On Artin Representations and Nearly Ordinary Hecke Algebras over Totally Real Fields

We prove many new cases of the strong Artin conjecture for two-dimensional, totally odd, insoluble (icosahedral) representations Gal(F/F ) → GL2(C) of the absolute Galois group of a totally real field F . 2010 Mathematics Subject Classification: 11G80, 11F33, 11F41, 14G22, 14G35


Introduction
Let K be a number field.Artin conjectured that the L-series of any continuous representation ρ : Gal(K/K) → GL n (C) of the absolute Galois group Gal(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(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(K/M ) of a quadratic extension M of Q, and Artin himself proved the conjecture (the holomorphy of L(ρ, s) = L(Ind GK GM 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(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(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(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(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(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 .

Documenta Mathematica 18 (2013) 997-1038
• 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(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 ρ : Gal(F /F ) → GL 2 (F p ) is modular, then ρ : Gal(F /F ) → GL 2 (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(F /F ) → GL 2 (O) be a continuous representation satisfying the following conditions.
• ρ = (ρ mod m) is absolutely irreducible when restricted to Gal(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| Gp 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.
• For every prime p of F above 2, ρ is unramified at p.
Then there exists an embedding ι : K ֒→ 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].

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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 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 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 ρ| Gal(F /FΣ) 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 ρ can be weaker, more precisely, it suffices to assume 'ρ is modular'.It is not necessary to make appeal to their results however.
The next two theorems are about modularity of ρ.
Theorem 4 Let F be a totally real field.Suppose that 5 is unramified in F .Let ρ : Gal(F /F ) → GL 2 (F 5 ) be a continuous representation of satisfying the following conditions.
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 ρ.
The idea is exactly the same as that of Taylor-to prove modularity of 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(F /F Σ ) on the 5-torsion points of E is isomorphic to ρ| Gal(F /FΣ) ; secondly one proves E modular, therefore ρ| Gal(F /FΣ) modular; and finally it follows from Khare-Wintenberger [18] and Kisin [20] that ρ| Gal(F /FΣ) 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](F Σ ) of Gal(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](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](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(F /F ) → GL 2 (F 5 ), and this allows us to work with the prime 2-proving modularity of ρ 2 : Gal(F /F ) → SL 2 (F 4 ) with proj 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 ) 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 ρ : Gal(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 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(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.
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(F /F ) Lemma 6 Let F be a totally real field.Suppose that 5 is unramified in F .Suppose that ρ : Gal(F /F ) → GL 2 (F 5 ) is a continuous representation satisfying the following conditions.
Then there is a finite soluble totally real field extension F Σ of F and an elliptic curve E over F Σ such that • E has good ordinary reduction at every prime of F above 3 and has potentially ordinary reduction at every prime of F above 5; • ρ E,5 : Gal(F /F Σ ) → Aut(E(F Σ ) [5]) is equivalent to a twist of ρ| Gal(F /FΣ) ; ) is absolutely irreducible.
Proof.Firstly, as in [39], find a biquadratic totally real extension K 1 ⊂ F of F , which is a quadratic totally real extension of F ( √ 5) in which √ 5 splits completely, such that proj ρ : Gal(F /K 1 ) → P SL 2 (F 5 ) ≃ A 5 lifts to a representation ρ 1 : Gal(F /K 1 ) → GL 2 (F 5 ) with determinant the mod 5 cyclotomic character ǫ.Choose, by class field theory, a finite soluble totally real extension ) 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 ρ Σ denote the restriction of ρ to Gal(F /F Σ ).As in section 1 of [33], let Y ρ Σ /F Σ (resp.X 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(F /F Σ ), Aut X 5 ) defined by an isomorphism 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 ρ Σ ≃ X 5 and Y ρ Σ is isomorphic over F Σ to a Zariski open subset of the projective line P 1 .In particular, Y ρ Σ has infinitely many rational points.Let Y ρ Σ ,0 (3) denote the degree 4 cover over Y ρ Σ which parameterises isomorphism classes of elliptic curves E equipped with an isomorphism E[5] ≃ ρ Σ taking the Weil pairing on E [5] to ǫ : ∧ 2 ρ Σ → µ 5 and a finite flat subgroup scheme C ⊂ E[3] of order 3. Let Y ρ Σ ,split (3) denote the étale cover over Y ρ Σ which parameterises isomorphism classes of elliptic curves E equipped with an isomorphism E [5] ≃ ρ Σ taking the Weil pairing on E [5] to ǫ : ∧ 2 ρ Σ → µ 5 and an unordered pair, fixed by Gal(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 ρ Σ ,split (3) and Y 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 ρ Σ (F Σ,p ) with good ordinary reduction, and we let U p ⊂ Y 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 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 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 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 ρ : Gal(F /F ) → GL 2 (F 5 ) be a continuous representation of satisfying the following conditions.
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 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 ρ E,3 of Gal(F /F Σ ) on E(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 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 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 ρ E,5 is modular, ρ| Gal(F /FΣ) is modular.Since F Σ is a soluble extension of F , ρ Σ remains absolutely irreducible when restricted to Gal(F /F Σ (ζ)).Furthermore, since 5 is unramfied in F , the kernel of proj 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 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(F /F ) Theorem 8 Let F be a totally real field.Suppose that [F (ζ 5 ) : F ] = 4. Let ρ : Gal(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 ρ.
Proof.By Theorem 3.4 in [33], there exists a principally polarised abelian surface A over F with real multiplication by Z[(1 + √ 5)/2] compatible with the polarisation such that the action of Gal(F /F ) on 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(F )[ √ 5] shows that detρ A, √ 5 is the mod 5 cyclotomic character.Since [F (ζ 5 ) : F ] = 4, the determinant is indeed surjective, and therefore ρ A, √ 5 is absolutely irreducible.If ρ A, √ 5 is irreducible at some place of F above 5, the absolute irreducibility of ρ A, √ 5 implies the absolute irreducibility of its restriction to Gal(F /F (ζ 5 )).Otherwise, ρ A, √ 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(F /F ( √ 5)) of ρ A, √ 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 ρ A, √ 5 ; more precisely, ρ A, √ 5 is strongly residually modular.The modularity of ρ A, √ 5 follows from Theorem 3.5.5 in [20] and [12].

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, 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 to matrices with first column (1, 0) (resp.the second row (0, 1)).Let 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 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 Q p , an isomorphism Q p ≃ C, and an embedding For a ray class character 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 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 0 O (n) to be the inverse limit with respect to r ∈ Z ≥1 of h 2 (np r ) Zp ⊗ Zp O. Let I np ∞ denote the inverse limit of the I np r and the diamond operators : One can also see as the action of (O 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 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 Hida [14] proves that h 0 O (n) is a torsion free Λ K -module and, for a character ψ : such that, for every prime ideal q of O F not dividing np, ρ m is unramified at q and trρ m (Frob e., ρ m as above is absolutely irreducible, then there is a continuous representation which is unramified at every prime ideal q of O F not dividing np and satisfies tr ρ FH (Frob q ) = T q and det ρ FH = (NS)•ǫ 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 FH is of the form where χ FH,p,1 is an unramified character of G p such that χ FH,p,1 Definition.Following [5], we call two Λ-adic eigenforms F H,1 and F H,2 :

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 Fq ) 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 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 2,ψ (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 be a continuous representation such that • ρ is not scalar at every place p above p, • there exists a holomorphic automorphic representation π of (D ⊗ F A F ) × generated by a cusp form in S D 2,ψ (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 ρ π ≃ ρ, • ρ ramifies at Σ and possibly at S P ; for every p in S P ρ| Gp ∼ * * 0 χ p with χ p unramified; and for q ∈ Σ ρ| Gq ∼ ǫχ q * 0 χ q with χ q unramified such that 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 .
Let R ,ord q ∈ Σ: let R ,ψ q denote the domain (see 2.6 in [20], or Proposition 2.12 and 3.3.4 in [18]) parameterising liftings of ρ| Gq of the form ǫχ ur q * 0 χ ur q with χ ur q an unramified lifting of χ q such that (χ ur q ) 2 = (ψ τ |∞: let R ,odd τ denote the formally smooth ring which represents the liftings of ρ| Gτ which, if p is odd, are odd ; and, if p = 2, the image of complex conjugation in 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.

Hecke algebras
Since ρ arises from a holomorphic cusp form in S D 2 (n Σ , ψ) O on the quaternion algebra D over F Σ by assumption, there exists a maximal ideal 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 The determinant of ρ S defines

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(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| Gp to the decomposition group G p at p is of the form 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 is absolutely irreducible when restricted to Gal(F /F (ζ p )), and if p = 2, ρ 2 : Gal(F /F ) → GL 2 (F p ) has insoluble image; • if p > 2 and if ǫ k1−2 χ p,2 = 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, ρ 2 is unramified at every prime of F above 2; • ρ 2 is the direct sum of the characters ǫ k2−1 χ p,1 and 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 , and the U p -eigenvalue of the p-stabilised new form is a lifting of χ p,1 (Frob p ).
Proof.For p > 2, this is a result of Gee (Theorem 2.1 [13]).Let p = 2; thus For clarity, let ρ denote ρ 2 ⊗ ǫ where ǫ is the mod 4 cyclotomic character.Clearly the twist of ρ 2 by the Teichmuller lift of ǫ defines a modular lifting of ρ potentially ordinary and potentially Barsotti-Tate at p.By class field theory, find a finite totally real soluble extension F Σ ⊂ 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; • ρ| Gal(F /FΣ) is ramified exactly at Σ and the infinite places, and , in particular, for every prime q ∈ Σ, ρ| Gal(F /FΣ) at q is an extension of an unramified character by the twist of the character by ǫ at q; • there exists a maximal open compact subgroup U ⊂ (D ⊗ FΣ A ∞ FΣ ) × such that U q = GL 2 (O FΣq ) for q ∈ S D and U q = O × Dq for q ∈ S D , and a holomorphic cuspidal automorphic representation π 2 of (D ⊗ FΣ A FΣ ) × with central character ψ such that ρ| Gal(F /FΣ) ≃ ρ π2 : Gal(F /F Σ ) → GL 2 (F p ) and detρ| Gal(F /FΣ) = ψǫ and such that π is unramified at every prime of F Σ above 2.

Documenta Mathematica 18 (2013) 997-1038
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 is a continuous representation satisfying • ρ ramifies at only finite many primes; • ρ = (ρ mod m) is absolutely irreducible when restricted to Gal(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| Gp 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; • 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 ⊂ F of F in which p splits completely; a finite set Σ of finite places of F Σ (at which ρ| GΣ , where ; an ideal n of O FΣ 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, 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 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 2 P (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(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| GΣ 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(F /FΣ) χ −1 P and ρ P denote (ρ P mod m).If we let ρ Σ denote the modular lifting of ρ, then ρ Σ ⊗ χ −1 P is a modular lifting of 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 −1 P 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 : , and a height one prime ℘ P of O L such that • (ρ FH 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 ) 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) .

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 → Q → Q p .Let K denote a finite extension of Q p which contains the image of F by every embedding of F into 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 , 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 • 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 ⊗ OF 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; . Definition.Let Y Γ1(n,t) (resp.Y Γ1(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 Γ1(n,t) is a smooth scheme over O of relative dimension [F : Q].If n does divide 2, or 3, we let Y Γ1(n,t) denote the O-scheme for an auxiliary ideal m of O F such that n|m and Γ 1 (m) small enough, i.e., torsion-free.
Let Y Γ1(n,t) denote the fibre over k of Y Γ1(n,t) ; and let Let Y Γ1(n,t)∩Iw denote the fibre over k of Y Γ1(n,t)∩Iw .It is a well-known result of Deligne-Ribet that the fibre Y Γ1(n,t) is irreducible (Corollary 4.6 in [9]).It is a result of local model theory by Pappas that Y Γ1(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 Γ1(n,t)∩Γ1(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 Γ1(n,t)∩Γ1(p r ),Iwp,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 Γ1(n,t)∩Γ1(p r ) × Spec OK Spec K, and D p is finite flat subgroup scheme of A[p] of rank |O F /p| which has trivial intersection with C.

Compactification
By an unramified cusp C of Y Γ1(n,t) over R, we shall mean a pair of fractional ideals M 1 , M 2 of F such that M 1 M −1 2 ≃ t which comes equipped with For brevity, let ] and S n ֒→ S n,σ denote the affine torus embedding (see Theorem 2.5 in [6]) corresponding to the cone σ and let 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 0 n,σ = U n,σ × Tn,σ T 0 n,σ .
The Mumford construction applied to the O F -linear 'period' map q : gives rise to a semi-abelian scheme Tate M1,M2 (q over the complete ring U n,σ with action of O F , whose pull-back, which we shall denote by Tate 0 M1,M2 (q) to U 0 n,σ , is naturally a HBAV, t-polarised with level Γ 1 (n)-structure, and which gives rise to a map We glue T /≃ σ∈ΣC U n,σ × Spec Z Spec O along the map to get a toroidal compactification X Γ1(n,t) over O of Y Γ1(n,t) ([26]).Similarly, one can define a compactification X Γ1(n,t)∩Iw over O of Y Γ1(n,t)∩Iw with its choice of a rational cone decomposition compatible with that of X Γ1(n,t) .

Let
Tate 0 M1,M2,S (q Then there is a 'connected-étale' exact sequence 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 0 M1,M2,S (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 Proof.This is essentially Proposition 13.6.2 in [17].
Let M = M 1 M 2 as above.Fix an integer r ≥ 1. Suppose that S is an 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 M1,M2,S (q)(S).Let η ét r denote the image of 1 by defining a point of Tate M1,M2 (q)(S) of exact order |O F /p r |.Let η ét r,τ denote its τ = τ p component.

Generic fibres
With n fixed, for every integer r ≥ 1, let 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 U.
11 p-adic classical Hilbert modular forms H 1 dR (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, where Lie ∨ (A/S) and H 1 dR (A/S) are locally free sheaves of O S -modules of rank 1 and 2 respectively.Following Hida [14], let )), we will often write L k for L (k,w) .We shall also let L (k,w) denote its extension to the compactification.

Documenta Mathematica 18 (2013) 997-1038
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.
Lemma 13 Let r be a rational number < p/(p + 1).Suppose that G is not ordinary.Then On the other hand,
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: 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 Proof.This can be proved as in Proposition 3.5 in [3].
13 p-adic overconvergent Hilbert modular forms respectively denote the rigid analytic spaces in the sense of Tate ( [2]) associated to the K-schemes Given a closed point of Y rig Γ1(n) , 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 Γ1(n) is of finite type, hence X rig Γ1(n) 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 rig Γ1(n) .Let U ∧ good denote the smooth formal scheme U ∧ ∩ Y ∧ Γ1(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 Γ1(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 rig good .Glue these functions on U rig 's, there is a rigid function on X rig Γ1(n) ≃ X an Γ1(n) that defines deg.
, or half open interval with endpoint in Q, define the rigid space X an Γ1(n),K I = t X an Γ1(n,t),K I to be the admissible open set of points whose degrees are all in the range I.
where by P ηKM , we mean the image of 1 by η KM .
Definition.Define (X an Γ1(n)∩Γ1(p r ),K I)I 1S 1 I 2S 2 to be the preimage by π of (X an Γ1(n)∩Iw,K I, For 0 ≤ r ≤ p/(p + 1), it follows from the previous section that The theory of canonical subgroups provides rigid sections: is finite flat of degree |O F /p|, and π 2,p : X an Γ1(n)∩Iw,K [0, r/p]−→X an Γ1(n),K [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 ), 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(ǫ • Art) k−2 ψ) and S acting by (ǫ • Art) k−2 ψ.Indeed I np ∞ -action defines the character of F H mod (ker(ǫ 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(ǫ • 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 ǫ such that 0 ≤ ǫ < 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 an Γ1(n)∩Γ1(p r ),K [0, ǫ], L k (cusps)) U , where X an Γ1(n)∩Γ1(p r ),K [0, ǫ] is the preimage by the forgetful morphism of X an Γ1(n),K [0, ǫ].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 where eH 0 (X an Γ1(n)∩Γ1(p r ),K [0, ǫ], L k (cusps)) 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 an Γ1(n)∩Γ1(p r ),K [0, ǫ], L k (cusps)) U .
Lemma 15 For any integer k, the p-adic eigenform F H mod (ker(ǫ • 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 an Γ1(n)∩Γ1(p r ),K of a characteristic zero lifting of a sufficiently large power of the Hasse invariant.• the Galois representation ρ P associated to f P is ρ| Gal(F /FΣ) ⊗ 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 FΣ , 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 for every ideal m coprime to np; • for p in P , the character of f P at p is χ P −{p} χ −1 P while for p ∈ S Σ,P − P , the character of f P at p is χ P ∪{p} χ −1 P ; • for p in P , (χ • for a place p of P , the U p -eigenvalue of f P is (χ p,1 χ −1 P )(Frob p ) while for p in S Σ,P − P , the U p -eigenvalue of f P is (χ p,2 χ −1 P )(Frob p ).

Analytic continuation of overconvergent eigenforms
Fix τ = τ p throughout the section (except the last two assertions).
Definition.Fix t.
Proof.Analogous to the proposition above.
2 )) for the pair M 1 , M 2 of the fractional ideals such that M 1 M −1 2 ≃ t be a family of HBAVs around a cusp of X Γ1(n,t)∩Γ1(p r ),n .Choose (noncanonically) once for all a basis of the pull-back by Max (O ⊗ Z Z((q M ))) → X an Γ1(n)∩Γ1(p r ),K of the line bundle L k , since a subgroup of Tate M1,M2 (q)[p] of order |O F /p|, disjoint from η r , is of the form ζη + q M2 where ζ ranges over the |O F /p| points of (GL where q η denotes a representative in q p −1 M2 of the class η ∈ q p −1 M2/M2 = q p −1 M2 /q M2 defined earlier; and t p represents the class of pt ≃ pM 1 M −1 2 . denote the admissible open subsets of X an Γ1(n,t)∩Γ1(p r ) defined in such a way that the non-cuspidal S-points of X Γ1(n,t)∩Γ1(p r ),τ 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 Γ1(n,t)∩Γ1(p r ),τ .
Theorem 20 If r is an integer ≥ 2 and suppose that f ∈ H 0 (X Γ1(n)∩Γ1(p r ),τ , L k ) is an eigenform for U p with non-zero eigenvalue.Then f extends to X Proof.This can be proved as Lemma 6.1 in [3] Corollary 21 If f ∈ H 0 (X an Γ1(n)∩Iw,K [0, ǫ], L k ) for some 0 < ǫ < 1 is an eigenform for every U p , p|p, with non-zero eigenvalue, then f extends to , for some 0 < ǫ < 1 is an eigenform for every U p , p|p, with non-zero eigenvalue, then g extends to H 0 (X Γ1(n)∩Γ1(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 an Γ1(n)∩Iw,K 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 an Γ1(n),K , 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 • 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 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 an Γ1(n)∩Iw,K [0, r], L k ).It follows from results in [29] that π * 1 f P extends to a section over X an Γ1(n)∩Iw,K [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 Γ1(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 on the one component and a finite flat morphism of degree |O F /p| on the other.We are going to glue f S and f S−{p} ; more precisely glue f Q S and f Q S−{p} .Let F denote the section and G denote the section Since one can show readily the q-expansions of π * 2,p F and G are equal at around C = (Tate M1,M2 (q), . . ., ζ 1 ), we shall glue π * 2,p F and G at 1] p and therefore to (X Γ1(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 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.

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 Let w P denote the composite of the w ζp for all p ∈ P .Note that and each (X Γ1(n,t)∩Γ1(p),Iwp,K [0, 1))(0, 1] S−P is connected since it is isomorphic to X Γ1(n,t)∩Γ1(p),OK [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 Γ1(n)∩Γ1(p),Iwp,K [0, 1))(0, 1] S−P denote the disjoint union over T of (X Γ1(n,t)∩Γ1(p),Iwp,K [0, 1))(0, 1] S−P .
For every subset P of S, let g P denote f P |w P ∈ H 0 ((X Γ1(n)∩Γ1(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 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 M2 of the class η ∈ q p −1 M2 /q M2 .Finally It suffices to show that |O F /p|α p g P −S(g P −{p} |w ζp ) is identically zero; in which case, one can glue g P and (|O F /p|α p ) −1 S(g P −{p} |w ζp ) as desired.Showing that it is identically zero is exactly as in [3].
) deformation rings R p|p: if p is odd, let R ,ord p (resp.R ,BT,ord p ) denote the O-algebra which represents the p-ordinary (resp.Barsotti-Tate p-ordinary) framed deformations of ρ| Gp of the form * * 0 χ ur p with an unramified lifting χ ur p of χ p (resp. and its determinant is ǫψ 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 ,ord p,± (resp.R ,BT,ord p,± ) denote the complete local noetherian O-algebra which represents the p-ordinary (resp.Barsotti-Tate pordinary) liftings of ρ| Gp of the form * * 0 χ ur p with an unramified lifting χ ur p of χ p , and with its determinant corresponding, by the local class field theory, to the norm O ord,ψ S if p is odd; and let h 2 (n Σ 4, ψψ P,− ) m− = h 2 (n Σ 4, ψψ P,− ) m− ⊗ R BT,ord,ψ S,− R ,BT,ord,ψ S,− h 0 (n S , ψ) −,m− = h 0 (n Σ , ψψ P ) −,m ⊗ R ord,ψ S,− R ,ord,ψ S,− if p = 2.It then follow from results of Kisin and Khare-Wintenberger that there is a natural surjection

S
Documenta Mathematica 18 (2013) 997-1038 and R ord,ψ S /ker(S − (ψψ P • ǫ cyclo )) ≃ R BT,ord,ψ S if p odd, and It follows from the theorem in the previous section that, given a p-adic representationρ : Gal(F /F ) → GL 2 (O)as in the main theorem, there are 1. a finite soluble totally real field extension F Σ ⊂ 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 |SΣ,P| characters χ P : Gal(F /F Σ ) → O × of finite order and 2 |SΣ,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 0 O (n) → K, with character ψ P = ψ SΣ,P P ψ P,SΣ,P of (O FΣ /n) × × (O FΣ /p) × ,t)∩Γ1(p r ),τ parameterises (A/S, i, λ, η) Documenta Mathematica 18 (2013) 997-1038 equipped with a point P ηKM 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 ηKM,p .For every 0 ≤ s ≤ r − 1, X [s]