Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions

Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of $p$-adic $L$-functions (of Bella\"iche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson-Kato elements over the eigencurve (including the neighborhoods of $\theta$-critical points). Along the way, we prove \'etale variants of Bella\"iche's results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these $p$-adic $L$-functions away from $\theta$-critical points.


Introduction
Let p ≥ 5 be a prime number and let us fix forever an embedding ι ∞ : Q → C as well as an isomorphism ι : C ∼ − → C p . Let us also put ι p := ι • ι ∞ . Let f = n a n (f )q n ∈ S k+2 (Γ 1 (N ) ∩ Γ 0 (p), ε) be a cuspidal eigenform (for all Hecke operators {T ℓ } ℓ∤N and {U ℓ , ℓ } ℓ|N p ) of weight k + 2 ≥ 2 with p ∤ N . When ord p (ι p (a p (f ))) < k + 1, Amice-Vélu in [AV75] and Višik in [Viš76] have given a construction of a p-adic L-function L p (f, s), which is characterized with the property that it interpolates the critical values of Hecke L-functions attached to (twists) of f .
The analogous result in the extreme case when ord p (ι p (a p (f ))) = k + 1 (in which case we say that f has critical slope) was established by Pollack-Stevens in [PS13] and Bellaïche [Bel12]. Note that if ord p (ι p (a p (f ))) = k + 1 then f is necessarily p-old unless k = 0. It is worthwhile to note that the p-adic Lfunctions of Pollack-Stevens and Bellaïche are not characterized in terms of their interpolation property, but rather via the properties of the f -isotypic Hecke eigensubspace of the space of modular symbols. The work of Pollack-Stevens assumes in addition that f is not in the image of the p-adic θ-operator θ k+1 := (q(d/dq)) k+1 on the space of overconvergent modular forms of weight −k (i.e., f is not θ-critical); Bellaïche's work removes this restriction.
Both constructions in [PS13] and [Bel12] take place in Betti cohomology. Our objective in this series of two articles, of which the present article is the first, is to recover these results in the context of p-adic (étale) cohomology. More precisely, we shall recast the results of Bellaïche and Pollack-Stevens in terms of the Beilinson-Kato elements and the triangulation over the Coleman-Mazur-Buzzard eigencurve. Along the way, we interpolate the Beilinson-Kato elements over a neighborhood on the eigencurve (including the neighborhoods of θ-critical points).
1.1. Set up. We put Γ p = Γ 1 (N ) ∩ Γ 0 (p) and let f α0 0 ∈ S k0+2 (Γ p ) be a p-stabilized cuspidal eigenform, where U p f α0 0 = α 0 f α0 0 and N is coprime to p. We let f 0 be the newform associated to f α0 0 (note that it may happen that f 0 = f α0 0 , in which case f α0 0 is not of critical slope). We fix a real number ν ≥ v p (α 0 ), where v p (·) is the p-adic valuation on Q p normalized so that v p (p) = 1.
We shall call Hom cts (Z × p , G m ) the weight space, which we think of as a rigid analytic space over Q p . Let W be a nice affinoid neighborhood (in the sense of [Bel12], Definition 3.5) about k 0 of the weight space, which is adapted to slope ν (in the sense of [Bel12], §3.2.4). We will adjust our choice of W on shrinking it as necessary for our arguments.
Let C be the Coleman-Mazur-Buzzard eigencurve and let x 0 ∈ C be the point that corresponds to f α0 0 . We let C W,ν ⊂ C denote an open affinoid subspace of the eigencurve that lies over W and U p acts by slope at most ν. By shrinking W as necessary, there is a unique connected component X ⊂ C W,ν that contains x 0 , which is an affinoid neighborhood of x 0 .
For any E-valued point x ∈ X (E) (where E is a sufficiently large extension of Q p , which contains the image of the Hecke field of f α0 0 under the fixed isomorphism ι) of classical weight w(x) ∈ Z ≥0 in the irreducible component X ⊂ C, we let f x ∈ S w(x)+2 (Γ) denote the corresponding p-stabilized eigenform. We let V ′ fx denote Deligne's representation attached to f x ; see §6.4.1 for its precise description. There is a natural free O X -module V ′ X of rank 2, which is equipped with an O X -linear continuous G Q -action such that (51) and (55) below). As in [Kat04,§5], let ξ denote either the symbol a(B) with a, B ∈ Z and B 1 or an element of SL 2 (Z). For each integer r and natural number n, we let c,d BK N,mp n (f x , j, r, ξ) ∈ H 1 (Q(ζ mp n ), V ′ fx (2 − r)) denote the Beilinson-Kato element associated to the eigenform f x , given as in Theorem 6.8(iv) (see also §6.2 for details).
For any abelian group G, let us denote by X(G) its character group. We also put Λ(G) := Z p [[G]] for its completed group ring. We shall denote by Λ(G) ι for the free Λ(G)-module of rank one on which G acts via the character g ι → g −1 ∈ Λ(G) × .
1.2. Main results. We will briefly overview of the results in this article. In a nutshell, our work has two threads: The first concerns the interpolation of Beilinson-Kato elements along the eigencurve C, the second concerns p-local aspects, such as the (properties of the) triangulation over C and the formalism of Perrin-Riou exponential maps. The first part of Theorem A below belongs to the first thread and corresponds to Theorem 6.8 below. The second part corresponds to Theorem 7.3 in the main body of our article and dwells on the second thread, granted the first.
Remark 1.2. We note that in the particular case when f 0 = f α0 0 is a newform of level Γ 0 (N p) and weight k 0 + 2 with α 0 = a p (f 0 ) = p k0/2 , the conclusions of Theorem 7.3 play a crucial role in [BBb].
1.2.1. θ-critical case. We conclude §1.2 with a brief summary of our results in our companion article [BBa], where we focus on the θ-critical scenario (i.e. in the situation when X is not étale over W) and give an étale construction of Bellaïche's p-adic L-function.
The first key ingredient in [BBa] is Theorem A(i), namely the construction of a big Beilinson-Kato class about a θ-critical point on the eigencurve. The local aspects turn out to be significantly more challenging when X fails to be étale over W. To prove that the Perrin-Riou style p-adic L-functions L ± p (X ; j, ξ), defined in an analogous way, has the required properties, we introduce in op. cit. a new local argument (called the "eigenspace transition via differentiation") and prove the following results (which we state in vague form to avoid digression and refer readers to [BBa] for details): Theorem B. Suppose that the ramification index of X over W at x 0 equals 1 to 2. As before, we let L ± p (X ; j, ξ) ∈ H OX (Γ) denote the images of the ±-parts of the class BK [X ] N (j, ξ) under the Perrin-Riou dual exponential map. There exists two pairs (j + , ξ + ) and (j − , ξ − ) with the following properties: i) We have L ± p (x 0 ; j ± , ξ ± , ρχ r ) = 0 for all integers 1 r k 0 + 1 and characters ρ ∈ X(Γ) of finite order. ii) We define the improved arithmetic p-adic L-functions at the critical point x 0 on setting ∈ H E (Γ) 1 In an unpublished note (see [Bel] Proposition 1), Bellaïche explains that a conjecture of Jannsen combined with Greenberg's conjecture (that "locally split implies CM") and the important of result of Breuil-Emerton ("θ-critical implies locally split") would yield e = 2. We are grateful to R. Pollack for bringing Bellaïche's note to our attention.
Here, as in [Bel12,§4.4], we denote by X a fixed choice of a uniformizer of X about x 0 and consider the p-adic L-functions L ± p (X ; j, ξ) in the neighborhood X of x 0 as the functions L ± p (X; j, ξ) with X in a neighborhood of 0. Then the improved arithmetic p-adic L-functions verify the same interpolation property that Bellaïche's improved p-adic L-functions do, up to the bad Euler factors E N (x 0 ) and constants that depend only on the choices of Shimura periods.
This theorem supplies us with a new construction of Bellaïche's p-adic L-functions (with Euler factors at primes dividing the tame conductor removed). One of the consequences of the étale construction of the p-adic L-functions is the leading term formulae for these p-adic L-functions. These will be explored in the sequels to the present article.
1.2.2. Layout. We close our introduction reviewing the layout of our article.
After a very general preparation in §2 (where we axiomatise various constructions in [Bel12], §4.3), we give a general overview of triangulations in §3. In §3.2, we also define the Perrin-Riou exponential map, which is one of the crucial inputs defining the "arithmetic" p-adic L-functions.
We then introduce Perrin-Riou-style (abstract) two-variable p-adic L-functions in §4 and study their interpolative properties. These results are later applied in §7 in the context of the Coleman-Mazur-Buzzard eigencurve and with the Beilinson-Kato element c,d BK [X ] N,m (j, ξ) of Theorem A. In §5, we review Bellaïche's results in [Bel12] on the local description of the eigencurve and prove variants (in §5.3) of these results involving slightly different local systems as coefficients and non-compactly supported cohomology (at a level of generality that covers also the neighborhoods of θ-critical point). We utilize these in §6 (together with the work of Andreatta-Iovita-Stevens [AIS15]) to deduce the required properties for the Galois representation V ′ X , where the interpolated Beilinson-Kato elements take coefficients in. The variants of Bellaïche's construction that we discuss in §5.3 allow us (through Proposition 6.5) to establish the properties of V X as an O X -module, where the images of Perrin-Riou exponential maps take coefficients in.
In §6.5, we introduce the interpolated Beilinson-Kato elements (which are denoted by c,d BK [X ] N,m (j, ξ) in Theorem A(i)) as part of Definition 6.7(iii) (see also §6.6 where we introduce their normalized versions). Our construction builds primarily on the ideas in [Kin15,KLZ17,LZ16]. As we have noted in Remark 1.1, our approach in this portions is, in some sense, a synthesis of the techniques of [LZ16] and [Han16] (which we crucially enhance to apply also about θ-critical points). In §7, we apply the general results in §4 to define the "arithmetic" p-adic L-functions and study their interpolation properties, proving our Theorem A(ii).
1.3. Acknowledgements. The first named author (D.B.) wishes to thank the second author (K.B.) for his invitation to University College Dublin in April 2019 and Boǧaziçi University of Istanbul in December 2019. This work was started during these visits. D.B. was also partially supported by the Agence National de Recherche (grant ANR-18-CE40-0029) in the framework of the ANR-FNR project "Galois representations, automorphic forms and their L-functions". We thank the anonymous referees for carefully reading our work and their very helpful comments, which guided us towards many technical and stylistic improvements to the earlier versions of our article.

Linear Algebra
In §2, we fix some notation and conventions from linear algebra, which will be used in the remainder of the paper (as well as in the companion paper [BBa]). We also axiomatize various constructions in [Bel12, §4.3].
2.1. Tate algebras. Let E be a finite extension of Q p . Fix an integer e 1 and denote by R the Tate algebra R = E Y /p re , where r 0 is some fixed integer. Let A = R[X]/(X e − Y ). Then A = E X/p r .
Set W = Spm(R) and X = Spm(A). We will consider W as a weight space in the following sense. Fix an integer k 0 2 and denote by D(k 0 , p re−1 ) = k 0 + p re−1 Z p the closed disk with center k 0 and radius 1/p re−1 . We identify each y ∈ D(k 0 , p re−1 ) with the point of W that corresponds to the maximal ideal We have a canonical morphism (1) w : X → W, which we will call the weight map. If x ∈ X , we say that y = w(x) is the weight of x. Let x 0 ∈ X denote the unique point such that w(x 0 ) = k 0 . Set 3.1.1. We set K n = Q p (ζ p n ), K ∞ = ∞ ∪ n=0 K n and Γ = Gal(Q p (ζ p ∞ )/Q p ). For any n 1 we set Γ n = Gal(K ∞ /K n ), and put G n = Gal(K n /Q p ). Let us fix topological generators γ n ∈ Γ n such that γ p n = γ n+1 for all n 1 and γ p−1 0 = γ 1 . For any group G and left G-module M we denote by M ι the right G-module whose underlying group is M and on which G acts by m · g = g −1 m.
If G is a finite abelian group, we denote by X(G) its group of characters. If E is a fixed field such that ρ ∈ X(E) takes values in E, we denote by 3.1.2. In this section, we review the construction of the Bloch-Kato exponential map for crystalline (ϕ, Γ)modules. Let E be a finite extension of Q p . For each n 0, we denote by R E the Robba ring of formal power series f (π) = m∈Z a m π m coverging on some annulus of the form r(f ) |π| p < 1.
] is the ring of formal power series converging on the open unit disk. We equip R E with the usual E-linear actions of the Frobenius operator ϕ and the cyclotomic Galois group Γ given by Let ψ denote the left inverse of ϕ defined by Then R + E ψ=0 is a H (Γ)-module of rank one, generated by 1 + π. The differential operator ∂ = (1 + π) d dπ is a bijection from (R + ) ψ=0 to itself. Furthermore, we have Let t = log(1 + π) denote the "additive generator of Z p (1)". Recall that Let A = E X/p r be a Tate algebra over E as above. We denote by R A = A ⊗ E R E the Robba ring with coefficients in A. The action of Γ, ϕ, ψ and ∂ can be extended to R A by linearity.
3.1.3. Recall that a (ϕ, Γ)-module over R A is a finitely generated projective module over R A equipped with commuting semilinear actions of ϕ and Γ and satisfying some additional technical properties which we shall not record here (see [BC08] and [KPX14] for details). The cohomology H i (K n , D) of D over K n is defined as the cohomology of the Fontaine-Herr complex is a finitely generated free A-module equipped with an A-linear frobenius ϕ and a decreasing filtration Fil i D cris (D) i∈Z .
3.1.5. Let V be a p-adic representation of G Qp with coefficients in A. The theory of (ϕ, Γ)-modules associates to V a (ϕ, Γ)-module D † rig,A (V ) over R A . The functor D † rig,A is fully faithul and we have functorial isomorphisms ). If V is crystalline in the sense of [BC08], we have a functorial isomorphism between the "classical" filtered Dieudonné module D cris (V ) associated to V and D cris (D † rig,A (V )). Moreover, the diagram where the upper row is the Bloch-Kato exponential map, commutes.
The Iwasawa cohomology H 1 Iw (Q p , D) of a (ϕ, Γ)-module D over R A is defined as the cohomology of the complex is the (ϕ, Γ)-module associated to a p-adic representation V over A, we then have a functorial isomorphism Iw (Q p , V ) denotes the classical Iwasawa cohomology with coefficients in V . The isomorphism (3) composed with the projections pr n coincide with the natural morphisms . For each integer m, the cyclotomic twist can be extended to a map 3.2. The Perrin-Riou exponential map.
3.2.1. In this subsection, we shall review fragments of Perrin-Riou's theory of large exponential maps. Define the operators Note that ∇ does not depend on the choice of γ 1 ∈ Γ 1 . It is easy to check by induction that Let V be a p-adic crystalline representation of G Qp with coefficients in E satisfying the following condition: Note that this assumption can be relaxed (see, for example, [PR94,Ben14]), but it simplifies the presentation.
In particular, it implies that H 0 (K ∞ , V ) = 0 and therefore also that H 1 Iw (Q p , V ) is torsion free over the Iwasawa algebra (c.f. [PR94], Lemme 3.4.3 and Proposition 3.2.1).

Let us set
For any α ∈ D(V ), the equation The following is the main result of [PR94]. i) For all n 0 the following diagram commutes: ii) Let us denote by e −1 := ε −1 ⊗ t the canonical generator of D cris (Q p (−1)). Then Let us put 3.2.2. Set G n = Gal(K n /K). Without loss of generality, we can assume that the characters of G n take values in E. Recall that Shapiro's lemma gives an isomorphism of E[G n ]-modules On taking the ρ-isotypic components, we obtain isomorphisms We shall make use of the following elementary lemma giving the ρ-isotypic component of the map Ξ V,n .
3.2.3. We recall the explicit construction of the Perrin-Riou exponential that was discovered by Berger in [Ber03]. Let, as before, Define It is not difficult to see that Ω V,h (α) ∈ D † rig,E (V ) ψ=1 and an explicit computation shows that Ω V,h satisfies properties i-iii) in Theorem 3.1.
See also [Nak14] for a generalization of this approach to de Rham representations.
3.2.4. In the remainder of §3.2, we shall review the construction of the Perrin-Riou exponential maps for families of (ϕ, Γ)-modules of rank one.
Set D + δ := R + A e δ . We say that D δ is of Hodge-Tate weight −m ∈ Z (sic!) if δ(u) = u m , ∀u ∈ Z × p . If D δ is of Hodge-Tate weight −m, then D cris (D δ ) is the free A-module of rank one generated by d δ = t −m e δ . It has the unique filtration break at −m, and ϕ acts on D cris (D δ ) as the multiplication by p −m δ(p) map.
3.2.5. Let D δ be a (ϕ, Γ)-module of rank one and Hodge-Tate weight −m ∈ Z. The direct analogue of the condition LE) for families of (ϕ, Γ)-modules is the following condition: For any point x ∈ Spm(A), let D δ,x denote the specialization of D δ at x. To facilitate a comparison with LE), we remark that the condition LE*) implies that Let us put D(D δ ) = R ψ=0 A ⊗ A D cris (D δ ). We shall explain the construction of a family of maps (which we will call Perrin-Riou exponential maps) modelled on the discussion in §3.2.3. Let us set α(π) := f (π) ⊗ d δ ∈ D(D δ ). The equation has a unique solution in R + A . It is easy to see that and set The following result can be extracted from [Nak17, Section 4] or proved directly using Berger's arguments.
ii) For any h m the following diagram commutes: has a unique solution F ∈ D cris (D δ ) ⊗ R + A and it verifies iv) Under the conditions and notation of iii), let us put Then the map Ξ D δ ,n : D(D δ ) → D cris (D δ ) ⊗ Qp K n is surjective, and the diagram commutes.

Triangulations in families.
3.3.1. In §3.3, we shall work in the setting of §2 and introduce various objects with which we shall apply the general constructions in §2. To that end, let us fix an integer e 1 and put A = R[X]/(X e − Y ) as before, where R = E Y /p re is a Tate algebra. Set W = Spm(R), X = Spm(A) and denote by w : X → W the weight map. We fix an integer k 0 2 and identify W with the closed disk D(k 0 , p re−1 ) as in Section 2.1. Let x 0 ∈ X denote the unique point such that w(x 0 ) = k 0 . We put We let V denote a free A-module of rank 2 which is endowed with a continuous action of the Galois group G Q,S . In accordance with the notation of Section 2, for any y ∈ W(E) and x ∈ X , we set 3.3.2. We shall assume that V verifies following conditions:

C1)
For each x ∈ X cl of integer weight w(x) 0 the restriction of V x on the decomposition group at p is semistable of Hodge-Tate weights (0, w(x) + 1).
On shrinking X as necessary, it follows from [KPX14] that one can construct a unique (ϕ, Γ)-submodule D ⊆ D † rig,A (V ) of rank one with the following properties: Let D sat x0 denote the saturation of the specialization D x0 of the (ϕ, Γ)-module D at x 0 . Then, as explained in the final section of [KPX14], we have D x0 = t m D sat x0 for some m 0. We will consider the following two scenarios 2 : , the condition ¬ Θ) implies that the Hodge-Tate weight of D st (V x0 ) ϕ=α(x0) is 0. Since the Hodge-Tate weight of D x0 is also 0, we deduce that in this case m = 0 and D x0 is saturated Suppose that Θ) holds. Then D cris (D sat x0 ) = Fil k0+1 D st (V x0 ). Therefore D sat x0 is of Hodge-Tate weight k 0 + 1 and m = k 0 + 1. Let β(x 0 ) denote the other eigenvalue of ϕ on D st (V x0 ). By the weak admissibility of D st (V x0 ), we infer that v p (α(x 0 )) k 0 + 1 , v p (β(x 0 )) 0 and that v p (α(x 0 )) + v p (β(x 0 )) = k 0 + 1. Thence, v p (α(x 0 )) = k 0 + 1 and v p (β(x 0 )) = 0, and the eigenspaces D st (V x0 ) ϕ=α(x0) and D st (V x0 ) ϕ=β(x0) are weakly admissible. They are therefore admissible and the restriction of V x0 to the decomposition group at p decomposes into a direct sum Then In the context of elliptic modular forms, the condition ¬ Θ) (resp., Θ)) translates into the requirement that the corresponding eigenform is non-θ critical (resp., is θ-critical) in the sense of Coleman.
We have We remark that D cris (D x0 ) and D cris (V (α) x0 ) are isomorphic as ϕ-modules but not as filtered modules: they have Hodge-Tate weights 0 and k 0 + 1, respectively.

Two variable p-adic L-functions: the abstract definitions
Suppose that we are given another free A-module V ′ of rank two which is equipped with a continuous G Q,S -action, together with a Galois equivariant R-linear pairing satisfying the condition Adj) of Section 2 : , which has the following explicit description. Let denote the cup-product pairings induced by (7). Recall from §3.1.6 the projection maps We then have (see [PR94], Section 3.6). In particular, for any finite character ρ ∈ X(Γ) of conductor p n , we have where ( , ) ρ,0 stands for the cup-product pairing 4.1.1. We apply the formalism of Section 2.4 to our situation. Equip the tensor product with the action of A through the first factor. We extend the pairing (8) by linearity to the pairing (10) , For any x ∈ X (E), the functoriality of the cup-products gives rise to the following commutative diagram: where the left and the right vertical maps are the specializations at w(x) and x respectively, and the middle vertical map is the specialization defined in Section 2.4.
By Lemma 2.5, for any x ∈ X (E) the pairing ( , ) induces a pairing The pairing in the bottom row of (11) therefore factors as

Note that we have a canonical injection
The Perrin-Riou exponential map Exp D,h can be extended by linearity to an A-linear map

Note that we have a canonical injection
Fix an A-module generator η ∈ D cris (D) and set We remark that for each x ∈ X (E), the specialization η x := sp x (η) is a generator of D cris (D)[x] by Lemma 2.3. Let us put η x := η x ⊗ (1 + π).
i) For each h 0 and cohomology class z ∈ H 1 Iw (Q p , V ′ (1)), we define ii) We define the two-variable p-adic L-function associated to z on setting For each x ∈ X cl (E), we similarly define the one-variable p-adic L-function iii) For any x ∈ X (E), finite character ρ ∈ X(Γ), we denote by Proof. This is an immediate consequence of the diagram (11), combined with the discussion in Remark 2.4.

Local description of the eigencurve
In Section 5, we review Bellaïche's results in [Bel12] on the local description of the eigencurve (most particularly, about a θ-critical eigenform). We also follow his exposition very closely here, particularly of Sections 2.1.1, 2.1.3, 3.1, 3.2 and 3.4 in op. cit.; and rely also on the notation set therein for the most part (e.g., unless we declare otherwise).

Modular symbols.
5.1.1. Let WS = Hom(Z × p , G m ) denote the weight space which we consider as a rigid analytic space over a finite extension E of Q p . If y ∈ WS(E), we denote by κ y : Z × p → E × the associated character. We consider Z as a subset of WS(E) identifying k ∈ Z with the character u → u k . Set Note that Z ⊂ W * . If U ⊂ W * is either an affinoid disk or a wide open disk we will write O • U for the ring of analytic functions on U that are bounded by 1 and set the universal weight character characterized by the property We remark that our assumption that U ⊂ W * is to guarantee that κ U lands in O •,× U .

Let
satisfying the following properties: where c m goes to zero when m → ∞. Note that in the scenario when U is wide open, we work with the m U -adic topology.
The monoid Σ 0 (p) acts naturally on Z × p × Z p and therefore on A(U ) and D(U ). Similarly, the monoid Σ ′ 0 (p) acts on Z p × Z × p and therefore on A ′ (U ) and D ′ (U ).
, Lemma 3.8). In particular, we have an injection For any y ∈ W(E), the natural map We refer the reader to [AIS15], [Bel12] and [LZ16, Section 4] for further details pertaining to these objects.

Put
; note that this congruence subgroup is denoted by Γ in [Bel12]. We denote by H * (Γ p , −) (resp., H * c (Γ p , −)) the cohomology of Γ p (resp., the cohomology with compact support). Note that H 0 c (Γ p , −) and H 2 (Γ p , −) vanish since Γ p is the fundamental group of an open curve. Lemma 5.1. Let W ⊂ W * be an affinoid disk. Then: ii) For any y ∈ W(E), the specialization at y gives rise to the short exact sequence iii) The exact sequence (15) induces an isomorphism ) and an injection Proof. The part i) is proved in [Bel12, Lemma 3.9] and the same arguments compute H 2 c with coefficients in D(U ).
The isomorphism (16) in part iii) follows from the long exact sequence associated to the specialization exact sequence and the vanishing of H 2 (Γ p , D † (W)). The inclusion (17) for the cohomology with compact support and the point iv) follow formally from i) (c.f. [Bel12], Theorem 3.10), and the proofs of these statements for cohomology with coefficients in D(U ) are the same.
For a positive real number ν, we denote by Symb Γp (D) ≤ν the space of modular symbols of slope bounded by ν and by the ±1-eigenspace for the involution ι given by the action of the matrix Section 3.2.4]).
Lemma 5.2. Let U be a wide open and W ⊂ U an affinoid disk. Then we have canonical isomorphisms Proof. Only in this proof, A shall denote a general ring. For any complex C • of A-modules over a ring A and a morphism of rings A → B, one has the spectral sequence Let us choose C • as the complex computing the cohomology with compact support with coefficients in D(U ) • , Then our spectral sequence induces an exact sequence is a finitely generated O E -module and therefore, there exists a natural number N such that for any n 1, the O Emodule On passing to projective limits in (19) and using [AIS15, Lemma 3.13], we infer that where C is a finitely generated module that is annihilated by p N . This concludes the proof of the first isomorphism. The proof of the second isomorphism is analogous (an even simpler because H 2 (Γ p , D(U ) • ) vanishes). 5.1.6. For a non-negative integer k, we let P k (E) ⊂ E[Z] denote the space of polynomials of degree less or equal to k, which is equipped with a left GL 2 (Z p )-action; see [Bel12, 3.2.5] for a description of this action. We let V k (E) denote the E-linear dual of P k (E), which also carries an induced GL 2 (Z p )-action.
Regarding elements P k (E) as analytic functions, we have an induced map We also have the non-compact version of this isomorphism, proved by Ash and Stevens [AS08, Theorem 1] :

5.2.
Local description of the Coleman-Mazur-Buzzard eigencurve. Our main objective in this subsection is to record Proposition 5.3 and Theorem 5.4 (which is due to Bellaïche; see the second paragraph in [Bel12, §1.5]).
5.2.1. We now briefly recall the Coleman-Mazur-Buzzard construction of the eigencurve. We retain the notation from Section 5.1 and continue following the exposition in [Bel12, §2.1]. Let H denotes the Hecke algebra generated over Z by the Hecke operators {T ℓ } ℓ∤N p , the Atkin-Lehner operator U p and diamond We fix a nice affinoid disk W = Spm(O W ) (in the sense of [Bel12], Definition 3.5) of the weight space WS adapted to slope ν in the sense of [Bel12], §3.2.4. Let us denote by M † (Γ p , W) Coleman's space of overconvergent modular forms of level Γ p and weight in W; and let M † (Γ p , W) ≤ν denote its O W -submodule on which U p acts with slope at most ν. We similarly let S † (Γ p , W) denote the space of cuspidal overconvergent modular forms of level Γ p and weight in the affinoid disk W; and S † (Γ p , W) ≤ν ⊂ S † (Γ p , W).
is an open affinoid subspace of the Coleman-Mazur-Buzzard eigencurve C (resp, cuspidal eigencurve C cusp ) that lies over W.
We let T ± W,ν denote the image of H W in End OW (Symb ± Γp (D(W)) ≤ν ). We then define the open affinoids C ± W,ν := Spm(T ± W,ν ). By [Bel12, Theorem 3.30], there exist canonical closed immersions The open affinoids C ± W,ν admissibly cover the Coleman-Mazur-Buzzard eigencurve C as (W, ν) varies (see [Bel12], §3). 5.2.2. In the remainder of this subsection, we fix a p-stabilization of a newform f 0 and denote by x 0 the corresponding point of C. The following proposition will allow us to apply the formalism of Sections 3-4 in the context of eigencurves. We denote by W an affinoid neighborhood of Proposition 5.3 (Bellaïche). Up to shrinking W and enlarging E, there exists an affinoid neighborhood X of x 0 ∈ C cusp W,ν such that a) X is a connected component of C W,ν ; b) There exist integers r, e 1 and an element a ∈ O X such that O W = E Y /p re and the map X → a induces an isomorphism of R-algebras Proof. By [Bel12, Proposition 4.11], there exists an affinoid neighborhood X of C ± W,ν which satisfies the condition (b). By [Bel12, Corollary 2.17], the eigencurves C and C cusp are locally isomorphic at x 0 . Together with (22), this concludes the proof of (a).

We set
where X is as in Proposition 5.3.
i) The O X -module Symb ± Γp (X ) ≤ν is free of rank one.
ii) The O X -module Symb Γp (X ) ≤ν is free of rank 2.
Proof. The first assertion follows as a consequence of the discussion in Section 4.2.1 in op. cit.; see also the first paragraph following the proof of [Bel12, Proposition 4.11]. The second assertion follows from the first one and the decomposition

A variant of
denote the canonical representations of H W . We define the ideals I ± 1 = ker(r ± 1 ) and I ± 2 = ker(r ± 2 ). Lemma 5.5. Let k 0 0 be an integer such that k 0 ∈ W ⊂ U . For sufficiently small W and U the map (13) together with Lemma 5.2 induce natural Hecke equivariant isomorphisms In particular, the morphism and denote by g : M 1 → M 2 the map induced from (13) together with Lemma 5.2. It follows from (18) and general results about the slope decomposition that M 1 and M 2 are finitely generated free O W -modules. By Lemma 5.1, we have a commutative diagram [k 0 ] are both either isomorphic to E or else vanish (depending on whether k 0 = 0 or not). Moreover, since the middle vertical arrow is an isomorphism, the right vertical arrow is surjective and therefore an isomorphism as well. We conclude that the left vertical map induced from the vertical isomorphism in the middle is an isomorphism: In particular, M 1 and M 2 are free O W -modules with the same rank. Let G ∈ O W denote the determinant of g in some bases of M 1 and M 2 . The isomorphism (24) shows that G(k 0 ) = 0 and hence, on shrinking the neighborhood W as necessary, G is non-vanishing on W. This concludes the proof that g is an isomorphism for sufficiently small W and U , as we have asserted in the statement of our lemma.
The second isomorphism in (23) can be established by the same argument, using the fact that the map
a) x 0 belongs to a unique connected component iii) Fix a wide open U as well as an affinoid neighborhood W ⊂ U of x 0 chosen so as to ensure that the conclusions of Part ii) are verified. If x 0 is cuspidal, then x 0 ∈ C + W,ν ∩ C − W,ν and X + = X − . Proof.
i) We first show that for every integer k ∈ U with k ≥ ν there exists a Hecke equivariant 3 isomorphism ) ≤ν as well as the natural surjection We are therefore reduced to proving that we have an Hecke equivariant isomorphism ) . This follows from the isomorphism of GL 2 (Z p )-modules (where det stands for the determinant character of GL 2 (Z p )), together with the fact that det |Γ p = 1. Here, det * is the isomorphism induced from the perfect pairing see the paragraph following [BSV20, Proposition 3.2] as well as Remark 3.3.2 in op. cit. This proves (25).
In view of [BSV20], Proposition 4.2.1, it follows from the isomorphisms (25) that the representations r 1 and r 2 verify the conditions of [Che05, Proposition 3.7]. This implies that I ± 1 = I ± 2 and the remaining assertions are immediate from this fact and Lemma 5.5.
ii) We will only prove the assertion when the coefficients are A ′ (U ), since the proof in the case of D(U ) is similar. We remark that one may alternatively deduce our claim that the O X ± -module H 1 c (Γ p , D(U )) ±,≤ν ⊗ HW O X ± is free of rank one from [Bel12, Proposition 4.5], using the isomorphism (18) of Ash-Stevens and Proposition 5.5.
We will prove that for U sufficiently small the O X ± -module H 1 c (Γ p , A ′ (U )) ±,≤ν ⊗ HW O X ± is free of rank one. It is clear that one may choose U so as to ensure that the condition a) holds. Let us fix such U . According to [BSV20, Proposition 4.2.1], the T ′,± U,ν -module H 1 c (Γ p , A ′ (U )) ±,≤ν OW is of finite rank as an O Wmodule (therefore also as a T ′,± OW ,ν -module). For any y ∈ U, the long exact sequence Γ p -cohomology induced from the short exact sequence of specialization at y Since H 0 c (Γ p , A ′ y (E)) = 0, this proves that H 1 c (Γ p , A ′ (U )) is torsion-free over O U and therefore, the O Wmodule H 1 c (Γ p , A ′ (U ))⊗ OU O W is torsion-free as well. We have proved that the O W -modules H 1 c (Γ p , A ′ (U )) ±,≤ν W are finitely generated and torsion-free (therefore free, since O W is a PID).
Set N ± = H 1 c (Γ p , A ′ (U )) ±,≤ν ⊗ HW O X ± . Now the arguments of Bellaïche apply verbatim. More precisely, since O X ± are PIDs, [Bel12,Lemma 4.1] tells us that for any x ∈ X ± , the localization N ± (x) is a free O X ± ,xmodule of finite rank. Moreover, for any classical point x = x 0 of weight k ≥ ν, the isomorphism (25) together with [Bel12,Lemma 2.8] show that N ± (x) is of rank one over O X ± ,x . By the local constancy of the rank, the same also holds true for N ± (x0) . We conclude that N ± is a free O X ± -module of of rank one, when U and W sufficiently small.
Corollary 5.7. Let us fix W that ensures the validity of the conclusions of Theorem 5.6(ii) and suppose x 0 is cuspidal, so that X + = X − according to Theorem 5.6(iii). Put X = X ± . Then both O X -modules Proof. Clear, thanks to Theorem 5.6.
5.3.4. We record the following proposition which will shall use in the proof of Proposition 6.5. It is well known to experts, but we prove it for reader's convenience.
Proposition 5.8. Let x 0 be a classical cuspidal point on the eigencurve C and k 0 = w(x 0 ). Then the following hold true.
i) The natural map ii) There exists an affine neighborhood W of k 0 and ν > 0 such that for the connected component X ⊂ C W,ν of x 0 ∈ C W,ν the map Proof.
i) Let us denote by Y the Borel-Serre compactification of the modular curve Y = Γ p \ H (and ∂ Y its boundary). The sheaf V k0 on Y extends to a sheaf on Y . It follows from [AS86, Proposition 4.2] that and the surjectivity of (26) follows once we verify that This follows from the properties 4 of the Eisenstein cohomology, since x 0 is cuspidal.
The proof that the map (26) is injective is similar, where one instead relies on the vanishing of the Our assertion on the dimension of these vector spaces is standard, c.f. [Bel12, Proposition 3.18].
ii) Fix ν k 0 + 1. Let W be an affinoid neighborhood of k 0 such that the conclusion of Theorem 5.4 hold for W. To simplify notation, set M c = H 1 c (Γ p , D † (W)) ≤ν ⊗ HW O X and M = H 1 (Γ p , D † (W)) ≤ν ⊗ HW O X . Then the O X -module M c is free of rank 2. The same argument (that Bellaïche utilizes to prove M c is free of rank 2) shows that M is also free of rank 2 over O X . Namely, on shrinking W if necessarly, we can assume that M is a finitely generated free O X -module. For any point x of weight w(x) we have Together with Lemma 5.1 this gives Now it follows from the classical Eichler-Shimura isomorphism and multiplicity-one that M/m x M has dimension 2 over E, and the O X -module M is free of rank 2 as well.
Let det j ∈ O X denote the determinant of j ⊗ id : M c → M with respect to fixed bases of M c and M. If we knew that det j (x 0 ) = 0, we could shrink W and the neighborhood X of x 0 ) to ensure that det j is non-vanishing on X and thereby conclude that j ⊗ id is an isomorphism, as required. We have therefore reduced to proving that det j (x 0 ) = 0.
Recall the isomorphism (29) for x = x 0 : The analogous isomorphism (28) for M c together with Lemma 5.1 shows that we have an injection M c /m x0 M c ֒→ H 1 c (Γ p , D † k0 (E)) ≤ν ⊗ H,x0 E, which is an isomorphism if k 0 = 0. As a matter of fact, it follows from [Bel12, Proposition 3.14] that this map is an isomorphism since x 0 is cuspidal. We therefore have the following commutative diagram: To conclude with the proof of the asserted isomorphism in our proposition with sufficiently small W and X , it suffices to prove that the right vertical map is an isomorphism. Observe that the following diagram commutes: , where B is also a positive integer. In Sections 6.3-6.5, we will be working over the modular curve Y (1, N (p)) of Γ 1 (N ) ∩ Γ 0 (p)-level. We note that the modular curve Y (1, N (p)) is denoted by Y (N, p) in [AIS15].
If Y denotes any one of the modular curves above, we denote by λ : E → Y the universal elliptic curve with the appropriate level structure (which depends on Y , but we suppress this dependence from our notation). We let T := R 1 λ * Z p (1) denote the pro-system (T n ) n≥1 of étale lisse sheaves T n := R 1 λ * µ p n on the open modular curves Y [1/p] given as in [Kat04, § §1-2]. The sheaf T has rank 2 and the Poincaré duality identifies it with the p-adic Tate module T p (E ) of E . We write T Qp for the associated sheaf of Q p -vector spaces.
For each non-negative integer k and a locally free sheaf F over Y [1/p], we let TSym k F denote the locally free sheaf on Y of symmetric k-tensors, given as in [  We refer the reader to [Kin15] where Kings develop these notions in a general framework.
If D = Y and i D = s : Y → E is a section of λ, we will write Λ(T s ) instead Λ(T D ). In particular, we denote by Λ(T ) = Λ(T 0 ) the sheaf associated to the identity section 0. For any section s, the sheaf Λ(T s ) is a sheaf of modules of rank one over the sheaf of Iwasawa algebras Λ(T ) (c.f. [Kin15], §2.4). 6.1.3. In the remainder of §6.1, we recall a number of notation and constructions from [KLZ17,LZ16]. Let g n : Y 1 (N p n ) → Y 1 (N ) denote the canonical projection. We define the pro-etale sheaf Λ N on Y 1 (N ) ét on setting Λ N = Λ N,n n 1 , Λ N,n = g n, * (Z/p n Z) .
If p | N, the moduli description of Y 1 (N p n ) shows that there exists a canonical isomorphism Y 1 (N p n ) ≃ E [p n ] s N , and therefore we have (c.f. the proof of [KLZ17], Theorem 4.5.1). In particular, we can apply the formalism developed in [Kin15] with the sheaf Λ N .
6.1.4. Assume that N is coprime to p and consider the modular curve Y (1, N (p)) equipped with the universal elliptic curve E N (p) and the associated sheaf T N (p) . Recall that Y (1, N (p)) is the moduli space for the triples (E, β N , C), where E is an elliptic curve, β N : Z/N Z → E is an injection and C ⊂ E is a (cyclic) subgroup of order N p that contains β N (1 + N Z).  N (p)), of degrees p 2 − p and p − 1, respectively, and we denote by Λ(T N (p) D ) and Λ(T N (p) D ′ ) the associated sheaves.
Since both D and D ′ are contained in E N (p) [p], the "multiplication-by-p" morphism induces the trace map N (p)). N (p)), which in turn induces the cartesian square For each positive integer r, we have a natural morphism where the schemes E ? [p r ] · are given as in [KLZ17, Definition 4.1.4]. The composition of this map with (32) gives a map , (see also [BH20, §4.2.5] for further details). • The canonical projections  where E is an elliptic curve and α : (Z/N p n Z) 2 ≃ E[N p n ] is an isomorphism, shows that the restriction of E M,N [p n ] ≃ T p (E M,N )/p n on Y (M p n , N p n ) has a canonical basis {e 1,n , e 2,n } given by e 1,n = α(M, 0) and e 2,n = α(0, N ). Analogously, the interpretation of Y (M p n , N p n ) as the moduli space of pairs (E, β), where E is an elliptic curve and β : Z/N p n Z → E[N p n ] is an injection, shows that the restriction of the sheaf T N on Y 1 (N p n ) has the canonical section β(N ) which we also denote by e 2,n to simplify notation.

One has an isomorphism of continuous Galois modules
The commutative diagrams (for each n) for each m dividing M .
Note that this map coincides with (37) if m = M, a = 0, B = 1 and L = N.

We write
for the Kummer map. For any integer j 0, consider the chain of maps where the first map is induced by the composition of ∂ n with the cup product in étale cohomology, and the very last map is deduced from the Hochschild-Serre spectral sequence.
-2] for their definition). As in op. cit., we put c,d z Mp n ,N p n := c g 1/Mp n ,0 ∪ d g 0,1/N p n ∈ K 2 (Y (M p n , N p n )). We recall that belongs to the source of the map Ch M,N,j (c.f. [Kat04], Proposition 2.3).
6.2.6. Let us fix a positive integer N . As in [Kat04,§5], let ξ denote either the symbol a(B) with a, B ∈ Z and B 1 or an element of SL 2 (Z). For each integer m 1, we denote by S the following set of primes: Let (c, d) be a pair of positive integers satisfying the following conditions: the composition of the maps (39) and (38). Here, the symbol ι on the final line means that the target cohomology group is equipped with a continuous action of Γ Q(ζm) = Gal(Q(ζ mp ∞ )/Q(ζ m )) induced by the right action of this group on Λ(Γ Q(ζm) ) via the canonical involution ι(g) = g −1 (c.f. §3.1.1).
The element ξ induces an automorphism of Y (Lp n ) = Y (Lp n , Lp n ). We consider the map (41) where Ch L,j,ξ is the map (41). Proof. The proof is similar to that of [Kat04,Proposition 8.7]. Let (M ′ , L ′ ) be another pair of integers that satisfy (40) and such that M | M ′ , N | N ′ . Since the trace map commutes with cup products, we have the following commutative diagram:  N,n can be described as the composition On the level of cohomology, one then obtains the commutative diagram where the bottom horizontal map is the trace map (c.f. [Kin15], Proposition 2.6.8).

For each r ∈ Z, we have the moment maps of sheaves on µ
The stalks of Λ cyc µ • m are isomorphic to the Γ-module Λ(Γ Q(ζm) ) ι at geometric points (c.f. [KLZ17], §6.3) and the moment map coincides with the map whereḡ n ∈ G n denotes the image of g ∈ Γ Q(ζm) under the natural projection Γ Q(ζm) → G n . Let us define . Recall the elements c,d z ∈ Note that in his construction Kato makes use of the dual sheaf T ∨ N together with the canonical isomorphism T ∨ N ≃ T N (−1) provided by duality. Proposition 6.3. For all integers j, k 0 and r ∈ Z, the induced map Proof. This is clear by the construction. 6.3. Overconvergent étale sheaves. Until the end of this article, we set Y = Y (1, N (p)) unless we state otherwise and assume that N is coprime to p.
6.3.1. We denote by F the canonical sheaf on Y. Throughout this section, W and U denote an affinoid and a wide open disk such that W ⊂ U ⊂ W * . In what follows, we will allow ourselves to shrink both U and W as necessary. We adopt the notation and conventions of Section 5. In particular, O • U denotes the ring of analytic functions on U that are bounded by 1. We let denote the composition of the cyclotomic character χ : G Q → Z × p with the canonical weight character We review the theory of overconvergent sheaves introduced in [AIS15]. See also [LZ16,§4] and [BSV20,§4] for further details concerning the material in this subsection.
The evaluation map We finally recall that Proposition 4.4.5 of [LZ16] supplies us with a morphism of sheaves on Y : 6.3.2. By GAGA, we have the canonical isomorphisms ii) All three O X -modules We remark that it is absolutely crucial that the affinoid neighborhood X of x 0 falls within the cuspidal eigencurve.
ii) It follows from (i) and Theorem 5.
Passing to the duals, we obtain a map ) and the associated morphism of sheaves: the map given by δ j (F ) := 1 j! i+m=j ∂ j F (x, y) ∂x i ∂y m ⊗ x i y m . On transposing this map, we obtain 6.4. Big Galois representations.
6.4.1. For the convenience of the reader, we review Deligne's construction of p-adic representations associated to eigenforms. Let T ∨ Qp be the dual sheaf of T Qp . Let f denote a p-stabilized eigenform of weight k + 2 2 and level Γ p , which is new away from p (but could be new or old at p). Deligne's representation associated to f is defined as the f -isotypic Hecke submodule for the Hecke operators {T ℓ } ℓ∤N p and {U ℓ } ℓ|N p , which are given as in [Kat04, §4.9] (but denoted in op. cit. by T (ℓ) for all ℓ). Recall that this is a two dimensional vector space equipped with a continuous action of G Q , which is de Rham at p (crystalline iff f is p-old) with Hodge-Tate weights (0, k + 1). By the theory of newforms, the Hecke module Paralleling the discussion in the previous paragraph, the

6.4.2.
For any normalized eigenform f = n≥1 a n q n ∈ S k+2 (Γ p ), we set f c = n≥1 a n q n .
Since f is an eigenform for the Hecke operators {T ℓ } ℓ∤N p and {U ℓ } ℓ|N p with eigenvalues {a ℓ }, it is also an eigenform for the dual operators {T ′ ℓ } ℓ∤N p and {U ′ ℓ } ℓ|N p , with eigenvalues {a ℓ }. The isomorphism T ∨ Qp (1) ≃ T Qp induces a Hecke equivariant map on the f -isotypic subspaces for the dual operators {T ′ ℓ } ℓ∤N p and {U ′ ℓ } ℓ|N p . With obvious modifications, all these constructions also make sense for modular forms of level Γ 1 (N ). In the remainder of this paper we will work with the O X -adic Galois representations By Proposition 6.5, both O X -modules have rank two.
We denote by 6.4.4. For any E-valued classical point x ∈ X cl (E) of X , we let f x denote the corresponding p-stabilized cuspidal eigenform of weight w(x) + 2. In particular, f x0 is our fixed p-stabilized eigenform f . By Propo- for some e 1, so the formalism of Section 2 applies in this context.
We recall that e = 1 except when the p-stabilized form f x0 is θ-critical.
Let P k (O E ) denote the space of homogeneous polynomials in two variables of degree k with coefficients in We have the associated morphisms of sheaves (53) We obtain the following morphisms by the definitions of M U (T ′ ) and N U (T ′ ): (1)) . On tensoring with O X and taking slope ≤ ν submodules, we deduce for any x ∈ X cl (E) that there are canonical morphisms It follows from definitions that , where e denotes the ramification degree of the weight map w : is the largest subspace of V ? X ,w(x) annihilated by X − X(x) ∈ O X ); and these are both E-vector spaces of dimension 2.
For any x ∈ X cl (E), let us also write ( , ) w(x) for the pairing −→ E induced from the pairing (52) by O W -linearity. We summarize the basic properties of these objects (that we shall make use of in what follows) in Proposition 6.6 below. Proposition 6.6. Suppose x ∈ X cl (E) is an E-valued classical point.
i) The Hecke-equivariant map is surjective. In particular, it induces an isomorphism is injective. In particular, it induces an isomorphism iii) The pairing ( , ) X verifies the property Adj).
iv) The restriction of ( , where the bottom pairing is the Poincaré duality (49), commutes. Namely, for all Proof.
i) The surjectivity of V ′ X [[x]] → V ′ fx follows from [Bel12, Corollary 3.19], where the analogous statement is proved for the spaces of modular symbols, combined with Proposition 6.5(i).
ii) This is clear by definitions.
iii) This portion follows from the [BSV20, §4.2] (see in particular the discussion following (69)). iv) This is a particular case of the factorization (2). v) This statement follows from the functoriality of the cup products. 6.5. Beilinson-Kato elements over the eigencurve.
6.5.1. We maintain previous notation and conventions in §5 and §6. Let x 0 ∈ C cusp (E) be a classical cuspidal point on the eigencurve. Fix a wide open U , an affinoid W and a slope ν 0 such that x 0 ∈ W ⊂ U and the conditions of Proposition 6.5 are satisfied. Let X denote the connected component of x 0 in C W,ν . Consider the following maps: • For each non-negative integer k ∈ U, the map (54) which we notate as • For each classical x ∈ X , the map (55), which we shall notate as Moreover, we have the following commutative diagram, which follows from the slope decomposition:  r)) . It follows from the commutativity of the diagram (56) that the maps (57) and (58) are compatible in the evident sense.

The map (44) induces a morphism
). The composition of this map with (35) gives rise to the map . More generally, for each integer j 0, the morphism (48) together with the trace map induce is the projection of the Beilinson-Kato element c,d BK N p,mp n (w(x) − j, j, r, ξ) introduced in §6.2.8 to the f c xisotypic eigenspace for the action of Hecke operators {T ℓ } ℓ∤N p and{U ℓ } ℓ|N p . Here we have used the canonical isomorphism (50) to identify c,d BK N,mp n (f c x , j, r, ξ) with an element of H 1 (Z[1/S, ζ mp n ], V ′ fx (1 − r)).

Proof.
i) Consider the following diagram: It follows from definitions that the map (60) decomposes as explained in the diagram. The factorization of the moment map mom ii-iii) These assertions follow immediately from (i).
iv) This portion follows from (iii) and Proposition 6.3. Corollary 6.9. We retain the notation and hypotheses of Theorem 6.8. We let ) denote the unique class which specializes to c,d BK N p,p n (f c x , j, r, ξ) ∈ H 1 (Z[1/S, ζ p n ], V f c x (w(x) + 2 − r)) for all n ∈ N under the evident morphisms. Then, c,d BK [X ] N,1 (j, ξ, x) = c,d BK N p,Iw (f c x , j, ξ) where c,d BK [X ] N,1 (j, ξ, x) is the specialization of c,d BK [X ] N,1 (j, ξ) to x.
Remark 6.10. We retain the notation and hypotheses of Theorem 6.8. Suppose that x ∈ X cl (E) is a classical point and f x is p-old, arising as the p-stabilization of a newform g of level N coprime to p, with respect to the root a p (f x ) of the Hecke polynomial of g at p. Then f c x is the p-stabilization of the newform g c with respect to the root a p (f x ) of the Hecke polynomial of g c at p. In this remark, we shall explain the relation between c,d BK N p,Iw (f c x , j, ξ) and the Beilinson-Kato element c,d BK N,Iw (g c , j, ξ) that one associates to the newform g c . The natural projection Y (1, N (p)) =: Y Pr −→ Y 1 (N ) induces the trace map , which restricts to an isomorphism (since this map commutes with the action of Hecke operators Let us set Λ X (Γ 1 ) = O X ⊗ Zp Λ(Γ 1 ) and put Λ X = O X ⊗ Zp Λ.
We start with the following auxiliary lemma.
Lemma 6.11. Let M be a finitely generated Λ X (Γ 1 )-module. Suppose that M x0 := M/XM is torsionfree as a Λ(Γ 1 )[1/p]-module and that M [X] = 0. Then there exist an affinoid disk X ′ ⊂ X such that 6.6.3. We consider the product as a two variable function in the weight and cyclotomic variables.
Lemma 6.13. For each classical point x and integer m, Proof. According to [Sai00, Corollary 2] (the Weight-Monodromy Conjecture for modular forms), a ℓ (f x ) is an algebraic integer with complex absolute value either ℓ 6.6.4. For an integer c, we let σ c ∈ Γ denote the element defined by the requirement that χ(σ c ) = c. We shall consider σ c also as an element of Λ in the obvious manner. Following Kato, we define the partial normalization factors Here, for any x ∈ X , we define d w( We also define the full normalization factor: 6.6.5. To simplify notation, we shall write c,d BK [X ] N (j, ξ) in place of c,d BK [X ] N,1 (j, ξ). When c, d and j are understood, we will write µ ? in place of µ ? (c, d, j, x) for ? ∈ {0, ∅}. We denote by Λ X [µ −1 0 ] ⊂ Q(Λ X ) the Λ Xsubalgebra generated by µ 0 (c, d, j, x) −1 . We also denote by Λ[µ −1 x0 ] ⊂ Q(Λ) the Λ[ 1 p ]-subalgebra generated by µ 0 (c, d, j, x 0 ) −1 . Definition 6.14. Let us fix W so that H 1 Iw (Z[1/S], V ′ X (1)) is torsion-free as a Λ X -module (we can choose such W thanks to our discussion in §6.6.2). Let us fix integers c ≡ 1 ≡ d mod N with (cd, 6p) = 1 and c 2 = 1 = d 2 , and an integer j ∈ [0, k 0 ]. We define the partially normalized Beilinson-Kato elements on setting BK [X ] The integrality properties of the partial normalizations of the Beilinson-Kato elements are established in Proposition 6.15, whose proof has been relegated to Appendix A. Proposition 6.15. Suppose c, d and j are as in Definition 6.14.
Proof. The first assertion is Proposition A.2, whereas the second is Proposition A.4 combined with (i). These elements are given as the Hecke equivariant projection of the elements c,d z (p) p n (k, k, j, ξ, prime(pN )) n≥1 recalled at the end of Section 6.2 to the g-isotypic Hecke eigenspace. He introduces the normalization factor Suppose x ∈ X cl (E) is a classical point. As we have explained in Corollary 6.9 (see also Remark 6.10), our big Beilinson-Kato element c,d BK [X ] N,1 (j, ξ) interpolates the elements which coincide with the f c x -isotypic projection of Kato's element c,d z p n (k + j + 2, 0, j + 1, ξ, prime(pN )) n≥1 , where w(x) = k + j. Kato's element c,d z p n (k + j + 2, 0, j + 1, ξ, prime(pN )) n≥1 differs from (67) c,d z p n (k + j + 2, k + j + 2, j + 1, ξ, prime(pN )) n≥1 by the twist by χ w(x)+2 = χ k+j+2 . Recall that we denote by Tw m : Λ → Λ the twisting morphism which is given on the group like elements by γ → χ −m (γ)γ.
Since f is new away from p, it follows from [Bel12, Lemma 2.7] that one can shrink X as necessary to ensure that f x is new away from p and p-old for all x ∈ X cl (E) \ {x 0 }. There are two scenarios: i) f x is a newform (so x = x 0 and f x = f , owing to our choice of X ). Then f c x is a newform too. ii) f x is the p-stabilization of a newform f • x =: g of level N coprime to p with respect to the root a p (f x ) of the Hecke polynomial of g at p. As we have explained in Remark 6.10, f c x is a p-stabilization of the newform g c and one naturally identifies c,d BK N p,Iw (f c x , j, ξ) with c,d BK N,Iw (g c , j, ξ). In this scenario, we have µ Kato (c, d, j, f c x ) = µ Kato (c, d, j, g c ). We then have, in both scenarios, In [Kat04, Section 13], Kato shows that there exists an element z(g c , j, ξ) ∈ H 1 Iw (Z[1/S], V g c (w(x) + 2)) such that c,d BK N,Iw (g c , j, ξ) = µ(c, d, j, x) z(g c , j, ξ). We remark that in Kato's notation, we have z(g c , j, ξ) = Tw w(x)+2 (z (p) γ ), where γ is the cohomology class associated to j and ξ. Combining this fact with the preceding remarks, we deduce that (68) BK [X ] N (j, ξ, x) = E N (x) z(g c , j, ξ).

"Étale" construction of p-adic L-functions
Our objective in §7 is to use the O X -adic Beilinson-Kato element we have introduced in Definition 6.7(iii) and 6.14 to give an "étale" construction of Stevens' two-variable p-adic L-function in neighborhoods of nonθ-critical cuspidal points on the eigencurve. In particular, we assume throughout §7 that X is étale over W (i.e. e = 1); see our companion article [BBa] for the treatment of the complementary case (which concerns the neighborhoods of θ-critical points on the eigencurve).
Recall the representations V X and V ′ X given as in (51). We will always assume that X is sufficiently small to satisfy the conditions of Proposition 6.15. For each classical x ∈ X cl (E), we denote by f x the corresponding eigenform. The O X -adic Beilinson-Kato elements BK [X ] N (j, ξ) take coefficients in the cohomology of the universal cyclotomic twist of V ′ X . Recall also that these representations are equipped with the O X -linear pairing . (52), bearing in mind that we have assumed e = 1), which satisfies the formalism of Section 2. We will use repeatedly the properties of this pairing summarized in Proposition 6.6.
Let f = f α0 0 be a p-stabilized non-θ-critical cuspidal eigenform of weight k 0 + 2 2 and as before, which corresponds to the point x 0 on the cuspidal eigencurve. The eigencurve is étale at x 0 over the weight space (c.f. [Bel12], Lemma 2.8). In other words, e = 1 and O X = O W . As we have explained in §3.3.2, the simplification of the local behaviour of the eigencurve in this scenario exhibits itself also in the the p-local study (namely, the properties of the triangulation over the eigencurve). The reader might find it convenient to identify X with a Coleman family f through f α0 0 , so that the overconvergent eigenform f x that corresponds to a point x ∈ X (E) is simply the specialization f(x) of the Coleman family f.
To simplify notation, we write R X for the relative Robba ring R OX and H X (Γ) for the relative Iwasawa algebra H OX (Γ). The p-adic representation V X (given as in (51)) verifies the properties C1)-C3) and therefore comes equipped with a triangulation D ⊂ D † rig,OX (V X ) over the relative Robba ring R X , verifying the properties ϕΓ 1 )-ϕΓ 3 ). Fix an O X -basis {η} of D cris (D) and let η x ∈ D cris (D x ) = D cris (D[x]) denote its specialization at x ∈ X (E) (equivalently, at weight w(x)).
Definition 7.2. For any x ∈ X (E), we put L ± p,η (f; j, ξ, x) = x • L ± p,η (f; j, ξ, x) ∈ H E (Γ). For a character ρ ∈ X(Γ), we set L ± p,η (f; j, ξ, x, ρ) := ρ • L ± p,η (f; j, ξ, x). For x above, we let f • x denote the newform associated to f x . We write α(x) for the U p -eigenvalue on f x , so that when f x is p-old, it is the p-stabilization of the newform f • x with respect to the root α(x) of the Hecke polynomial of f • x at p. We let L p,α(x) (f • x ) denote the Manin-Višik p-adic L-function associated to an unspecified choice of (a pair of) Shimura periods (see, however, item c) in the proof of Theorem 7.3 below, where this choice is made explicit).
The following result (in varying level of generality) has been previously announced in the independent works of Hansen, Ochiai and Wang.
Theorem 7.3. There exists a neighborhood X of x 0 such that the following hold true.
p,η (f; j ± , ξ ± , x 0 ) agree with the one variable p-adic L-functions of Pollack-Stevens [PS13] up to multiplication by D ± E N (x), where D ± ∈ E × is a constant. iv) Let L p (f, Φ ± ) denote the two-variable p-adic L-functions of Bellaïche and Stevens associated to modular Proof. Let x ∈ X cl (E) be a classical point as in the statement of the theorem. Let g denote the unique newform which admits f x as a p-stabilization if f x is p-old and g = f x if f x is new. By (68), the specialization of the big Beilinson-Kato element at x can be compared with the normalized Beilinson-Kato element for g: BK [X ] N (j, ξ, x) = E N (x) z(g c , j, ξ). Since Perrin-Riou's exponential map commutes with base-change, the specializations L ± p,η (f; j, ξ, x) of p-adic L-functions L ± p,η (f; j, ξ) coincide with one variable Perrin-Riou's L-functions constructed using the element BK [X ] N (j, ξ, x): L + p,η (f; j, ξ, x) := res p BK [X ],ǫ(k0−1) N (j, ξ, x) , c • Exp Dx,0 ( η x ) ι x ∈ H E (Γ) , L − p,η (f; j, ξ, x) := res p BK [X ],ǫ(k0) N (j, ξ, x) , c • Exp Dx,0 ( η x ) ι x ∈ H E (Γ) .
We next prove Part (iv). Using the Amice transform, we can consider the functions L p (f, Φ ± ) as elements of H X (Γ). It follows from [Bel12, Theorem 3] for classical non-critical points x ∈ X (E) that L p (f, Φ + , x) coincides, up to multiplication by a non-zero constant, with the Manin-Višik p-adic L-function. We infer from (i) that for classical non-critical points x ∈ X cl (E), we have L + p,η (f; j + , ξ + , x) = u x E N (x)L p (f, Φ + , x) for some constant u x ∈ E, with u x0 = 0 by the choice of (j + , ξ + ). On expressing L + p,η (f; j + , ξ + , x) and E N (x)L p (f, Φ + , x) as power series with coefficients in O X , we immediately deduce that there exists a function u + (x) ∈ O X such that u + (x 0 ) = 0 and L + p,η (f; j + , ξ + , x) = u + (x)E N (x)L p (f, Φ + , x).
On shrinking X as necessary, we can ensure that u + (x) does not vanish on X . The same argument proves that L − p,η (f; j − , ξ − , x) = u − (x)L p (f, Φ − , x) for some u − (x) ∈ O X . This concludes the proof of (iv). Part (iii) is a direct consequence of (iv) (as per the definition of the Pollack-Stevens p-adic L-function) and our theorem is proved.
We note that in the particular case when f is the Coleman family passing through a newform f 0 = f α0 0 of level Γ 0 (N p) and weight k 0 + 2 with α 0 = a p (f 0 ) = p k0/2 , the conclusions of Theorem 7.3 play a crucial role in [BBb]. When f = f α0 0 has critical slope but not θ-critical, it is also crucially used in the proof of a conjecture of Perrin-Riou in [BPS].

Appendix A. Integrality of normalizations
For any x ∈ X , let us define d w(x) := d k0 d w(x)−k0 . We define the partial normalization factor A.1. The normalization factors revisited. Let χ cyc : G Q ։ Γ ֒→ Λ × denote the universal cyclotomic character, so that χ −1 cyc gives the action on Λ ι . We also define the universal cyclotomic character parametrized by the weight space χ wt as the compositum Let γ ∈ Γ denote a fixed topological generator.
Lemma A.1. Suppose c and d are integers coprime to p.
iii) Suppose c and d are primitive roots modulo p 2 . For each y ∈ X cl (E) with w(y) = 2j, the elements µ 1 (c, j) and µ 2 (d, j, y) of Λ ⊗ Zp O E are coprime.
A.2. Dependence on c and d. In this subsection, we prove the first part of Proposition 6.15.
Before we proceed with the proof of Proposition A.2, we record the following auxiliary lemma. Lemma A.3. Suppose R is an integral domain and M is a finitely generated torsion-free R-module. Let {P i } i∈I be an infinite collection of prime ideals of R such that i∈I P i = {0}. Then, Re k of rank r, and M/M 0 is annihilated by multiplication by some a ∈ R. Let m ∈ i∈I P i M be any element. Then am belongs to i∈I P i M 0 . Since am can be written in a unique way in the form am = r k=1 α k e k , α k ∈ R, we see that α k ∈ i∈I P i = {0} for all k.

A.3. Integrality of partial normalizations.
A.3.1. We will next analyze the regularity of the partially normalized Beilinson-Kato element BK [X ] N (j, ξ). This amounts to an analysis of the divisibility of c,d BK [X ] N (j, ξ) by µ 0 (c, d, j, x). In view of Proposition A.2, we may (and henceforth will) work with c and d which are both primitive roots mod p 2 , without any loss.
The uniqueness of BK [X ] N (j, ξ) follows from Lemma 6.12 (on shrinking X as necessary). The existence will be proved in the §A.3.2 and §A.3.2 below.