universal isogeny to the semi-abelian schemes over Y 0 (P) tor .The theorem then follows from the compatibility under π 1 between the Kodaira-Spencer isomorphisms on Y 0 (P) and Y (see [Dia21b, Prop.3.2.3]), the completion of the desired isomorphism along each connected component of Z 0 (P) being the one described at the start of the proof (or more precisely, its quotient by (O × F ∩ U ) 2 ).5.2.Cohomological vanishing.Now we explain how the cohomological vanishing results (and consequences) of [Dia21b, §5.3] extend to toroidal compactifications.Recall from [Dia21b, Cor.5.3.2,5.3.5] that the coherent sheaves vanish for i > 0 and are locally free if i = 0, so the same holds with Y 0 (P) (resp.π 1 ) replaced by Y 0 (P) (resp.π 1 ).We claim that the same holds for Y 0 (P) replaced by Y 0 (P) tor (and π 1 by its extension).
To that end, first note that by Grothendieck-Serre duality, it suffices to prove the assertions for O Y0(P) tor (see for example the proof of [Dia21b, Cor.5.3.4]).Furthermore in view of the result for the restriction to Y 0 (P), it suffices to prove the assertions after localization at every point of Z, and hence by the Theorem on Formal Functions (and faithful flatness of completion), we are reduced to proving the analogous assertions for the morphisms ζ : S 0 (P) → S. Applying the Theorem on Formal Functions again, we may replace ζ by the associated morphisms of toric varieties f : , and Σ is its refinement in the chosen decomposition of (M * ⊗ R) ≥0 .To prove the assertions for f , write it as the composite R (as usual omitting the subscripts for base-changes of morphisms).Finally we describe the effect of the saving trace on completions along Z.To that end, let [ H 1 ] be a cusp in C and ξ : S → Y tor the formal scheme as in (eqn:canonical), so that are in bijection with factorizations P = q 1 q 2 , where we define q 1 and q 2 by α(I 1 ) = q 2 I 2 and α(J 1 ) = q 1 J 2 , Thus ξ * ( π 1, * π * 2 ω) is comprised of 2 t summands of the form , where ξ 0 : S 0 (p) → Y 0 (p) tor and ζ : S 0 (p) → S are as above and t = #{p|P}.It follows from [Dia21b, Prop.3.2.2]that the saving trace pulls back via ξ to the sum of the morphisms corresponding under the above isomorphisms to the maps β ⊗ Nm(q 1 ) −1 tr : where tr is the trace relative to the finite flat morphism ζ * O S0(P) → O S and β = Nm(q 2 ) −1 ∧ d α * (writing α * : I −1 2 → I −1 1 for the map induced by α).Note that β is an isomorphism and the second factor may be described locally on S as Nm(q 1 ) −1 times the completion of the trace of the finite flat morphism h : T M,σ → T M1,σ .In particular on global sections the resulting map is given by (writing α also for the induced map N −1 M 1 → N −1 M ).Finally it follows that the base-change st R is given by the same formula with O replaced by R. We conclude: Proposition 5.3.1.The pull-back to Y R of the saving trace (i.e., [Dia21b,(51)]) extends to a morphism st R : whose completion along the complement of Y R is the morphism whose effect on global sections is induced by the maps

Hecke operators on q-expansions
Recall that in [Dia21a], we computed the effect on q-expansions of all the weightshifting operators defined there, namely partial Θ and Frobenius operators. 8We do the same here for Hecke operators.
For Hecke operators associated to primes not dividing p, it is well-known that the same computation as for classical Hilbert modular forms, i.e. the case R = C, carries over to arbitrary bases.This is explained in [DS22] in the case that p is unramified in F , and we do the same here in general for completeness.
Hecke operators at primes dividing p are also considered in [DS22], but in addition to the assumption that p be unramified in F , constraints are placed on the weight to allow for a simpler more ad hoc definition of the operator T p .A more general, indeed optimal, construction of T p is given in [Dia21b] (see also [FP21] for p unramified in F and [ERX17] for work in this direction allowing ramfication at p).The effect of T p on q-expansions is also considered in [DW20] and [DM20], premised on partial results in [ERX17]; we complete the analysis here using the operators defined in [Dia21b].
8 and for completeness, the simpler and well-known effect of multiplication by partial Hasse invariants 6.1.Hilbert modular forms.For k, m ∈ Z Θ , a Noetherian O-algebra R and (sufficiently small) U such that χ k+2 m,R = 1 on O × F ∩ U , we define the space of Hilbert modular forms of weight ( k, m) and level U over R to be ).Note that the hypotheses imply that either • p n R = 0 for some n > 0, or • k θ + 2m θ is independent of θ.We assume henceforth that one of these holds.Note that in either case M k, m (U ; R) is defined for all sufficiently small U (containing GL 2 (O F,p )), so we may define where the limit is taken over all such U with respect to the (injective) morphisms Recall that for g ∈ GL 2 (A (p) F,f ) and U ′ ⊂ gU g −1 (where U and U ′ are sufficiently small), we have the morphisms M k, m (U ; R) → M k, m (U ′ ; R) defined as det(g) times the composite ), the transition maps in the definition of M k, m,R being the special case where g = 1.
These satisfy the usual compatibilities and hence define an action of GL 2 (A U ; R) for any normal subgroup U ′ of U (whenever both have already been defined), we may extend the definition to arbitrary open compact R ′ ; moreover this is an isomorphism whenever R ′ is flat over R, or indeed if both are flat over O.
We remark also that obvious variants of the above construction yield analogous assertions with M k, m (U ; R) replaced by M ), and we denote the resulting limits M 0 (P) induced by the morphisms π i , and a natural GL 2 (A (p) F,f )-equivariant action of (O/P) × on M 1 (P) ♭ k, m,R under which the invariants may be identified with M 0 (P) where the completion is along Z c,R and the inclusion depends on a choice of splitting H ∼ −→ J × I.The notation here is as in §3.7, so in particular M = d −1 I −1 J and D k, m = (I −1 ) ⊗k θ θ ⊗ (d(IJ) −1 ) ⊗m θ θ depend on c, and N is such that U (N ) ⊂ U and ζ N ∈ O.The above definition of q c assumes a priori that U is sufficiently small, but we may drop this assumption by letting q c (f ) = q c ′ (f ) for any c ′ ∈ C U(N ) in the preimage of c under the natural projection.The resulting q-expansion is independent of the choice of such a c ′ (as a special case of Proposition 3.7.1(3));furthermore the definition of q c is independent (in the obvious sense) of the choice of N such that U (N ) ⊂ U .
More generally for any S ⊂ C, we define as the direct product of the maps q c .We then have the following q-expansion Principle, which can be proved exactly as in [Rap78, Thm.6.7] if U is sufficiently small; the assumption can then removed by applying the result to sufficiently small U ′ normal in U and taking invariants under U/U ′ .Proposition 6.2.1.Suppose that S meets every connected component of Y min , or equivalently, the restriction to S of the map More generally for any cusp c ∈ C 0 (P), we may define q-expansion maps for i = 1, 2, where now M = d −1 I −1 1 J 2 , and more generally q (i) S for any S ⊂ C 0 (P).However since irreducible components of Y 0 (P) min R need not contain cusps, the analogue of Proposition 6.2.1 fails.
Recall also from Proposition 4.5.2(1) that connected components of the complement of Y 1 (P) in Y 1 (P) min correspond to clasps cℓ = [H, Ξ] ∈ C 1/2 (P), and by Proposition 4.6.1,we have q-expansion maps , where M = d −1 I −1 J, q = Ann OF (Ξ), r = q −1 P and T r is the finite flat O-algebra representing generators of µ p ⊗ I/rI.
(U 0 (P); R) have the obvious effect on qexpansions.More precisely q ) and the inclusion of power series rings obtained from the maps I 1 ֒→ I i and J i ֒→ J 2 (one of which is the identity and the other is induced by α; see Theorem 3.5.1(3)).
We may similarly describe the effect of M where H 1 = H and q −1 J ∼ −→ J 2 (via α).We may thus identify D (i) k, m (resp.M ) for c with D k, m (resp.q −1 M ) for cℓ, and q cℓ (h * (f )) is obtained from q c (f ) by tensoring with the structure map O → T r .
The effect on q-expansions of the action of d ∈ (O/P) × is given simply by denote the R-linear (evaluation) map sending f to the constant coefficient of q c (f ), and we similarly define for any S ⊂ C. We define the space of cuspidal Hilbert modular forms of weight ( k, m) and level U over R to be S k, m (U ; R) := ker(e C ).Thus f ∈ M k, m (U ; R) is cuspidal if e c (f ) = 0 for all c ∈ C, in which case we also refer to f as a cusp form (of weight ( k, m) and level U over R).
Note that e c is independent of the choice of splittings in the definition of q c , and that it takes values in In particular, if R is flat over O, then a = 0 for all cusps c ∈ C, and hence S k, m (U ; R) = M k, m (U ; R), unless k and m are parallel in the sense that the pair (k θ , m θ ) is independent of θ.On the other hand if (k θ , m θ ) is independent of θ, or if p n R = 0 for some n > 0, then for sufficiently small U , we have a = R for all c ∈ C. Note that f ∈ M k, m (U ; R) is cuspidal if and only if its image in M k, m (U ′ ; R) for some (hence all) U ′ ⊂ U .Furthermore Proposition 3.7.1(3)(in the case P = O F ) implies that the subspace for which assertions analogous to those for S k, m,R hold.Furthermore S k, m,R is the preimage of S 0 (P) under the injection M k, m,R ֒→ M 0 (P) , and the action of (O/P) × on M 1 (P) ♭ k, m,R restricts to one on S 1 (P) ♭ k, m,R with invariants S 0 (P) • η is defined by η(x, y) = (t −1 x, y).Note that Proposition 6.2.1 holds with S as the set of cusps at ∞.
For simplicity, we assume throughout this section U = U 1 (n) or U (n) for some n prime to p.We let V n = ker(( O (p) Note that in the case of U 1 (n), we have c t = c t ′ if and only if t and t ′ define the same strict ideal class of F .Similarly in the case of U (n), the cusps at ∞ are indexed by the strict ray class group of conductor n.Note that in either case, there is a unique cusp at ∞ on each connected component of Y min .
Recall that R is a Noetherian O-algebra and k, m ∈ Z Θ are such that either p n R = 0 for some n > 0, or k θ + 2m θ is independent of θ.If U = U 1 (n) and f ∈ M k, m (U ; R), then Proposition 3.7.1(2)implies (for sufficiently small n, and hence for any n) that the q-expansion 9 q ct (f ) takes values in the P t -module  are induced (in the latter case thanks to (6.2)) by the inclusions I −1 2 ֒→ I −1 1 and J −1 2 ֒→ J −1 1 , and the last morphism by the saving trace.The desired conclusion is therefore immediate from Proposition 5.3.1.6.7.The operator S p .As in the case of primes v ∤ p, the expression for its effect on q-expansions will involve the operator S p , which may also be defined on M k, m (n; R) for arbitrary R under hypotheses on ( k, m).More precisely, consider the automorphism ρ of Y defined in terms of the universal object (A, ι, λ, η, F • ) by the data (A ′ , ι ′ , λ ′ , η ′ , F ′,• ), where • A ′ = A ⊗ OF p −1 ; • ι ′ is the O F -action compatible with the isogeny σ : each cusp in C yields an open immersion Y − ֒→ Y tor − , where Y tor − is an infinite disjoint union of flat projective schemes 2 over O. Let us assume furthermore that O contains the N th roots of unity.The connected components of the complement of Y − in Y tor − are then in canonical bijection with C, and the completion of Y tor − along the component corresponding to [H, I, λ, [η]] is described explicitly as the quotient by a free action of (O × F ∩ U ) 2 on a certain locally Noetherian formal scheme S, namely the completion of the complement of Spec (O[N −1 M ]) in the torus embedding defined by the chosen (O × F ∩U ) 2 -invariant cone decomposition of Hom (M, R) ≥0 , where M = d −1 I −1 J.
where the diagonal morphisms are open immersions, and the vertical morphism is projective.Furthermore the reduced complement of Y − in Y min − is étale over O with geometric connected components indexed by C, and the Koecher Principle (in the form of [Lan17, Thm.8.7]) applies to give an explicit description of the completion along this complement, as in [Dia21a, (23)].2.6.Minimal compactification of Y − .The action of O × F,(p),+ /(U ∩ O × F ) 2 on Y − extends uniquely to an action on Y min − , and we define Y min − to be the quotient.Furthermore for any sufficiently small level U prime to p, we may choose N ≥ 3 (prime to p) so that U ′ := U (N ) ⊂ U and define Y min − to be the quotient of Y ′ min − by the (unique extension of the) natural action of U , where Y ′ min − is defined as above.The resulting scheme Y min − is then normal, flat and projective over O, and independent of the choice of N .Furthermore the reduced complement of the image of the open immersion Y − ֒→ Y min − is étale over O with geometric connected For k, m ∈ Z Θ and R a Noetherian O-algebra, let A k, m,R be the associated line bundle on Y R (see [Dia21a, §3.2]).The extension of the (semi-)abelian scheme A to A tor over Y tor yields an extension of A k, m,R to Y R which is formally canonical in the sense of [Lan17, Def.8.5] (see [Dia21a, (20)]).Therefore the Koecher Principle ([Lan18, Thm.4.4.10])applies to show that j * A k, m,R is coherent, where j denotes the open immersion Y R ֒→ Y min R , and that its completion along the complement of Y R is described as in [Dia21a, (22)].2.10.Minimal compactification of Y .For U = U (N ), we define Y min to be the quotient of Y min by the (unique extension of the) action of O × F,(p),+ /(U ∩ O × F ) 2 .More generally for any sufficiently small level U prime to p, we let Y min to be the quotient of Y ′ min by the (unique extension of the) action of U , where Y ′ min is defined using U ′ = U (N ) for suitable N .Just as for Y − , the resulting scheme Y min is normal, flat and projective over O, and independent of the choice of N .Furthermore the reduced complement of the image of the open immersion Y ֒→ Y min , and the completion along it, are the same as for Y − ֒→ Y min − .Recall that if χ k+2 m,R = 1 on O × F ∩ U , then the line bundle A k, m,R on Y R descends to one on Y R , which we denote A k, m,R .Letting i denote the open immersion Y R ֒→ Y min R , it follows from the analogous statements over Y ′ min R that i * A k, m,R is coherent and that its completion along the complement of the image of i is given by [Dia21a, Prop.7.2.1].

p
of the form a ⊕ d −1 qa for an invertible O F,p -submodule a of F p and an ideal q of O F containing P. (See [Dia21b, §2.4], especially (3) for the condition in [Lan18, Def.2.3.3(4)].)Just as for Y , it follows from [Lan18, Cor.2.4.10] that Y 0 (P) is an infinite disjoint union of schemes of the form M spl H,O where g is the projective morphism M spl,tor H,Σ,O → M spl,min H,O , and the morphism M spl,min H,O → Y min is finite over the ordinary locus, so we may identify Y 0 (P) minor with the ordinary locus of the infinite disjoint union of the M spl,min H,O .3.5.Minimal compactification of Y 0 (P).The action of O × F,(p),+ /(U ∩ O × F ) on Y 0 (P) again extends to Y 0 (P) min , and we define Y 0 (P) to be the quotient.Theorem 3.5.1.There is a normal scheme Y 0 (P) min over O and an open immersion Y 0 (P) ֒→ Y 0 (P) min with the following properties: ,R , where j denotes the open immersion Y 0 (P) R ֒→ Y 0 (P) min R .First assume U = U (N ) and consider the line bundle

Corollary 4.5. 1 .
For sufficiently small U , the morphism ρ = π 1 • h is flat in a neighborhood of each component of the complement of Y 1 (P) in Y 1 (P) min corresponding to a clasp of the form [H, 0].Similarly the composite π 2 • h is flat, and in fact étale, in a neighborhood of each component corresponding to a clasp [H, Ξ] such that O F • Ξ = J/PJ.

SS
k, m,R := lim −→U S k, m (U ; R) ⊂ M k, m,R is stable under the action of GL 2 (A (p) F,f ), and that S k, m (U ; R) = S U p k, m,R for all U = U p GL 2 (O F,p ).Note also that the definition of S k, m,R is functorial in R, and that S k, m,R ′ = S k, m,R ⊗ R R ′ if R ′ is flat over R (or both are flat over O).More generally for c ∈ C 0 (P), we have the evaluation maps e for S ⊂ C 0 (P), and letS (i) k, m(U 0 (P); R) = ker(e (i) C0(P) ).For cℓ ∈ C 1/2 (P) the evaluation maps take the forme cℓ : M ♭ k, m (U 1 (P); R) −→ D k, m ⊗ O T r ⊗ O Rand e S for S ⊂ C 1/2 (P), and we let S ♭ k, m (U 1 (P); R) = ker(e C 1/2 (P) ).The same considerations as above yield GL 2 (A (p) F,f )-submodules S 0 (P) (i) k, m,R ⊂ M 0 (P) (i) k, m,R and S 1 (P) ♭ k, m,R ⊂ M 1 (P) ♭ k, m,R Cusps at ∞.We call c ∈ C = C U a cusp at ∞ if it is of the form c t := B(O F,(p) ) + t 0 0 1 U p for some t ∈ (A (p) F,f ) × .Thus the data H = (H, I, [λ], [η]) corresponding to c t is given by • H = J × I, where I = O F and J = J t = t −1 O F ∩ F , • λ is the identification J (p) = O F,(p) (or equivalently, multiplication by any element of O × F,(p),+ ),
We assume O is sufficiently large that the components are defined over O, and refer to them also as cusps.The completion of Y min − (S) and for each p ∈ S p and τ : O Toroidal compactification of Y .Again assuming U is of the form U (N ) for some N ≥ 3, the toroidal compactification Y ֒→ Y tor is defined similarly to that of Y − (see §2.3).Thus Y tor is an infinite disjoint union of projective schemes 4 over O, and the forgetful morphism Y → Y − extends to a projective morphism Y tor → Y tor − .Furthermore the universal abelian scheme A over Y extends to a semi-abelian scheme A tor over Y tor .
Minimal compactification of Y .Letting Y minor denote the ordinary locus in Y min − , defined as in §2.8, we have Y min − = Y − ∪ Y minor , and we construct the minimal compactification Y min by gluing Y to Y minor along Y ord ∼ −→ Y ord − .The scheme Y min is thus normal and independent of choice of cone decompositions, its connected components are flat and projective over O, and we have a commutative diagram of morphisms where d * r is defined by the action of (O/P) × on T r ; thus if t = t ξ (in the notation of Proposition 4.5.2(2)),then d * r t = ξ(d)t ξ .6.3.Cusp forms.For c ∈ C, we let e