Analytic continuation of overconvergent Hilbert eigenforms in the totally split case

Shu Sasaki

Abstract

We firstly generalise results by Buzzard-Taylor, Buzzard and Kassaei about analytic continuation of p-adic overconvergent eigenforms over Q to the case of p-adic overconvergent Hilbert eigenforms over totally real fields F, with the assumption that p splits completely in F. This includes weight 1 forms, and has applications to generalisations of the main theorem of Buzzard-Taylor in the formthcoming work of the author. Secondly, we follow an idea of Kassaei’s to generalise the Coleman’s well-known theorem “an overconvergent U_{p}-eigenform of small slope is classical” to the case of p-adic overconvergent Hilbert eigenforms of the Iwahori level

1. Introduction

Let p ⩾ 5 be a prime and O be the ring of integers of a finite extension of Q_{p} with maximal ideal λ. Let ρ : G_{Q} → GL_{2}(O) be a continuous two-dimensional representation of the absolute Galois group G_{Q} = Gal(\overline{Q}/Q) of Q. If it satisfies the following conditions

(1) ρ ramifes at only finite many primes

(2) ρ is unramified at p and ρ(Frob_{p}) has eigenvalues α and β in O which are distinct modulo λ.

(3) ρ mod λ is absolutely irreducible and “modular”, then Buzzard and Taylor in [5] prove that such a ρ arises from a holomorphic eigenform of weight 1, in the sense of Deligen-Serre [15]. The arguments in [5] rely crucially on two key ingredients. Firstly one needs “companion forms theorem”, due to Gross and Coleman-Voloch, about congruences between ordinary forms of low weight, and secondly it is necessary to have results about “analytic continuation of overconvergent eigenforms”– an overconvergent eigenform extends to the non-ordinary locus of modular curves– which are crucial when “gluing” weight one liftings (=overconvergent eigenforms of weight one and slope 0) of mod p companion forms on their overlap.

Our underlying motivation is to generalise the theorem to the Hilbert case, and follow Taylor’s strategy (outlined in [36]) to prove “insoluble” cases of the strong Artin conjecture for totally odd, continuous representations G_{F} → GL_{2}(C) of the absolute Galois group of totally real fields F. Since the Jacquet-Langlands correspondence does not transfer weight 1 forms for GL_{2} over F to forms on Shimura curves over F of the type considered by Carayol [6], one has to directly work with Hilbert modular varieties, and develop a theory of p-adic (overconvergent) modular forms on Hilbert modular varieties.

The companion forms theorems by Gross and Coleman-Voloch have been generalised by Gee in his thesis [19] to the setting of Hilbert modular forms over totally real fields F, with the assumption that p splits completely; but p-adic analytic continuation theorem for overconvergent Hilbert eigenforms, for example for weight one specialisations of Hida families in the Hilbert case, remained

2000 Mathematics Subject Classification 11G18, 11F33, 11F41, 14G22, 14G35 Keywords: p-adic modular forms, Hilbert modular varieties

The author is supported by a EPSRC project grant

Shu Sasaki

unsolved. We solve this in this paper. In the forthcoming work [35] of the author which will shortly appear elsewhere, he will prove a modular lifting theorem for weight 1 Hilbert modular forms, analogous to the main theorem of Buzzard [3], and Buzzard and Taylor will prove many new cases of the strong Artin conjecture for totally odd, continuous representations G_{F} → GL_{2}(C), with the local condition that some p, in particular, splits completely in F.

The technique to prove that certain overconvergent p-adic Hilbert eigenforms of level p are classical p-adic Hilbert eigenforms is analogous to [5], [3] and [24]. In particular, Buzzard [3] proves the optimal overconvergence results, and this is what we follow. The new idea we put in is that we write down a function more formally, which control the overconvergence, on the generic fibre of the level p HMV and this generalise the Buzzard’s ad hoc v′ in [3].

Given an overconvegent eigenform of weight k ⩾ 1 and level Γ_{1}(N)∩Γ_{0}(p), Kassaei [24] observes that, even when its companion form is not assumed, one can explicitly write down (and then glue) another (over)convergent modular form if the valuation of the eigenvalue of U_{p} is strictly less than k −1. The key observation in [24] is that, whilst an overconvergent eigenform f such that f|U_{p} = αf overconverges to “the far end” of the supersingular annuli in (X_{Γ_{1}(N)∩Γ_{0}(p)})^{an} but not any further, f −(1/αp)f|Frob_{p} does extend (See [20] for Frob_{p}), if we remove the “factor” which precisely causes the problem. Based on this observation, [24] writes down an infinite sum of overconvergent modular forms and endeavours to glue it back onto f. In order to do this, [24] proves a general “gluing lemma” in p-adic “integral” geometry and uses critically the overconvergence of the Frobenius operator, i.e., the application of Frob_{p} makes supersingular elliptic curves “more supersingular” (“opposite” of U_{p} which makes them “less supersingular”). However, as noted in Gouvea (See the remark before Corollary II.2.5 in [20]), the Frobenius operator Frob_{p} is not “integral” (but up to a power of the Hasse invariant) and precisely because of this, Kassaei needs to do a delicate calculation in Lemma 3.3 in [24] to ensure the convergence of the infinite sum.

We generalise his result and idea to the setting of Hilbert modular forms and prove an analogue in the Hilbert case of the Coleman’s theorem [10] in the 90s that an overconvergent p-adic U_{p}-eigenform of small slope is classical. More precisely,

Theorem. Let F be a totally real field with [F : Q] = g > 1 and p be a prime which we assume to split completely in F. Let f be an overconvergent Hilbert modular form of weight (k_{1}, …, k_{g}) and level Γ_{1}(N) ∩ Γ_{0}(p) which is an eigenvector of U_{v_{i}} (v_{i}|p) with non-zero eigenvalue a_{i}. Assume that v_{p}(a_{i}) < k_{i} − 1 for all 1 ⩽ i ⩽ g then f is a classical Hilbert modular form.

The constraint we really have to submit ourselves is that a prime p, which we fix, has to split completely in F but we are currently trying to remove this assumption.

Following closely the original construction of Coleman and Mazur [11], Kisin and Lai construct in [29] an “eigenvariety”, the term coined by Buzzard in [4], for overconvergent Hilbert modular forms for GL_{2} over a totally real field F. They substitute the Eisenstein series of weight p−1, whose p-adic variation property allows Coleman [11] to p-adically vary the weights of classical modular forms over Q, for a classical Hilbert modular form lifting the full Hasse invariant of sufficiently large parallel weight. Since only full Hasse invariants of parallel weight lift to classical Hilbert modular forms in characteristic zero, their construction is 1-dimensional; one should expect the weight space of the eigenvariety for Hilbert modular forms over F to be [F : Q]. Buzzard, however, constructs an eigenvariety over the [F : Q]-dimensional weight space by defining overconvergent Hilbert modular forms on a totally definite quaternion algebra D over F, i.e, on a zero-dimensional “Hida variety”.

Analytic continuation of overconvergent Hilbert eigenforms in the totally split case

The missing input in the construction of Kisin-Lai [29] is that the locus of parallel weight classical Hilbert modular forms is Zariski dense; for the Coleman-Mazur eigencurve, this follows from the main theorem of Coleman [10] (see [9] for example). Our main theorem proves this. As a corollary, one can use the argument of [9] and p-adically vary the classical Jacquet-Langlands correspondence for GL_{2} over F and D over F, for example. Application of our results to a conjecture of Fontaine and Mazur as in the work of Kisin [28] may also be possible. Results of this kind will appear shortly elsewhere. See also [35].

The author would like to thank his Ph.D supervisor Kevin Buzzard for useful conversations on numerous occasions, Gaetan Chenevier, Payman Kassaei, Jacques Tilouine and Alain Genestier for inspirational discussions and Teruyoshi Yoshida for moral support. He would also like to apologise for the delay in the write-up and thank Gaetan Chenevier and Payman Kassaei for unflagging encouragement over years, without which the writing process would have taken longer.

2. Hilbert modular varieties

Let F be a totally real field with [F : Q] = g > 1. Denote by 𝔡 its different. Let 𝔠 be a fractional ideal of F with a notion of positivity (“ordered” in [37]), i.e., the choice, for each embedding τ : F ↪ R of an element of Aut(𝔠 ⊗_{τ:F ↪R} R) ≃ R^{×}. The choice corresponds to the orientation of a onedimensional vector space 𝔠 ⊗ R. The isomorphism classes of such a thing correspond precisely to the narrow ideal class group, the quotient of the group of fractional ideals in F by the principal ideals generated by totally positive elements. For fractional ideals 𝔞 and 𝔟, if 𝔞𝔟^{−1} is generated by a totally positive element in F, then we write 𝔞 ∼ 𝔟.

By a HBAV over a scheme S, we shall mean an abelian variety A over S equipped with real multiplication O_{F} ↪ End(A). Note that its dual A^{∨} is naturally a HBAV.

We shall denote by (A/S, ı, ȷ) a triple consisting of

(1) a HBAV A over S of relative dimension g

(2) a Γ_{1}(N)-level structure on the HBAV over S, that is, an embedding ı : (O_{F} /NO_{F})(1) = 𝔡^{−1} ⊗_{Z} μ_{N} ↪ A[N]

(3) an O_{F} -linear homomorphism ȷ : 𝔠 → P(A) where P(A) is the rank one projective O_{F} -module of O_{F} -linear symmetric (f^{∨} = f) homomorphisms f : A → A^{∨} such that

• the cone of totally positive elements 𝔠^{+} in 𝔠 maps to the cone P(A)^{+} of polarisations in P(A)

• the induced morphism of sheaves (on the large étale site of S) A ⊗_{O_{F}} 𝔠 → A ⊗ P(A) → A^{∨} sending a ⊗ x ↦ a ⊗ ȷ(x) ↦ ȷ(x)(a) is an isomorphism.

If N ⩾ 4, the functor which associates, to a Z[1/N]-scheme S, the set of isomorphism classes of triples (A, ı, ȷ) is represented [18] by a scheme over Z[1/N], which we shall henceforth denote by Y_{Γ_{1}(N;𝔠),Z[1/N]}.

By calculating its local model, Deligne and Pappas [13] show that, the fibre over a prime dividing the discriminant Δ is a singular in codimension 2 closed subscheme but, when Δ is invertible, Corollaire 2.9 in [13] shows that LieA is a locally free O_{F} ⊗_{Z} O_{S}-module of rank one and thus Y_{Γ_{1}(N;𝔠),Z[1/NΔ]} coincides with the Rapoport’s smooth moduli space.

We choose a set of representatives {𝔠_{1}, …, 𝔠_{h^{+}}} for the narrow class group. We let

Y_{Γ_{1}(N),Z[1/N]} ≃ ∐_{𝔠∈{𝔠_{1},...,𝔠_{h^{+}}}}Y_{Γ_{1}(N;𝔠),Z[1/N]}

and let A_{Γ_{1}(N)} = ∐_{𝔠∈{𝔠_{1},...,𝔠_{h^{+}}}} A_{Γ_{1}(N;𝔠)}, where A_{Γ_{1}(N;𝔠)} is the universal HBAV over Y_{Γ_{1}(N;𝔠),Z[1/N]}, denote the “universal” HBAV over Y_{Γ_{1}(N),Z[1/N]}. The HBAV A_{Γ_{1}(N)} comes equipped with the sheaf of relative differentials Ω_{A_{Γ_{1}(N)}/Y_{Γ_{1}(N),Z[1/N]}} and we shall denote by ω_{Y_{Γ_{1}(N),Z[1/N]}}, or simply ω if there is no confusion anticipated, the pull-back of Ω_{A_{Γ_{1}(N)}/Y_{Γ_{1}(N),Z[1/N]}} by the identity section. Following Rapoport and Deligne as attributed in [33], we shall define, in the next section, a Hilbert modular form of weight k⃗ ∈ Z_{⩾0}^{{τ:F ↪R}} and level Γ ⊂ SL_{2}(F) to be a section of a certain invertible sheaf ωk⃗ over the Hilbert modular variety Y_{Γ}. Note that “congruence subgroups” are subgroups of SL_{2}(F), rather than GL_{2}(F) or its subgroup GL_{2}^{+}(F) of matrices in GL_{2}(F) with totally positive determinants as one might expect from the classical theory of Hilbert modular forms. This is because of polarisations. The complex HBAVs, which canonically come equipped with “polarisations faible” ([12] 4.4), are parameterised by the quotient by GL_{2}^{+}(F) of [F : Q] copies of the complex upper half plane. Since GL_{2}^{+}(F) acts on polarisations by determinant, the moduli space for complex HBAVs with a “polarisation homogène” ([12] 4.3) is parameterised by the quotient by the subgroup {M ∈ GL_{2}^{+}(F)|detM ∈ Q} = SL_{2}(F).

3. Hilbert modular forms

Let K ⊂ R be the Galois closure of F over Q containing all the conjugates of F and O_{K} be its ring of integers. For brevity we let ω denote ω_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]}. Since the discriminant Δ is inverted, this is a locally free O_{F} ⊗_{Z} O_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]}-module of rank 1. We think

of Z^{Hom(F,\overline{Q})} as the (algebraic) character group Hom\overline{Q}(Res_{F/Q}GL_{1} × \overline{Q}, GL_{1} × \overline{Q}) by identifying k⃗ = (k_{1}, …, k_{g}) = ∑_{τ∈Hom(F,\overline{Q})} k_{τ} τ with the character which sends x ∈ F^{×} to ∏_{τ∈Hom(F,\overline{Q})}(τ(x))^{k_{τ}}

in \overline{Q}^{×}. The character corresponding to k⃗ gives rise to an invertible sheaf ωk⃗ on Y_{Γ_{1}(N),Z[1/N]} × O_{K}[1/(NΔ)], which we define to be ⨂_{τ∈Hom(F,\overline{Q})}(ω_{τ})^{⊗k_{τ}}, where by ω_{τ} we mean the invertible sheaf of O_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]}-module obtained by tensoring ω with O_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]} by

O_{F} ⊗_{Z} O_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]} ≃ O_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]}^{Hom(F,\overline{Q})} →^{τ} O_{Y_{Γ_{1}(N),Z[1/N]}×O_{K}[1/(NΔ)]}.

Definition. For an O_{K}[1/(ΔN)]-algebra R, an element of H^{0}(Y_{Γ_{1}(N),Z[1/N]}×_{O_{K}[1/(NΔ)]}R, (ω×_{O_{K}[1/(ΔN)]} R)k⃗) is called a Hilbert modular form defined over R of weight k⃗ and of level Γ_{1}(N).

Fix a prime p not dividing N and assume that it splits completely in F. Let v_{1}, …, v_{g} denote the prime ideals in F above p which we may think of as the (finite) places defining F ↪ \overline{Q}_{p}. Fix an embedding ı : \overline{Q} ↪ \overline{Q}_{p} once for all.

Let K_{0} be the compositum of the images of F by all the field embeddings F ↪ \overline{Q} ↪^{ı} \overline{Q}_{p}. Since p splits completely in F, it is Q_{p}. The composition with ı naturally defines a surjective map Hom(F, \overline{Q}) = \{τ\} → Hom(F, \overline{Q}_{p}) = \{v|p\}, which is bijective since p splits completely in F. We assume that τ_{i} : F ↪ K ↪ R corresponds to v_{i}|p for every 1 ⩽ i ⩽ g.

For k⃗ ∈ Z^{Hom(F,\overline{Q})}, define ık⃗ ∈ Z^{Hom(F,\overline{Q}_{p})} by (ık⃗)_{v} = k_{ı^{−1}(v)}, which corresponds to the character

which sends x ∈ (F ⊗ Q_{p})^{×} = Res_{F/Q}GL_{1}(Q_{p}) to ∏_{v|p} x_{v}^{(ık⃗)_{v}}.

4. Hilbert modular varieties with the Iwahori level structure

Let Y_{Γ_{1}(N;𝔠)∩Γ_{0}(p),Z[1/N]} be the scheme over Z[1/N] in [32], which represents the functor which sends a Z[1/N]-scheme S to O_{F} -linear isogenies α : (A, ı, ȷ) → (A′, ı′, ȷ′) of degree p^{g} such that

(1) (A, ı, ȷ) ∈ Y_{Γ_{1}(N;𝔠),Z[1/N]}(S) and (A′, ı′, ȷ′) ∈ Y_{Γ_{1}(N;𝔠),Z[1/N]}(S),

(2) the kernel of α : A → A′ is annihilated by p

(3) for every x ∈ 𝔠, we have α^{∨} ∘ ȷ′(x) ∘ α = pȷ(x)

(4) the level structures are compatible, more precisely ı′ = α ∘ ı.

Remark It does not seem possible to work with the “Γ_{0}(p)-level structure” as in [27] (3.4). The proof in [27] of its representability certainly does not work in the higher-dimensional case, and we work instead with a “moduli space of isogenies” as in [27] (6.5).

Lemma 1. Let v_{1}, …, v_{g} be the prime ideals in O_{F} lying above p. Giving an degree O_{F} -linear degree p^{g} isogeny α : (A, ı, ȷ) → (A′, ı′, ȷ′) over S as above is equivalent to giving (A, ı, ȷ, C) where C is an O_{F} -stable finite flat subgroup scheme of A[p] over S of order p^{g}, and by the action of O_{F} it decomposes as C = ∏_{1⩽i⩽g} C_{i} where C_{i} is a finite flat subgroup of A[v_{i}] of order the cardinality of O_{F} /v_{i}.

Proof. This follows from [32]. □

For each prime v_{i} above p, we choose a uniformiser ϖ_{i} of the integers of the completion of F_{v_{i}}. For each 1 ⩽ i ⩽ g and 1 ⩽ j ⩽ h^{+}, ϖ_{i}𝔠_{j} ∼ 𝔠_{j′} for some j′ ∈ [1, h^{+}] depending on ϖ_{i} and j and we shall fix it. With these choices being made, we let

Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} ≃ ∐_{𝔠∈{𝔠_{1},...,𝔠_{h^{+}}}}Y_{Γ_{1}(N;𝔠)∩Γ_{0}(p),Z[1/N]}.

Having defined them, unless necessary, we shall cease to mention 𝔠-polarisations.

There are canonically defined (representable) morphisms of schemes π_{1} : Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} → Y_{Γ_{1}(N),Z[1/N]} which takes (A, C, ı) to (A, ı), and π_{2,i} : Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} → Y_{Γ_{1}(N),Z[1/N]} for each 1 ⩽ i ⩽ g which takes (A, C, ı) to (A/C_{i}, ı mod C_{i}), where by (ı mod C_{i}), we mean the composition of ı with the isogeny A → A/C_{i}.

For every 1 ⩽ i ⩽ g, we have an automorphisms w_{i} of Y_{Γ_{1}(N)∩Γ_{0}(p)} × Z_{p}, which takes (A, C = ∏_{1⩽j⩽g} C_{j}, ı) to (A/C_{i}, C′, mod C_{i}) where C′ is the O_{F} -stable subgroup of (A/C_{i})[p] of order p^{g} defined by C′_{j} = (C_{j} + C_{i})/C_{i} for all j ≠ i and C′_{i} = A[v_{i}]/C_{i}. Note that (C_{i} + C_{i})/C_{i} is nothing other than the schematic closure in A[p]/C_{i} of the image of C_{j} under the map A[p] → A[p]/C_{j}. It is now clear that π_{2,i} = π_{1} ∘ w_{i}.

For each prime v above p, we let Y_{Γ_{1}(N)∩Γ_{0}(p)∩Γ^{0}(v),Z[1/(Np)]} denote the Hilbert modular variety parameterising A, C, ı and ȷ as above, equipped with a finite flat subgroup D ⊂ A[v] of order p which, locally f.p.p.f, admits a O_{F} /v-generator in the sense of [27], and which has trivial intersection with C. One can prove this is relatively representable to Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} × Z[1/(Np)] as in the proof of Theorem 3.7.1 in [27] and therefore defines a scheme in characteristic prime to Np. There are natural maps π_{1} : Y_{Γ_{1}(N)∩Γ_{0}(p)∩Γ^{0}(v),Z[1/(Np)]} → Y_{Γ_{1}(N)∩Γ_{0}(p)} × Z[1/(Np)] (resp. π_{2,v} : Y_{Γ_{1}(N)∩Γ_{0}(p)∩Γ^{0}(v),Z[1/(Np)]} → Y_{Γ_{1}(N)∩Γ_{0}(p)} ×Z[1/(Np)]) which forgets D (resp. which quotients out by D). With these maps, we define a Hecke operator U_{v} on Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} × Z[1/(Np)] as in (1.11) in [29], or 2.4 in [17]:

For (k⃗, w⃗) ∈ Z^{Hom(F,\overline{Q})}×Z^{Hom(F,\overline{Q})} such that k_{τ} +2w_{τ} is independent of τ and w =^{def} k_{τ} +2w_{τ} ⩾ 0 (note that Hida’s w (resp. n+2v) in the notation of [21] is our −w⃗ (resp. −w)), we define an invertible sheaf of O_{F} ⊗_{Z} O_{Y_{Γ_{1}(N)∩Γ_{0}(p)}×O_{K}[1/N]}-module ω^{(k⃗,w⃗)} to be

Ω_{A_{Γ_{1}(N)∩Γ_{0}(p)}/Y_{Γ_{1}(N)∩Γ_{0}(p)}×O_{K}[1/N]} ⊗ (⨂_{τ} ⋀^{2}(R^{1}π_{∗}Ω_{A_{Γ_{1}(N)∩Γ_{0}(p)}/Y_{Γ_{1}(N)∩Γ_{0}(p)}×O_{K}[1/N]}^{•})_{τ}^{⊗w_{τ}} ⊗ ω_{τ}^{k⃗_{τ} −2})

where by π we mean A_{Γ_{1}(N)∩Γ_{0}(p)} → Y_{Γ_{1}(N)∩Γ_{0}(p)} × O_{K}[1/N] and by “_{τ} ” we mean tensoring with

O_{Y_{Γ_{1}(N)∩Γ_{0}(p)}×O_{K}[1/N]} by O_{F} ⊗_{Z} O_{Y_{Γ_{1}(N)∩Γ_{0}(p)}×O_{K}[1/N]} →^{τ} O_{Y_{Γ_{1}(N)∩Γ_{0}(p)}×O_{K}[1/N]}. Following (1.11)

in [29], we use the degeneracy maps π_{1} and π_{2,v} as above to define a Hecke operator U_{v} on H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]}×O_{K}[1/(Np)], ω^{(k⃗,w⃗)}), which is compatible, by the projection map “averaging the action of the totally positive units” as (1.11.8) in [29], with the classical U_{v} on the Hilbert modular forms on Res_{F/Q}GL_{2}. Base-changing from O_{K}[1/(Np)] to K and choose a (canonical) trivialisation of R^{1}π_{∗}Ω_{A_{Γ_{1}(N)∩Γ_{0}(p)}/Y_{Γ_{1}(N)∩Γ_{0}(p)}}^{•}, we have an isomorphism H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} × K, ω^{(k⃗,w⃗)}) ≃

H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} ×K, ωk⃗), and define U_{v} on H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} ×K, ωk⃗) by composing with the isomorphism.

5. Generic fibres

As in [14] and [34], an (unramified) Γ_{1}(N)-cusp C of Y_{Γ_{1}(N;𝔠)} over a ring R consists of the following data up to isomorphisms

(1) projective rank 1 O_{F} -modules 𝔞, 𝔟 such that 𝔟^{−1}𝔞 ≃ 𝔠

(2) an O_{F} -linear isomorphism N^{−1}O_{F} /O_{F} ≃ N^{−1}𝔞^{−1}/𝔞^{−1}

(3) an O_{F} ⊗ R linear isomorphism 𝔞^{−1} ⊗ R ≃ O_{F} ⊗ R

Fix an (unramified) Γ_{1}(N)-cusp C. Let U_{N} ⊂ O_{F}^{×} denote the group of units in O_{F} which are

congruent to 1 mod N. Let X = 𝔞𝔟 and let X^{∗} denote the dual Hom_{O_{F}} (X, 𝔡^{−1}) =^{tr_{F/Q}} Hom_{Z}(M, Z) and X^{∗,+} denote its subset of totally positive elements. One can choose a “Γ_{1}(N)-admissible” smooth polyhedral cone decomposition {σ} of X^{∗,+} ⊗_{Z} R ∪ {0} depending on C such that {σ} is invariant under the action of U_{N}^{2}, and {σ}/U_{N}^{2} is finite.

Let X_{N} =^{def} N^{−1}X. Let S_{N}^{0} = SpecZ[X_{N}] ↪ S_{N}({σ}) be the torus embedding of a split torus over Z, corresponding (Theorem 2.5 , Chapter IV in [8]) to \{σ\}, and { \atop S_{N}(\{σ\})} be the formal completion of S_{N}(\{σ\}) along S_{N}(\{σ\})\S_{N}^{0}. Note that { \atop S_{N}(\{σ\})} has a covering by open formal subschemes of the form { \atop S_{N}(σ)} =^{def} SpfR_{σ} where R_{σ} is the completion of Z[q^{ξ}]_{ξ∈X_{N} ∩σ̌}, where by σ̌ we mean the dual cone of a cone σ, along the ideal ∩_{ξ∈X_{N} ∩σ̌}(q^{ξ}). We set R_{σ}^{0} = R_{σ}[1/q^{ξ}]_{ξ∈σ̌∩X_{N}}.

By the “Mumford construction”, there exists a semi-abelian scheme with the action of O_{F} over SpecR_{σ} extending a HBAV over R_{σ}^{0}, and it is the main theorem of [33] for the full level N structure (and of [16] for the Γ_{1}(N)-level structure, following [33]), that one can construct an algebraic space by “gluing” a disjoint union of finitely many “good algebraic models (Definition 4.5, Chapter IV in [8])” ∐_{C} ∐_{σ} SpecR_{σ} over Z[1/N, ζ_{N}], where by ζ_{N} we mean the group scheme of Nth roots of unity, to Y_{Γ_{1}(N;𝔠),Z[1/N]} × Z[1/N, ζ_{N}], and get a proper scheme X_{Γ_{1}(N;𝔠),Z[1/N,ζ_{N}]} over Z[1/N, ζ_{N}]. In the Hilbert case, the Q-rank is zero, in other words, we only deal with the “totally degenerate cusps” and the subtle analysis from p.104 to p.106 in Section 3, Chapter IV in [8] is not necessary. In particular, it is smooth over the base. Also it follows from (1.6.5) in [29] that it descends to X_{Γ_{1}(N;𝔠),Z[1/N]} over Z[1/N]. We remark that since the torus embeddings are fibrewise open dense by definition, Y_{Γ_{1}(N;𝔠),Z[1/N]} is fibrewise open dense in X_{Γ_{1}(N;𝔠),Z[1/N]}.

Let X_{Γ_{1}(N)} × R denote a smooth toroidal compactification ∐_{𝔠∈{𝔠_{1},...,𝔠_{h^{+}}}} X_{Γ_{1}(N;𝔠)} ×_{Z[1/N]} R of Y_{Γ_{1}(N),Z[1/N]} × R for any Z[1/N]-algebra R as defined in [33], or more precisely in [16]. It depends on our choice of polyhedral cone decompositions but we suppress it from the notation.

Let X_{Γ_{1}(N;𝔠)∩Γ_{0}(p),Z[1/N]} be a “partial” toroidal compactification of Y_{Γ_{1}(N;𝔠)∩Γ_{0}(p)} × Z[1/N] as in 4.5.2 in [7]. The partial compactification X_{Γ_{1}(N;𝔠)∩Γ_{0}(p),Z[1/N,ζ_{N}]} is obtained from Y_{Γ_{1}(N;𝔠)∩Γ_{0}(p),Z[1/N,ζ_{N}]} by compactifying at the (isomorphism classes of) unramified Γ_{1}(N) cusps of Y_{Γ_{1}(N),Z[1/N,ζ_{N}]}, and it descends to a proper scheme over Z[1/N] as argued in [29] 1.6.5. We remark that [29] uses the partial compactification of level Γ_{1}(Np^{r}) Hilbert modular varieties to construct a eigenvariety for Hilbert modular forms. We then let X_{Γ_{1}(N)∩Γ_{0}(p)} × R denote denote the toroidal partial compactification ∐_{𝔠∈{𝔠_{1},...,𝔠_{h^{+}}}} X_{Γ_{1}(N;𝔠)∩Γ_{0}(p)} ×_{Z[1/N]} R of Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/N]} × R for any Z[1/N]-algebra R

The morphisms π_{1}, π_{2,i} (1 ⩽ i ⩽ g), w_{i} (1 ⩽ i ⩽ g) and the Hecke operators U_{v_{i}} (1 ⩽ i ⩽ g) on Y_{Γ_{1}(N)∩Γ_{0}(p),Z[1/(Np)]} × R naturally extend to the cusps. Think of them as correspondences and take their schematic closures for example.

For Γ = Γ_{1}(N) or Γ_{1}(N)∩Γ_{0}(p), let ({ \atop Y_{Γ} × Z_{p}})^{rig} (resp. ({ \atop X_{Γ} × Z_{p}})^{rig}) be the generic fibre, in the sense of Raynaud [1], of the formal completion { \atop Y_{Γ} × Z_{p}} (resp. { \atop X_{Γ} × Z_{p}}) of Y_{Γ} × Z_{p} (resp. X_{Γ} × Z_{p}) along its special fibre, and let (Y_{Γ} × Q_{p})^{an} (resp. (X_{Γ} × Q_{p})^{an}) be the rigid space, in the sense of [2], associated to the generic fibre Y_{Γ} × Q_{p} (resp. X_{Γ} × Q_{p}). One may think of (Y_{Γ} × Q_{p})^{an} as an admissible open subset of ({ \atop X_{Γ} × Z_{p}})^{rig} containing ({ \atop Y_{Γ} × Z_{p}})^{rig} by the canonical isomorphism ({ \atop X_{Γ_{1}(N)} × Z_{p}})^{rig} ≃ (X_{Γ_{1}(N)} × Q_{p})^{an} ↩ (Y_{Γ_{1}(N)} × Q_{p})^{an} (Proposition 0.3.5 in [1]).

We shall use the same notation π_{1} (resp. π_{2,i}, resp. w_{i}) for (π_{1})^{an} (resp. (π_{2,i})^{an}, resp. (w_{i})^{an}).

Definition For Γ = Γ_{1}(N) or Γ_{1}(N)∩Γ_{0}(p), the sheaf ω on Y_{Γ} ×Q_{p} is a locally free O_{F} ⊗_{Z} O_{Y_{Γ}×Q_{p}}- module of rank 1 and, for k⃗ ∈ Z^{Hom(F,R)}, we define ωk⃗ to be the invertible sheaf corresponding to ık⃗, i.e., ⨂_{v|p} ω_{v}^{⊗(ık⃗)_{v}} where by ω_{v} we mean the invertible sheaf of O_{Y_{Γ}×Q_{p}}-module obtained by tensoring

ω with O_{Y_{Γ}×Q_{p}} with O_{F} ⊗_{Z} O_{Y_{Γ}×Q_{p}} ≃ O_{Y_{Γ}×Q_{p}}^{Hom(F,Q_{p})} →^{v} O_{Y_{Γ}×Q_{p}}. Because of Koecher’s principle, we

may call an element of H^{0}((Y_{Γ} × Q_{p})^{an}, ωk⃗) a classical (p-adic) Hilbert modular form of weight k⃗ and of level Γ.

6. Overconvergent Hilbert modular forms of level Γ_{1}(N)

We shall write down a function v on (X_{Γ_{1}(N)} × Q_{p})^{an}, and define overconvergence in terms of v. Let x ∈ (X_{Γ_{1}(N)} × Q_{p})^{an} be a point . If it is not a cusp, it then corresponds to a closed point of Y_{Γ_{1}(N)} × Q_{p}, and there is a finite extension L of Q_{p} such that x corresponds to (A, ı, ȷ) over L. Consider a model over the ring of integers R of L, which comes equipped with a norm which we normalised such that |p| = 1/p. Suppose it has good reduction. It follows from the Serre-Tate theorem that its formal completion Â along the identity section on the closed fibre is smooth over R and is equivalent to the connected component of the associated p-divisible group; in particular it comes with the action of O_{F}. The underlying ring is a formal group in g parameters and decomposes as the product of g one parameter formal groups R[[Y_{i}]], 1 ⩽ i ⩽ g. If we let X_{i} = ∑_{ζ∈μ}[ζ]Y_{i}/ζ where μ is the group of (p − 1)st roots of unity and [−] denote the action of O_{F_{v}} ≃ Z_{p}, we have [ζ]X_{i} = ζX_{i} and isomorphisms R[[X_{i}]] ≃ R[[Y_{i}]] for all i. We then define v(x) to be the g-tuple (v_{i}(x))_{1⩽i⩽g} where v_{i}(x) is the minimum of 1 and the (normalised) p-adic variation of the coefficient of X_{i}^{p} in [p]X_{i}. Note that, whilst the coefficient itself depends on a choice of parameters, its p-adic valuation does not, when less than 1 and depends only on the HBAV.

If x does not have good reduction, define v_{i}(x) to be 0 for all 1 ⩽ i ⩽ g.

The definition works in families. Let A be the universal HBAV over Y_{Γ_{1}(N)} × Z_{p} and let Â → { \atop Y_{Γ_{1}(N)}} be the induced map of formal completions along special fibres. Since X_{Γ_{1}(N)} × F_{p} is of finite type, one may choose its finite affine covering {U = SpecR}. For each U, we let Û be an open formal affine subscheme of { \atop X_{Γ_{1}(N)} × Z_{p}} satisfying U = Û ∩(X_{Γ_{1}(N)} ×F_{p}). It is then clear that \{Û = SpfR\} is a finite formal affine covering of { \atop X_{Γ_{1}(N)} × Z_{p}} and if we denote sp^{−1}(Û) = Sp(R ⊗_{Z} Q_{p}) simply by U^{rig}, it follows from Proposition 1.1.14 in [1] that {U^{rig}} is an admissible covering of (X_{Γ_{1}(N)}×Q_{p})^{an}. In fact ]U[= (X_{Γ_{1}(N)} × Q_{p})^{an} ∩ U^{rig} by identifying (X_{Γ_{1}(N)} × Q_{p})^{an} with the Raynuad generic fibre of the formal completion { \atop X_{Γ_{1}(N)} × Z_{p}} along its special fibre (See Proposition 0.3.5 in [1]).

Let Û′ = Û ×{ \atop X_{Γ_{1}(N)}×Z_{p}} { \atop Y_{Γ_{1}(N)} × Z_{p}}. Shrink Û if necessary, we then assume \{Û′ = SpfR′\} is

an affine formal covering of { \atop Y_{Γ_{1}(N)} × Z_{p}} and \{U′^{rig} = sp^{−1}(Û′)\} is an admissible covering of the Raynaud generic fibre ({ \atop Y_{Γ_{1}(N)} × Z_{p}})^{rig} (See Proposition 0.2.3 (iii) in [1]). Shrinking the covering, if necessary, to assume that Lie(Â) is trivialised on each Û′. Then the formal group associated to Â/{ \atop Y_{Γ_{1}(N)} × Z_{p}}, when restricted to Û′, gives a formal group in g variables with coefficients in R′. Considering the action of O_{F}, it is isomorphic to the product of the g one parameter formal groups R′[[X_{i}]] where X_{i} is normalised as before. The coefficient in R′ of X_{i}^{p} in [p]X_{i} = pX_{i} + ⋯ can be thought of as a function on Û′ and call it h_{i} ∈ OÛ′(Û′). One may think of this as a lift of partial Hasse invariant. It follows from Koecher’s principle that h_{i} extends to the cusps and gives an element of OÛ(Û) which we shall denote by h_{i} again.

A point x ∈ (Y_{Γ_{1}(N)}×Q_{p})^{an} corresponds to a map x : SpL → (Y_{Γ_{1}(N)}×Q_{p})^{an} ↪ ({ \atop X_{Γ_{1}(N)} × Z_{p}})^{rig} for a finite extension L over Q_{p}, and factors through U^{rig} ↪ ({ \atop X_{Γ_{1}(N)} × Z_{p}})^{rig} for some U^{rig} in the covering. We then alternatively define v(x) to be the g-tuple (v_{i}(x))_{1⩽i⩽g} where v_{i}(x) = min{1, v_{p}(x^{∗}h_{i}^{rig})} where h_{i}^{rig} is the rigid analytic function on U^{rig} defined as the image of h_{i} ∈ OÛ(Û) as above by the map Γ(Û, O{ \atop X_{Γ_{1}(N)}×Z_{p}} ⊗_{Z_{p}} Q_{p}) → Γ(U^{rig}, (X_{Γ_{1}(N)} × Q_{p})^{an}) (See (0.2.3.4) in [1]).

One can check that this does not depend on the choice of the covering {Û}.

For a vector ([0, r_{i}])_{1⩽i⩽g} of g intervals where r_{i} ∈ p^{Q} and r_{i} ∈ [0, 1) for all i, we define a rational subdomain U^{rig}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]) of U^{rig} for each U^{rig} in the aforementioned covering of (X_{Γ_{1}(N)} × Q_{p})^{an} to be the set of x ∈ U^{rig} such that v(h_{i}^{rig}(x)) ∈ [0, r_{i}] for every 1 ⩽ i ⩽ g. We then define X_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]) by gluing U^{rig}([0, r_{1}], …, [0, r_{g}]). By construction, this is clearly an admissible subset of (X_{Γ_{1}(N)}×Q_{p})^{an}. Note that, as remarked in [3], this construction does not depend on choices of parameters for the formal groups. We define Y_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]) to be X_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]) ×_{(X_{Γ_{1}(N)}×Q_{p})^{an}} (Y_{Γ_{1}(N)} × Q_{p})^{an}.

Definition. Let r⃗ = (r_{i})_{1⩽i⩽g}. We define an element of H^{0}(Y_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ωk⃗) to be a r⃗-overconvergent (p-adic) Hilbert modular forms of weight k⃗ and of level Γ_{1}(N). We shall denote the sections H^{0}(Y_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ωk⃗) by Mk⃗(Γ_{1}(N))([0, r_{1}], … [0, r_{g}]). By Koecher’s principle (Lemma 4.1.4 in [29]), the sections of ωk⃗ extend to sections over the quasi-compact X_{Γ_{1}(N)}([0, r_{1}], …, [0, r_{g}]) and therefore it is naturally a Banach space. See the remark in (2.4) in [29]. Alternatively, for 1 ⩽ i ⩽ g, the sections H^{0}(Y_{Γ_{1}(N)}([0, r_{1}], …, [0, r_{g}]), ω_{i}^{k_{i}}) which we think of it as H^{0}(X_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ω_{i}^{k_{i}}), has a natural norm (in the natural equivalence class) which makes it a Banach space. More precisely, for f ∈ M_{k_{i}}(Γ_{1}(N))([0, r_{1}], …, [0, r_{g}])), set |f|r⃗ = sup_{x∈X_{Γ_{1}(N)}([0,r_{1}],...,[0,r_{g}])}|f|_{x} where |f|_{x} is the p-adic norm of (f/h_{i})(x) where h_{i} is as in the definition of overconvergent modular forms above; we are deriving a Banach space structure by mapping the space into the Banach space of “overconvergent functions on X_{Γ_{1}(N)}([0, r_{1}], …, [0, r_{g}]). Finally, since all H^{0}(X_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ω_{i}^{k_{i}}) are complete, Mk⃗(Γ_{1}(N))([0, r_{1}], …, [0, r_{g}]) = ⨂ H^{0}(X_{Γ_{1}(N)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ω_{i}^{k_{i}}) rather than their completed tensor product which makes it a Banach space.

The union of all r⃗-overconvergent modular forms of weight k⃗ and level N for all r⃗ satisfying r_{i} ∈ (0, 1) for all i is called the space of overconvergent (p-adic) Hilbert modular forms of weight k⃗ and level Γ_{1}(N).

7. Overconvergent Hilbert modular forms of level Γ_{1}(N) ∩ Γ_{0}(p)

In this section, we define functions v′ = (v′_{i})_{1⩽i⩽g} : (X_{Γ_{1}(N)∩Γ_{0}(p)} × Q_{p})^{an} → [0, 1]^{g} and define overconvergent modular forms of level Γ_{1}(N) ∩ Γ_{0}(p) in terms of v′.

If a point x ∈ (X_{Γ_{1}(N)∩Γ_{0}(p)}×Q_{p})^{an} is not a cusp, it corresponds to a closed point of Y_{Γ_{1}(N)∩Γ_{0}(p)}× Q_{p}, and therefore there is a finite extension L of Q_{p}, equipped with the valuation normalise such that |p| = 1/p, such that x corresponds to a L-valued point of Y_{Γ_{1}(N)∩Γ_{0}(p)} × Q_{p}. Suppose it has good reduction. Let A be the model over the ring of integers R of L of the corresponding HBAV over L, which comes equipped with the action of O_{F} lifting the action on the generic fibre. By taking the schematic closure in A[p], A has an O_{F} -stable finite flat group scheme C over R. For brevity, we shall denote the quotient A/C by B. The isogeny A → B induces a map of locally free sheaves ĽieB → ĽieA. It is O_{F} -linear and decomposes as the sum of morphisms (ĽieB)_{i} → (ĽieA)_{i} where (ĽieA)_{i} and (ĽieB)_{i} are both locally free O_{SpecR}-module of rank 1 and O_{F} acts by τ_{i}. We may assume that (ĽieA)_{i} and (ĽieB)_{i} are simultaneously trivialised on SpecR. In which case, for each 1 ⩽ i ⩽ g, the Fitting ideal ([31]) of its cokernel is generated by one element, say, g_{i} ∈ R. We then define v′(x) to be the g-tuple (1 − v(g_{i}))_{1⩽i⩽g}.

The isogeny A → B induces a homomorphism of the formal groups over R; it is O_{F} -linear, and decomposes as the product of g homomorphisms R[[X_{i}]] → R[[Y_{i}]] of one parameter formal groups with the normalised parameters X_{i} and Y_{i} as before. It in fact sends X_{i} to Y_{i} = ∏_{x} G(X_{i}, x) where x is a point in C_{i} and G denotes the group law [30] on R[[X_{i}]]. The annihilator of the cokernel of the induced map of the R-modules of the invariant differential forms on the formal groups is the first derivative at X_{i} = 0 of the homomorphism with respect to X_{i}, and its p-adic valuation therefore is (p − 1)v(c) with c ∈ \{c_{0} = 0, c_{1}, …, c_{p−1}\} = C_{i}(\overline{L}). As a result, we have v′_{i}(x) = 1 − (p − 1)v(c).

A more conceptual way of thinking about v′ is as follows. One can check that, locally on U = SpecO_{U}, the Fitting ideal of the cokernel of (ĽieB)_{i} → (ĽieA)_{i} is isomorphic to the Fitting ideal of (ĽieC)_{i} and it equals to the Fitting ideal of R/δ_{i} if C_{i}, over U, is of the form SpecO_{U}[T]/(T^{p} −δ_{i}T); in other words, if δ_{i} is the Oort-Tate, or more generally Raynaud, parameter of C_{i}. Note that (ĽieC)_{i} ≃ ĽieC_{i}.

If the model A over R does not have good reduction, in other words, if x is a cusp, we define (v′_{i}(x))_{1⩽i⩽g} to be the same as that of any ordinary point in the same component.

More generally and possibly more amenable to generalisations, one may paraphrase this in terms of cotangent complexes. Let A and B = A/C be HBAVs over SpecR as above; in particular they are smooth and local complete intersections over R. Let L_{C/SpecR} denote the cotangent complex of C over SpecR in the derived category D(O_{F} ⊗_{Z} O_{SpecR}) ([22] and [23]) of complexes of O_{F} ⊗_{Z} O_{SpecR}- modules. If we let e be the unit section SpecR → C, the complex l_{C/SpecR} := Le^{∗}L_{C/SpecR} of O_{F} ⊗_{Z}O_{SpecR}-modules, locally free as O_{SpecR}-modules, is perfect and concentrated in degrees [−1, 0]. It turns out that it is isomorphic to the two term complex 0 → ĽieB → ĽieA → 0 and, by taking “determinant divisors”, one can deduce the same result.

For (A, C) as above, we fix i ∈ [0, g] and let A′ = A/C_{i} and C′ ⊂ A′ be the O_{F} -stable subgroup of A[p] of order p^{g} defined by C′_{j} = (C_{j} + C_{i})/C_{i} for all j ≠ i and C′_{i} = A[v_{i}]/C_{i}. Note that Fit_{R}(ĽieC_{i}) · Fit_{R}(ĽieC′_{i}) ⊂ Fit_{R}((ĽieA[p])_{i}) (c.f. Appendix in [31]) and since Ľie(A[p]) is a locally free (O_{F} /p) ⊗_{Z} O_{SpecR}-module of rank 1, the right hand side is the R-module generated by p, and we have v′_{i}(A, C) + v′_{i}(A′, C′) = 1.

For a vector ([0, r_{i}])_{1⩽i⩽g} of g intervals where r_{i} ∈ p^{Q} and r_{i} ∈ [0, 1) for all i, we define an admissible subset Y_{Γ_{1}(N)∩Γ_{0}(p)}([0, r_{1}], …, [0, r_{g}]) (resp. X_{Γ_{1}(N)∩Γ_{0}(p)}([0, r_{1}], …, [0, r_{g}])) of (Y_{Γ_{1}(N)∩Γ_{0}(p)})^{an} (resp. (X_{Γ_{1}(N)∩Γ_{0}(p)})^{an})to be the component, containing the points whose v′_{i} are all zero, of the inverse image by π_{1} of Y_{Γ_{1}(N)}([0, r_{1}], …, [0, r_{g}]) (resp. X_{Γ_{1}(N)}([0, r_{1}], …, [0, r_{g}])). One can check that it coincides with the set of points x ∈ (Y_{Γ_{1}(N)∩Γ_{0}(p)})^{an} such that v′_{i}(x) ∈ [0, r_{i}] for all 1 ⩽ i ⩽ g.

Definition. Let r⃗ = (r_{i})_{1⩽i⩽g}. We define an element of H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ωk⃗) to be a r⃗-overconvergent (p-adic) Hilbert modular forms of weight k⃗ and of level Γ_{1}(N) ∩ Γ_{0}(p). We shall denote the sections H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p)}([0, r_{1}], [0, r_{2}], …, [0, r_{g}]), ωk⃗) by Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([0, r_{1}], … [0, r_{g}]). Similarly this is a Banach space. The union of all r⃗-overconvergent modular forms of weight k⃗ and level N for all r⃗ satisfying r_{i} ∈ (0, 1) for all i is called the space of overconvergent (p-adic) Hilbert modular forms of weight k⃗ and level Γ_{1}(N) ∩ Γ_{0}(p).

We remark that if we let ({ \atop Y_{Γ_{1}(N)∩Γ_{0}(p)} × Z_{p}})^{rig}([0, r_{1}], …, [0, r_{g}]) be the admissible open subset of the points x ∈ ({ \atop Y_{Γ_{1}(N)∩Γ_{0}(p)} × Z_{p}})^{rig} such that v′_{i}(x) ∈ [0, r_{i}] for all 1 ⩽ i ⩽ g, we have natural

morphisms H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([0, r_{1}], …, [0, r_{g}]), ωk⃗) → H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p)}([0, r_{1}], …, [0, r_{g}]), ωk⃗) → H^{0}(({ \atop Y_{Γ_{1}(N)∩Γ_{0}(p)} × Z_{p}})^{rig}([0, r_{1}], …, [0, r_{g}]), ωk⃗),

and they are isomorphisms. To check this, follow the proof of Lemma 4.1.4 in [29]. We may henceforth extend overconvergent modular forms to the cusps tacitly.

8. Canonical subgroups of Hilbert-Blumenthal abelian varieties

In this section, we extend the notion of canonical subgroups of elliptic curves over p-adically complete rings as in the work of Katz [26] to the case of abelian varieties with the action of real multiplication O_{F}.

Definition Let A over a p-adically complete Z_{p}-algebra R. Assume that 0 < v_{i}(A) < p/(p + 1) for 1 ⩽ i ⩽ g. Then the identity component of the p-divisible group A[v_{i}^{∞}] is formally smooth of dimension 1, and we let R[[X_{i}]] denote the underlying formal group with one variable X_{i} normalised as before. We define the canonical subgroup H_{i} of A to be the finite flat group scheme of order p in the p-torsion subgroup of R[[X_{i}]], defined by the equation X_{i}^{p} − t_{can}X_{i} as in the Kaz’s paper [26]. If v_{i}(A) = 0, then h_{i}(A) is invertible and the i-th component of the kernel of Frobenius in the special fibre of A gives a finite flat group scheme of order p. Since its dual is étale, one can lift the dual to R by Hensel’s lemma and the dual of such a thing is the canonical subgroup H_{i} in this case.

The following lemma will be needed. For a HBAV A over R as above such that v_{i}(A) ∈ [0, p/(p+ 1)), we follow [26] to denote p/h_{i}(A) by r_{1}. Then r_{1} ∈ R and v(r_{1}) > 1/(p + 1), and

Lemma 2. Modulo r_{1}, the canonical subgroup H_{i} of A is the kernel of Frobenius.

Proof. Modulo r_{1}, the canonical subgroup is defined by X_{i}^{p} in ker[p] (In the notation of [26], r_{1} = 0 in R/r_{R} implies that t_{0} = 0 and thus t_{can} = 0 mod r_{1}). □

Definition Let L be a finite extension of Q_{p} and let A be a HBAV over L. We say that A over L is not too v_{i}-(non-ordinary) if, after base changing to a finite extension L′ of L if necessary, v_{i}(A ×_{SpecL} SpecL′) < p/(p + 1).

Since it is essentially calculations with formal groups, one can readily generalise the argument of Katz in Theorem 3.10.7 in [26] to associate a canonical subgroup H_{i} to not too v_{i}-(non-ordinary) HBAV A. It is then easy to deduce Lemma 3. Let A be a HBAV over a finite extension L of Q_{p}.

(1) If v_{i}(A) = 0, then the canonical subgroup H_{i} of A is the finite étale subgroup of A[v_{i}].

(2) If v_{i}(A) < 1/(p + 1), then A/H_{i} is not too v_{i}-(non-ordinary) and v_{i}(A/H_{i}) = pv_{i}(A) whilst v_{j}(A/H_{i}) = v_{j}(A) for all 1 ⩽ j ⩽ g not equal to i.

(3) If v_{i}(A) = 1/(p + 1) then A/H_{i} is too v_{i}-(non-ordinary).

(4) If 1/(p + 1) < v_{i}(A) < p/(p + 1), then A/H_{i} is not too v_{i}-(non-ordinary) and v_{i}(A/H_{i}) = 1−v_{i}(A) whilst v_{j}(A/H_{i}) = v_{j}(A) for all 1 ⩽ j ⩽ not equal to i. Furthermore the canonical subgroup of A/H_{i} is A[v_{i}]/H_{i}.

(5) If v_{i}(A) < p/(p + 1) and C_{i} ⊂ A[v_{i}] is a finite subgroup ≠ H_{i} of order p, then v_{i}(A/C_{i}) = v_{i}(A)/p whilst v_{j}(A/C_{i}) = v_{j}(A) for all 1 ⩽ j ⩽ g not equal to i. The canonical subgroup of A/C_{i} is A[v_{i}]/C_{i}.

(6) If p/(p + 1) ⩽ v_{i}(A) and C_{i} ⊂ A[v_{i}] is a finite flat subgroup of order p, then v_{i}(A/C_{i}) = 1/(p + 1) and the canonical subgroup of A/C_{i} is A[v_{i}]/C_{i}.

Remark. Let (A, C, ı) over R correspond to a point of (X_{Γ_{1}(N)∩Γ_{0}(p)} × Q_{p})^{an}. One can check that v′_{i}(A, C) = v_{i}(A) if A has the canonical subgroup and C_{i} = H_{i}. In fact, this follows from [26].

Note that it can not happen that C_{i} ⊂ A[v_{i}] and A[v_{i}]/C_{i} ⊂ A[p]/C_{i} are not canonical at the same time; and therefore if v′_{i}(A, C) < 1/(p+1), A has to have the canonical subgroup and C_{i} = H_{i}; on the other hand, if v′_{i}(A, C) > p/(p + 1), then C_{i} cannot be the canonical subgroup.

9. Analytic continuation of overconvergent eigenforms

For I⃗ = (I_{i})_{1⩽i⩽g} where each I_{i} ⊆ [0, 1] is an interval of the form [0, r_{i}) or [0, r_{i}] for 0 < r_{i} ⩽ 1, we let X_{Γ_{1}(N)∩Γ_{0}(p)}(I⃗) denote the set of points x ∈ (X_{Γ_{1}(N)∩Γ_{0}(p)} × Q_{p})^{an} such that v_{i}(x) ∈ I_{i} for all 1 ⩽ i ⩽ g and Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))(I⃗) the space H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}(I⃗), ωk⃗) of overconvergent modular forms of weight k⃗.

Let f be an overconvergent Hilbert modular form over Q_{p} of weight k⃗ and level Γ_{1}(N) ∩ Γ_{0}(p). Then there exists a g-tuple r⃗ = (r_{i})_{1⩽i⩽g} such that f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, r_{1}], …, [0, r_{g}]).

9.1 Analytic continuation of overconvergent U_{v}-eigenforms I Fix 1 ⩽ i ⩽ g, and we shall let X_{Γ_{1}(N)∩Γ_{0}(p)}(I_{i}) mean X_{Γ_{1}(N)∩Γ_{0}(p)}(I⃗) and Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))(I_{i}) to mean Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))(I⃗). Similarly for X_{Γ_{1}(N)}.

Proposition 4. The X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n−1}(p + 1))]) for all n ∈ Z_{⩾0} form an admissible covering of X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1)).

Proof. It essentially follows from Maximum Modulus Principle [2] that they form an admissible covering of the admissible open X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1)). See for example 9.1.4, Proposition 5 in [2].□

For π_{1} and π_{2,v} : (X_{Γ_{1}(N)∩Γ_{0}(p)∩Γ^{0}(v)} × Q_{p})^{an} → (X_{Γ_{1}(N)∩Γ_{0}(p)} × Q_{p})^{an} as above, if U and V are admissible open subsets of (X_{Γ_{1}(N)∩Γ_{0}(p)} ×Q_{p})^{an} such that (π_{1})^{−1}(U) ⊆ (π_{2,v})^{−1}(V), one can define “the trace map” tr_{v} : H^{0}(V, ωk⃗) → H^{0}(U, ωk⃗) as in Section 5 of [3] and, if U and V furthermore satisfy V ⊆ U, define U_{v} to be p^{−1}tr_{v} followed by the restriction map H^{0}(U, ωk⃗) → H^{0}(V, ωk⃗).

Proposition 5. If f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, r]) is an eigenform for U_{v} with non-zero eigenvalue a, then it extends to an eigenform in Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1))

In order to prove this, we need two lemmas:

Lemma 6. Let (A, C, ı) correspond to a point x ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n}(p + 1))]). Let D_{i} be a subgroup of A[v_{i}] of order p which does not meet C non-trivially. If we let y denote the point corresponding to (A/D_{i}, (C + D_{i})/D_{i}, ı mod D_{i}), then y ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n−1}(p + 1))]).

Proof. It follows from case-by-case calculations as in Lemma 4.2 in [3]

Lemma 7. If f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, r_{i}]) is an eigenform for U_{v_{i}} with non-zero eigenvalue a_{i}, then it extends to an eigenform in Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1 − 1/(p^{−1}(p + 1)))

Proof. If 0 < v′_{i}(A, C) < p/(p + 1), then v′_{i}(A, C) > v′_{i}(A/D_{i}, (C + D_{i})/D_{i}) = v′_{i}(A, C)/p for a subgroup D_{i} of A[v_{i}] of order p such that D_{i} ≠ C_{i}. To prove this, observe that since C_{i} = H_{i} we have v′_{i}(A, C) = v_{i}(A) (resp. v′_{i}(A/D_{i}, (C + D_{i})/D_{i}) = v′_{i}(A/D_{i})) and it follows from Lemma 3(5) that v_{i}(A) > v_{i}(A/D_{i}) = v_{i}(A)/p. If r_{i} ⩾ 1/(p+1), there is nothing to prove, so we assume r_{i} < 1/(p+1). Suppose that f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, t_{i}]) for some r_{i} ⩽ t_{i} < 1/(p + 1) and that it is an eigenform for U_{v_{i}} with eigenvalue a_{i}. Then (f|p^{−1}tr_{v_{i}})/a_{i} ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, pt_{i}]) since for (A, C, ı) ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([0, pt_{i}]), it is defined that ((f|p^{−1}tr_{v_{i}})/a_{i})(A, C, ı) = 1/(a_{i}p) ∑_{D_{i}}(pr_{i})^{∗}f(A/D_{i}, (C + D_{i})/D_{i}, ı mod D_{i}) where pr_{i} denotes A → A/D_{i}. Since f is an eigenform for U_{v_{i}}, the restriction to X_{Γ_{1}(N)∩Γ_{0}(p)}([0, t_{i}]) of f|p^{−1}tr_{v_{i}} is a_{i}f, in other words, the restriction of (f|p^{−1}tr_{v_{i}})/a_{i} to X_{Γ_{1}(N)∩Γ_{0}(p)}([0, t_{i}]) is f, and therefore f|p^{−1}tr_{v_{i}} extends f. Since the trace map and the restriction map “commute” as observed in Lemma 2.18 (1) in [25], one checks that f|p^{−1}tr_{v_{i}} is also an eigenform for U_{v_{i}} with eigenvalue a_{i}. Repeat the argument. □

Remark In [3], analytic continuation of overconvergent eigenforms are typically proved by firstly showing that admissible open subsets, over which overconvergent modular forms are defined, are connected and then use the q-expansion principle at cusps to ensure that f|p^{−1}tr does in fact extend f as they have the same q-expansions. In calculating q-expansions, p^{−1}tr is not any different from U_{p}. Kassaei, however, observes in [25] Section 3 that, in fact, establishing that admissible subsets are connected is not necessary, and this substantially simplifies the argument in [3]; in fact one does not need to compute q-expansions! Though almost all admissible subsets we see in this paper are undoubtedly connected, we shall follow his approach.

Proof of the proposition. It follows from the preceding lemma that the Hecke operator p^{−1}tr_{v_{i}} defines a map of sections:

p^{−1}tr_{v_{i}} : H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1−1/(p^{n−1}(p+1))]), ωk⃗) → H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1−1/(p^{n}(p+1))]), ωk⃗).

More explicitly, for an element f_{n} ∈ H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n−1}(p + 1))]), ωk⃗) = Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1 − 1/(p^{n−1}(p + 1))]) and for (A, C, ı) ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n}(p + 1))]), f_{n}|p^{−1}tr_{v_{i}} ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1 − 1/(p^{n}(p + 1))]) is defined by

(f_{n}|p^{−1}tr_{v_{i}})(A, C, ı) = 1/p ∑_{D_{i}≠C_{i}}(pr_{i})^{∗}f_{n}(A/D_{i}, (C + D_{i})/D_{i}, ı mod D_{i}).

By the preceding lemma, f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, r_{i}]) extends to a U_{v_{i}}-eigenform f_{0} ∈ Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([0, 1/(p+1)]). Inductively define f_{n+1} ∈ Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([0, 1−1/(p^{n}(p+1))]) by (f_{n}|p^{−1}tr_{v_{i}})/a_{i} for all n ∈ Z_{⩾0}, which is an U_{v_{i}}-eigenform with eigenvalue a_{i} when restricted to X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n−1}(p + 1))]). Since f_{n} is an eigenform for U_{v_{i}}, the restriction of f_{n+1} to X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − 1/(p^{n−1}(p + 1))]) therefore is f_{n}, and hence f_{n+1} extends f_{n}. Since the {X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1−1/(p^{n−1}(p+1))]}_{n∈Z_{⩾0}} form an admissible covering of X_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1)), this inductive construction gives an element of Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1)) extending f; it is still an eigenform for U_{v_{i}} with eigenvalue a_{i}. □

Corollary 8. If f is an overconvergent Hilbert modular form of weight k⃗ which is an eigenform for U_{v} with non-zero eigenvalue for all v|p, then it extends to an eigenform in Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1), …, [0, 1))

9.2 Analytic continuation of overconvergent U_{v}-eigenforms II Fix any 0 ⩽ t_{i} < 1/(p + 1). Since the following construction applies to both, for convenience, we use Y_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)]) to mean Y_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)) or Y_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1]). Similarly for X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)]).

Define s_{i} : Y_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{1}/p, 1)]) → Y_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{1}, 1)]) which takes (A, C, ı) to (A/H_{i}, (C+ H_{i})/H_{i}, ı mod H_{i}) where H_{i} ⊂ A[v_{i}] is the canonical subgroup of A. Note that it is important to have C_{i} ≠ H_{i}. See the end of the last section. Note that this extends to X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p)]) → X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i})]). At a cusp with a choice of polyhedral cone decomposition {σ}, for an O_{F} -stable semi-abelian scheme G_{σ} over Spec(R_{σ} ⊗ Z_{p}), extending a 𝔠-polarised HBAV A_{σ} over Spec(R_{σ}^{0} ⊗ Z_{p}), s_{i} corresponds to the isogeny A_{σ} → A_{σ}/H_{σ,i} where H_{σ,i} is the kernel of the morphism of p-divisible groups A_{σ}[v_{i}^{∞}] → A_{σ}[v_{i}^{∞}] corresponding to q ↦ q^{p} in the base. Since X_{N} ∩ σ̌ ∋ ξ ↦ ξp leaves X_{N} ∩ σ̌ stable, the isogeny extends to G_{σ} and consequently s_{i} extends to X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p)]) → X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i})]) which we shall gain denote by s_{i}.

Define id : X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)]) → X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)]) which takes (A, C, ı) to itself.

We shall define the operator V_{i} : Mk⃗(Γ_{1}(N)∩Γ_{0}(p))[1−t_{i}, 1)] → Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([1−t_{i}/p, 1)]) in the following. Let p_{i}^{∗} denote the natural morphism of sheaves s_{i}^{∗}ω → (id)^{∗}ω on Y_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1− t_{i}/p]) defined by taking the universal HBAV over Y_{Γ_{1}(N)∩Γ_{0}(p)}([0, 1 − t_{i}/p]) to its quotient by the canonical subgroup in the v_{i}-torsion points. It clearly extends to the cusps. We shall use the same notation for the variant s_{i}^{∗}ωk⃗ → (id)^{∗}ωk⃗. Define V_{i} now to be the composite

V_{i} : Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))[1 − t_{i}, 1)]  →^{s_{i}^{∗}} H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)])), s_{i}^{∗}ωk⃗)
→^{p^{−k_{i}}p_{i}^{∗}} H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)])), (id^{∗})ωk⃗) → Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))[1 − t_{i}/p, 1)].

Let x ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{i}/p, 1)]) be a point corresponding to SpL → X_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{i}/p, 1)]) for some finite extension L of Q_{p}. Denote by x_{1} (resp. x_{2}) the composite

SpL →^{x} X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)]) ⟶^{id(resp. s_{i})} X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)])(↪ (X_{Γ_{1}(N)∩Γ_{0}(p)} × Q_{p})^{an}).

By specialisation to x, the map p_{i}^{∗} induces a map of the sections p_{i}^{∗} : H^{0}(SpL, x_{2}^{∗}ωk⃗) → H^{0}(SpL, x_{1}^{∗}ωk⃗) and one can check that, for f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([1 − t_{i}, 1)]), f|V_{i} ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([1 − t_{i}/p, 1)]) satisfies (f|V_{i})(x_{1}) = p^{−k_{i}}p_{i}^{∗}(x_{2}^{∗}f) = p^{−k_{i}}p_{i}^{∗}f(x_{2}), in other words, if x correspond to a point (A, C, ı), then (f|V_{i})(A, C, ı) = p^{−k_{i}}pr_{i}^{∗}f(A/H_{i}, (C + H_{i})/H_{i}, ı mod H_{i}), where by pr_{i}^{∗} we mean the pull back by the isogeny pr_{i} : A → A/H_{i}.

Lemma 9. Let x ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)]) be a point corresponding to SpL → X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)]) for some finite extension L over Q_{p}. Let h_{i} be an element of L such that |h_{i}| = |h_{i}(π_{1}(x))| and let x_{1} ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)]) (resp. x_{2} ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)])) to be the image of x by id (resp. s_{i}). Then for f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([1 − t_{i}, 1)]), we have |(f|V_{i})(x_{i})| ⩽ |f(x_{2})||h_{i}|^{−k_{i}} with respect to the normalised norm on L.

Proof. If x is a point in X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1]), its image in (X_{Γ_{1}(N)} × Q_{p})^{an} by π_{2,i} has v_{i}(x) satisfying 0 ⩽ v_{i}(x) ⩽ t_{i}/p < p/(p + 1). Let x correspond to a HBAV over a finite extension L of Q_{p}, with residue field k, which has a model A over the ring of integers O_{L}. Assume firstly that A has good reduction. Then A has the canonical subgroup H_{i} ⊂ A[v_{i}] and we let B_{i} = A/H_{i}. Observe that p_{i}^{∗} : s_{i}^{∗}ωk⃗ → (id)^{∗}ωk⃗ induces the map of sections p_{i}^{∗} : H^{0}(B_{i}, Ωk⃗ ; B_{i}) → H^{0}(A, Ωk⃗ ; A) of sheaves of relative differentials over SpecO_{L}. We claim that the “unit ball” sections

H^{0}(B_{i}[v_{i}^{∞}], Ω_{B_{i}[v_{i}^{∞}],v_{i}}^{k_{i}}) ⊂ H^{0}(B_{i}[p^{∞}], Ω_{B_{i}[p^{∞}],v_{i}}^{k_{i}}) ⊂ H^{0}(B_{i}, Ωk⃗ ; B_{i}) of the sheaf of relative differen-

tials of the p-divisible group B_{i}[v_{i}^{∞}] over SpfO_{L} on which O_{F} acts by O_{F} ↪ O_{F_{v_{i}}} →O_{L}, map by p_{i}^{∗} to (p/h_{i})^{k_{i}}H^{0}(A[v_{i}^{∞}], Ω_{A[v_{i}^{∞}],v_{i}}^{k_{i}}) ⊂ H^{0}(A, Ω_{A/S}^{k_{i}}). This follows from observing that the map

Ω_{B_{i}[v_{i}^{∞}],v_{i}} → Ω_{A[v_{i}^{∞}],v_{i}} reduces modulo p/h_{i} (Lemma 2) to φ^{∗}Ω_{(A[v_{i}^{∞}]×k)^{(p)},v_{i}} → Ω_{A[v_{i}^{∞}]×k,v_{i}}^{k_{i}}, where φ

is the relative Frobenius, which is zero. For f = ⨂_{j=1}^{g} f_{j} ∈ H^{0}(Y_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{i}/p, 1]), ⨂_{j=1}^{g} ω_{v_{j}}^{k_{j}}), if |f_{i}(x_{2})| ⩽ 1 then f_{i}(x_{2}) ∈ H^{0}(B_{1}[v_{1}^{∞}], Ω_{B[v_{i}^{∞}],v_{i}}^{k_{1}}) and therefore |(f_{i}|V_{i})(x_{1})| = |p^{−k_{i}}p_{i}^{∗}f_{i}(x_{1})| =

|p^{−k_{i}}(p/h_{i})^{k_{i}}f_{i}(x_{1})| ⩽ |h_{i}|^{−k_{i}} since f_{i}(x_{1}) ∈ H^{0}(A[v_{i}^{∞}], Ω_{A[v_{i}^{∞}],v_{i}}^{k_{i}}) and therefore |f_{i}(x_{1})| ⩽ 1, and this suffices as in [26].

Secondly if A does not have good reduction, by definition, we have v′_{i}(x) = 1. We let (Y_{Γ_{1}(N)} × F_{p})^{v_{i}ord} (resp. (X_{Γ_{1}(N)} × F_{p})^{v_{i}ord}) denote the open subscheme of Y_{Γ_{1}(N)} × F_{p} (resp. X_{Γ_{1}(N)} × F_{p}) where (resp. the extension, by Koecher’s principle, of) the i-th partial Hasse invarinat does not vanish, and let { \atop (Y_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}} (resp. { \atop (X_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}}) denote the formal completion of Y_{Γ_{1}(N)} × Z_{p} (resp. X_{Γ_{1}(N)} × Z_{p}) along (Y_{Γ_{1}(N)} × F_{p})^{v_{i} ord} (resp. (X_{Γ_{1}(N)} × F_{p})^{v_{i} ord}).

Let x_{1} (resp. x_{2}) again denote the composition

SpL →^{x} Y_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]) →^{id(resp.s_{i})} Y_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]) →^{π_{1}} X_{Γ_{1}(N)}([0, 0]) ≃ ({ \atop (X_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}})^{rig},

let x̂_{1} (resp. x̂_{2}) denote its formal model SpfO_{L} → { \atop (X_{Γ_{1}(N)} × Z_{p})^{ord}} and let \overline{x}_{1} (resp. \overline{x}_{2}) denote its specialisation in (X_{Γ_{1}(N)} × F_{p})^{ord}; note that \overline{x}_{1} and \overline{x}_{2} lie in the complement of Y_{Γ_{1}(N)} × F_{p} in X_{Γ_{1}(N)} × F_{p}, in other words, the cusps of Y_{Γ_{1}(N)} × F_{p}.

The p_{i}^{∗} induces x_{2}^{∗}ω → x_{1}^{∗}ω by specialising to x, and it suffices to establish H^{0}(SpfO_{L}, (x̂_{2})^{∗}ω) ⊂ H^{0}(SpfO_{L}, (x̂_{2})^{∗}ω)⊗L ≃ H^{0}((SpfO_{L})^{rig}, x_{2}^{∗}ω) ≃ H^{0}(SpL, x_{2}^{∗}ω) maps by p_{i}^{∗} to pH^{0}(SpfO_{L}, (x̂_{1})^{∗}ω) ⊂ H^{0}(SpL, x_{1}^{∗}ω). This follows from observing that p_{i}^{∗} induces zero morphism on { \atop (Y_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}}×F_{p}

by the preceding argument and, since { \atop (Y_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}} × F_{p} is open dense in { \atop (X_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}} ×

F_{p}, p_{i}^{∗} has to be zero on { \atop (X_{Γ_{1}(N)} × Z_{p})^{v_{i} ord}} × F_{p}. The rest is analogous to the first case. □

We shall prove that if f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1)) is an eigenvactor for U_{v_{i}} with non-zero eigenvalue a_{i} such that v(a_{i}) < k_{i} − 1, then f extends to Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1]). Fix t_{i} ∈ (0, p/(p+1)). For brevity, we typically use f to mean the restriction of f ∈ Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([0, 1)) to X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)]). Let b_{i} = p^{k_{i}−1}/a_{i} for brevity. We then have

Lemma 10. f−b_{i}(f|V_{i}) ∈ Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([1−t_{i}/p, 1)) extends to Mk⃗(Γ_{1}(N)∩Γ_{0}(p))([1−t_{i}/p, 1]).

Proof. On the non-cuspidal points, simply define f′ ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([1 − t_{i}/p, 1]) to be

f′((A, C, ı)) = 1/(a_{i}p) ∑_{D_{i}≠H_{i},≠C_{i}} p_{i}^{∗}f((A/D_{i}, (C + D_{i})/D_{i}, ı mod D_{i}))

and follow the remark at the end of Section 7 to extend to the cusps. Then for (A, C, ı) ∈ X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)), we have f′(A, C, ı) = (1/a_{i})(f|U_{v_{i}})(A, C, ı) − (1/(a_{i}p))f(A/H_{i}, (C + H_{i})/H_{i}, (ı mod H_{i})) = (f − (p^{k_{i}−1}/a_{i})f|V)(A, C, ı). □

For n ∈ Z_{⩾1}, define g_{n} ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([1 − t_{i}/p^{n}, 1]) to be ∑_{j=0}^{n−1} b_{i}^{j}(f′|V_{i}^{j}), where by f′|V_{i}^{j} we mean the iterated (… ((f′|V_{i})|V_{i}) …)|V_{i}. One can check that, since the restriction of f′ to X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p, 1)) is f − b_{i}(f|V_{i}) by definition, the restriction of g_{n} to X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p^{n}, 1)) is f|_{X_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{i}/p^{n},1))} − b_{i}^{n}(f|V_{i}^{n}). Since v(b_{i}) = v(p^{k_{i}−1}/a_{i}) = (k_{i} − 1) − v(a_{i}) > 0 and since it follows from the Lemma 9 that, for all j, |f′|V_{i}^{j}| ⩽ |f′| < ∞ on X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]), g_{n} converges to give a section of ωk⃗ over X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]), which we shall denote by g. This is the (over)convergent form that we will glue to f. Unravelling the definition, one checks that g − b_{i}^{n}(g|V_{i}^{n}) in fact equals to ∑_{j=0}^{n−1} b_{i}^{j}(f|V_{i}^{j}) for any n ∈ Z_{⩾1}. However, observe that the former is defined over X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]) and therefore the latter is nothing other than the restriction of g_{n} to X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]).

In order to apply Kassaei’s gluing lemma, we need the following lemma.

Lemma 11. The g|V_{i}^{n} on X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]) and the f|V_{i}^{n} on X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{1}/p^{n}, 1)) are both uniformly bounded.

Proof. For any n ∈ Z_{⩾1}, |(g|V_{i}^{n})|_{X_{Γ_{1}(N)∩Γ_{0}(p)}([1,1])} ⩽ |g|_{X_{Γ_{1}(N)∩Γ_{0}(p)}([1,1])} < ∞. This follows from Lemma 9 and the quasi-compactness of X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]). See Lemma 4.4 and 4.5 in [24] for the second assertion; in [24], Lemma 3.3 is critical to the proof of Lemma 4.5 but we have its generalisation in Lemma 9 and one can argue similarly to prove the assertion. □

To summarise, for f ∈ H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}, 1)), ωk⃗), we have g_{n} ∈ H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([1 − t_{i}/p^{n}, 1]), ωk⃗) and g ∈ H^{0}(X_{Γ_{1}(N)∩Γ_{0}(p)}([1, 1]), ωk⃗) which satisfy |f − g_{n}|_{X_{Γ_{1}(N)∩Γ_{0}(p)}([1−t_{i}/p^{n},1))} → 0 and |g_{n} − g|_{X_{Γ_{1}(N)∩Γ_{0}(p)}([1,1])} → 0 as n → ∞. As a result, we have

Proposition 12. Let f ∈ Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1)). If it is eigenform for U_{v_{i}} with non-zero eigenvalue a_{i} such that v(a_{i}) < k_{i} − 1, then f extends to Mk⃗(Γ_{1}(N) ∩ Γ_{0}(p))([0, 1])

Proof. It follows from Lemma 3.2 in [24]; apply the lemma to affinoid coverings of X_{Γ_{1}(N)∩Γ_{0}(p)}([1− t_{i}, 1]). □

Essentially we can repeat the argument for the remaining indices.

Theorem 13. Let f be an overconvergent Hilbert modular form of weight k⃗ = (k_{i})_{1⩽i⩽g} and of level Γ_{1}(N) ∩ Γ_{0}(p). If it is an eigenform for U_{v_{i}} with non-zero eigenvalue a_{i} satisfying v(a_{i}) < k_{i} − 1 for all 1 ⩽ i ⩽ g, then f is a classical Hilbert eigenform.

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Shu Sasaki s.sasaki.03@cantabgold.net Max Planck Institut fur Mathematik, Vivatsgasse 7, 5311 Bonn, Germany