On Artin Representations

and Nearly Ordinary Hecke Algebras

over Totally Real Fields

Shu Sasaki

Received: May 29, 2012 Revised: May 19, 2013

Communicated by Don Blasius

Abstract. We prove many new cases of the strong Artin conjecture for two-dimensional, totally odd, insoluble (icosahedral) representations Gal(\overline{F} /F) → GL_{2}(C) of the absolute Galois group of a totally real field F.

2010 Mathematics Subject Classification: 11G80, 11F33, 11F41, 14G22, 14G35 Keywords and Phrases: Galois representations, p-adic modular forms, Hilbert modular varieties

1 Introduction

Let K be a number field. Artin conjectured that the L-series of any continuous representation ρ : Gal(\overline{K}/K) → GL_{n}(C) of the absolute Galois group Gal(\overline{K}/K) of K is holomorphic except a possible pole at s = 1 when the trivial representation is a constituent of ρ. A result of Brauer (See [36]) about finite groups immediately implies that L(ρ, s) has meromorphic continuation and satisfies a certain functional equation relating the values at s and 1 − s. Any such complex representation is semi-simple, and because Artin showed that L(ρ_{1} + ρ_{2}) = L(ρ_{1}, s)L(ρ_{2}, s), the conjecture immediately follows from the case where ρ is irreducible. In the case where ρ is irreducible, the strong form of this conjecture, known as the strong Artin conjecture, asserts that there is a cuspidal automorphic representation π of GL_{n}(A_{K}) such that L(π, s) = L(ρ, s), and Artin conjecture follows from the strong Artin conjecture (See [22], Theorem 8.8 with its proof (p.286) attributed to Ramakrishnan).

When n = 2 and the image of the projective representation proj ρ : Gal(\overline{K}/K) → P GL_{2}(C) = GL_{2}(C)/C^{×} is dihedral (D_{2n} for some n ≥ 2), ρ is induced from a character χ of the absolute Galois group Gal(\overline{K}/M) of a quadratic extension M of Q, and Artin himself proved the conjecture (the holomorphy of L(ρ, s) = L(Ind_{G_{M}}^{G_{K}} χ, s) = L(χ, s) follows from earlier work of Hecke). When n = 2 and the image of proj ρ is tetrahedral (A_{4}) and when n = 2, K = Q, ρ odd, and the projective image of ρ is octahedral (S_{4}), Langlands [23], using his theory of (cyclic) base change, “deduced” the strong Artin conjecture from the dihedral case. Tunnell, building on work of Langlands, completed the octahedral case n = 2 and general K. In the octahedral case, in order to “descend” a cuspidal automorphic representation Π of GL_{2}(A_{E}) such that L(Π, s) = L(ρ|_{Gal(\overline{K}/E)}, s) to a cuspidal automorphic representation π of GL_{2}(A_{K}), where E is the quadratic extension of K corresponding to the unique index 2 subgroup (≃ A_{4}) of S_{4}, Langlands uses a theorem of Deligne-Serre (and therefore K = Q and ρ should be necessarily odd) whilst Tunnell uses cubic base change to match up, for all but finitely many places v of K, the restriction of ρ to the decomposition group at v and the local representation π_{v}. The icosahedral (A_{5}) case had remained largely intractable until Buzzard-Dickinson-Shepherd-Barron-Taylor [4] proved many new cases of the strong Artin conjecture for odd ρ : Gal(\overline{Q}/Q) → GL_{2}(C).

[4] follows the program of Taylor ([37]), which may be succinctly described as an approach to deduce results about weight one forms from results about weight two forms (more specifically the idea of Wiles in [42]), and it is a culmination of a series of work: “R = T theorem for 2-adic ordinary finite flat representations” by Dickinson [10], “modularity of mod 2 icosahedral representations” by Shepherd-Barron and Taylor [33], and “modular lifting theorem for two-dimensional p-adic Artin representations unramifed at p (for any prime p)” by Buzzard and Taylor [5]. Buzzard [3] later extended [5] to treat almost all two-dimensional p-adic Artin representations potentially unramifed at p (the image of the inertia group at p is finite) and subsequently it led to modularity of two-dimensional “5-adic” icosahedral Artin representations by Taylor [39]. The strong Artin conjecture for odd two-dimensional representations of Gal(\overline{Q}/Q) is now completely proved by work of Khare-Wintenberger and Kisin on Serre’s conjecture for odd two-dimensional mod p representations of Gal(\overline{Q}/Q). In this paper, we push through Taylor’s program and generalise them to treat new cases of the strong Artin conjecture for two-dimensional, totally odd, icosahedral Artin representations of the absolute Galois group of a totally real field. More precisely, we prove the following theorems.

Theorem 1 Let F be a totally real field. Suppose that 5 splits completely in F. Suppose that ρ : Gal(\overline{F} /F) → GL_{2}(C) is a totally odd, irreducible, continuous representation satisfying the following conditions.

• The image of the projective representation proj ρ of ρ is A_{5}.

Artin Representations and T^{ord} over Totally Real … 999

• The projective image of the decomposition group at every place of F above 5 has order 2.

Then ρ arises from a holomorphic cuspidal Hilbert modular eigenform of weight 1 and the Artin L-function L(ρ, s) is entire.

Theorem 2 Let F be a totally real field. Suppose that 2 splits completely in F and that [F (ζ_{5}) : F] = 4. Suppose that ρ : Gal(\overline{F} /F) → GL_{2}(C) is a totally odd, irreducible, continuous representation satisfying the following conditions.

• The image of the porjective representation proj ρ of ρ is A_{5}.

• At every place 𝔭 of F above 2, the projective representation of ρ is unramified, and the image of Frob_{𝔭} has order 3.

Then ρ arises from a holomorphic cuspidal Hilbert modular eigenform of weight 1 and the Artin L-function L(ρ, s) is entire.

These are corollaries of the following theorems, first of which is about “if \overline{ρ} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{p}) is modular, then ρ : Gal(\overline{F} /F) → GL_{2}(\overline{Q}_{p}) ≃ GL_{2}(C) is modular”:

Theorem 3 Let p be a rational prime. Let K be a finite extension of Q_{p} with ring O of integers and maximal ideal 𝔪. Let F be a totally real field. Suppose that p splits completely in F. Let ρ : Gal(\overline{F} /F) → GL_{2}(O) be a continuous representation satisfying the following conditions.

• ρ ramifies at only finitely many primes.

• \overline{ρ} = (ρ mod 𝔪) is absolutely irreducible when restricted to Gal(\overline{F} /F (ζ_{p})), and has a modular lifting which is potentially ordinary and potentially Barsotti-Tate at every prime of F above p.

• For every prime 𝔭 of F above p, the restriction ρ|_{G_{𝔭}} to the decomposition group G_{𝔭} at 𝔭 is the direct sum of 1-dimensional characters χ_{𝔭,1} and χ_{𝔭,2} of G_{𝔭} such that the images of the inertia subgroup at 𝔭 are finite and (χ_{𝔭,1} mod 𝔪) ≠ (χ_{𝔭,2} mod 𝔪).

If p = 2, assume moreover the following conditions.

• The image of the complex conjugation, with respect to every embedding of F into R, is not the identity matrix.

• \overline{ρ} has insoluble image.

• For every prime 𝔭 of F above 2, ρ is unramified at 𝔭.

Then there exists an embedding ι : K ↪ \overline{Q}_{p} ≃ C and a classical holomorphic cuspidal Hilbert modular eigenform f of weight 1 such that ι ∘ ρ is isomorphic to the representation associated to f by Rogawski-Tunnell [28].

In proving the theorem, we shall firstly establish R = T theorems for Hida p-ordinary families over a finite soluble totally real extension F_{Σ} of F in which p ≥ 2 remains split completely–for lack of reference we shall prove them. Since \overline{ρ} has a potentially p-Barsotti-Tate and potentially p-ordinary modular lifting, one can deduce R = T in p-adic families from Kisin’s R = T theorems in the p-Barsotti-Tate case. Note that, unfortunately, it is not possible to make appeal to Geraghty’s R = T theorems in p-ordinary families as they assume that p > 2 and that \overline{ρ} is trivial at every prime of F above p. This is because one can not eliminate the possibility that, upon ‘soluble’ base-changing to F_{Σ} to set \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} trivial at every prime of F_{Σ} above p, F_{Σ} may no longer be split at p, which is crucial in constructing weight one forms in our approach. In the light of [1], the condition about the existence of a potentially ordinary Barsotti-Tate lifting of \overline{ρ} can be weaker, more precisely, it suffices to assume ‘\overline{ρ} is modular’. It is not necessary to make appeal to their results however.

The next two theorems are about modularity of \overline{ρ}.

Theorem 4 Let F be a totally real field. Suppose that 5 is unramified in F. Let \overline{ρ} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{5}) be a continuous representation of satisfying the following conditions.

• \overline{ρ} is totally odd.

• \overline{ρ} has projective image A_{5}.

The there exists a cuspidal Hilbert modular eigenform of weight 2 such that its associated 5-adic Galois representation is potentially Barsotti-Tate and potentially ordinary at every prime of F above 5, and its associated mod 5 Galois representation is isomorphic to \overline{ρ}.

The idea is exactly the same as that of Taylor–to prove modularity of \overline{ρ}, one firstly finds an elliptic curve E over a finite soluble totally real field extension F_{Σ} of F such that the action of Gal(\overline{F} /F_{Σ}) on the 5-torsion points of E is isomorphic to \overline{ρ}|_{Gal(\overline{F} /F_{Σ})}; secondly one proves E modular, therefore \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} modular; and finally it follows from Khare-Wintenberger [18] and Kisin [20] that \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} has a characteristic zero lifting which is modular. The ‘automorphic descent’ works as in [39]. In proving E is modular, we make some technical improvements on a ‘naive’ generalisation over totally real fields of the main theorem of Taylor in [39] by making appeal to the main result of Kisin [20] rather than the main result of Skinner-Wiles [35]. While Taylor/Skinner-Wiles requires the mod 3 representation E[3](\overline{F}_{Σ}) of Gal(\overline{F} /F_{Σ}) to be reducible with distinct characters on the diagonal at every prime of F_{Σ} above 3, we no longer requires this and consequently remove the ‘3-distinguishedness condition’ in the main theorems of [39]. The key observation is that the weight 2 specialisation F_{H,2} of the Hida family F_{H}, whose weight 1 specialisation F_{H,1} renders E[3](\overline{F}_{Σ}) modular by Langlands-Tunnell, does indeed render the 3-adic Barsotti-Tate representation T_{3}E ‘strongly residually modular’ in the sense of Kisin [20] if E[3](\overline{F}_{Σ}) is unramified at every prime above 3.

As is clear from its proof, what we are proving indeed is modularity of general mod 5 representations Gal(\overline{F} /F) → GL_{2}(F_{5}), and this allows us to work with the prime 2–proving modularity of \overline{ρ}_{2} : Gal(\overline{F} /F) → SL_{2}(F_{4}) with proj \overline{ρ}_{2} ≃ A_{5}–instead of the prime 5, going back to the original approach of Buzzard-Dickinson-Shepherd-Barron-Taylor; in [4] one firstly finds an abelian surface A over F with real multiplication Z[(1 + \sqrt{5})/2] such that A(\overline{F})[2] ≃ \overline{ρ}_{2}; secondly proves the mod 5 representation Gal(\overline{F} /F) → GL_{2}(A(\overline{F})[\sqrt{5}]) ≃ GL_{2}(F_{5}) is modular; and deduce A is modular by a modular lifting theorem.

Theorem 5 Let F be a totally real field. Suppose that [F (ζ_{5}) : F] = 4. Let \overline{ρ} : Gal(\overline{F} /F) → SL_{2}(F_{4}) be a continuous representation. Then there exists a cuspidal Hilbert modular eigenform of weight 2 such that its associated 2-adic Galois representation is potentially Barsotti-Tate and potentially unramified at every prime of F above 2 and its associated mod 2 Galois representation is isomorphic to \overline{ρ}.

Lastly it might come in useful comparing our work and others. After the first draft of this paper was written in 2010, Kassaei announced a result proving an analogue of the main theorem 3 in the case when p is odd, p is unramified in F, and χ_{𝔭,1}/χ_{𝔭,2} and χ_{𝔭,2}/χ_{𝔭,1} are both unramified at every prime 𝔭 of F above p. Pilloni, on the other hand, seems to have proved a slightly stronger analogue in which p is allowed to ramify a little in F. The fundamental ideas in both works and ours are essentially the same and are due to Buzzard, more specifically to Buzzard’s Theorem 9.1 in [3]. In forthcoming joint work with Kassaei and Tian, we extend Kassaei’s work to the case where χ_{𝔭,1}/χ_{𝔭,2} and χ_{𝔭,2}/χ_{𝔭,1} are of conductor 𝔭 for every prime 𝔭 of F above p (unramified in F) and prove many new cases of the strong Artin conjecture for ρ : Gal(\overline{F} /F) → GL_{2}(C) in the insoluble case as above. To prove an analogue of the main theorem 3 in the case where χ_{𝔭,1}/χ_{𝔭,2} and χ_{𝔭,2}/χ_{𝔭,1} are of conductor 𝔭^{r} with r > 1 for every prime 𝔭 of F above p, one needs to know precise geometry of Hilbert modular varieties of level p^{r} and, unless p splits completely in F which we solve, this may not even be possible. Calculating q-expansions at cusps to glue weight one forms does not seem to depend on the ramification of p in F and, for that, this work is very useful in general. On the other hand, in order to prove the general case (p ramifies arbitrarily in F), the author [30] considers new moduli spaces of Hilbert-Blumenthal abelian varieties; and he expects to make progress in the general case in his forthcoming work.

Acknowledgements The author would like to thank Kevin Buzzard, Fred Dimanond, and Payman Kassaei for extremely helpful comments whilst the author was preparing this paper. He would also like to thank David Geraghty and Wansu Kim for their help. Finally, he would like to thank the referee for very helpful comments which led to many improvements in the paper. The author was supported financially by EPSRC (UK) whilst he was preparing the manuscript. He would like to thank their support. He would also like to thank Institut Henri Poincaré (Paris, France) and King’s College London (London, UK) for the financial support and hospitality they offered.

2 Modularity of mod 5 icosahedral representations of Gal(\overline{F} /F)

Lemma 6 Let F be a totally real field. Suppose that 5 is unramified in F. Suppose that \overline{ρ} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{5}) is a continuous representation satisfying the following conditions.

• \overline{ρ} is totally odd.

• \overline{ρ} has projective image A_{5}.

Then there is a finite soluble totally real field extension F_{Σ} of F and an elliptic curve E over F_{Σ} such that

• F (\sqrt{5}) ⊂ F_{Σ} ⊂ \overline{F}, and \sqrt{5} splits completely in F_{Σ};

• E has good ordinary reduction at every prime of F above 3 and has potentially ordinary reduction at every prime of F above 5;

• \overline{ρ}_{E,5} : Gal(\overline{F} /F_{Σ}) → Aut(E(\overline{F}_{Σ})[5]) is equivalent to a twist of \overline{ρ}|_{Gal(\overline{F} /F_{Σ})};

• \overline{ρ}_{E,3}|_{Gal(\overline{F} /F_{Σ}(ζ_{3}))} : Gal(\overline{F} /F_{Σ}(ζ_{3})) → Aut(E(\overline{F}_{Σ})[3]) is absolutely irreducible.

Proof. Firstly, as in [39], find a biquadratic totally real extension K_{1} ⊂ \overline{F} of F, which is a quadratic totally real extension of F (\sqrt{5}) in which \sqrt{5} splits completely, such that proj \overline{ρ} : Gal(\overline{F} /K_{1}) → P SL_{2}(F_{5}) ≃ A_{5} lifts to a representation \overline{ρ}_{1} : Gal(\overline{F} /K_{1}) → GL_{2}(F_{5}) with determinant the mod 5 cyclotomic character ϵ. Choose, by class field theory, a finite soluble totally real extension K_{2} ⊂ \overline{F} of K_{1} such that \overline{ρ}_{1}|_{Gal(\overline{F} /K_{2})} is trivial when restricted to the decomposition group at every prime of K_{2} above 3. Let F_{Σ} denote the Galois closure of K_{2} over F. Let \overline{ρ}_{Σ} denote the restriction of \overline{ρ} to Gal(\overline{F} /F_{Σ}). As in section 1 of [33], let Y_{\overline{ρ}_{Σ}}/F_{Σ} (resp. X_{\overline{ρ}_{Σ}}/F_{Σ}) denote the twist of the (resp. compactified) modular curve Y_{5} (resp. X_{5}) with full level 5 structure by the cohomology class in H^{1}(Gal(\overline{F} /F_{Σ}), Aut X_{5}) defined by an isomorphism \overline{ρ}_{Σ} ≃ (Z/5Z) × μ_{5} of the F_{5}-vector spaces. As proved in Lemma 1.1 in [33], the ‘twist’ cohomology class is indeed trivial, and therefore X_{\overline{ρ}_{Σ}} ≃ X_{5} and Y_{\overline{ρ}_{Σ}} is isomorphic over F_{Σ} to a Zariski open subset of the projective line P^{1}. In particular, Y_{\overline{ρ}_{Σ}} has infinitely many rational points. Let Y_{\overline{ρ}_{Σ},0}(3) denote the degree 4 cover over Y_{\overline{ρ}_{Σ}} which parameterises isomorphism classes of elliptic curves E equipped with an isomorphism E[5] ≃ \overline{ρ}_{Σ} taking the Weil pairing on E[5] to ϵ : ∧^{2}\overline{ρ}_{Σ} → μ_{5} and a finite flat subgroup scheme C ⊂ E[3] of order 3. Let Y_{\overline{ρ}_{Σ},split}(3) denote the étale cover over Y_{\overline{ρ}_{Σ}} which parameterises isomorphism classes of elliptic curves E equipped with an isomorphism E[5] ≃ \overline{ρ}_{Σ} taking the Weil pairing on E[5] to ϵ : ∧^{2}\overline{ρ}_{Σ} → μ_{5} and an unordered pair, fixed by Gal(\overline{F} /F_{Σ}), of finite flat subgroup schemes C, D ⊂ E[3] of order 3 which intersect trivially. Then it follows from Lemma 12 in [27] that Y_{\overline{ρ}_{Σ},split}(3) and Y_{\overline{ρ}_{Σ},0}(3) has only finitely many rational points. For every prime 𝔭 of F_{Σ} above 3, the elliptic curve y^{2} = x^{3} + x^{2} − x defines an element of Y_{\overline{ρ}_{Σ}}(F_{Σ,𝔭}) with good ordinary reduction, and we let U_{𝔭} ⊂ Y_{\overline{ρ}_{Σ}}(F_{Σ,𝔭}) denote a (non-empty) open neighbourhood (for the 3-adic topology) of the point, consisting of elliptic curves with good ordinary reduction at 𝔭. For every prime 𝔭 of F_{Σ} above 5, we define a non-empty open subset (for the 5-adic topology) U_{𝔭} ⊂ Y_{\overline{ρ}_{Σ}}(F_{Σ,𝔭}) as in the proof of Lemma 2.3 in [39]. By Hilbert irreducibility theorem (Theorem 1.3 in [11]; see also Theorem 3.5.7 in [32]), we may then find a rational point in Y_{\overline{ρ}_{Σ}}(F_{Σ}) which lies in U_{𝔭} for every 𝔭 of F_{Σ} above either 3 or 5 and does not lie in the images of Y_{\overline{ρ}_{Σ},0}(3)(F_{Σ}) → Y_{\overline{ρ}_{Σ}}(F_{Σ}) and Y_{\overline{ρ}_{Σ},split}(3)(F_{Σ}) → Y_{\overline{ρ}_{Σ}}(F_{Σ}). The elliptic curve over F_{Σ} corresponding to the rational point is what we are looking for. □

Theorem 7 Let F be a totally real field. Suppose that 5 is unramified in F. Let \overline{ρ} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{5}) be a continuous representation of satisfying the following conditions.

• \overline{ρ} is totally odd.

• \overline{ρ} has projective image A_{5}.

Then there exists a cuspidal Hilbert modular eigenform of weight 2 such that its associated 5-adic Galois representation is potentially Barsotti-Tate and potentially ordinary at every prime of F above 5 and its associated mod 5 Galois representation is isomorphic to \overline{ρ}.

Proof. Choose an elliptic curve over a finite soluble totally real extension F_{Σ} of F as in the lemma. Replace F_{Σ} by its finite soluble totally real extension if necessary to assume that the mod 3 representation \overline{ρ}_{E,3} of Gal(\overline{F} /F_{Σ}) on E(\overline{F}_{Σ})[3] is unramified when restricted to the decomposition subgroup at every prime of F_{Σ} above 3. By the Langlands-Tunnell theorem, there exists a weight 1 holomorphic cuspidal Hilbert modular eigenform f_{1} which gives rise to \overline{ρ}_{E,3}. By 3-adic Hida theory [14], we may find a holomorphic cuspidal Hilbert modular eigenform f_{2} of weight 2 and level prime to 3, ordinary at every prime of F_{Σ} above 3, which gives rise to \overline{ρ}_{E,3}. As E is ordinary at 3, ρ_{E,3} is strongly residually modular in the sense of Kisin [20] (3.5.4), and it follows from Theorem 3.5.5 in [20] that T_{3}E is modular. By Faltings’ isogeny theorem, E is therefore modular. As \overline{ρ}_{E,5} is modular, \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} is modular. Since F_{Σ} is a soluble extension of F, \overline{ρ}_{Σ} remains absolutely irreducible when restricted to Gal(\overline{F} /F_{Σ}(ζ)). Furthermore, since 5 is unramfied in F, the kernel of proj \overline{ρ}_{Σ} does not fix F_{Σ}(ζ_{5}). It then follows from results of Khare-Wintenberger [18] and Kisin [20] that there exists a modular lifting of \overline{ρ}_{Σ}. The ‘soluble descent’ to F is exactly as in [39]. □

Remark. In the forthcoming work with Kassaei and Tian, we remove the assumption that 5 is unramified in F in Lemma 6, and thereby in Theorem 7. Essentially the same argument works.

3 Modularity of mod 2 icosahedral representations of Gal(\overline{F} /F)

Theorem 8 Let F be a totally real field. Suppose that [F (ζ_{5}) : F] = 4. Let \overline{ρ} : Gal(\overline{F} /F) → SL_{2}(F_{4}) be a continuous representation. Then there exists a cuspidal Hilbert modular eigenform of weight 2 such that its associated 2-adic Galois representation is potentially Barsotti-Tate and potentially unramified at every prime of F above 2 and its associated mod 2 Galois representation is isomorphic to \overline{ρ}.

Proof. By Theorem 3.4 in [33], there exists a principally polarised abelian surface A over F with real multiplication by Z[(1+\sqrt{5})/2] compatible with the polarisation such that the action of Gal(\overline{F} /F) on A(\overline{F})[2] ≃ F_{4}^{2} is equivalent to \overline{ρ}; and the action of Gal(\overline{F} /F) on A(\overline{F})[\sqrt{5}] ≃ F_{5}^{2} is given via a homomorphism

\overline{ρ}_{A,\sqrt{5}} : Gal(\overline{F} /F) → GL_{2}(F_{5})

which is surjective and whose image contains SL_{2}(F_{5}). It suffices to prove that A is modular. Firstly, the Weil pairing on A(\overline{F})[\sqrt{5}] shows that det\overline{ρ}_{A,\sqrt{5}} is the mod 5 cyclotomic character. Since [F (ζ_{5}) : F] = 4, the determinant is indeed surjective, and therefore \overline{ρ}_{A,\sqrt{5}} is absolutely irreducible. If \overline{ρ}_{A,\sqrt{5}} is irreducible at some place of F above 5, the absolute irreducibility of \overline{ρ}_{A,\sqrt{5}} implies the absolute irreducibility of its restriction to Gal(\overline{F} /F (ζ_{5})). Otherwise, \overline{ρ}_{A,\sqrt{5}} is reducible at every place of F above 5; in which case, it is also equally easy to check that its restriction to Gal(\overline{F} /F (\sqrt{5})) of \overline{ρ}_{A,\sqrt{5}} is absolutely irreducible (See Proposition 7 in [27], for example). It follows from results of Khare-Wintenberger [16] and Kisin [20] that it is possible to construct a modular lifting of \overline{ρ}_{A,\sqrt{5}}; more precisely, \overline{ρ}_{A,\sqrt{5}} is strongly residually modular. The modularity of ρ_{A,\sqrt{5}} follows from Theorem 3.5.5 in [20] and [12]. □

4 Holomorphic Hilbert modular forms and Hida theory of modular Galois representations

Let F be a totally real field. We let O_{F} denote the ring of integers, 𝔡_{F} the different of F, A_{F} = A_{F}^{∞} × F_{∞}, and O_{F}^{∧} denote O_{F} ⊗_{Z} Z^{∧} ⊂ A_{F}^{∞}. Let S_{∞} denote the set of infinite places of F. For an ideal 𝔫 of O_{F}, let F_{𝔫} denote the strict ray class field of conductor 𝔫S_{∞}. For an ideal 𝔫, let U^{1}(𝔫) (resp. U_{1}(𝔫)) denote the open compact subgroup of GL_{2}(O_{F}^{∧}) consisting of matrices which are congruent modulo 𝔫O_{F}^{∧} to matrices with first column (1, 0) (resp. the second row (0, 1)). Let I_{𝔫} denote A_{F}^{×}/(F^{×}(A_{F}^{∞×} ∩ U^{1}(𝔫))F_{∞}^{+×}). For k ∈ Z and an open compact subgroup U of GL_{2}(O_{F}^{∧}) , let S_{k}(U) denote the space, S_{k,k/2}(U) in the sense of Hida [14], of cuspidal holomorphic Hilbert modular forms f of parallel weight k and level U with the Fourier coefficient c(𝔫, f) ∈ Z for all ideals 𝔫 of O_{F}. The spaces S_{k}(U^{1}(𝔫)) and S_{k}(U_{1}(𝔫)) for an ideal 𝔫 of O_{F} come equipped with an action of I_{𝔫} via the diamond operator ⟨ ⟩, and Hecke operators T_{𝔮} for every prime 𝔮 of O_{F} not dividing 𝔫 and U_{𝔮} for every prime of 𝔮 dividing 𝔫. Let h_{k}(𝔫) denote the sub Z-algebra of End(S_{k}(U^{1}(𝔫)) generated over Z by all these operators (See Proposition 2.3, Theorem 4.10, and Theorem 4.11 of [14]). For every prime 𝔮 not dividing 𝔫, let S_{𝔮} = (N_{F/Q}𝔮)^{k−2}⟨𝔮⟩ ∈ h_{k}(𝔫); this corresponds to the action of the scalar matrix with a uniformiser of O_{F} at 𝔮 on the diagonal. Following [14], for every ideal 𝔪 of O_{F}, we may define T_{𝔪} ∈ h_{k}(𝔫).

Let p be a rational prime and let S_{P} denote the set of prime ideals of O_{F} dividing p. Fix an algebraic closure \overline{Q}_{p}, an isomorphism \overline{Q}_{p} ≃ C, and an embedding \overline{Q} → \overline{Q}_{p}.

For a ring R ⊂ \overline{Q}_{p}, we shall let S_{k}(U_{1}(𝔫))_{R} denote S_{k}(U_{1}(𝔫)) ⊗_{Z} R and h_{k}(𝔫)_{R} denote h_{k}(𝔫) ⊗_{Z} R; there is a pairing ( , ) : h_{k}(𝔫)_{R} × S_{k}(U_{1}(𝔫))_{R} → R defined by (T, f) = c(O_{F} , T f).

For a ray class character ψ : I_{𝔫} → \overline{Q}_{p}^{×} mod 𝔫S_{∞}, let S_{k,ψ}(U_{1}(𝔫))_{Z_{p}[ψ]} denote the submodule of S_{k}(U_{1}(𝔫_{1}))_{Z_{p}[ψ]} consisting of cuspidal Hilbert modular forms of parallel weight k and level U_{1}(𝔫) with central character ψ–S_{𝔮} acts via ψ at 𝔮; the forms in S_{k,ψ}(U_{1}(𝔫))_{Z_{p}[ψ]} may be thought of as |I_{𝔫}|- tuple of classical Hilbert modular forms of ‘level Γ_{1}(𝔫)’ on the |I_{𝔫}|-copies of (GL_{2}(R)/(R^{×}SO_{2}(R))^{Hom(F,R)} with ‘Dirichlet character mod 𝔫’.

Fix an ideal 𝔫 of O_{F} coprime to p. For a finite extension K of Q_{p} with ring O of integers, Hida [14] defines the idempotent e and we set h_{O}^{0}(𝔫) to be the inverse limit with respect to r ∈ Z_{≥1} of h_{2}(𝔫p^{r})_{Z_{p}} ⊗_{Z_{p}} O. Let I_{𝔫p^{∞}} denote the inverse limit of the I_{𝔫p^{r}} and the diamond operators ⟨ ⟩ : I_{𝔫p^{r}} → eh_{2}(𝔫p^{r})_{O} induce

⟨ ⟩ : I_{𝔫p^{∞}} → h_{O}^{0}(𝔫)^{×}.

One can also see ⟨ ⟩ as the action of (O_{F} /𝔫)^{×} ×(O_{F} ⊗_{Z} Z_{p})^{×} by the composite:

(O_{F} /𝔫)^{×} × (O_{F} ⊗_{Z} Z_{p})^{×} → I_{𝔫p^{∞}} ⟶^{⟨ ⟩} h_{O}^{0}(𝔫).

We let Tor_{𝔫p^{∞}} (resp. Fr_{𝔫p^{∞}}) denote the torsion subgroup (resp. the maximal Z_{p} free subgroup of rank 1 + δ with δ = 0 if the Leopoldt conjecture holds) of I_{𝔫p^{∞}}; let Λ denote the completed group algebra over Z_{p} of Fr_{𝔫p^{∞}} and Λ_{K} = Λ ⊗_{Z_{p}} O. Then h_{O}^{0}(𝔫) is a Λ_{K}-module by ⟨ ⟩. We will let

Art : A_{F}^{×}/\overline{F^{×}F_{∞}^{+×}} ≃ Gal(\overline{F} /F)^{ab}

denote the (global) Artin map, normalised compatibly with the local Artin maps normalised to take uniformisers to arithmetic Frobenius elements. By abuse of notation, we shall let Art also denote the induced homomorphism I_{𝔫p^{∞}} → Gal(F_{𝔫}(μ_{p^{∞}})/F) and let ϵ denote the cyclotomic character ϵ : Gal(F_{𝔫}(μ_{p^{∞}})/F) → Z_{p}^{×}.

Hida [14] proves that h_{O}^{0}(𝔫) is a torsion free Λ_{K}-module and, for a character ψ : I_{𝔫p^{∞}} → K which factors through I_{𝔫p^{r}} for r ∈ Z_{≥1}, if k ≥ 2, then h_{O}^{0}(𝔫)_{ker((ϵ∘Art)^{k−2}ψ)} is isomorphic to the subspace of eS_{k}(U_{1}(𝔫p^{r}))_{O} where ⟨ ⟩ = ψ on Fr_{𝔫p^{∞}}. We will let ϵ^{cyclo} denote the character

Gal(\overline{F} /F) ↠ Gal(\overline{F} /F)^{ab} ↠ I_{𝔫p^{∞}} ↪ O[[I_{𝔫p^{∞}}]]^{×} = Λ_{K}[Tor_{𝔫p^{∞}}]^{×}.

Note that 𝔮 ↦ N𝔮S_{𝔮} extends to NS : (O_{F} /𝔫)^{×} × (O_{F} ⊗_{Z} Z_{p})^{×} → I_{𝔫p^{∞}} → h_{O}^{0}(𝔫)^{×}. Let NS_{Σ} (resp. NS^{P}) denote the Σ (resp. the prime to S_{P}) part ∏_{𝔭∈Σ} O_{F_{𝔭}}^{×} → h_{O}^{0}(𝔫) (resp. (O_{F} /𝔫)^{×} → h_{O}^{0}(𝔫)) for s subset Σ of S_{P}.

If 𝔪 is a maximal ideal of h_{O}^{0}(𝔫) with residue field k_{𝔪}, there is a continuous representation

\overline{ρ}_{𝔪} : G_{F} = Gal(\overline{F} /F) → GL_{2}(k_{𝔪})

such that, for every prime ideal 𝔮 of O_{F} not dividing 𝔫p, \overline{ρ}_{𝔪} is unramified at 𝔮 and tr\overline{ρ}_{𝔪}(Frob_{𝔮}) = T_{𝔮}. Set S_{O}^{0}(𝔫) = Hom_{Λ_{K}} (h_{O}^{0}(𝔫), Λ_{K}). For a finite field extension L of the field Frac Λ_{K} of fractions of Λ_{K} with its integral closure O_{L} of Λ_{K} in L, Buzzard-Taylor [5] calls a Λ_{K}-algebra homomorphism F_{H} ∈ S_{O}^{0}(𝔫) ⊗_{Λ_{K}} L = Hom_{Λ_{K}} (h_{O}^{0}, L) a Λ-adic eigenform (of level 𝔫). If the unique maximal ideal 𝔪 above ker F_{H} ⊂ h_{O}^{0}(𝔫) is non-Eisenstein, i.e., \overline{ρ}_{𝔪} as above is absolutely irreducible, then there is a continuous representation

ρ_{F_{H}} : G_{F} → GL_{2}(h_{O}^{0}(𝔫)_{𝔪}) →^{F_{H}} GL_{2}(O_{L})

which is unramified at every prime ideal 𝔮 of O_{F} not dividing 𝔫p and satisfies tr ρ_{F_{H}}(Frob_{𝔮}) = T_{𝔮} and det ρ_{F_{H}} = (NS)∘ϵ^{cyclo}. Moreover, it is a result of Wiles [43] that, for every place 𝔭 of F above p, the restriction to the decomposition group G_{𝔭} at 𝔭 of ρ_{F_{H}} is of the form

(\begin{matrix}χ_{F_{H},𝔭,2} & ∗ \\ 0 & χ_{F_{H},𝔭,1} \end{matrix})

where χ_{F_{H},𝔭,1} is an unramified character of G_{𝔭} such that χ_{F_{H},𝔭,1}(Frob_{𝔭}) = U_{𝔭} and χ_{F_{H},𝔭,1}χ_{F_{H},𝔭,2} = (F_{H} ∘ NS) ∘ ϵ^{cyclo}|_{G_{𝔭}}.

Definition. Following [5], we call two Λ-adic eigenforms F_{H,1} and F_{H,2} : h_{O}^{0}(𝔫) → O_{L} of level 𝔫 Λ-adic companion form with respect to primes ℘_{1} and ℘_{2} of O_{L} which do not divide p, if there are embeddings ι_{1} : O_{L}/℘_{1} ↪ \overline{Q}_{p} and ι_{2} : O_{L}/℘_{2} ↪ \overline{Q}_{p} such that, for every ideal 𝔪 of O_{F} not dividing p, there exists a subset Σ of S_{P} such that (F_{H,2}(T_{𝔪}) mod ℘_{2}) = (F_{H,1}(T_{𝔪}(NS_{Σ})(𝔪)^{−1}) mod ℘_{1}); and such that, for every place 𝔭 in Σ, (F_{H,2}(U_{𝔭}) mod ℘_{2}) = (F_{H,1}(U_{𝔭}^{−1}(NS^{P})(𝔭)) mod ℘_{1}) while, for every 𝔭 in S_{P} − Σ, (F_{H,2}(U_{𝔭}) mod ℘_{2}) = (F_{H,1}(U_{𝔭}) mod ℘_{1}).

5 Deformation rings and Hecke algebras

Let F be a totally real field of even degree in which p is unramified; if p = 2 assume furthermore that 2 splits completely in F. If p is odd, suppose p ≥ 5. Let D be the quaternion algebra over F which ramifies exactly at a finite set Σ of finite places of F not dividing p and the infinite places S_{∞} of F. Let O_{D} denote a maximal order and fix an isomorphism O_{D𝔮} ≃ M_{2}(O_{F_{𝔮}}) for 𝔮 not in Σ. Let S denote the disjoint union of Σ, the set S_{P} of places of F above p, and the infinite places of F. For a topological Z_{p}-algebra R, let ψ : A_{F}^{∞,×}/F → R^{×} be a continuous character such that ψ|_{O_{F𝔭}^{×}} is trivial for every place 𝔭 of F above p, and, for an open

compact subgroup U = ∏_{𝔮} U_{𝔮} ⊂ ∏_{𝔮} O_{D𝔮}^{×}, let S_{2,ψ}^{D}(U)_{R} denote the R-module of R-valued modular forms on D^{×}\(D ⊗_{F} A_{F}^{∞})^{×} of weight 2 and of level U in the sense of Taylor [40]. Let 𝔫_{Σ} denote the square-free product of the primes in Σ and define U_{Σ} ⊂ (D ⊗_{F} A_{F}^{∞})^{×} by U_{Σ,𝔮} = GL_{2}(O_{F 𝔮}) for 𝔮 not in Σ; and U_{Σ,𝔮} = O_{D𝔮}^{×} for 𝔮 ∈ Σ.

We shall write S_{2,ψ}^{D}(𝔫_{Σ}) for S_{2,ψ}^{D}(U_{Σ}) and h_{2,ψ}^{D}(𝔫_{Σ})_{R} for the R-subalgebra of End_{R}(S_{2,ψ}^{D}(𝔫_{Σ})_{R}) generated by T_{𝔮} and S_{𝔮} for all 𝔮 not in S; and T_{𝔭} and S_{𝔭} for all 𝔭 in S_{P}.

Let K be a finite extension of Q_{p} and O be the ring of integers with maximal ideal 𝔪 ad residue field k. Let

ρ : Gal(\overline{F} /F) → GL_{2}(O)

be a continuous representation such that

• \overline{ρ} = (ρ mod 𝔪) is unramified outside S_{P},

• \overline{ρ} is not scalar at every place 𝔭 above p,

• if p is odd, the restriction of \overline{ρ} to Gal(\overline{F} /F (ζ_{p})) is absolutely irreducible; if p = 2, \overline{ρ} has insoluble image, • there exists a holomorphic automorphic representation π of (D ⊗_{F} A_{F})^{×}

generated by a cusp form in S_{2,ψ}^{D}(𝔫_{Σ})_{O} such that π_{𝔮} is unramified for every 𝔮 not in Σ ∪ S_{P}, π_{𝔭} is ordinary at every 𝔭 in S_{P}, for every 𝔮 ∈ Σ, π_{𝔮} corresponds by the local Jacquet-Langlands correspondence to a special representation of conductor 𝔮, and such that \overline{ρ}_{π} ≃ \overline{ρ},

• ρ ramifies at Σ and possibly at S_{P}; for every 𝔭 in S_{P}

ρ|_{G_{𝔭}} ∼ (\begin{matrix}∗ & ∗ \\ 0 & χ_{𝔭} \end{matrix})

with χ_{𝔭} unramified; and for 𝔮 ∈ Σ

ρ|_{G_{𝔮}} ∼ (\begin{matrix}ϵχ_{𝔮} & ∗ \\ 0 & χ_{𝔮} \end{matrix})

with χ_{𝔮} unramified such that χ_{𝔮}^{2} = (ψ ∘ Art)|_{G_{𝔮}}.

Let A_{F}^{×} = A_{F}^{×S} × A_{F S}^{×} for a finite subset S of the places of F. Let ψ be a character of A_{F}^{×S_{P}}. For p = 2 let ψ_{P,±} denote the Z_{p}-linear extension of the norm N : (O_{F} ⊗_{Z} Z_{2})^{×} → Z_{2}^{×} followed by the character Z_{2}^{×} → Z_{p}^{×} whose restriction to (Z/4)^{×} = {±1} sends −1 to ∓1 and whose restriction to (1 + 4Z_{2})^{×} is trivial. For p odd, let ψ_{P} denote the norm followed by the trivial character on Z_{p}^{×}.

5.1 (Framed) deformation rings R

\underline{𝔭|p}: if p is odd, let R_{𝔭}^{□,ord} (resp. R_{𝔭}^{□,BT,ord}) denote the O-algebra which represents the 𝔭-ordinary (resp. Barsotti-Tate 𝔭-ordinary) framed deformations of \overline{ρ}|_{G_{𝔭}} of the form

(\begin{matrix}∗ & ∗ \\ 0 & χ_{𝔭}^{ur} \end{matrix})

with an unramified lifting \overline{χ}_{𝔭}^{ur} of χ_{𝔭} (resp. and its determinant is ϵψ_{P}); if p = 2, we shall write ‘±’ in shorthand to mean two independent cases– ‘+’ corresponds to the 2-old case while ‘−’ corresponds to the 2-new case in the sense to be made precise below, and let R_{𝔭,±}^{□,ord} (resp. R_{𝔭,±}^{□,BT,ord}) denote the complete local noetherian O-algebra which represents the 𝔭-ordinary (resp. Barsotti-Tate 𝔭ordinary) liftings of \overline{ρ}|_{G_{𝔭}} of the form

(\begin{matrix}∗ & ∗ \\ 0 & χ_{𝔭}^{ur} \end{matrix})

with an unramified lifting χ_{𝔭}^{ur} of χ_{𝔭}, and with its determinant corresponding,

by the local class field theory, to the norm O_{F_{𝔭}}^{×} ↪ (O_{F} ⊗_{Z} Z_{2})^{×} →^{N} Z_{2}^{×} followed by the character Z_{2}^{×} → Z_{2}^{×} whose restriction to (Z/4)^{×} = {±1} sends −1 to

Artin Representations and T^{ord} over Totally Real … 1009 ∓1 (resp. with its determinant ϵψ_{P,±}). Let R_{P}^{□,ord} = ⨂_{𝔭∈S_{P}}^{∧} R_{𝔭}^{□,ord} (resp. R_{P}^{□,BT,ord} = ⨂_{𝔭∈S_{P}}^{∧} R_{𝔭}^{□,BT,ord}) if p is odd; and R_{P,±}^{□,ord} = ⨂_{𝔭∈S_{P}}^{∧} R_{𝔭,±}^{□,ord} (resp. R_{P}^{□,BT,ord} = ⨂_{𝔭∈S_{P}}^{∧} R_{𝔭,±}^{□,BT,ord}) if p = 2. \underline{𝔮 ∈ Σ}: let R_{𝔮}^{□,ψ} denote the domain (see 2.6 in [20], or Proposition 2.12 and 3.3.4 in [18]) parameterising liftings of \overline{ρ}|_{G_{𝔮}} of the form

(\begin{matrix}ϵχ_{𝔮}^{ur} & ∗ \\ 0 & χ_{𝔮}^{ur} \end{matrix})

with χ_{𝔮}^{ur} an unramified lifting of \overline{χ}_{𝔮} such that (χ_{𝔮}^{ur})^{2} = (ψ ∘ Art^{−1})|_{G_{𝔮}}. Let R_{Σ}^{□,ψ} denote the completed tensor product ⨂_{𝔮∈Σ}^{∧} R_{𝔮}^{□,ψ}. \underline{τ|∞}: let R_{τ}^{□,odd} denote the formally smooth ring which represents the liftings of \overline{ρ}|_{G_{τ}} which, if p is odd, are odd ; and, if p = 2, the image of complex conjugation in G_{τ} ≃ Gal(C/R) is not the identity matrix. Let R_{∞}^{□,odd} denote the completed tensor product ⨂_{τ|∞}^{∧} R_{τ}^{□,odd} Fix a k-basis of \overline{ρ} and let

ρ^{□S} : G_{F} → GL_{2}(R^{□S})

denote the S-framed universal deformation ring. Let R_{S}^{□} denote the completed tensor product of the local framed deformation rings at places in S. Let

R_{S}^{□,ord,ψ} = R^{□S} ⊗_{R_{S}^{□}}^{∧} (R_{P}^{□,ord} ⊗^{∧} R_{Σ}^{□,ψ} ⊗^{∧} R_{∞}^{□,odd})
R_{S}^{□,BT,ord,ψ} = R^{□S} ⊗_{R_{S}^{□}}^{∧} (R_{P}^{□,BT,ord} ⊗^{∧} R_{Σ}^{□,ψ} ⊗^{∧} R_{∞}^{□,odd})

if p is odd; and

R_{S,±}^{□,ord,ψ} = R^{□S} ⊗_{R_{S}^{□}}^{∧} (R_{P,±}^{□,ord} ⊗^{∧} R_{Σ}^{□,ψ} ⊗^{∧} R_{∞}^{□,odd})
R_{S,±}^{□,BT,ord,ψ} = R^{□S} ⊗_{R_{S}^{□}}^{∧} (R_{P,±}^{□,BT,ord} ⊗^{∧} R_{Σ}^{□,ψ} ⊗^{∧} R_{∞}^{□,odd})

if p = 2. Let R_{S}^{ord,ψ} (resp. R_{S}^{ord,BT,ψ}) denote the subring of R_{S}^{□,ord,ψ} (resp. R_{S}^{□,BT,ord,ψ}) generated by the traces of ρ^{□S}. Similarly define R_{S,±}^{ord,ψ} and R_{S,±}^{□,BT,ord,ψ}.

Documenta Mathematica 18 (2013) 997–1038

5.2 Hecke algebras

Since \overline{ρ} arises from a holomorphic cusp form in S_{2}^{D}(𝔫_{Σ}, ψ)_{O} on the quaternion algebra D over F_{Σ} by assumption, there exists a maximal ideal 𝔪^{D} ⊂ h_{2}^{D}(𝔫_{Σ}, ψψ_{P})_{O} if p odd (resp. 𝔪^{D} ⊂ h_{2}^{D}(𝔫_{Σ}, ψψ_{P,+})_{O} if p = 2). It then follows that there exists a maximal ideal 𝔪 ⊂ h_{2}(𝔫_{Σ}p, ψψ_{P})_{O} such that

h_{2}(𝔫_{Σ}p, ψ)_{𝔪} ≃ h_{2}^{D}(𝔫_{Σ}, ψψ_{P})_{𝔪^{D}}

if p odd (resp. 𝔪_{+} ⊂ h_{2}(𝔫_{Σ}2, ψψ_{P,+})_{O} such that

h_{2}(𝔫_{Σ}2, ψψ_{P,+})_{𝔪_{+}} ≃ h_{2}^{D}(𝔫_{Σ}, ψψ_{P,+})_{𝔪^{D}}

if p = 2). When p = 2, there also exists 𝔪_{−} ⊂ h_{2}(𝔫_{Σ}4, ψψ_{P,−}) such that

h_{2}(𝔫_{Σ}4, ψψ_{P,−})_{𝔪_{−}}/(2) ≃ h_{2}(𝔫_{Σ}2, ψψ_{P,+})_{𝔪_{+}}/(2)

This can be proved exactly as in the proof of Lemma 3.2 in [4]; instead use the 0-dimensional Shimura variety corresponding to D over F_{Σ}. For p = 2 define e_{BDST,±} and let h^{0}(𝔫)_{±} = e_{BDST,±}h^{0}(𝔫). Let

h_{2}^{□}(𝔫_{Σ}p, ψψ_{P})_{𝔪} = h_{2}(𝔫_{Σ}p, ψψ_{P})_{𝔪} ⊗_{R_{S}^{BT,ord,ψ}} R_{S}^{□,BT,ord,ψ}
h^{0□}(𝔫_{Σ}, ψ)_{𝔪} = h^{0}(𝔫_{S}, ψψ_{P})_{𝔪} ⊗_{R_{S}^{ord,ψ}} R_{S}^{□,ord,ψ}

if p is odd; and let

h_{2}^{□}(𝔫_{Σ}4, ψψ_{P,−})_{𝔪_{−}} = h_{2}(𝔫_{Σ}4, ψψ_{P,−})_{𝔪_{−}} ⊗_{R_{S,−}^{BT,ord,ψ}} R_{S,−}^{□,BT,ord,ψ}
h^{0□}(𝔫_{S}, ψ)_{−,𝔪_{−}} = h^{0}(𝔫_{Σ}, ψψ_{P})_{−,𝔪} ⊗_{R_{S,−}^{ord,ψ}} R_{S,−}^{□,ord,ψ}

if p = 2. It then follow from results of Kisin and Khare-Wintenberger that there is a natural surjection

R_{S}^{□,BT,ord,ψ} → h_{2}^{□}(𝔫_{Σ}p, ψψ_{P})_{𝔪}

if p odd and

R_{S,−}^{□,BT,ord,ψ} → h_{2}^{□}(𝔫_{Σ}4, ψψ_{P,−})_{𝔪_{−}}

if p = 2, which induce isomorphisms

R_{S}^{□,BT,ord,ψ}[1/p] ≃ h_{2}^{□}(𝔫_{Σ}p, ψψ_{P})_{𝔪}[1/p]

if p odd and

R_{S,−}^{□,BT,ord,ψ}[1/2] ≃ h_{2}^{□}(𝔫_{Σ}4, ψψ_{P,−})_{𝔪_{−}}[1/2].

The determinant of ρ^{□S} defines

NS : I_{𝔫_{Σ}p^{∞}} → R_{S}^{ord,ψ}

and R_{S}^{ord,ψ}/ker(S − (ψψ_{P} ∘ ϵ^{cyclo})) ≃ R_{S}^{BT,ord,ψ} if p odd, and

NS : I_{𝔫_{Σ}p^{∞}} → R_{S,−}^{ord,ψ}

induces R_{S,−}^{ord,ψ}/ker(S − (ψψ_{P,−} ∘ ϵ^{cyclo})) ≃ R_{S,−}^{BT,ord,ψ} if p = 2. On the other hand, h^{0}(𝔫_{Σ}, ψ)_{𝔪}/ker(S − (ψψ_{P} ∘ ϵ^{cyclo})) ≃ h_{2}(𝔫_{Σ}p, ψψ_{P})_{𝔪} and h^{0}(𝔫_{Σ}, ψ)_{−,𝔪_{−}}/ker(S − (ψψ_{P,−} ∘ ϵ^{cyclo})) ≃ h_{2}(𝔫_{Σ}4, ψψ_{P,−})_{𝔪_{−}}. Then the surjective Λ-algebra homomorphisms

R_{S}^{□,ord,ψ} → h^{0□}(𝔫_{Σ}, ψ)_{𝔪}

if p odd and

R_{S,−}^{□,ord,ψ} → h^{0□}(𝔫_{Σ}, ψ)_{−,𝔪_{−}}

if p = 2 induce the isomorphisms

R_{S}^{□,ord,ψ}[1/p] ≃ h^{0□}(𝔫_{Σ}, ψ)_{𝔪}[1/p]

and

R_{S,−}^{□,ord,ψ}[1/2] ≃ h^{0□}(𝔫_{Σ}, ψ)_{−,𝔪_{−}}[1/2].

6 Companion forms mod p

Let F be a totally real field and p be a rational prime. Suppose that [F (ζ_{p}) : F] > 3 if p > 3 and that 2 splits completely in F if p = 2. Let f_{2} be a holomorphic cuspidal Hilbert eigenform of weight 2 ≤ k_{2} ≤ p and of level prime to p. Assume that the associated p-adic representation ρ_{2} of Gal(\overline{F} /F) is crystalline and ordinary at every prime 𝔭 of F above p. It is a well-known theorem of Wiles (Theorem 2.1.4 in [43]) that, for every prime 𝔭 of F above p, the restriction ρ|_{G_{𝔭}} to the decomposition group G_{𝔭} at 𝔭 is of the form

ρ|_{G_{𝔭}} ≃ (\begin{matrix}ϵ^{k_{2}−1}χ_{𝔭,1} & ∗ \\ 0 & χ_{𝔭,2} \end{matrix})

where χ_{𝔭,1} and χ_{𝔭,2} are unramified characters of G_{𝔭}, and χ_{𝔭,2}(Frob_{𝔭}) is a unit U_{𝔭}-eigenvalue of the p-stabilised newform of f_{2}.

Theorem 9 Let f_{2} be a holomorphic cuspidal Hilbert eigenform of weight 2 ≤ k_{2} ≤ p and of level prime to p as above. Let k_{1} =^{def} p if k_{2} = p and k_{1} =^{def} p+1−k_{2} if 2 ≤ k_{2} < p. Suppose that

• if p > 2, the associated mod p representation \overline{ρ}_{2} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{p}) is absolutely irreducible when restricted to Gal(\overline{F} /F (ζ_{p})), and if p = 2, \overline{ρ}_{2} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{p}) has insoluble image;

• if p > 2 and if \overline{ϵ}^{k_{1}−2}\overline{χ}_{𝔭,2} ≠ \overline{χ}_{𝔭,1}, the ramification index of F_{𝔭} is strictly less than p − 1 for every prime 𝔭 of F above p, and if p = 2, \overline{ρ}_{2} is unramified at every prime of F above 2; • if p > 2, \overline{ϵ}^{k_{2}−1}\overline{χ}_{𝔭,1} ≠ \overline{χ}_{𝔭,2} and if p = 2, \overline{χ}_{𝔭,2} ≠ \overline{χ}_{𝔭,1}

• \overline{ρ}_{2} is the direct sum of the characters \overline{ϵ}^{k_{2}−1}\overline{χ}_{𝔭,1} and \overline{χ}_{𝔭,2} at every prime 𝔭 of F above p.

Then there exists a holomorphic cuspidal Hilbert eigenform of weight 2 ≤ k_{1} ≤ p and of level prime to p with its associated mod p representation ρ_{1} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{p}) satisfying \overline{ρ}_{1} ≃ \overline{ρ}_{2} ⊗ \overline{ϵ}^{k_{1}−1} if p > 2 and \overline{ρ}_{1} ≃ \overline{ρ}_{2} if p = 2, and the U_{𝔭}-eigenvalue of the p-stabilised new form is a lifting of \overline{χ}_{𝔭,1}(Frob_{𝔭}).

Proof. For p > 2, this is a result of Gee (Theorem 2.1 [13]). Let p = 2; thus k_{1} = k_{2} = 2. For clarity, let \overline{ρ} denote \overline{ρ}_{2} ⊗ \overline{ϵ} where \overline{ϵ} is the mod 4 cyclotomic character. Clearly the twist of ρ_{2} by the Teichmuller lift of \overline{ϵ} defines a modular lifting of \overline{ρ} potentially ordinary and potentially Barsotti-Tate at p. By class field theory, find a finite totally real soluble extension F_{Σ} ⊂ \overline{F} of F of even degree in which 2 remains split completely, and satisfies the following conditions:

• there exists a quaternion algebra D over F_{Σ} ramified exactly at a finite set Σ of finite primes of F_{Σ} not dividing 2;

• \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} is ramified exactly at Σ and the infinite places, and , in particular, for every prime 𝔮 ∈ Σ, \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} at 𝔮 is an extension of an unramified character by the twist of the character by ϵ at 𝔮;

• there exists a maximal open compact subgroup U ⊂ (D ⊗_{F_{Σ}} A_{F_{Σ}}^{∞})^{×} such that U_{𝔮} = GL_{2}(O_{F_{Σ}𝔮}) for 𝔮 ∉ S^{D} and U_{𝔮} = O_{D𝔮}^{×} for 𝔮 ∈ S^{D}, and a holomorphic cuspidal automorphic representation π_{2} of (D ⊗_{F_{Σ}} A_{F_{Σ}})^{×} with central character ψ such that \overline{ρ}|_{Gal(\overline{F} /F_{Σ})} ≃ \overline{ρ}_{π_{2}} : Gal(\overline{F} /F_{Σ}) → GL_{2}(\overline{F}_{p}) and det\overline{ρ}|_{Gal(\overline{F} /F_{Σ})} = \overline{ψ}\overline{ϵ} and such that π is unramified at every prime of F_{Σ} above 2.

It then follows from work of Khare-Wintenberger (See Corollary 4.7 and Theorem 10.1 in [18]) that there is a lifting ρ : Gal(\overline{F} /F_{Σ}) → GL_{2}(\overline{Q}_{p}) of \overline{ρ}|_{Gal(\overline{F} /F_{Σ})}, unramified outside S_{Σ,P} ∐ Σ ∐ S_{Σ,∞} with detρ = ψϵ such that, for every prime 𝔭 of F_{Σ} above 2, ρ is ordinary at 𝔭 and Barsotti-Tate and is of the form

(\begin{matrix}ϵχ̃_{𝔭,2} & ∗ \\ 0 & χ̃_{𝔭,1} \end{matrix})

where χ̃_{𝔭,1} and χ̃_{𝔭,2} are unramified liftings of \overline{χ}_{𝔭,1}|_{Gal(\overline{F} /F_{Σ})} and \overline{χ}_{𝔭,2}|_{Gal(\overline{F} /F_{Σ})} respectively. It then follows from the main theorem of Kisin [19] and Khare-Wintenberegr [18], and by soluble descent that there exists a holomorphic cuspidal Hilbert eigenform f_{1} of weight k_{1} = 2 and of level prime to 2 such that ρ_{f_{1}}|_{Gal(\overline{F} /F_{Σ})} ≃ ρ. □

7 Λ-adic companion forms

Theorem 10 Let p be a rational prime. Let F be a totally real field. Suppose that p splits completely in F. Let K be a finite extension of Q_{p} with ring of integers O and residue field k = O/𝔪. Suppose that

ρ : Gal(\overline{F} /F) → GL_{2}(O)

is a continuous representation satisfying

• ρ ramifies at only finite many primes;

• \overline{ρ} = (ρ mod 𝔪) is absolutely irreducible when restricted to Gal(\overline{F} /F (ζ_{p}), and has a modular lifting which is potentially ordinary and potentially Barsotti-Tate at every prime of F above p;

• for every prime ideal 𝔭 of F above p, the restriction ρ|_{G_{𝔭}} to the decomposition group G_{𝔭} at 𝔭 is the direct sum of characters χ_{𝔭,1} and χ_{𝔭,2} : G_{𝔭} → O^{×} such that the images of the inertia subgroup at 𝔭 are finite and (χ_{𝔭,1} mod 𝔪) ≠ (χ_{𝔭,2} mod 𝔪);

If p = 2, assume furthermore that

• the image of the complex conjugation with respect to every embedding of F into R is not the identity matrix;

• \overline{ρ} has insoluble image;

• for every 𝔭 of F above p, ρ is unramified at 𝔭.

Then there is a finite totally real soluble extension F_{Σ} ⊂ \overline{F} of F in which p splits completely; a finite set Σ of finite places of F_{Σ} (at which ρ|_{G_{Σ}}, where G_{Σ} =^{def} Gal(\overline{F} /F_{Σ}) is ramified of conductor 𝔫_{Σ}); an ideal 𝔫 of O_{F_{Σ}} coprime to p which 𝔫_{Σ} divides; and, for every subset P of the set S_{Σ,P} of places of F_{Σ} above p,

1. a character

χ_{P} : G_{Σ} → O^{×}

of finite order, unramified outside a finite set of places containing S_{Σ,P}, such that the restriction to the inertia subgroup of G_{Σ} at 𝔭 of χ_{P} equals that of χ_{𝔭,1} (resp. χ_{𝔭,2}) for all 𝔭 in P (resp. S_{Σ,P} − P);

2. a finite extension L of Frac Λ_{K} and a Λ-adic form

F_{Hida,P} : h_{O}^{0}(𝔫_{Σ}) → L;

3. a homomorphism f_{P} : h_{O}^{0}(𝔫) → O if p > 2 while f_{P} : h_{O}^{0}(𝔫)_{−} → O if p = 2 satisfying • f_{P} (T_{𝔮}) = tr ρ(Frob_{𝔮})/χ_{P} (Frob_{𝔮}) for all 𝔮 not dividing 𝔫p;

• f_{P} (N𝔮S_{𝔮}) = det ρ(Frob_{𝔮})/χ_{P}^{2} (Frob_{𝔮}) for all 𝔮 not dividing 𝔫p;

• f_{P} (U_{𝔮}) = 0 for 𝔮 dividing 𝔫 but not dividing p;

• f_{P} (U_{𝔭}) = (χ_{𝔭,1}/χ_{P})(Frob_{𝔭}) for every 𝔭 in P and f_{P} (U_{𝔭}) = (χ_{𝔭,2}/χ_{P})(Frob_{𝔭}) for every 𝔭 in S_{Σ,P} − P.

Proof. Choose a finite soluble totally real extension F_{Σ} of F in which p splits completely such that the restriction of ρ is absolutely irreducible when restricted to Gal(\overline{F} /F_{Σ}(ζ_{p})), unramified outside a finite set Σ ∐ S_{Σ,P} ∐ S_{Σ,∞} of finite places 𝔮 of F_{Σ} such that ρ|_{G_{Σ}} is of of conductor 1 or 𝔮 at 𝔮, and arises from–by Jacquet-Langlands–a cuspidal automorphic representation, nearly ordinary at every 𝔭 ∈ S_{Σ,P} and special at 𝔮 ∈ Σ, of the quaternion algebra D_{Σ} over F_{Σ} as in the previous section. For every P ⊆ S_{Σ,P}, it follows from class field theory that one can choose χ_{P}, of conductor 1 away from a finite set of places containing the set of places above p, as asserted in the theorem. Let ρ_{P} denote ρ ⊗_{Gal(\overline{F} /F_{Σ})} χ_{P}^{−1} and \overline{ρ}_{P} denote (ρ_{P} mod 𝔪). If we let ρ_{Σ} denote the modular lifting of \overline{ρ}, then ρ_{Σ} ⊗ χ_{P}^{−1} is a modular lifting of \overline{ρ}_{P}; in fact it is ordinary at every 𝔭 ∈ S_{Σ,P} by Jarvis’ level lowering results [15]–by which one shows ρ_{Σ} ⊗ χ_{P}^{−1} is crystalline at 𝔭– followed by Fontaine-Laffaille theory. Let 𝔪_{P} denote the corresponding maximal ideal of eh_{2}(𝔫_{Σ}p)_{O} if p > 2 and eh_{2}(𝔫_{Σ}4)_{O,−} if p = 2. It then follows from Hida theory [14] and results from preceding sections that there exists a finite extension L in an algebraic closure of Frac Λ_{K} which we fix; and, for every P ⊆ S_{Σ,P}, a Λ-adic eigenform F_{H,P} : h_{O}^{0}(𝔫_{Σ}) → h_{O}^{0}(𝔫_{Σ})_{𝔪_{P}} → O_{L}, and a height one prime ℘_{P} of O_{L} such that

• (ρ_{F_{H}} mod ℘_{P}) ∼ ρ_{P}

• for every distinct subsets P and Q, (F_{H,P} , ℘_{P}) and (F_{H,Q}, ℘_{Q}) are in companion; more precisely, for every 𝔮 not dividing 𝔫_{Σ}p, (F_{H,Q}(T_{𝔮}) mod ℘_{Q}) = (F_{H,P} (T_{𝔮}S_{(P −(P ∩Q))∪(Q−(P ∩Q))}(𝔮)^{−1}) mod ℘_{P}); for 𝔭 in (P ∩ Q) ∪ ((S_{Σ,P} − P) ∩ (S_{Σ,P} − Q)), (F_{H,Q}(U_{𝔭}) mod ℘_{Q}) = (F_{H,P} (U_{𝔭}) mod ℘_{P}), while for 𝔭 in (P ∩ (S_{Σ,P} − Q)) ∪ ((S_{Σ,P} − P) ∩ Q), (F_{H,Q}(U_{𝔭}) mod ℘_{Q}) = (F_{H,P} (U_{𝔭}^{−1}S^{P} (𝔭)) mod ℘_{P}).

Let f_{P} be the composite h_{O}^{0}(𝔫_{Σ}) → O_{L} → O_{L}/℘_{P} ≃ O; if the image of U_{𝔮} for 𝔮 dividing 𝔫_{Σ} but not dividing p is not zero, we may increase the level at 𝔮 if necessary to assume the image of indeed zero (See [34] for example) . □

8 Models of Hilbert modular varieties

Let F be a totally real field–F_{Σ} in the preceding section–of degree d = [F : Q] with ring of integers O_{F}. Fix a rational prime p and an ideal 𝔫 of O_{F} prime to p. For every integer r ≥ 1, fix a p^{r}-th primitive root ζ_{p^{r}} of unity. For a prime 𝔭 of F above p, let F_{𝔭} denote the completion of F with respect to the absolute value corresponding to 𝔭, k_{𝔭} the residue field of F_{𝔭}, f_{𝔭} the residue class degree, and e_{𝔭} the ramification index. Fix embeddings Q → \overline{Q} → \overline{Q}_{p}. Let K denote a finite extension of Q_{p} which contains the image of F by every embedding of F into \overline{Q}_{p}; and let O denote its ring of integers and k denote the residue field. For a fractional ideal I of F canonically ordered, let I^{+} denote the totally positive elements. Fix a set T of representatives in A_{F}^{×} of the strict ideal class group A_{F}^{×}/(F^{×}(O_{F} ⊗ Z^{∧})^{×}F_{∞}^{+×}), and we shall let t also mean the fractional ideal t𝔡 corresponding to a representative t in T.

Definition. A t-polarised Hilbert-Blumenthal abelian variety (henceforth abbreviated as HBAV) with level Γ_{1}(𝔫)-structure over a O-scheme S is an abelian variety A over S of relative dimension d together with

• i : O_{F} → End(A/S);

• a homomorphism λ : (t, t^{+}) → (Sym(A/S), Pol(A/S)) of ordered invertible O_{F} -modules, where Sym(A/S) (resp. Pol(A/S)) denotes the invertible O_{F} -module (via i) of symmetric homomorphisms (resp. polarisations), such that A ⊗_{O_{F}} t → A^{∨}, induced by λ, is an isomorphism of HBAVs–it is shown in [41] that this is equivalent to the condition that there exists a prime-to-p polarisation A → A^{∨}; and to the ‘determinant condition’ on Lie(A) in the sense of Kottwitz;

• an O_{F} /𝔫-module morphism η : (O_{F} /𝔫)^{∨} = (GL_{1} ⊗ 𝔡_{F}^{−1})[𝔫] → A[𝔫].

Definition. Let Y_{Γ_{1}(𝔫,t)} (resp. Y_{Γ_{1}(𝔫,t)∩Iw}) denote the O-scheme representing the functor which sends an O-scheme S to the set of isomorphism classes (A, i, λ, η) (resp. (A, i, λ, η, C)) of t-polarised HBAVs with level Γ_{1}(𝔫)-level structure (resp. and a finite flat subgroup scheme C of A[p] with compatible O_{F} -action locally free of rank ∑_{𝔭} |O_{F} /𝔭|). It follows from [27] and [8] that if 𝔫 does not divide 2, nor 3, Y_{Γ_{1}(𝔫,t)} is a smooth scheme over O of relative dimension [F : Q]. If 𝔫 does divide 2, or 3, we let Y_{Γ_{1}(𝔫,t)} denote the O-scheme

(Γ_{1}(𝔫, t)/Γ_{1}(𝔪, t))\Y_{Γ_{1}(𝔪,t)}

for an auxiliary ideal 𝔪 of O_{F} such that 𝔫|𝔪 and Γ_{1}(𝔪) small enough, i.e., torsion-free.

Let \overline{Y}_{Γ_{1}(𝔫,t)} denote the fibre over k of Y_{Γ_{1}(𝔫,t)}; and let Let \overline{Y}_{Γ_{1}(𝔫,t)∩Iw} denote the fibre over k of Y_{Γ_{1}(𝔫,t)∩Iw}. It is a well-known result of Deligne-Ribet that the fibre \overline{Y}_{Γ_{1}(𝔫,t)} is irreducible (Corollary 4.6 in [9]). It is a result of local model theory by Pappas that \overline{Y}_{Γ_{1}(𝔫,t)∩Iw} is normal (Corollary 2.2.3 in [25]).

Suppose that p splits completely in F. In which case, the p-divisible group of a HBAV over the ring of integers of a finite extension of Q_{p} decomposes as the product of [F : Q] one-dimensional p-divisible groups, one for each prime 𝔭 of F above p, and this allows us to define ‘Katz-Mazur-Drinfeld’ higher level structures at p by defining level structures at 𝔭 on the ‘𝔭-divisible group’ for each 𝔭.

Definition. Let r be an integer ≥ 1. Define Y_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r})} to be the O-scheme representing the functor which sends an O-scheme S to the set of isomorphism classes of the sextuples (A, i, λ, η, C, η_{KM}) over S where (A, i, λ, η) is a t-polarised HBAV over S with Γ_{1}(𝔫)-level structure, C is a finite flat subgroup scheme of A[p^{r}] locally free of finite rank |O_{F} /p^{r}| = ∑_{𝔭} |O_{F} /𝔭^{r}| with compatible action of O_{F}, and an O_{F} -linear group homomorphism η_{KM} : O_{F} /p^{r} → Mor(S, C) ⊂ Mor(S, A[p^{r}]) such that the image of η_{KM} defines a ‘full set of sections’ in the sense of Katz-Mazur [17] (See 1.10.5 and 1.10.10 in [17]).

Definition. For every prime 𝔭 of F above p, let Y_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),Iw_{𝔭},K} denote the fine moduli space over K of the septuples (A, i, λ, η, C, η_{KM}, D_{𝔭}) where the sextuple (A, i, λ, η, C, η_{KM}) defines a point of Y_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r})} ×_{Spec O_{K}} Spec K, and D_{𝔭} is finite flat subgroup scheme of A[𝔭] of rank |O_{F} /𝔭| which has trivial intersection with C.

9 Compactification

By an unramified cusp C of Y_{Γ_{1}(𝔫,t)} over R, we shall mean a pair of fractional ideals M_{1}, M_{2} of F such that M_{1}M_{2}^{−1} ≃ t which comes equipped with

• an O_{F} ⊗_{Z} R-linear isomorphism λ : M_{1}^{−1} ⊗_{Z} R ≃ O_{F} ⊗_{Z} R;

• an O_{F} -linear embedding η : O_{F} /𝔫 → 𝔫^{−1}M_{2}^{−1}/M_{2}^{−1}.

For brevity, let M = M_{1}M_{2}, M^{∨} = Hom_{Z}(M, Z) = Hom_{O_{F}} (M, 𝔡_{F}^{−1}) ≃ 𝔠M_{2}^{−1}𝔡_{F}^{−1}, and M^{∨+} ⊂ M^{∨} of the totally positive elements in (𝔠M_{2}^{−1}𝔡_{F}^{−1})^{+}. Choose a rational polyhedral cone decomposition Σ_{C} of (M^{∨+} ⊗_{Z} R) ∪ {0}. For a cone σ ⊂ (M^{∨+} ⊗_{Z} R), we let σ^{∨} ⊂ M ⊗_{Z} R denote the dual cone. Let S_{𝔫} = Spec Z[q^{𝔫^{−1}M}] and S_{𝔫} ↪ S_{𝔫,σ} denote the affine torus embedding (see Theorem 2.5 in [6]) corresponding to the cone σ and let S_{𝔫,σ}^{∧} = Spf Z[[q^{𝔫^{−1}M∩σ^{∨}}]] denote the formal completion of S_{𝔫,σ} along the boundary

S_{𝔫,σ}^{∞} =^{def} S_{𝔫,σ} − S_{𝔫}.

Let T_{𝔫,σ} = Spec Z[[q^{𝔫^{−1}M∩σ^{∨}}]] and T_{𝔫,σ}^{0} = T_{𝔫,σ} − S_{𝔫,σ}^{∞} = Spec Z[[q^{𝔫^{−1}M∩σ^{∨}}, q^{−𝔫^{−1}M∩σ^{∨}}]]. The henselisation of (S_{𝔫,σ}, S_{𝔫,σ}^{∞}) projects onto an affine étale scheme U_{𝔫,σ} over S_{𝔫,σ} which approximates S_{𝔫,σ}^{∧} in the sense of Artin, and let U_{𝔫,σ}^{0} = U_{𝔫,σ} ×_{T_{𝔫,σ}} T_{𝔫,σ}^{0}.

The Mumford construction applied to the O_{F} -linear ‘period’ map q : M_{2} → GL_{1}(U_{𝔫,σ}) ⊗_{Z} 𝔡_{F}^{−1}M_{1}^{−1} gives rise to a semi-abelian scheme

Tate_{M_{1},M_{2}}(q) =^{def} (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/q^{M_{2}}

over the complete ring U_{𝔫,σ} with action of O_{F}, whose pull-back, which we shall denote by Tate_{M_{1},M_{2}}^{0}(q) to U_{𝔫,σ}^{0}, is naturally a HBAV, t-polarised

Tate_{M_{1},M_{2}}(q) ⊗_{O_{F}} M_{1}M_{2}^{−1} ≃ (GL_{1} ⊗_{Z} 𝔡^{−1}M_{2}^{−1})/q^{M_{1}}
||
Tate_{M_{2},M_{1}}(q)  ≃ Tate_{M_{1},M_{2}}(q)^{∨},

with level Γ_{1}(𝔫)-structure, and which gives rise to a map

U_{𝔫,σ}^{0} ×_{Spec Z} Spec O → Y_{Γ_{1}(𝔫,t)}.

We glue ∐_{T/≃} ∐_{σ∈Σ_{C}} U_{𝔫,σ} ×_{Spec Z} Spec O along the map to get a toroidal compactification X_{Γ_{1}(𝔫,t)} over O of Y_{Γ_{1}(𝔫,t)} ([26]). Similarly, one can define a compactification X_{Γ_{1}(𝔫,t)∩Iw} over O of Y_{Γ_{1}(𝔫,t)∩Iw} with its choice of a rational cone decomposition compatible with that of X_{Γ_{1}(𝔫,t)}.

Let

Tate_{M_{1},M_{2},S}^{0}(q) =^{def} Tate_{M_{1},M_{2}}(q) ×_{Spec Z[[q^{M} ,q^{−M}]]} S

for a Z[[q^{M}, q^{−M}]]-scheme S; it is t-polarised. Let S be a O ⊗_{Z} Z[[q^{M}, q^{−M}]]- scheme. Then there is a ‘connected-étale’ exact sequence

0 → (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})[p^{r}] → Tate_{M_{1},M_{2},S}(q)[p^{r}] → (1/p^{r})M_{2}/M_{2} → 0

of (O_{F} /p^{r})-modules schemes over S.

Lemma 11 Fix an integer r ≥ 1. Let S be a connected O ⊗_{Z} Z[[q^{M}, q^{−M}]]- scheme. Suppose that C is an O_{F} -stable finite flat subgroup scheme of Tate_{M_{1},M_{2},S}^{0}(q)[p^{r}] of order |O_{F} /p^{r}|. Then for every τ = τ_{𝔭}, there exists a unique pair of non-negative integers ρ_{τ,1}, ρ_{τ,2} such that ρ_{τ,1} +ρ_{τ,2} = r and such that

C_{𝔭} ∩ (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})[p^{r}] ≃ (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})[𝔭^{ρ_{τ,1}}]

and the image of C_{𝔭} in (1/p^{r})M_{2}/M_{2} is isomorphic to 𝔭^{−ρ_{τ,2}}M_{2}/M_{2}.

Proof. This is essentially Proposition 13.6.2 in [17]. □

By a cusp of C of Y_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r})} over R, we shall mean a pair of fractional ideals M_{1}, M_{2} of F such that M_{1}M_{2}^{−1} ≃ t which comes equipped with

• an O_{F} ⊗_{Z} R-linear isomorphism λ : M_{1}^{−1} ⊗_{Z} R ≃ O_{F} ⊗_{Z} R;

• an O_{F} -linear embedding η : O_{F} /𝔫 → 𝔫^{−1}M_{2}^{−1}/M_{2}^{−1}.

• an O_{F} -linear isomorphism η_{KM} : O_{F} /p^{r} ≃ p^{−r}M_{2}/M_{2}.

Let M = M_{1}M_{2} as above. Fix an integer r ≥ 1. Suppose that S is an O ⊗_{Z} Z[ζ_{p^{r}}][[q^{(1/p^{r})M,−(1/p^{r})M}]]-scheme. Definition Let ζ_{r} denote the image of 1 by

ζ_{KM,r} : (O_{F} /p^{r}) ≃ 𝔡^{−1}/p^{r}𝔡^{−1} ≃ GL_{1}[p^{r}] ⊗_{Z} 𝔡^{−1} ≃ (GL_{1} ⊗_{Z} 𝔡^{−1})[p^{r}]

and ζ_{r,τ} denote its τ = τ_{𝔭} component. We often allow ζ_{r} and ζ_{r,τ} to mean their images in (GL_{1} ⊗_{Z} 𝔡^{−1})(S) and Tate_{M_{1},M_{2},S}(q)(S). Let η_{r}^{ét} denote the image of 1 by

η_{KM,r}^{ét} : (O_{F} /p^{r}) ≃^{η_{KM}} p^{−r}M_{2}/M_{2} →^{q} q^{p^{−r}M_{2}/M_{2}}

defining a point of Tate_{M_{1},M_{2}}(q)(S) of exact order |O_{F} /p^{r}|. Let η_{r,τ}^{ét} denote its τ = τ_{𝔭} component.

Lemma 12 Fix an integer r ≥ 1. Let S be a connected O ⊗_{Z} Z[[q^{M}, q^{−M}]]- scheme. Suppose that C is an O_{F} -stable finite flat subgroup scheme of Tate_{M_{1},M_{2},S}^{0}(q)[p^{r}] of order |O_{F} /p^{r}|. Suppose that C is of type ρ = (ρ_{τ,1}, ρ_{τ,2})_{τ}. Let P_{KM} ∈ C(S) denote a point of exact order |O_{F} /p^{r}|. Then for every τ = τ_{𝔭}, P_{KM,τ} is of the form ζ_{r,τ}^{σ_{τ,1}}η_{r,τ}^{ét,σ_{τ,2}} for a pair of integers 0 ≤ σ_{τ,1} ≤ ρ_{τ,1} and 0 ≤ σ_{τ,2} ≤ ρ_{τ,2} such that both σ_{τ,1} and σ_{τ,2} are coprime to p.

proof. This is essentially 13.6.3 in [17]. □

10 Generic fibres

With 𝔫 fixed, for every integer r ≥ 1, let \overline{U}_{r} denote the quotient group of the totally positive units of F by the subgroup of elements which are squares of elements in O_{F} which are congruent to 1 mod 𝔫p^{r}. If r = 0, we simply write \overline{U}.

Let Y_{Γ_{1}(𝔫)}, X_{Γ_{1}(𝔫)}, Y_{Γ_{1}(𝔫)∩Iw}, X_{Γ_{1}(𝔫)∩Iw}, Y_{Γ_{1}(𝔫)∩Γ_{1}(p^{r})}, Y_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),Iw_{𝔭},K} respectively denote the disjoint unions, t ranging over T, of Y_{Γ_{1}(𝔫,t)}, X_{Γ_{1}(𝔫,t)}, Y_{Γ_{1}(𝔫,t)∩Iw}, X_{Γ_{1}(𝔫,t)∩Iw}, Y_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r})}, Y_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),Iw_{𝔭},K}.

Let X_{Γ_{1}(𝔫),K}, X_{Γ_{1}(𝔫)∩Iw,K} respectively denote the generic fibres over K of the O_{K}-schemes X_{Γ_{1}(𝔫)}, X_{Γ_{1}(𝔫)∩Iw}.

Let X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}, X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),Iw_{𝔭},K} respectively denote the toroidal compactifications of the K-schemes Y_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}, Y_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),Iw_{𝔭},K}.

Let Y_{Γ_{1}(𝔫,t)}^{∧}, X_{Γ_{1}(𝔫,t)}^{∧}, Y_{Γ_{1}(𝔫,t)∩Iw}^{∧}, X_{Γ_{1}(𝔫,t)∩Iw}^{∧} respectively denote the formal completions of Y_{Γ_{1}(𝔫,t)}, X_{Γ_{1}(𝔫,t)}, Y_{Γ_{1}(𝔫,t)∩Iw}, X_{Γ_{1}(𝔫,t)∩Iw} along their closed k-fibres \overline{Y}_{Γ_{1}(𝔫,t)}, \overline{X}_{Γ_{1}(𝔫,t)}, \overline{Y}_{Γ_{1}(𝔫,t)∩Iw}, \overline{X}_{Γ_{1}(𝔫,t)∩Iw}. Let Y_{Γ_{1}(𝔫)}^{∧}, X_{Γ_{1}(𝔫)}^{∧}, Y_{Γ_{1}(𝔫)∩Iw}^{∧}, X_{Γ_{1}(𝔫)∩Iw}^{∧} denote their disjoint unions over T.

Finally, let Y_{Γ_{1}(𝔫)}^{rig}, X_{Γ_{1}(𝔫)}^{rig}, Y_{Γ_{1}(𝔫)∩Iw}^{rig}, X_{Γ_{1}(𝔫)∩Iw}^{rig} respectively denote the Raynaud rigid generic fibres of Y_{Γ_{1}(𝔫)}^{∧}, X_{Γ_{1}(𝔫)}^{∧}, Y_{Γ_{1}(𝔫)∩Iw}^{∧}, X_{Γ_{1}(𝔫)∩Iw}^{∧}.

11 p-adic classical Hilbert modular forms

Suppose that (k = ∑_{τ∈Hom(F,K)} k_{τ} τ, w = ∑_{τ∈Hom(F,K)} w_{τ} τ) ∈ Z^{Hom(F,K)} × Z^{Hom(F,K)} is such that w = 2w_{τ} − k_{τ} is independent of τ (this is Taylor’s μ in [38]). For S ∈ {Y_{Γ_{1}(𝔫),K}, Y_{Γ_{1}(𝔫)∩Iw,K}, Y_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}}, let Lie^{∨}(A/S) (resp. H_{dR}^{1}(A/S)) denote the pull-back by the identity section of the sheaf of relative differentials of the universal HBAV A over S (resp. the higher direct image of the relative de Rham complex). By the decomposition,

O_{F} ⊗_{Z} O ≃ ∏_{τ∈Hom(F,K)} O_{τ}

where O_{τ} is O into which F embeds by τ, we have

Lie^{∨}(A/S) = ⨁_{τ∈Hom(F,K)} Lie^{∨}(A/S)_{τ} , H_{dR}^{1}(A/S) = ⨁_{τ∈Hom(F,K)} H_{dR}^{1}(A/S)_{τ}

where Lie^{∨}(A/S) and H_{dR}^{1}(A/S) are locally free sheaves of O_{S}-modules of rank 1 and 2 respectively. Following Hida [14], let

L_{(k,w)} = ⨂_{τ∈Hom(F,K)}(⋀ H_{dR}^{1}(A/S)_{τ})^{⊗w/2} ⊗_{O_{S}} (Lie^{∨}(A/S))_{τ}^{⊗k_{τ}}

If k is parallel, more precisely, if (k, w) = ((k, …, k), (k/2, …, k/2)), we will often write L_{k} for L_{(k,w)}. We shall also let L_{(k,w)} denote its extension to the compactification.

Let π_{1} (resp. π_{2,𝔭}) denote the degeneracy map

X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),Iw_{𝔭},K} ⟶ X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}

defined, on the non-cuspdail points, by

(A, i, λ, η, C, η_{KM}, D_{𝔭}) ↦ (A, i, λ, η, C, η_{KM})

(resp. (A/D_{𝔭}, (i mod D_{𝔭}), (λ mod D_{𝔭}), (η mod D_{𝔭}), (η_{KM} mod D_{𝔭})).

12 Canonical subgroups for one-dimensional formal groups

Let L be a finite extension of K, and let val_{L} be a valuation on L normalised so that val_{L}(p) = 1. Let G be a one-dimensional principally polarised p-divisible/Barsotti-Tate group over O_{L}.

Definition. The identity component G^{∧} of G is a one-dimensional formal group, and define Ha(G) to be val_{L}(a) for a as defined in Proposition 3.6.6,

[16] (see also [29]).

By definition, G is ordinary if and only if Ha(G) = 0.

Let C be a finite flat subgroup scheme of G[p] of order p.

Definition. Define deg(G, C) to be 1 − val_{L}(Ann(coker(Lie^{∨}(G/C) → Lie^{∨}(G)))).

It follows immediately from the definition that deg(G, C)+deg(G/C, G[p]/C) = 1.

Suppose that deg(G, C) < p/(p + 1). Then there exists a canonical subgroup H(G) of G. If C = H(G), then deg(G, C) = Ha(G). To see this, note that H(G)(L) consists of 0 and p − 1 points P of the formal group G^{∧} of valuation (1 − Ha(G))/(p − 1) (Theorem 3.10.7, [16]). Since deg(G, C) = 1 − ∏_{P} val(P) (Lemma 1.3 [24]), deg(G, C) = Ha(G).

Lemma 13 Let r be a rational number < p/(p + 1). Suppose that G is not ordinary. Then

{(G, C) | Ha(G) ≤ r}

divides into two disjoint subsets, namely

{(G, C) | C = H(G) and deg(G, C) ∈ (0, r]}

and

{(G, C) | C ≠ H(G) and deg(G, C) ∈ [1 − r/p, 1)}.

On the other hand,

{(G, C) | Ha(G/C) ≤ r}

divides into two disjoint subsets, namely

{(G, C) | deg(G, C) ∈ (0, r/p], C = H(G), and Ha(G) < 1/(p + 1) }
∪ {(G, C) | deg(G, C) ∈ (0, r/p], C ≠ H(G), and Ha(G) < p/(p + 1) }

and

{(G, C) | deg(G, C) ∈ [1 − r, 1), C = H(G), 1/(p + 1) < Ha(G) < p/(p + 1)}

∪ {(G, C) | deg(G, C) ∈ [1 − r, 1), C ≠ H(G), Ha(G) ≥ p/(p + 1)}.

Proof. This follows from canonical subgroup theorem in [29]. □

Fix an integer n ≥ 1 and suppose furthermore that deg(G, C) ≤ p^{1−n}/(p+1) < p/(p + 1). Then define subgroup H_{n} = H_{n}(G) of G order p^{n} inductively as follows: If n = 1, set H_{1} = D. If n > 1, then let H_{n} to be the pre-image by the map G → G/H(G) of H_{n−1}(G/H(G)) ⊂ G/H(G).

Proposition 14 Suppose that one-dimensional principally polarised pdivisible group G over O_{L} has a subgroup H_{n}(G) as defined above. Suppse that m ≥ 1 is an integer. Suppose that C_{m} is a subgroup of G of order p^{m} such that H_{n}(G) ∩ C_{m} = {0}, and suppose that D_{m+n} is a cyclic subgroup of G of order p^{m+n} such that H_{n}(G) ⊆ D_{m+n}. Then deg(G/C_{m}) < p^{1−(m+n)}/(p + 1) and G/C_{m} has the subgroup H_{m+n}(G/C_{m}). Indeed, H_{m+n}(G/C_{m}) = (D_{m+n} + C_{m})/C_{m}.

Proof. This can be proved as in Proposition 3.5 in [3]. □

13 p-adic overconvergent Hilbert modular forms

Let X_{Γ_{1}(𝔫),K}^{an}, X_{Γ_{1}(𝔫)∩Iw,K}^{an}, X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an} respectively denote the rigid analytic spaces in the sense of Tate ([2]) associated to the K-schemes X_{Γ_{1}(𝔫),K}, X_{Γ_{1}(𝔫)∩Iw,K}, X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}.

Given a closed point of Y_{Γ_{1}(𝔫)}^{rig}, it corresponds to a point (A, λ, η) defined over the integer O_{L} of a finite extension L of K. We then define deg_{τ} (A), for τ = τ_{𝔭} for a place 𝔭 of F above p, to be ‘deg’ as in the previous section with the (one-dimension) Barsotti-Tate group of 𝔭-power torsions of A in place of ‘G’.

The O-scheme X_{Γ_{1}(𝔫)} is of finite type, hence X_{Γ_{1}(𝔫)}^{rig} is quasi-compact. There exists a finitely many sufficiently small affine formal schemes U^{∧} such that their generic fibres U^{rig} form an admissible covering of X_{Γ_{1}(𝔫)}^{rig}. Let U_{good}^{∧} denote the smooth formal scheme U^{∧} ∩Y_{Γ_{1}(𝔫)}^{∧} and let i : U_{good}^{∧} ↪ U^{∧}. On each U_{good}^{∧}, there is a function whose corresponding rigid function has its valuation deg_{𝔭}; indeed, apply the construction to the formal completion of the ‘universal’ semi-abelian scheme over X_{Γ_{1}(𝔫)} along the underlying scheme of U_{good}^{∧}. We may think of the function on U_{good}^{∧} as a lift of the Hasse invariant at 𝔭, and it follows from Kocher’s principle that i_{∗}O_{U_{good}^{∧}} = O_{U^{∧}}, i.e., the function extends to U^{∧}. The valuation of its induced function on the generic fibre U^{rig} extends the function on U_{good}^{rig}. Glue these functions on U^{rig}’s, there is a rigid function on X_{Γ_{1}(𝔫)}^{rig} ≃ X_{Γ_{1}(𝔫)}^{an} that defines deg.

Definition. If I ⊂ [0, 1) is a closed, open, or half open interval with endpoint in Q, define the rigid space X_{Γ_{1}(𝔫),K}^{an}I = ∐_{t} X_{Γ_{1}(𝔫,t),K}^{an}I to be the admissible open set of points whose degrees are all in the range I.

For every t, X_{Γ_{1}(𝔫,t),K}^{an}I is connected; this follows from the fact that X_{Γ_{1}(𝔫,t),K}^{rig} is connected (since \overline{X}_{Γ_{1}(𝔫,t)} is irreducible) and its ordinary locus is open, dense, and connected.

Similarly, given a closed point of Y_{Γ_{1}(𝔫)∩Iw}^{rig}, it corresponds to a point (A, λ, η, C) defined over the integer O_{L} of a finite extension L of K. Let B = A/C and S = Spec O_{L}; let val_{S} denote the valuation on L normalised such that val_{S}(p) = 1. Then the O_{F} -equivariant map of O_{S}-modules

Lie^{∨}(B/S) ⟶ Lie^{∨}(A/S)

decomposes into

Lie^{∨}(B/S)_{τ} ⟶ Lie^{∨}(A/S)_{τ}

for every τ ∈ Hom(F, K), and, for the unique prime 𝔭 of F above p corresponding to τ, let deg_{𝔭}((A, C)) denote 1 − val_{S}(Ann(Coker(Lie^{∨}(B/S)_{τ} → Lie^{∨}(A/S)_{τ}))). Applying the construction to the universal HBAV over Y_{Γ_{1}(𝔫)∩Iw}^{rig}, we locally have functions on Y_{Γ_{1}(𝔫)∩Iw}^{rig} whose valuations define the degrees. As for deg(A), Kocher’s principle allows us to extend the function to X_{Γ_{1}(𝔫)∩Iw}^{rig} ≃ X_{Γ_{1}(𝔫)∩Iw}^{an}

Definition. If S_{1} and S_{2} are disjoint subsets of Hom(F, K) and if I, I_{1}, I_{2} ⊆ [0, 1] are closed, open, or half open intervals with endpoints in Q, define the rigid space (X_{Γ_{1}(𝔫)∩Iw,K}^{an}I), I_{1}_{S_{1}}I_{2}_{S_{2}} to be the admissible open set of points whose degree at τ ∈ Hom(F, K) − S_{1} − S_{2} (resp. S_{1}, resp. S_{2}) is in the range I (resp. I_{1}, resp. I_{2}).

Definition. Let π_{1} (resp. π_{2,𝔭}) denote the degeneracy map

X_{Γ_{1}(𝔫)∩Iw,K}^{an} ⟶ X_{Γ_{1}(𝔫),K}^{an}

which, on the non-cuspidal points, is defined by

(A, i, λ, η, C) ↦ (A, i, λ, η)
(resp. (A, i, λ, η, C) ↦ (A/C_{𝔭}, (i mod C_{𝔭}), (λ mod C_{𝔭}), (η mod C_{𝔭}))

Definition. Let π denote the degeneracy map

X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an} ⟶ X_{Γ_{1}(𝔫)∩Iw,K}^{an}

which, on the non-cuspidal points, is defined by

(A, i, λ, η, η_{KM}) ↦
(A/⟨pP_{η_{KM}}⟩, (i mod ⟨pP_{η_{KM}}⟩), (λ mod ⟨pP_{η_{KM}}⟩), (η mod ⟨pP_{η_{KM}}⟩).

where by P_{η_{KM}}, we mean the image of 1 by η_{KM}.

Definition. Define (X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}I)I_{1}_{S_{1}}I_{2}_{S_{2}} to be the preimage by π of (X_{Γ_{1}(𝔫)∩Iw,K}^{an}I,)I_{1}_{S_{1}}I_{2}_{S_{2}}.

For 0 ≤ r ≤ p/(p + 1), it follows from the previous section that

π_{1}^{−1}(X_{Γ_{1}(𝔫),K}^{an}[0, r]) ≃ X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r] ∐ X_{Γ_{1}(𝔫)∩Iw,K}^{an}[1 − r/p, 1];

and for τ = τ_{𝔭}

π_{2,𝔭}^{−1}(X_{Γ_{1}(𝔫),K}^{an}[0, r])
≃ (X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r])_{τ} [0, r/p] ∐(X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r])_{τ} [1 − r, 1].

The theory of canonical subgroups provides rigid sections:

π_{1} : X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r] ⟶^{≃} X_{Γ_{1}(𝔫),K}^{an}[0, r]

and

π_{2,𝔭} : X_{Γ_{1}(𝔫)∩Iw,K}^{an}[1 − r, 1] ⟶^{≃} X_{Γ_{1}(𝔫),K}^{an}[0, r].

On the other hand,

π_{1} : X_{Γ_{1}(𝔫)∩Iw,K}^{an}[1 − r/p, 1]⟶X_{Γ_{1}(𝔫),K}^{an}[0, r]

is finite flat of degree |O_{F} /p|, and

π_{2,𝔭} : X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r/p]⟶X_{Γ_{1}(𝔫),K}^{an}[0, r]

is finite flat of degree |O_{F} /𝔭|.

Hida [14] proves (Theorem 5.6 in [14]) that, for a character ψ : Fr_{𝔫p^{∞}} → K^{×} which factors through I_{𝔫p^{r}} and k ≥ 2, an element F_{H} : h_{O}^{0}(𝔫) → L of S_{O}^{0}(𝔫) ⊗ L defines, modulo(ker(ϵ ∘ Art)^{k−2}ψ), a cusp eigenform of weight k and level Γ_{1}(𝔫p^{r}) which is an eigenform with its T_{𝔪}-eigenvalue F_{H}(T_{𝔪}) mod (ker(ϵ ∘ Art)^{k−2}ψ) and S acting by (ϵ ∘ Art)^{k−2}ψ. Indeed I_{𝔫p^{∞}}-action defines the character of F_{H} mod (ker(ϵ ∘ Art)^{k−2}ψ), i.e,

(F_{H} mod (ker(ϵ ∘ Art)^{k−2}ψ))(⟨ ⟩)
= ψ(ψ_{F_{H}} mod (ker(ϵ ∘ Art)^{k−2}ψ))(ψ_{T} ∘ ϵ^{2−k})

where ψ_{F_{H}} is the composite Tor_{𝔫p^{∞}} ↪ I_{𝔫p^{∞}} →^{⟨ ⟩} h_{O}^{0}(𝔫) followed by F_{H} : h_{O}^{0}(𝔫) → L; and ψ_{T} is the ‘Teichmuller character’, the projection from Z_{p}^{×} to its torsion subgroup of finite order. We shall prove that the specialisation F_{H} mod ker(ϵ ∘ Art)^{k−2}ψ defines a p-ordinary overconvergent eigenform of weight k and of level Γ_{1}(𝔫p^{r}) for any k = 1.

For ϵ such that 0 ≤ ϵ < 1/(p^{r−2}(p + 1)), the theory of canonical subgroups in [29] (see also Proposition 2.3.1 and 2.4.1 in [21]) shows that U_{p} =^{def} ∏_{𝔭} U_{𝔭} defines a completely continuous endomorphism on H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}(cusps))\overline{U}, where X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ] is the preimage by the forgetful morphism of X_{Γ_{1}(𝔫),K}^{an}[0, ϵ]. We remark that, when F = Q, this is proved in [4] Lemma 2.3 as a result of calculations with q-expansions.

By Serre’s theory [31], there is an idempotent e commuting with U_{p} by which we may write

H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}(cusps))\overline{U}
= eH^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}(cusps))\overline{U}
+(1 − e)H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}(cusps))\overline{U}

where eH^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}(cusps))\overline{U} is finite-dimensional K-vector space and all the generalised eigenvalues of U_{p} are units, while U_{p} is topologically nilpotent on the complement. It is well-known that e = e|H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}(cusps))\overline{U}.

Lemma 15 For any integer k, the p-adic eigenform F_{H} mod (ker(ϵ∘Art)^{k−2}ψ) as above is overconvergent of weight k and of level Γ_{1}(𝔫p^{r}).

Proof. This can be proved as in Lemma 1 in [5]; replace the Eisenstein series ‘E’ of weight (p − 1) therein by the pull-back to X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an} of a characteristic zero lifting of a sufficiently large power of the Hasse invariant. □

It follows from the theorem in the previous section that, given a p-adic representation

ρ : Gal(\overline{F} /F) → GL_{2}(O)

as in the main theorem, there are

1. a finite soluble totally real field extension F_{Σ} ⊂ \overline{F} of F in which p splits completely,

2. a finite set S = Σ ∐ S_{Σ,P} ∐ S_{Σ,∞} of places in F_{Σ}, where S_{Σ,P} denotes the set of places of F above p and S_{Σ,∞} denotes the set of infinite places of F_{Σ},

3. an ideal 𝔫 of O_{F} divisible by 𝔫_{Σ} = ∏_{𝔮∈Σ} 𝔮,

4. 2^{|S_{Σ,P}|} characters χ_{P} : Gal(\overline{F} /F_{Σ}) → O^{×} of finite order and 2^{|S_{Σ,P}|} weight one p-ordinary overconvergent cuspidal Hilbert modular eigenforms f_{P} of ‘tame level’ 𝔫, one for every subset P of S_{Σ,P}, such that:

• f_{P} is the weight one specialisation of the Λ-adic companion form F_{Hida,P} : h_{O}^{0}(𝔫) → K, with character ψ_{P} = ψ_{P}^{S_{Σ,P}}ψ_{P,S_{Σ,P}} of (O_{F_{Σ}}/𝔫)^{×} × (O_{F_{Σ}}/p)^{×}

• the Galois representation ρ_{P} associated to f_{P} is ρ|_{Gal(\overline{F} /F_{Σ})} ⊗ χ_{P}^{−1},

• ρ_{P} is unramified outside S and ordinary at every place in S_{Σ,P},

and the f_{P} ’s are ‘in companion’ in the sense that

• c(O_{F_{Σ}}, f_{P}) = 1, and c(𝔪, f_{P}) = 0 if 𝔪 is not coprime to 𝔫;

• c(𝔮, f_{P}) = tr ρ(Frob_{𝔮})/χ_{P} (Frob_{𝔮}) for every prime ideal 𝔮 not dividing 𝔫p;

• for 𝔭 in P, c(𝔪, f_{P})(χ_{P} ∘ Art)(𝔪) = c(𝔪, f_{P −{𝔭}})(χ_{P −{𝔭}} ∘ Art)(𝔪) for every ideal 𝔪 coprime to 𝔫p;

• for 𝔭 in P, the character of f_{P} at 𝔭 is χ_{P −{𝔭}}χ_{P}^{−1} while for 𝔭 ∈ S_{Σ,P} − P, the character of f_{P} at 𝔭 is χ_{P ∪{𝔭}}χ_{P}^{−1};

• for 𝔭 in P, (χ_{P}^{S_{Σ,P}} ∘ Art)(𝔭) = (ψ_{P −{𝔭}}^{S_{Σ,P}} ∘ Art)(𝔭);

• for a place 𝔭 of P, the U_{𝔭}-eigenvalue of f_{P} is (χ_{𝔭,1}χ_{P}^{−1})(Frob_{𝔭}) while for 𝔭 in S_{Σ,P} − P, the U_{𝔭}-eigenvalue of f_{P} is (χ_{𝔭,2}χ_{P}^{−1})(Frob_{𝔭}).

14 Analytic continuation of overconvergent eigenforms

Fix τ = τ_{𝔭} throughout the section (except the last two assertions).

Definition.  Fix t.  For brevity, let X_{Γ_{1}(𝔫,t)∩Iw,τ} denote

(X_{Γ_{1}(𝔫,t)∩Iw,K}[0, r])[0, 1)_{τ}; and for an integer n ≥ 0, let X_{Γ_{1}(𝔫,t)∩Iw,τ,n} denote (X_{Γ_{1}(𝔫,t)∩Iw,K}[0, r])[0, 1 − 1/p^{n}(p + 1)]_{τ}. Let X_{Γ_{1}(𝔫)∩Iw,τ} (resp. X_{Γ_{1}(𝔫)∩Iw,τ,n}) denote the disjoint union over T of X_{Γ_{1}(𝔫,t)∩Iw,τ} (resp. X_{Γ_{1}(𝔫,t)∩Iw,τ,n}).

Proposition 16 For every integer n ≥ 0, X_{Γ_{1}(𝔫)∩Iw,τ,n} is an admissible open subset of X_{Γ_{1}(𝔫)∩Iw,τ}, and the X_{Γ_{1}(𝔫)∩Iw,τ,n} form an admissible covering of X_{Γ_{1}(𝔫)∩Iw,τ}. For every t and every n ∈ Z_{≥0}, X_{Γ_{1}(𝔫,t)∩Iw,τ,n} is connected.

Proof. Clear. □

Definition. Let X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ} (resp. X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ,n}) denote the preimage by the degeneracy morphism

π : X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),K}^{an} → X_{Γ_{1}(𝔫,t)∩Iw,K}^{an}

of X_{Γ_{1}(𝔫,t)∩Iw,τ} (resp. X_{Γ_{1}(𝔫,t)∩Iw,τ,n}). Let X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ} (resp. X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ,n}) denote the disjoint union over T of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ} (resp. X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ,n}).

Proposition 17 For every integer n ≥ 0, X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ,n} is an admissible open subset of X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ}, and the X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ,n} form an admissible covering of X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ}. For every t and an integer n ≥ 0, X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ,n} is connected.

Proof. Analogous to the proposition above. □

Corollary 18 We have π_{1}^{−1}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ,n+1}) ⊂ π_{2,𝔭}^{−1}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ,n}).

Proof. This follows from [29]. □

Let (Tate_{M_{1},M_{2}}(q) = (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/q^{M_{2}}, i, λ, η, η_{KM} : 1 ↦ ζ_{r}) over O ⊗ Z((q^{M_{1}M_{2}^{−1}})) for the pair M_{1}, M_{2} of the fractional ideals such that M_{1}M_{2}^{−1} ≃ t be a family of HBAVs around a cusp of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),n}. Choose (noncanonically) once for all a basis of the pull-back by Max (O ⊗_{Z} Z((q^{M}))) → X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an} of the line bundle L_{k}, since a subgroup of Tate_{M_{1},M_{2}}(q)[𝔭] of order |O_{F} /𝔭|, disjoint from η_{r}, is of the form ζη + q^{M_{2}} where ζ ranges over the |O_{F} /𝔭| points of (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})(S)[𝔭] and η = η_{1,𝔭}^{ét} ∈ q^{𝔭^{−1}M_{2}/M_{2}}, U_{𝔭}(f)(Tate_{M_{1},M_{2}}(q), i, λ, η, η_{KM}) is:

|O_{F} /𝔭|^{−1} ∑_{ζ} f(((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/q^{M_{2}})/(ζη))

= |O_{F} /𝔭|^{−1} ∑_{ζ} f((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/(ζq_{η})^{𝔭^{−1}M_{2}})
= |O_{F} /𝔭|^{−1} ∑_{ζ} |O_{F} /t_{1}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(𝔭M^{−1}ν, f)(ζq_{η})^{ν}
= |O_{F} /𝔭|^{−1}|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}}(∑_{ζ} ζ^{ν})c(𝔭M^{−1}ν, f)q_{η}^{ν}
= |O_{F} /𝔭|^{−1}|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈M^{+}} |O_{F} /𝔭|c(𝔭M^{−1}ν, f)q^{ν}

where q_{η} denotes a representative in q^{𝔭^{−1}M_{2}} of the class η ∈ q^{𝔭^{−1}M_{2}/M_{2}} = q^{𝔭^{−1}M_{2}}/q^{M_{2}} defined earlier; and t_{𝔭} represents the class of 𝔭t ≃ 𝔭M_{1}M_{2}^{−1}.

Theorem 19 Suppose that f ∈ H^{0}((X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K})_{τ} [0, ϵ], L_{k}) is an eigenform for U_{𝔭} with non-zero eigenvalue, then f extends to X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ} = (X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r})})_{τ} [0, 1).

Definition. Let

X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[0]} ⊂ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[1]} ⊂ ⋯ ⊂ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[r−1]}
||
X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}

denote the admissible open subsets of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r})}^{an} defined in such a way

that the non-cuspidal S-points of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[s]} parameterises (A/S, i, λ, η) equipped with a point P_{η_{KM}} of exact of order ∑_{𝔭} |O_{F} /𝔭^{r}| where A/S is either 𝔭non-ordinary, or it is 𝔭-ordinary and H_{r−s}(A[𝔭]) equals the subgroup generated by |O_{F} /𝔭|^{s}P_{η_{KM},𝔭}. For every 0 ≤ s ≤ r − 1, X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[s]} is connected since it is the pre-image of a closed subset of the union of irreducible components intersecting precisely at the 𝔭-non-ordinary locus of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}.

Theorem 20 If r is an integer ≥ 2 and suppose that f ∈ H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ} , L_{k}) is an eigenform for U_{𝔭} with non-zero eigenvalue. Then f extends to X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[r−1]} .

Proof. This can be proved as Lemma 6.1 in [3] □

Corollary 21 If f ∈ H^{0}(X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, ϵ], L_{k}) for some 0 < ϵ < 1 is an eigenform for every U_{𝔭}, 𝔭|p, with non-zero eigenvalue, then f extends to H^{0}(X_{Γ_{1}(𝔫)∩Iw,K}[0, 1), L_{k}). Similarly, if f ∈ H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}[0, ϵ], L_{k}), for some 0 < ϵ < 1 is an eigenform for every U_{𝔭}, 𝔭|p, with non-zero eigenvalue, then g extends to H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}[0, 1), L_{k}).

15 Gluing eigenforms

15.1 The Iwahori case

Definition. For every subset P of the set of places of F above p, let w_{P} denote the automorphism of X_{Γ_{1}(𝔫)∩Iw,K}^{an} defined by a composite (independent of ordering) of the w_{𝔭} for all 𝔭 in P.

Theorem 22 For every subset P of the set S = S_{P} of places of F above p, suppose f_{P} ∈ H^{0}(X_{Γ_{1}(𝔫),K}^{an}, L_{k}) is an overconvergent modular form of parallel weight k = ∑_{τ∈Hom(F,K)} kτ ∈ Z and of level Γ_{1}(𝔫). Assume furthermore that

• the Fourier coefficient c(f_{P} , O_{F}) = 1;

• for every place 𝔭 of F above p,there exist α_{𝔭}, β_{𝔭} ∈ K such that α_{𝔭} ≠ β_{𝔭} and such that, for every P, f_{P} is an eigenform for U_{𝔭} with eigenvalue α_{𝔭} if 𝔭 ∈ P whilst with eigenvalue β_{𝔭} if 𝔭 ∉ P;

• for all ideal 𝔪 of O_{F} coprime to p, c(𝔪, f_{P}) are equal for every P.

Then every f_{P} is a classical Hilbert modular eigenform of weight k and of level Γ_{1}(𝔫) ∩ Iw.

Proof. By the isomorphism

π_{1} : X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r] ⟶^{≃} X_{Γ_{1}(𝔫),K}^{an}[0, r]

for r < p/(p + 1) given by the canonical subgroups theorem [29], we may think of f_{P} as an element of H^{0}(X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, r], L_{k}). It follows from results in [29] that π_{1}^{∗}f_{P} extends to a section over X_{Γ_{1}(𝔫)∩Iw,K}^{an}[0, 1). For brevity, we shall only show that f_{P}, with P the (full) set S of places of F above p, is classical; the general case follows by changing the roles of α_{𝔭} and β_{𝔭}. Choose a rational number r ∈ Q with 1/2 < r < p/(p + 1). Suppose that f_{S} extends to a section of L_{k} over (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[0, 1]_{S−P} for some P ⊆ S. Fix a prime 𝔭 ∈ P. It suffices to show that f_{S} extends to (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[0, 1]_{S−(P −{𝔭})}. For f ∈ H^{0}(X_{Γ_{1}(𝔫),K}[0, r], L_{k}) and for every subset Q ⊆ S − P, let f^{Q} denote the restriction of f to (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}[0, r]_{(S−P)−Q} by the map π_{1} ∘ w_{Q} which defines an isomorphism

(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}, [0, r]_{(S−P)−Q} ≃ X_{Γ_{1}(𝔫),K}[0, r]
||
(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}  .

The pre-image by π_{2,𝔭} ∘ w_{Q} of (X_{Γ_{1}(𝔫),K}[0, r])(0, r)_{𝔭} is the union of two components

(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}(1 − r, 1)_{𝔭} ∐(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}(0, r/p)_{𝔭}

and it induces an isomorphism

(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}(1 − r, 1)_{𝔭} ≃ (X_{Γ_{1}(𝔫),K}[0, r])(0, r)_{𝔭}

on the one component and a finite flat morphism of degree |O_{F} /𝔭|

(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}(0, r/p)_{𝔭} ⟶ (X_{Γ_{1}(𝔫),K}[0, r])(0, r)_{𝔭}

on the other. We are going to glue f_{S} and f_{S−{𝔭}}; more precisely glue f_{S}^{Q} and f_{S−{𝔭}}^{Q}. Let F denote the section

(α_{𝔭}f_{S}^{Q} − β_{𝔭}f_{S−{𝔭}}^{Q})/(α_{𝔭} − β_{𝔭}) ∈ H^{0}((X_{Γ_{1}(𝔫),K}[0, r])(0, r)_{𝔭}, L_{k})

and G denote the section

|O_{F} /𝔭|⟨𝔭⟩((f_{S}^{Q}−f_{S−{𝔭}}^{Q})/(α_{𝔭}−β_{𝔭}) ∈ H^{0}((X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1−r, 1]_{Q}(0, r)_{𝔭}, L_{k})

Since one can show readily the q-expansions of π_{2,𝔭}^{∗}F and G are equal at around C = (Tate_{M_{1},M_{2}}(q), …, ⟨ζ_{1}⟩), we shall glue π_{2,𝔭}^{∗}F and G at (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r]_{Q}(0, r/p)_{𝔭} to construct an extension F′ of F to a section over (X_{Γ_{1}(𝔫),K}[0, r])[0, 1)_{𝔭}; this extension constructs an extension of f_{S}^{Q} (and f_{S−{𝔭}}^{Q}) to (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r, 1]_{Q}[0, r]_{(S−P)−Q}[0, 1]_{𝔭} and therefore to (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[0, 1]_{S−(P −{𝔭})} (by assumption, there is an extension ‘over [0, 1]’ at S − P)

Gluing of π_{2,𝔭}^{∗}F and G is analogous to [3] since we have a commutative diagram

(X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r]_{Q}(0, r/p)_{𝔭} ⟶ (X_{Γ_{1}(𝔫)∩Iw,K}[0, r])[1 − r]_{Q}(0, 1)_{𝔭}
↓  ↓
(X_{Γ_{1}(𝔫),K}[0, r])(0, r)_{𝔭}  ⟶  (X_{Γ_{1}(𝔫),K}[0, r])(0, 1)_{𝔭}

where the vertical arrows are π_{2,𝔭} ∘ w_{Q} but of degree |O_{F} /𝔭| on the left and 1 + |O_{F} /𝔭| on the right.□

15.2 The Γ_{1}(p) case

For evert t in T and for every subset P of the set S = S_{P} of places of F above p, we well let

(X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1))(0, 1]_{S−P}
=^{def} π_{1}^{−1}(X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),K}[0, 1)(0, 1]_{S−P}) ⊂ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}^{an}.

Let w_{P} denote the composite of the w_{ζ_{𝔭}} for all 𝔭 ∈ P. Note that

(X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1))(0, 1]_{S−P} →^{≃} w_{S−P}^{−1} X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1)

and each (X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1))(0, 1]_{S−P} is connected since it is isomorphic to X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),O_{K}} [0, 1) and the latter is connected since it is the pre-image of a connected component in the Zariski topology of the closed fibre. Let (X_{Γ_{1}(𝔫)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1))(0, 1]_{S−P} denote the disjoint union over T of (X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1))(0, 1]_{S−P}.

Definition. If f is a section of L_{k} over X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1), then w_{S−P}^{∗} f is a section of w_{S−P}^{∗} L_{k} over w_{S−P}^{−1} (X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}, [0, 1)) = (X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1))(0, 1]_{S−P}. Because p is inverted, the natural morphism of invertible sheaves

L_{k}|(X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1))(0, 1]_{S−P} →^{≃} w_{S−P}^{∗} (L_{k}|X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1))

is an isomorphism and we let f|w_{P} denote the section of L_{k} over (X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1))(0, 1]_{P −S} corresponding to w_{S−P}^{∗} f by the isomorphism.

Theorem 23 For every subset P of S = S_{P}, let f_{P} ∈ H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1), L_{k})^{\overline{U}^{1}} be an overconvergent Hilbert modular form of parallel weight k = ∑_{τ∈Hom(F,K)} kτ ∈ Z and of level Γ_{1}(𝔫p). For every subset P of S, suppose that f_{P} has a Hecke character

ψ_{P} =^{def} ψ_{P}^{S} ψ_{S,P} : (O_{F} /𝔫p)^{×} ≃ (O_{F} /𝔫)^{×} × (O_{F} /p)^{×} ⟶ O^{×};

and that ψ_{P}^{S} (𝔭) = ψ_{P −{𝔭}}^{S}(𝔭) for every 𝔭 in P. Suppose that • the Fourier coefficients c(O_{F} , f_{P}) = 1 and c(𝔪, f_{P}) = 0 if 𝔪 and 𝔫 are

not coprime,

• for every 𝔭 ∈ P, c(𝔪, f_{P}) = ψ_{P,𝔭}(𝔪)c(𝔪, f_{P −{𝔭}}) for every ideal 𝔪 coprime to 𝔫p, where by ψ_{P,𝔭} we mean the 𝔭-component of ψ_{P,S} which we assume non-trivial,

• for every 𝔭 in S, f_{P} is an U_{𝔭}-eigenform with non-zero eigenvalue α(𝔭, f_{P}), and for every 𝔭 ∈ P, α(𝔭, f_{P})α(𝔭, f_{P −{𝔭}}) = ψ_{P}^{S} (𝔭)|O_{F} /𝔭|^{k−1} = ψ_{P −{𝔭}}^{S}(𝔭)|O_{F} /𝔭|^{k−1}.

Then f_{P} is a section of L_{k} over X_{Γ_{1}(𝔫)∩Γ_{1}(p),O_{K}}^{an} .

Proof. For every subset P of S, let g_{P} denote f_{P} |w_{P} ∈ H^{0}((X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1))(0, 1]_{S−P} , L_{k}). Clearly g_{S} = f_{S}. We shall prove that f_{S} is classical. Fix an integer 0 ≤ n ≤ |S| and suppose that the g_{P} with P ⊆ S such that |P | ≥ n glue together to define sections, which will again be denoted by g_{P}, over

⋃_{P ⊂S,|P |≥n}(X_{Γ_{1}(𝔫)∩Γ_{1}(p),K}[0, 1))(0, 1]_{P −S}.

Fix a subset P ⊆ S with #P = n and fix 𝔭 ∈ P. It suffice to show that g_{P} (≃ w_{S−P}^{∗} f_{P}) and (a constant multiple of) g_{P −{𝔭}}(≃ w_{S−(P −{𝔭})}^{∗}f_{P −{𝔭}} = w_{{𝔭}}^{∗}w_{S−P}^{∗} f_{P −{𝔭}}) glue. Let α_{𝔭} (resp. β_{𝔭}) denote the U_{𝔭}-eigenvalue α(𝔭, f_{P}) (resp. α(𝔭, f_{P −{𝔭}})). Fix a p-th root ζ_{1} of unity. Let (Tate_{M_{1},M_{2}}(q), …, η_{KM} : 1 ↦ ζ_{1}). be a point around a cusp C. By abuse of notation, we call it C. There is a morphism

π_{1} : X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}^{an} ⟶ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),K}^{an}

defined, on the non-cuspidal points, by

(A, i, λ, η, η_{KM}, D_{𝔭}) ↦ (A, i, λ, η, η_{KM})

and, for γ ∈ (O_{F} /𝔭)^{×},

π_{2,γ} : X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}^{an} ⟶ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),K}^{an}

defined, on the non-cuspidal points, by

(A, i, λ, η, η_{KM}, D_{𝔭}) ↦ (A/(γη_{KM}(1)_{𝔭} +D_{𝔭}), …, (η_{KM} mod (γη_{KM}(1)_{𝔭} +D_{𝔭}))).

To single out, let

π_{2,𝔭} : X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}^{an} ⟶ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),K}^{an} denote the morphism ‘γ = 0 in O_{F} /𝔭’ which takes (A, i, λ, η, η_{KM}, D_{𝔭}) to (A/D_{𝔭}, …, (η_{KM} mod D_{𝔭})).

By abuse of notation, let C also denote the pre-image

(Tate_{M_{1},M_{2}}(q), …, η_{KM} : 1 ↦ ζ_{1}, ⟨η_{1}^{et}⟩) ∈ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}^{an}

by π_{1} above of C = (Tate_{M_{1},M_{2}}(q) …, ζ_{1}) for (M_{1}, M_{2}) = (O_{F} , t^{−1}) and M = M_{1}M_{2} = t^{−1}; and let C_{P} ∈ (X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}[0, 1))(0, 1]_{S−P} denote the cups w_{S−P}^{∗} C. Then

(g_{P} |π_{1})(C_{P}) = pr^{∗}f(Tate_{M_{1},M_{2}}(q), …, ζ_{1}) = |O_{F} /t|^{−1} ∑_{ν∈M^{+}} c(νM^{−1}, f_{P})q^{ν}

On the other hand, for γ ∈ (O_{F} /𝔭)^{×},

(g_{P} |π_{2,γ})(C_{P})
= (f_{P} |π_{2,γ})(Tate_{M_{1},M_{2}}(q), …, ζ_{1})
= pr^{∗}f_{P} (Tate_{M_{1},M_{2}}(q)/(ζ^{γ}η), …) where η := η_{1,𝔭}^{et} and ζ := ζ_{1,𝔭}
= pr^{∗}f_{P} ((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/(ζ^{γ}q_{η})^{𝔭^{−1}M_{2}}, …)
= |O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P})ζ^{γν}q_{η}^{ν}

where t_{𝔭} is one of the (fixed) representatives of the narrow class group of F representing the class of t𝔭, and where q_{η} denote a representative in q^{𝔭^{−1}M_{2}} of the class η ∈ q^{𝔭^{−1}M_{2}}/q^{M_{2}}. Finally

\begin{aligned}(g_{P} |π_{2,𝔭})(C_{P}) & = & pr^{∗}(f_{P} |π_{2,𝔭})(Tate_{M_{1},M_{2}}(q), …, ζ_{p}) \\ & = & |O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P})q_{η}^{ν} \end{aligned}

For brevity, let S denote the ‘Gauss sum’

S =^{def} ∑_{γ∈(O_{F} /𝔭)^{×}} ζ^{γ}ψ_{P,𝔭}(γ)

for (M_{1}, M_{2}) = (O_{F} , t^{−1}). Then for ν ∈ (𝔭^{−1}M)^{+} such that ν𝔭M^{−1} ⊂ O_{F}^{+} is not divisible by 𝔭,

S = ∑_{γ∈(O_{F} /𝔭)^{×}} ζ^{γν𝔭M^{−1}}ψ_{P,𝔭}(γν𝔭M^{−1}) = ψ_{P,𝔭}(ν𝔭M^{−1}) ∑_{γ∈(O_{F} /𝔭)^{×}} ζ^{γν}ψ_{P,𝔭}(γ).

1032 Shu Sasaki It then follows that

∑_{γ∈(O_{F} /𝔭)^{×}} ψ_{P,𝔭}(γ)(g_{P} |π_{2,γ})(C_{P})

= ∑_{γ∈(O_{F} /𝔭)^{×}} ψ_{P,𝔭}(γ)|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P})ζ^{γν}q_{η}^{ν}
= |O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P})q_{η}^{ν} ∑_{γ∈(O_{F} /𝔭)} ψ_{P,𝔭}(γ)ζ^{γν}
= S|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+},𝔭∤ν𝔭M^{−1}} c(ν𝔭M^{−1}, f_{P})ψ_{P,𝔭}^{−1}(ν𝔭M^{−1})q_{η}^{ν}
= S|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+},𝔭∤ν𝔭M^{−1}} c(ν𝔭M^{−1}, f_{P −{𝔭}})q_{η}^{ν}
= S|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P −{𝔭}})q_{η}^{ν}
−S|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+},𝔭|ν𝔭M^{−1}} c(ν𝔭M^{−1}, f_{P −{𝔭}})q_{η}^{ν}
= S|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P −{𝔭}})q_{η}^{ν}
−S|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈M^{+}} c(νM, f_{P −{𝔭}})q_{η}^{ν}
= S(|O_{F} /t_{𝔭}|^{−1} ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P −{𝔭}})q_{η}^{ν} − (U_{𝔭}g_{P −{𝔭}}|π_{1})(C_{P}))
= S(g_{P −{𝔭}}|π_{2,𝔭} − β_{𝔭}g_{P −{𝔭}}|π_{1})(C_{P})

By the connectedness of (X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),K}[0, 1))(0, 1]_{S−P},

∑_{γ∈(O_{F} /𝔭)^{×}} ψ_{P,𝔭}(γ)(g_{P} |π_{2,γ}) = S(g_{P −{𝔭}}|π_{2,𝔭} − β_{𝔭}g_{P −{𝔭}}|π_{1})

on (X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),K}[0, 1))(0, 1]_{S−P}. Let (A, i, λ, η, η_{KM}, D_{𝔭} = ⟨Q_{𝔭}⟩) be a non-cuspidal point of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p),Iw_{𝔭},K}^{an} and let P = η_{KM}(1) = P^{𝔭} × P_{𝔭} and Q = P^{𝔭} × Q_{𝔭}. Then

|O_{F} /𝔭|α_{𝔭}g_{P} (A, i, λ, η, Q) − pr^{∗}g_{P} (A/⟨P_{𝔭}⟩, …, \overline{η}, \overline{Q}))

where \overline{η} := η mod ⟨P_{𝔭}⟩, and \overline{Q} := Q mod ⟨P_{𝔭}⟩ is:

= |O_{F} /𝔭|U_{𝔭}g_{P} (A, i, λ, η, Q) − pr^{∗}g_{P} (A/⟨P_{𝔭}⟩, …, \overline{η}, \overline{Q})
= ∑_{C_{𝔭}⊂A[𝔭],C_{𝔭}≠⟨P_{𝔭}⟩,⟨Q_{𝔭}⟩} pr^{∗}g_{P} (A/C_{𝔭}, …, (Q mod C_{𝔭}))
= ∑_{C_{𝔭}=⟨γP_{𝔭}+Q_{𝔭}⟩,γ∈(O_{F} /𝔭)^{×}} pr^{∗}g_{P} (A/⟨γP_{𝔭} + Q_{𝔭}⟩, …, (Q mod ⟨γP_{𝔭} + Q_{𝔭}⟩))
= ∑_{γ∈(O_{F} /𝔭)^{×}} ψ_{P,𝔭}(−γ)pr^{∗}g_{P} (A/⟨γP_{𝔭} + Q_{𝔭}⟩, …, (P^{𝔭} × P_{𝔭} mod ⟨γP_{𝔭} + Q_{𝔭}⟩)
= ψ_{P,𝔭}(−1) ∑_{γ∈(O_{F} /𝔭)^{×}} ψ_{𝔭,P} (γ)(g_{P} |π_{2,γ})(A, …, η, P)
= ψ_{P,𝔭}(−1)S(g_{P −{𝔭}}|π_{2,𝔭} − β_{𝔭}g_{P −{𝔭}}|π_{1})(A, …, η, P)
= ψ_{P,𝔭}(−1)S(pr^{∗}g_{P −{𝔭}}(A/⟨Q_{𝔭}⟩, …, P mod ⟨Q_{𝔭}⟩) − β_{𝔭}g_{P −{𝔭}}(A, …, P)
= ψ_{P,𝔭}(−1)S((g_{P −{𝔭}}|w_{ζ_{𝔭}})(A, …, P^{𝔭} × (−Q_{𝔭}))
−α_{𝔭}^{−1}ψ_{P}^{S} (𝔭)|O_{F} /𝔭|pr^{∗}(g_{P −\{𝔭\}}|w_{ζ_{𝔭}})(A/⟨P_{𝔭}⟩, …, 𝔭^{−1}\overline{η}, (P^{𝔭} × (−Q_{𝔭}) mod ⟨P_{𝔭}⟩))
= S((g_{P −\{𝔭\}}|w_{ζ_{𝔭}})(A, …, Q) − |O_{F} /𝔭|^{−1}α_{𝔭}^{−1}pr^{∗}(g_{P −\{𝔭\}}|w_{ζ_{𝔭}})(A/⟨P_{𝔭}⟩, …, \overline{Q}).

Therefore

\begin{aligned}& (|O_{F} /𝔭|α_{𝔭}g_{P} − S(g_{P −\{𝔭\}}|w_{ζ_{𝔭}}))(A, …, Q) \\ = & (|O_{F} /𝔭|α_{𝔭})^{−1}pr^{∗}(|O_{F} /𝔭|α_{𝔭}g_{P} − S(g_{P −\{𝔭\}}|w_{ζ_{𝔭}}))(A/⟨P_{𝔭}⟩, …, (Q mod ⟨P_{𝔭}⟩)) \end{aligned}

It suffices to show that |O_{F} /𝔭|α_{𝔭}g_{P} −S(g_{P −{𝔭}}|w_{ζ_{𝔭}}) is identically zero; in which case, one can glue g_{P} and (|O_{F} /𝔭|α_{𝔭})^{−1}S(g_{P −{𝔭}}|w_{ζ_{𝔭}}) as desired. Showing that it is identically zero is exactly as in [3]. □

Documenta Mathematica 18 (2013) 997–1038

15.3 The Γ_{1}(p^{r}), r ≥ 2, case

Theorem 24 Let S denote the set S_{P} of places of F above p. For any set P ⊆ S, let f_{P} ∈ H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}[0, 1), L_{k})^{\overline{U}^{r}} be an overconvergent modular form of weight k = ∑_{τ∈Hom(F,K)} kτ ∈ Z and of level Γ_{1}(𝔫p^{r}). Suppose that, for every P ⊆ S, f_{P} has a character ψ_{P}^{S} ψ_{S,P} of (O_{F} /𝔫p^{r})^{×} ≃ (O_{F} /𝔫)^{×}×(O_{F} /p^{r})^{×}. Suppose furthermore that f_{P} is an eigenform for U_{𝔭} with non-zero eigenvalue for every 𝔭 ∈ S. Suppose finally that, for every P ⊆ S,

• c(O_{F} , f_{P}) = 1;

• c(𝔪, f_{P}) = 0 if 𝔪 and 𝔫 are not coprime;

• for every 𝔭 ∈ P, c(𝔪, f_{P}) = ψ_{𝔭,P} (𝔪)c(𝔪, f_{P −{𝔭}}) for every ideal 𝔪 coprime to 𝔫p, where ψ_{𝔭,P} is the 𝔭-component of ψ_{S,P}.

Then the f_{P} are classical Hilbert modular forms in H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}^{an}, L_{k})^{\overline{U}^{r}}.

Proof. As in the previous subsection, we shall prove the theorem by induction. For every subset P of S, let g_{P} denote f_{P} |w_{P} ∈ H^{0}(X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}[0, 1))(0, 1]_{S−P} , L_{k}). We shall prove that f_{S} is classical. Fix an integer 0 ≤ n ≤ |S| and suppose that the g_{P} with P ⊆ S such that |P | ≥ n glue together to define sections, which will again be denoted by g_{P}, over

⋃_{P ⊂S,|P |≥n} X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),K}[0, 1)(0, 1]_{S−P} .

Fix a subset P ⊆ S with #P = n, and fix 𝔭 ∈ P. It suffice to show that g_{P} and (a constant multiple of) g_{P −{𝔭}} glue. Let C denote a point (Tate_{M_{1},M_{2}}(q), i, λ, η, P) around a cusp (M_{1}, M_{2}) = (O_{F} , t^{−1}) where

P = η_{KM}(1) = P^{𝔭} × P_{𝔭} ∈ Tate_{M_{1},M_{2}}(q)(O ⊗_{Z} Z((q^{M})))

where P^{𝔭} =^{def} ∏_{𝔮|p,𝔮≠𝔭} ζ_{1,𝔮} and P_{𝔭} =^{def} ζ_{1,𝔭}η_{1,𝔭}^{et}.

For brevity, let μ denote ζ_{r}^{p^{r−1}}, and μ_{𝔭} its 𝔭-component. We shall compute q-expansions of g_{P} and g_{P −{𝔭}} at the cusp C_{P} =^{def} w_{S−P}^{∗} C. Let α_{𝔭} denote the U_{𝔭}-eigenvalue of f_{P}.

|O_{F} /𝔭|α_{𝔭}g_{P} (C_{P})
= |O_{F} /𝔭|U_{𝔭}g_{P} (C_{P})
= ∑_{C_{𝔭}⊂Tate(q)[𝔭],C_{𝔭}≠⟨μ_{𝔭}⟩} pr^{∗}f_{P} (Tate_{M_{1},M_{2}}(q)/C_{𝔭}, …, (P mod C_{𝔭}))
= ∑_{γ∈(O_{F} /𝔭)^{×}} pr^{∗}f_{P} ((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/(μ_{𝔭}^{γ}q_{η})^{𝔭^{−1}M_{2}}, …, P)
= ∑_{γ} pr^{∗}f_{P} ((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/(μ_{𝔭}^{γ}q_{η})^{𝔭^{−1}M_{2}}, …, P^{𝔭} × {ζ_{r,𝔭}μ_{𝔭}^{−γ}})
= ∑_{γ} pr^{∗}f_{P} ((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/(μ_{𝔭}^{γ}q_{η})^{𝔭^{−1}M_{2}}, …, P^{𝔭} × {ζ_{r,𝔭}^{1−p^{r−1}γ}})
= ∑_{γ} ψ_{𝔭,P} ((1 − p^{r−1}γ)𝔭M^{−1})pr^{∗}f_{P} ((GL_{1} ⊗_{Z} 𝔡^{−1}M_{1}^{−1})/(μ_{𝔭}^{γ}q_{η})^{𝔭^{−1}M_{2}}, …, ζ_{r})
= ∑_{γ} ψ_{𝔭,P} ((1 − p^{r−1}γ)𝔭M^{−1}) ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P})(μ_{𝔭}^{γ}q_{η})^{ν}
= ∑_{ν∈(𝔭^{−1}M)^{+}} c(ν𝔭M^{−1}, f_{P})q_{η}^{ν} ∑_{γ} ψ_{𝔭,P} ((1 − p^{r−1}γ)𝔭M^{−1})μ_{𝔭}^{γν}.

We know that ψ_{S,P} has a conductor p^{r}, and hence ψ_{S,P} ((1+p^{r−1})𝔭M^{−1}) = μ^{ν_{1}} for some integer 0 < ν_{1} < p (not that 1 + p^{r−1} is thought of as an element of

𝔭^{−1}M_{2} ↠^{𝔭M_{2}^{−1}M_{1}} M_{1} ↠ M_{1}/𝔭^{r}M_{1} ≃ O_{F} /𝔭^{r}); it therefore follows that ψ_{𝔭,P} ((1 +

p^{r−1})𝔭M^{−1}) = μ_{𝔭}^{ν_{1}}. In particular, ψ_{𝔭,P} ((1 − p^{r−1}γ)𝔭M^{−1}) = μ_{𝔭}^{−γν_{1}}. Hence

∑_{γ} ψ_{𝔭,P} ((1 − p^{r−1}γ)𝔭M^{−1})μ_{𝔭}^{γν} = ∑_{γ} μ_{𝔭}^{γ(ν−ν_{1})} = \{ \begin{matrix}|O_{F} /𝔭| if 𝔭|(ν − ν_{1}) \\ 0 otherwise \end{matrix}

Therefore,

g_{P} (C_{P}) = (α_{𝔭}|O_{F} /𝔭|)^{−1}|O_{F} /𝔭| ∑_{ν∈(𝔭^{−1}M)^{+},𝔭|(ν−ν_{1})} c(ν𝔭M^{−1}, f_{P})q_{η}^{ν}.

We now calculate the q-expansion of g_{P −{𝔭}}|w_{ζ_{𝔭}} at C_{P}. Firstly, note that

(g_{P −{𝔭}}|w_{ζ_{𝔭}})(C_{P}) = pr^{∗}g_{P −{𝔭}}(Tate_{M_{1},M_{2}}(q)/(ζ_{r,𝔭}η_{1,𝔭}^{et}), …, P^{𝔭} × Q_{𝔭})

where Q_{𝔭} is defined by ⟨ζ_{r,𝔭}η_{1,𝔭}^{et}, Q_{𝔭}⟩ = ζ_{𝔭}. Tensoring over O_{F} with 𝔭^{r−1} on GL_{1} ⊗ 𝔡^{−1}M_{1}^{−1} induces an isomorphism

(GL_{1} ⊗ 𝔡^{−1}M_{1}^{−1})/⟨q^{M_{2}}, ζ_{r,𝔭}q⟩
≃ (GL_{1} ⊗ 𝔡^{−1}M_{1}^{−1})/⟨q^{𝔭^{r−1}M_{2}}, ζ_{r−(r−1),𝔭}q^{𝔭^{r−2}M_{2}}⟩
≃ (GL_{1} ⊗ 𝔡^{−1}M_{1}^{−1})/(ζ_{1,𝔭}q)^{𝔭^{r−2}M_{2}}

The HBAV

Tate_{M_{1},𝔭^{r−2}M_{2}}(ζ_{1,𝔭}q) =^{def} (GL_{1} ⊗ 𝔡^{−1}M_{1}^{−1})/(ζ_{1,𝔭}q)^{𝔭^{r−2}M_{2}}

is naturally t𝔭^{2−r}(≃ (𝔭^{r−2}M_{2})^{−1}M_{1})-polarised, and comes equipped with the level structure ⟨𝔭⟩^{r−1}η and the point P^{𝔭} × {η_{1,𝔭}^{et}} of order |O_{F} /p^{r}|; it defines a point of X_{Γ_{1}(𝔫)∩Γ_{1}(p^{r}),τ}^{[r−1]} . For a point (A, i, λ, η, P) of X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[r−1]} ,

(|O_{F} /𝔭|β_{𝔭})^{r−1}g_{P −{𝔭}}(A, i, λ, η, P)
= (|O_{F} /𝔭|U_{𝔭})^{r−1}g_{P −{𝔭}}(A, i, λ, η, P)
= ∑_{C_{𝔭}⊂A[𝔭],|C_{𝔭}|=|O_{F} /𝔭^{r−1}|,C_{𝔭}∩P ={1}} pr^{∗}g_{P −{𝔭}}(A/C_{𝔭}, …, (P mod C_{𝔭}))

In which case, observe that (A/C_{𝔭}, …, (P mod C_{𝔭})) ∈ X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[0]} and

that it allows one to extend finite slope U_{𝔭}-eigenforms over X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[0]} to

U_{𝔭}-eigenforms over X_{Γ_{1}(𝔫,t)∩Γ_{1}(p^{r}),τ}^{[r−1]} by ‘analytic continuation’. If we let

(A, i, λ, η, P) = (Tate_{M_{1},𝔭^{r−2}M_{2}}(ζ_{1,𝔭}q), …, ⟨𝔭⟩^{r−1}η, P^{𝔭} × {η_{1,𝔭}^{et}})

then the cyclic subgroups C_{𝔭} of order |O_{F} /𝔭^{r−1}|, disjoint from the subgroup of order |O_{F} /𝔭| generated by η_{1,𝔭}^{et}, are of the form (ζ_{r,𝔭}ζ_{r−1,𝔭}^{ν_{r−1}}η_{1,𝔭}^{et,𝔭^{−1}M_{2}})/(ζ_{1,𝔭}q^{𝔭^{r−2}M_{2}}) for ν_{r−1} ∈ M_{1}/𝔭^{r−1}M_{1} ≃ O_{F} /𝔭^{r−1}. Then

((GL_{1} ⊗ 𝔡^{−1}M_{1}^{−1})/(ζ_{1,𝔭}q^{𝔭^{r−2}M_{2}}))/((ζ_{r,𝔭}ζ_{r−1,𝔭}^{ν_{r−1}}η_{1,𝔭}^{et,𝔭^{−1}M_{2}})/(ζ_{1,𝔭}q^{𝔭^{r−2}M_{2}}))

≃ (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1})/(ζ_{r,𝔭}ζ_{r−1,𝔭}^{ν_{r−1}}η_{1,𝔭}^{et,𝔭^{−1}M_{2}})
≃ (GL_{1} ⊗_{Z} 𝔡^{−1}M_{1})/(ζ_{r,𝔭}^{1+pν_{r−1}}q_{η})^{𝔭^{−1}M_{2}},

where q_{η} is a representative in q^{𝔭^{−1}M_{2}} of η_{1,𝔭}^{et} ∈ q^{𝔭^{−1}M_{2}}/q^{M_{2}}, is naturally (𝔭^{−1}M_{2})^{−1}M_{1} ≃ 𝔭t-polarised. Then there exists a non-zero constant κ_{1} such that

(gP −{𝔭}|wζ𝔭)(CP )

= ∑νr−1 pr∗gP −\{𝔭\}((GL1 ⊗Z 𝔡−1M1)/ζr,𝔭ζ{νr−1 \atop r−1,𝔭}η{et,𝔭−1M2 \atop 1,𝔭}), ..., ⟨𝔭⟩r−1η, P 𝔭 × \{η{et \atop 1,𝔭}\}))

= ψS (𝔭r−1) ∑νr−1 pr∗gP −\{𝔭\}((GL1 ⊗Z 𝔡−1M1)/(ζ{1+pνr−1 \atop r,𝔭} η{et,𝔭−1M2 \atop 1,𝔭}), ..., η, P 𝔭 × \{q{et,−1 \atop 1,𝔭}\}))

= ψS (−𝔭r−1) ∑νr−1 pr∗gP −\{𝔭\}((GL1 ⊗Z 𝔡−1M1)/(ζ{1+pνr−1 \atop r,𝔭} η{et,𝔭−1M2 \atop 1,𝔭}), ..., η, P 𝔭 × \{ζ{1+pνr−1 \atop r,𝔭} \}))

= κ1 ∑νr−1 ψ𝔭,P ((1 + pνr−1)𝔭M−1)−1pr∗gP −\{𝔭\}((GL1 ⊗Z 𝔡−1M1)/(ζ{1+pνr−1 \atop r,𝔭} η{et,𝔭−1M2 \atop 1,𝔭}), ..., ζr))

= κ1 ∑νr−1 ψ𝔭,P ((1 + pνr−1)𝔭M−1)−1 ∑ν∈(𝔭−1M)+ c(ν𝔭M−1, fP −\{𝔭\})(ζ{1+pνr−1 \atop r,𝔭} qη)ν

= κ1 ∑ν c(ν𝔭M−1, fP −\{𝔭\})qν η ∑νr−1 ψ𝔭,P ((1 + pνr−1)𝔭M−1)−1ζ{(1+pνr−1)ν \atop r,𝔭}

where ν_{r−1} ranges over ∈ (O_{F} /𝔭^{r−1}). For brevity, let S_{ν} =^{def} ∑_{ν_{r−1}∈(O_{F} /𝔭^{r−1})} ψ_{𝔭,P} ((1 + pν_{r−1})𝔭M^{−1})^{−1}ζ_{r,𝔭}^{(1+pν_{r−1})ν}. As in the proof of Theorem 11.1 in [3], one can deduce that

S_{ν} = μ_{𝔭}^{ν−ν_{1}}S_{ν}

where ν_{1} is defined by ψ_{𝔭,P} ((1 + p^{r−1})𝔭M^{−1}) = μ_{𝔭}^{ν_{1}}, and therefore S_{ν} = 0 unless 𝔭|(ν − ν_{1}); one also deduces that, for ν ∈ (𝔭^{−1}M)^{+} such that 𝔭|(ν − ν_{1}),

S_{νν′} = ψ_{𝔭,P} (ν′𝔭M^{−1})S_{ν}

for ν′ ∈ (𝔭^{−1}M)^{+} such that ν′𝔭M^{−1} ≡ 1 mod 𝔭 and therefore S_{νν′}/ψ_{𝔭,P} ((ν𝔭M^{−1})(ν′𝔭M^{−1})) = S_{ν}/ψ_{𝔭,P} (ν𝔭M^{−1}). Consequently, there is a non-zero constant κ_{2} such that

\begin{matrix}(g_{P −\{𝔭\}}|w_{ζ_{𝔭}})(C_{P}) = & κ_{1} ∑_{ν∈(𝔭^{−1}M)^{+},𝔭|(ν−ν_{2})} c(ν𝔭^{−1}M, f_{P −\{𝔭\}})S_{ν}q_{η}^{ν} \\ = & κ_{2} ∑_{ν,𝔭|(ν−ν_{2})} c(ν𝔭^{−1}M, f_{P −\{𝔭\}})ψ_{𝔭,P} (ν𝔭M^{−1})q_{η}^{ν} \\ = & κ_{2} ∑_{ν,𝔭|(ν−ν_{2})} c(ν𝔭^{−1}M, f_{P})q_{η}^{ν}. \end{matrix}

Therefore α_{𝔭}g_{P} and κ_{2}^{−1}g_{P −{𝔭}} agree at C_{P} and hence f_{P} and (α_{𝔭}κ_{2})^{−1}g_{P −{𝔭}} glue together. □

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Shu Sasaki

Universität Duisburg-Essen Fakultät für Mathematik Thea-Leymann-Str. 9 45127 Essen, Germany s.sasaki.03@cantabgold.net