Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an \'etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a $p$-stabilised newform of weight $k \geq 2$ without CM by $K$. Suppose $f$ has finite slope at $p$ and its base-change $f_{/K}$ to $K$ is $p$-regular. Then: (1) We construct a two-variable $p$-adic $L$-function attached to $f_{/K}$ under assumptions on $f$ that conjecturally always hold, in particular with no non-critical assumption on $f/K$. (2) We construct three-variable $p$-adic $L$-functions over the eigenvariety interpolating the $p$-adic $L$-functions of classical base-change Bianchi cusp forms. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space.

Our main result, Thm. A, is the variation of p-adic L-functions in a p-adic family through f /K , via the construction of a three-variable p-adic L-function. Our construction is unconditional when f /K is non-critical, and more generally is valid under hypotheses on f /K that conjecturally always hold. This generalises previous constructions for Hida families when p is split in K, and when p is inert or ramified, we believe this construction to be entirely new, even for Hida families.
We apply this in Thm. B to construct the 'missing' p-adic L-functions attached to p-regular base-change forms. Suppose f as above has critical slope. Under the same (conjecturally automatic) hypotheses, we construct a p-adic L-function attached to f /K . We make no non-criticality assumptions and allow arbitrary p. This p-adic L-function naturally has two (cyclotomic and anticyclotomic) variables; note that it is difficult to construct this directly from p-adic L-functions attached to f , as we would not see any anticyclotomic variation.
After restricting to the cyclotomic line, in Thm. C we relate our constructions to the p-adic Lfunctions attached to the original form f via a p-adic analogue of the factorisation given by classical Artin formalism. This generalises a result that is important in the Iwasawa theory of elliptic curves, and that was previously known only under an even more restrictive slope condition that excludes cases of important arithmetic interest.
To prove these results, we build on work of the second author, who gave constructions of padic L-functions for (non-critical slope) Bianchi modular forms, under no base-change assumption, in [Wil17]. The p-adic L-function was shown to be the Mellin transform of a class in overconvergent cohomology/modular symbols, as introduced by Stevens in the classical setting [Ste94]. This class, and hence the p-adic L-function, is canonical up to p-adic scalar. The main input of the current paper is a pairing of this construction with a systematic study of the eigenvariety parametrising Bianchi modular forms. The use of overconvergent cohomology in constructing eigenvarieties -generalising the pioneering work of Hida in the ordinary setting -was known to Stevens, later explored by Ash-Stevens [AS08], Urban [Urb11] and more recently by Hansen [Han17] and the authors [BSW21].

Applications to p-adic L-functions.
We summarise our main applications. Let f ∈ S k+2 (Γ 1 (N )) satisfy (a), (b) and (c) above, and suppose f is decent and f /K is Σ-smooth. Let X (Cl K (p ∞ )) be the two-dimensional rigid space of p-adic characters on Cl K (p ∞ ), the ray class group of K of conductor p ∞ . Let φ be a finite order Hecke character of K of conductor prime to pO K . Let x f be the point in the Coleman-Mazur eigencurve C corresponding to f , and let V Q be a neighbourhood of x f in C. If y ∈ V Q is a classical point, write f y for the corresponding classical eigenform. For a Zariski-dense set of classical y ∈ V Q , the base-change f y/K is non-critical (Def. 2.8). In §6.1, in the language of distributions, we prove: Theorem A. Up to shrinking V Q , and for sufficiently large L ⊂ Q p , there exists a unique rigidanalytic function L φ p : V Q × X (Cl K (p ∞ )) −→ L such that at any classical point y ∈ V Q (L) of weight k y + 2 with non-critical base-change, and any Hecke character ϕ of K with conductor f|p ∞ and infinity type (0, 0) ≤ (q, r) ≤ (k y , k y ), we have where c y ∈ L × is a p-adic period at y, ϕ p−fin ∈ X (Cl K (p ∞ )) is the p-adic avatar of ϕ, Z p (ϕ) is an Euler-type factor, A(f y/K , ϕ) is an explicit non-zero scalar, and Λ(f y/K , −) is the L-function of f y/K , all of which are defined in §2.4. For a fixed set {c y } of p-adic periods, this L φ p is unique.
In particular, for y ∈ V Q as in the theorem, the specialisation of L φ p at y is precisely the twovariable p-adic L-function (twisted by φ) attached to f y/K . Thus L φ p is the three-variable p-adic L-function attached to V Q and φ. The meat of the proof is in producing a canonical class (up to scaling) in the overconvergent cohomology over the eigenvariety, interpolating the overconvergent classes of [Wil17] at classical points. We do this in §4-6 using the local geometry of the parallel weight eigenvariety. Given this class, L φ p is defined as its (twisted) Mellin transform. We comment briefly on the 'choice' of periods c y . We do have some control over them; under a non-vanishing hypothesis, which is satisfied for f /K non-critical and conjecturally for all f /K , any two systems of periods {c y }, {c ′ y } for which there exists a three-variable p-adic L-function are of the form c y = α(y)c ′ y , where α ∈ O(V Q ) × . We show this in Prop. 6.15. When p splits in K, using the strategy and results developed in the present paper, in [BW] Thm. A is proved in the p-irregular case (i.e. when (c) fails).

Remark:
It is natural to ask if there are analogues of Thm. A for Bianchi modular forms that are not base-change. A general Bianchi modular form F still varies in a 1-dimensional p-adic family V of overconvergent Bianchi modular symbols (Thm. 3.8) over a curve Σ in the (2-dimensional) Bianchi weight space. When F is non-critical and Σ is smooth at the weight of F , then we prove V is étale over Σ (Thm. 4.5, Cor. 4.8). In this case we prove existence of a rigid function L φ p : V × X (Cl K (p ∞ )) → L satisfying the interpolation (1.1) of Thm. A.
If V is a classical family (i.e. the classical points in V are Zariski-dense), then Σ is parallel, hence smooth. We thus unconditionally construct three-variable p-adic L-functions around noncritical points in every classical family.
In general the nature of V is mysterious; it might contain only finitely many classical points, so (1.1) could be empty outside F ! We give a possible arithmetic interpretation in this case, and the construction of L φ p in general, in Rem. 6.11. Now suppose f /K is critical and Σ-smooth, so we cannot use [Wil17]. Using Thm. A, we define the 'missing' p-adic L-function for f /K to be the specialisation L φ p (f /K , −) . . = L φ p (x f , −). This is a two-variable p-adic L-function attached to f /K satisfying the expected growth property and which we prove is canonical up to a p-adic scalar (corresponding to the p-adic period of f ). In Thm. 6.14, we prove: Theorem B. Suppose f /K is critical and Σ-smooth. Then we have for all ϕ of conductor f|p ∞ and infinity type (0, 0) ≤ (q, r) ≤ (k, k).
Thm. B gives at least a conjectural construction for every p-regular finite slope f /K , so it goes much further than [Wil17] (see Rem. 2.14).
The L-functions of f and f /K are related by Artin formalism L(f /K , s) = L(f, s) · L(f, χ K/Q , s), where χ K/Q is the quadratic character associated to K/Q. For the p-adic L-functions, such a factorisation does not make sense on the nose, since L p (f /K ) is two-variabled whilst L p (f ) and L χ K/Q p (f ) are both one-variabled (valued on X (Cl + Q (p ∞ )) ∼ = X (Z × p )). We fix this by letting L cyc p (f /K ) denote the restriction of L p (f /K ) to the cyclotomic line. In Thm. 7.5, we then prove: Theorem C. Suppose f /K is Σ-smooth and that L cyc p (f /K ) and L p (f )L This is a GL 2 -analogue of Gross's (GL 1 ) p-adic Artin formalism relating Katz and Kubota-Leopoldt p-adic L-functions [Gro80]. A factorisation relating Rankin and symmetric square p-adic L-functions, again mimicking classical Artin formalism, has been obtained by Dasgupta [Das16].
We remark that since our periods are only defined up to an algebraic scalar, this is really an equality of one-dimensional lines in the (infinite-dimensional) space O(X (Cl + Q (p ∞ ))). We explain this more fully in §7.2. When f has sufficiently small slope -namely, slope h < (k + 1)/2 -this theorem is automatic from classical Artin formalism, since both sides satisfy a growth property that renders this line unique with respect to their interpolation properties. For more general slopes, this result is far from obvious, as the growth and interpolation properties are satisfied by an infinite number of distinct lines in O(X (Cl + Q (p ∞ ))), and we really require the additional input of our three-variable p-adic L-function to see the equality.
Ideally, we would also be able to control the (p-adic and archimedean) periods integrally to pin down an equality of lattices within this line, but this seems an extremely subtle question; studying integral period relations in the Bianchi base-change case was already the subject of [TU], even before studying this in the context of p-adic L-functions. We comment further in §7.
The non-vanishing condition is automatically satisfied when f and f /K are non-critical. Under a conjecture of Greenberg, which says that all critical elliptic modular forms are CM, we expect that L p (f ) and L χ K/Q p (f ) can be related to Katz p-adic L-functions, and are always non-zero; work in this direction is explained in [Bela]. We conjecture that L cyc p (f /K ) is similarly never zero. A case of particular interest where this theorem applies is the following. Let E/Q be an elliptic curve with good supersingular reduction at odd p, and let f α be a p-stabilisation of the corresponding weight 2 classical modular form corresponding to a root α of the Hecke polynomial at p. Since h = v p (α) = 1/2 and k = 0, this is outside the range where Thm. C is automatic. Suppose p splits in K. Then the base-change f α/K has (non-critical) slope 1/2 at each of the primes above p. Since the L-function of f α/K corresponds to a p-depleted L-function for E/K, we get a factorisation of the p-adic L-function of E/K in terms of the p-adic L-functions of E and its quadratic twist by χ K/Q . In the ordinary case this factorisation was required in Skinner and Urban's proof of the Iwasawa main conjecture (see [SU14]). Finally, we remark that modulo the existence of anticyclotomic p-adic L-functions in Coleman families, the same methods also apply to restriction to the anticyclotomic line. In this case, under the same non-vanishing hypothesis, we obtain L anti p (f /K ) = L anti p (f ) 2 (where the anticyclotomic p-adic L-function exists). We leave the details to the interested reader. Note that anticyclotomic p-adic L-functions do not yet exist in the case where f is critical. The above suggests that a good candidate for (the square of) an anticyclotomic p-adic L-function in this case is the restriction to the anticyclotomic line of the p-adic L-function attached to f /K in this paper.
These carry an action of U 1 (n), and the corresponding local systems on Y 1 (n) will be denoted by D( * ).
We will use f for a classical modular form and F a Bianchi modular form. For X an affinoid in a rigid space, O(X) will denote the ring of rigid functions on V . We will write V (resp. V Q ) for affinoids in the Bianchi (resp. Coleman-Mazur) eigenvariety. If y is a classical point in an eigenvariety, we will write F y or f y for the corresponding (Bianchi or classical) normalised modular form of minimal level, which will always be uniquely defined by our running assumptions.
2.1. Bianchi modular forms, L-functions and cohomology. Let λ = (k, v) be a weight, where k = (k 1 , k 2 ) and v = (v 1 , v 2 ) are two elements of Z 2 . There is a finite-dimensional C-vector space S λ (U 1 (n)) of Bianchi cusp forms of weight λ and level U 1 (n), which are vector-valued functions on GL 2 (A K ) satisfying suitable transformation, harmonicity and growth conditions. These objects are defined precisely in e.g. [Wil17, Def. 1.2]. If k 1 = k 2 , then S λ (U 1 (n)) = 0 (see [Har87]), so we will restrict to parallel weight k 1 = k 2 = k ≥ 0; and in this case, we can always twist the central character by a power of the norm to assume that v 1 = v 2 = 0 as well. For the rest of this section, we fix λ = [(k, k), (0, 0)], and we will write this as λ = (k, k) without further comment. For . The double coset operators T q , U q , v are all independent of choices of representatives and act on S λ (U 1 (n)). An eigenform is a simultaneous eigenvector. Attached to an eigenform F is a character ǫ F : Definition 2.1. Let H n,p denote the Z p -algebra generated by the Hecke operators Our theorems require n to be divisible by each prime p above p. If p ∤ N and F ∈ S λ (U 1 (N)) is an eigenform, let a p (F ) denote the T p eigenvalue of F , and let α p and β p denote the roots of the Then F αp and F βp are eigenforms of level U 1 (Np) with U p -eigenvalues α p and β p .
Throughout the paper, we work with the following Bianchi modular forms:

Conditions 2.2[K]
. Let λ = (k, k) and n ⊂ O F divisible by each p|p. Let F ∈ S λ (U 1 (n)) be a finite slope p-regular p-stabilised newform, in the sense that: (C1) F is an eigenform, and for each p|p, we have U p F = α p F with α p = 0; (C2) there exist S ⊂ {p|p}, N prime to S, and a newform F new ∈ S λ (U 1 (N)) such that n = N p∈S p and F is obtained from F new by p-stabilising for p ∈ S; Note newforms of level n themselves satisfy (C2),(C3) with S = ∅.
Let F satisfy Conditions 2.2[K] and let Λ(F , ϕ) denote the (completed) L-function of F , normalised as in [Wil17]. Here ϕ runs over Hecke characters of K. By [Hid94a, Thm. 8.1], we see that there exists a period Ω F ∈ C × and a number field E containing the Hecke eigenvalues of F such that if ϕ is an algebraic Hecke character of infinity type 0 ≤ (q, r) ≤ (k, k) with q, r ∈ Z, we have where E(ϕ) ⊂ Q is the extension of E generated by the values of ϕ.

Base-change.
Let f new ∈ S k+2 (Γ 1 (N )) be a classical cuspidal newform of nebentypus ǫ fnew , generating an automorphic representation π of GL 2 (A Q ). Let BC(π) be the base-change of π to GL 2 (A K ) (see [Lan80] If p ∤ N , let α p , β p be the roots of X 2 − a p (f new )X + ǫ fnew (p)p k+1 , and for p|p, let α p , β p be the roots of X 2 − a p (F new )X + ǫ Fnew (p)N (p) k+1 . If p is split or ramified in K, then we can take α p = α p , β p = β p ; and if p is inert, then we may take α p = α 2 p , β p = β 2 p . If f α (resp. f β ) is the pstabilisation of f new corresponding to α p (resp. β p ), we define its base-change to be the p-stabilisation F αα (resp. F ββ ) of F new corresponding to α p (resp. β p ) for all p|p.
We will consider the following classical analogue of Conditions 2.2[K]: (C2 ′ ) f is new or the p-stabilisation of a newform f new of level prime to p; . This space has a natural left action of GL 2 (R) 2 induced by the action of GL 2 (R) on each factor by a b c d · P (z) = (a + cz) k P b+dz a+cz , inducing a right action on the dual V λ (R) * . .= Hom(V λ (R), R). When R is a K-algebra, this gives Let R be an (O K ⊗ Z Z p )-algebra, and L a finite extension of Q p . Definition 2.6. Let A(R) be the space of locally analytic functions O K ⊗ Z Z p → R. When R = L, we equip this space with a weight λ action of the semigroup for A(L) with this action. As n is divisible by each prime above p, U 1 (n) acts on A λ (L) by projection to p.
Definition 2.7. Let D(R) . .= Hom cts (A(R), R) be the space of R-valued locally analytic distributions on O K ⊗ Z Z p . When R = L as above, we write D λ (L) for this space equipped with the weight λ right action of Σ 0 (p) given by µ|γ(ζ) = µ(γ · ζ). Then D λ (L) gives rise to a local system on Y 1 (n), which we denote by D λ (L). In [BSW19b,Def. 4.2] this local system is denoted L 2 (D λ (L)).
There is a natural map D λ (L) → V λ (L) * given by dualising the inclusion of V λ (L) into A(L). For each i, this induces a specialisation map Definition 2.8. Let F ∈ S λ (U 1 (n)) be an eigenform. We say F is non-critical if ρ λ becomes an isomorphism (for each i) upon restriction to the generalised eigenspaces of the Hecke operators at F . If F is non-critical, let Ψ F ∈ H 1 c (Y 1 (n), D λ (L)) denote the unique lift of φ F /Ω F considered with L-coefficients, where we assume L contains all embeddings of the fields K and E (from (2.1)).
Definition 2.9. If F ∈ S λ (U 1 (n)) is an eigenform, we say F has small slope (or non-critical slope) if v p (α p ) < (k + 1)/e p for all p|p, where U p F = α p F and e p is the ramification degree of p.

Modular symbols and Mellin transforms.
Let ∆ 0 . . = Div 0 (P 1 (K)) denote the space of 'paths between cusps' in H 3 , and let V be any right Σ 0 (p)-module. For a discrete subgroup Γ ⊂ Σ 0 (p) ∩ SL 2 (K), define the space of V -valued modular symbols for Γ to be the space From the above this induces a (non-canonical) decomposition When V is a Σ 0 (p)-module, there is a natural action of the Hecke algebra H n,p on the direct sum, defined as in [Wil17, §3.3], and (2.2) is Hecke-equivariant.
Let R be an (O K ⊗ Z Z p )-algebra such that D(R) carries a right action of U 1 (n), hence giving rise to a local system on Y 1 (n). Let Ψ ∈ H 1 c (Y 1 (n), D(R)), and write Ψ = (Ψ 1 , ..., Ψ h ) with each Ψ i ∈ Symb Γi (D(R)). Define, for i, j ∈ Cl K , a distribution µ i (Ψ j ) ∈ D(Cl i K (p ∞ ), R) as follows. We have a distribution under the identification above. Then define the Mellin transform of Ψ to be the (R-valued) locally analytic distribution on Cl K (p ∞ ) given by A simple check identical to the arguments given in [BSW19b,Prop. 9.7] shows that the distribution Mel(Ψ) is independent of the choice of class group representatives.
Definition 2.11. Let F be a non-critical cuspidal Bianchi eigenform of level U 1 (n) with associated overconvergent class Ψ F . The p-adic L-function of F is the Mellin transform L p (F ) . .= Mel(Ψ F ) ∈ D(Cl K (p ∞ ), L).
Given an algebraic Hecke character ϕ of K of conductor f = p|p p rp |(p ∞ ), there is a natural associated character  [Wil17,Defs. 5.10,6.14]. If F has small slope, L p (F ) is unique with these interpolation and growth properties.
Remark 2.13: Let φ be a finite order Hecke character of K. We obtain the twisted p-adic Lfunctions L φ p of the introduction by using twisted Mellin transforms. If φ has principal conductor (c) prime to p, Remark 2.14: Suppose pO K = pp is split. Let F new be base-change of weight λ = (k, k) and level N prime to p. The Hecke polynomials at p and p coincide; let α, β be the roots, ; then only f αα has small slope, and [Wil17] does not give p-adic L-functions for F αβ , F βα or F ββ . For p inert, the Hecke roots α, β satisfy v p (αβ) = 2(k + 1); so at least one of v p (α), v p (β) is ≥ k + 1, and there is always a missing p-adic L-function.

The Bianchi eigenvariety
We summarise results on the Bianchi eigenvariety, following [Han17]. Hansen's results are stated for singular cohomology, but in [Han17, §3.3] he gives tools to produce (identical) proofs for cohomology with compact support.

Distributions over the weight space.
Definition 3.1. The Bianchi weight space of level n is the rigid analytic space W K,n whose L-points, for L ⊂ C p any sufficiently large extension of Q p , are This can be identified with Z(GL 2 (K)) ∩ U 1 (n), hence this invariance ensures the existence of non-trivial 'weight λ' local systems on Y 1 (n). Since the level will typically be clear from context, we will usually drop the subscript n from the notation.
A weight λ ∈ W K (L) is classical if it can be written in the form ǫλ alg , where ǫ is a finite order character and λ alg (z) Remark 3.2: This is a slightly smaller space of 'null weights' than that considered in Hansen, who uses the characters on the torus T (Z p ) of diagonal matrices in GL 2 (O K ⊗ Zp Z p ). The two spaces are essentially the same after twisting by a power of the norm, and the smaller space allows clearer comparison with the Coleman-Mazur eigencurve.
For each λ ∈ W K (L), as before one can define a weight λ action of Σ 0 (p) on A(L) by γ · λ f (z) = λ(a + cz)f b+dz a+cz , and hence a dual action on D(L). We can vary these action in families over W K . Let Ω ⊂ W K be an affinoid, equipped with a tautological character such that for any λ ∈ Ω(L), the homomorphism λ : where the second map is evaluation at λ. We can thus equip A Ω . . = A(O(Ω)) with a 'weight Ω' action of Σ 0 (p) given by Dually we get an action on D Ω .

The eigenvariety and base-change functoriality.
One of the main results of [Han17] specialises, in our setting, to the following. Recall from above that his results apply also to compactly supported cohomology, and recall H n,p from Def. 2.1. We write H * c for total cohomology.

Theorem 3.3 (Hansen).
There exists a separated rigid analytic space E n , and a morphism w : The level n will often be clear from context, so we usually drop the subscript. Let W Q denote the (null) weight space for GL 2 /Q, that is, the rigid analytic space whose L-points For the rest of the paper, let C be the Coleman-Mazur eigencurve of tame level Γ 1 (N ′ ), for N ′ the prime-to-p part of N . Abusing notation, we call the weight map w : C → W Q .
A point y ∈ E n (resp. C) is classical if there is a Bianchi (resp. classical) eigenform F y (resp. f y ) of weight w(y) such that tF y = ψ y (t)F y (resp. tf y = ψ y (t)f y ) for all t ∈ H n,p (resp. for all classical Hecke operators t  This is based on an idea of Chenevier [Che05]. We actually require a refined version of this result, defined locally, giving more precise control over the level. Let x be a classical point in C satisfying ) be the attached Galois representation over V Q (from e.g. [BC09,§4]) where G Q = Gal(Q/Q). For a classical point y ∈ V Q , the tame level of f y depends only on [ρ V Q | I ℓ ](y) for ℓ = p, where I ℓ is the inertia subgroup at ℓ (see [Ser70]) defined using the embedding Q ֒→ Q ℓ .
Note ρ V Q | I ℓ is trivial unless ℓ|N . By considering the attached family of Weil-Deligne representations, one sees that the conductor of ρ V Q | I ℓ is locally constant at x (see [Sah17,Thm. 3.1], noting that the Weil-Deligne representation attached to the classical cuspidal point x is pure). Hence we may shrink so that the conductor at each ℓ|N -and hence the tame level -is constant over V Q . Then every classical point in V Q satisfies Conditions 2.2[Q].
The Galois representation attached to Thus if f x base-changes to level n, then so does f y for every nearby classical y ∈ C. Applying [JN19b, Thm. 3.2.1] we get the required map Remark 3.6: For clarity of argument, in the remainder of the paper, we will assume that if x ∈ C satisfies Conditions 2.
. This is always the case, for example, if the tame level of x is coprime to d. Since the proofs in the sequel are all local in nature, all of the results can be proved without this assumption by working in E n for some n|N O K and using Prop. 3.5. We shall henceforth always drop the N and n and just write BC and E.

The dimensions of irreducible components.
Proposition 3.7. Let F ∈ S λ (U 1 (n)) be a finite slope cuspidal Bianchi eigenform. There is a point can be identified as a subspace of a degree 3 overconvergent cohomology group (see [BSW19b, §9.3]); but an analysis as in [Bel12,Lem. 3.9] shows that cuspidal eigensystems do not appear in such spaces. In particular, after restricting to the generalised eigenspace at F , the map ρ 2 is surjective. Thus H * c (Y 1 (n), D λ (L)) (F ) = 0, and there exists a Hecke eigenclass with the same eigenvalues as F in this space, as required.
For our purposes, if F is critical it suffices to assume F is base-change, whence such a point x F arises in the image of BC. Proof. In the ordinary case, this is due to Hida [Hid94b]. In general, [Han17, Prop. B.1] shows that I has dimension at least 1. The following was pointed out to us by Hansen. Suppose I is a 2-dimensional irreducible component passing through x F , and let Ω = w(I). Let ρ I be the twodimensional Galois pseudocharacter ρ I over I of [JN19a]. Let I ss be the set of points y of I such that w(y) is classical non-parallel in Ω and y has small slope. This set is Zariski-dense in I. Each y ∈ I ss necessarily corresponds to a classical form by [BSW19b,Thm. 8.7], and this classical form must be Eisenstein, as classical cuspidal forms exist only at parallel weights. Hence the specialisation of ρ I at y is reducible. Reducibility is a Zariski-closed condition, so ρ I and its specialisation at x F are reducible. As F is cuspidal, its attached Galois representation is irreducible, so we get a contradiction.

Families of modular symbols
In §3, we gave results about p-adic families using the total cohomology. The p-adic L-functions of [Wil17] arise only from H 1 c , however, so in this section we refine the above to show that padic families can be realised in H 1 c through modular symbols. Unlike in classical settings, this is obstructed by the contribution of a cuspidal Bianchi eigenform F to classical cohomology in degrees 1 and 2. Counter-intuitively, a key step in overcoming this is the 'purity' result of Lem. 4.2, which implies that F appears in H i c (Y 1 (n), D Ω ) -for two-dimensional affinoids Ω -only for i = 2. This control allows us to isolate families of modular symbols (in H 1 c ) over certain curves in Ω and prove that they are free of rank one over a Hecke algebra.

Families in H
Definition 4.1. For a slope-adapted affinoid (Ω, h), let (using total cohomology). We define a local piece of the eigenvariety by The affinoids E Ω,h glue to give E. We have a bijection between eigensystems ψ x : The following is an unpublished result of David Hansen.

Lemma 4.2. (i) The spaces H
, these spaces carry actions of Σ 0 (p) exactly as in Def. 2.6 and (3.1) respectively, and we thus get attached local systems D 0 on Y 1 (n).
We first prove that H 0 , so that µ(z → b) = 0 and hence µ is zero on the constant functions. Suppose µ is zero on functions that are polynomial of degree less than r − 1. Then consider any monomial z → θ j,r+1 (z) . . = z j z r+1−j of degree r + 1. We have for all b ∈ I i , where the lower terms vanish by assumption. Taking 0 = b ∈ Z gives jµ(θ j−1,r ) + (r + 1 − j)µ(θ j,r ) = 0; and taking 0 = b ∈ √ −dZ gives jµ(θ j−1,r ) − (r + 1 − j)µ(θ j,r ) = 0. Solving, we conclude that µ(θ j−1,r ) = µ(θ j,r ) = 0, and since we can work with arbitrary 0 ≤ j ≤ r, we conclude that µ vanishes on all monomials of degree r. We obtain a short exact sequence 0 → D Ω → D Ω → D Σ → 0, where the first map is multiplication by r. By truncating the associated long exact sequence at the first degree 2 term, and localising at x, we obtain a short exact sequence By Lem. 4.2, the first term vanishes, so the second map is an isomorphism. Minimal primes of T Ω,h are in bijection with associated primes in H 2 c (Y 1 (n), D Ω ) ≤h by [Mat89, Thm. 6.5] and Lem. 4.2, so the system of eigenvalues corresponding to x is P x -torsion. Thus P λ is an associated prime of H 2 c (Y 1 (n), D Ω ) ≤h mx , i.e. it annihilates a non-zero element. Thus H 2 c (Y 1 (n), D Ω ) ≤h mx [r] is non-zero, from which we conclude.
We were unable to find a proof of this proposition that used only the short exact sequence 0 → D Σ → D Σ → D λ → 0, due to the presence of classes in both degree 1 and 2; the additional input from D Ω , via the 'purity' of Lem. 4.2, appears to be necessary to obtain sufficient control. Recall Note that smoothness of Σ is satisfied in base-change components. We give the proof after two lemmas, and thank Adel Betina, who contributed to Lem. 4.6.
Proof. The localisation H 1 c (Y 1 (n), D Σ ) ≤h mx is a finite Λ m λ -module by general facts on slope decompositions. Thus we may freely use Nakayama's lemma.
From the short exact sequence of distribution spaces given by the natural surjection sp λ : D Σ → D λ , we obtain a long exact sequence of cohomology, which (since m λ is principal by smoothness) we truncate to an exact sequence Since localising is exact, we deduce the existence of an exact sequence The last term is the generalised eigenspace corresponding to the system of eigenvalues attached to x. (At this point, we are assuming that we have extended the base field of Λ so that x is defined over Λ/m λ ). As x is non-critical, this is isomorphic to the generalised eigenspace of F for H n,p in the classical cohomology, and by assumption (4.1), this is one-dimensional. Suppose the first term is 0; then by Nakayama's lemma, we must have H 1 c (Y 1 (n), D Σ ) ≤h mx = 0, which contradicts Prop. 4.4. Hence the first term is one-dimensional and there is an isomorphism Now we use Nakayama again. A generator of H 1 c (Y 1 (n), D Σ ) ≤h mx ⊗ Λm λ Λ m λ /m λ lifts to a generator of H 1 c (Y 1 (n), D Σ ) ≤h mx over Λ m λ , which completes the proof. For any R, the map Symb Γi (D(R)) ֒→ D(R) J , Ψ → (Ψ(δ j )) j∈J is an injective R-module map. By passing to the direct sum over all i ∈ Cl K , we obtain a Λ-module embedding of H 1 c (Y 1 (n), D Σ ) into a finite direct sum of copies of D Σ . But D Σ is a torsion-free Λ-module since Λ is a domain. Hence H 1 c (Y 1 (n), D Σ ) ≤h is finite torsion-free over Λ. Thus Tor Λ i (H 1 c (Y 1 (n), D Σ ) ≤h , Λ/m λ ) = 0 for all i > 0 and λ ∈ Σ; for i = 1, this is by torsion-freeness, and for i ≥ 2, this follows by smoothness of Σ (so Λ is regular of dimension 1) and [ Proof. (Thm. 4.5). By Lems. 4.6 and 4.7, H 1 c (Y 1 (n), D Σ ) ≤h mx is free of rank 1 over Λ m λ . As is non-zero by our assumption on Σ, we must have (T Σ,h ) mx ∼ = Λ m λ . As the actions of T Σ,h and Λ on H 1 c (Y 1 (n), D Σ ) ≤h are compatible, we see H 1 c (Y 1 (n), D Σ ) ≤h mx is free of rank 1 over (T Σ,h ) mx , completing the proof of Thm. 4.5.   (Y 1 (n), D Σ ) ≤h x is free of rank one over (T Σ,h ) x , which is free of rank one over Λ λ . Thus we are in the situation of [BSDJ, Lem. 2.10] (over the rigid space Σ), so that -possibly shrinking Σ -we may choose V ⊂ Sp(T Σ,h ) such that T = O(V ) is free of rank one over Λ = O(Σ). A second application of the same lemma to the second and third equations, over the rigid space Sp(T Σ,h ), now shows that, after potentially shrinking Σ and V again, we have H 1 c (Y 1 (n), D Σ ) ≤h ⊗ T Σ,h T free of rank one over T , as required.

The parallel weight eigenvariety
We describe a 'parallel weight eigenvariety' E par ⊂ E, using H 1 c over the parallel weight line, that contains the base-change image and is better behaved than the whole space E. This bears comparison with the 'middle-degree eigenvariety' of [BH]. We show smoothness at certain classical points in the base-change image. (ii) L par is the union of the irreducible components of L that lie above W K,par , which is itself a Fredholm hypersurface;

Definition and basic properties. In
(iii) M 1 par is the coherent sheaf on L par such that for any slope ≤ h affinoid L par Σ,h lying above Σ ⊂ W K,par , we have The classical weights correspond to classical weights in W Q , so are very Zariski-dense in W K,par . Let (Σ, h) be a slope-adapted affinoid for Σ ⊂ W K,par containing a classical weight. Note Σ contains a very Zariski-dense set of classical weights such that h is a small slope, and every point in Sp(T Σ,h ) above these weights is classical by Thm. 2.10. Thus the classical points are very Zariski-dense in Sp(T Σ,h ) (cf. [Belb, Prop. II.8.6]); and gluing, the same is true in E par .

Proposition 5.2. The parallel weight eigenvariety E par is reduced.
Proof. We closely follow [Belb, Thm. II.8.8]. We first give a Zariski-dense set of y ∈ E par with reduced local rings. For a slope-adapted (Σ, h) containing a classical weight, let R = Λ and M = H 1 c (Y 1 (n), D Σ ) ≤h ; then M is finite projective over R (Lem. 4.7). Let Z ⊂ Σ = Sp(Λ) be the set of classical weights κ = (k, k) such that h < (k + 1)/2; this set is Zariski-dense. Let Y = w −1 (Z) ⊂ Sp(T Σ,h ). Then: (1) each y ∈ Y is a non-critical slope classical point, and (2) F y is either new at p|p or a regular p-stabilisation (as irregular stabilisations have slope (k + 1)f p /2, for f p the inertia index of p|p).
By (2), U p acts semisimply on H 1 c (Y 1 (n), V * κ ) (Fy) for each p|p. The operators T q for q ∤ n act semisimply, as they commute with their adjoints under the natural Petersson inner product [Hid94a, (3.4a) and before (8.2a)]. Finally, the operators v act semisimply as they have finite order. We deduce H n,p acts semisimply on H 1 c (Y 1 (n), V * κ ) (Fy) , hence on H 1 c (Y 1 (n), D κ ) (Fy) by (1). Similarly to (4.2), for each κ ∈ Z we have a Hecke-equivariant inclusion Proof. By reducedness and [JN19b, Thm. 3.2.1], it suffices to check an inclusion of a very Zariskidense set of points. As every classical point x ∈ E par corresponds to a system of eigenvalues that appears in H 1 c (Y 1 (n), D λ ) for some λ ∈ W K,par , the conditions of the theorem are satisfied, giving the required closed immersion. , there is a closed immersion C K ֒→ E par , and the map BC is the composition C → C K ֒→ E par . It suffices to show that BC ′ is locally an isomorphism at x f .

The base-change eigenvariety and smoothness. By
Since f is non-critical, after localising and base-extending Λ, by [Bel12] we know that O(C Σ,h ) mx is free of rank one over Λ m λ . Since O(C K Σ,h ) m BC ′ (x) is a Λ m λ -subalgebra containing 1, it must be isomorphic to O(C Σ,h ) mx , and BC ′ is locally an isomorphism at x f . As C is smooth at x f (see [Bel12,Thm. 2.16]), we deduce that C K is smooth at BC ′ (x f ), as required.
We now consider the analogue of Prop. 5.4 when f is critical. We need an additional mild hypothesis, following [Bel12, §1.4]: Definition 5.5. We say f is decent if f is non-critical, or f has vanishing adjoint Selmer group  [All16]. Now suppose f is critical and decent. Let x . .= BC(x f ) and denote by t x the tangent space of E bc at x. As E bc is a curve, we know dim L t x ≥ 1; so to prove E bc is smooth at x, it suffices to show dim L t x ≤ 1. We use deformations of Galois representations, adapting [Bel12, Thm. 2.16].
Notation 5.6: Let S be the union of the infinite place with the set of places of Q supporting N , and S K the set of places of K lying over S. We let G Q,S and G K,SK be the Galois groups of the maximal algebraic extension of Q (resp. K) ramified only at the places S (resp. S K ).
Note ρ f factors through G Q,S ; from now on we consider ρ f as defined on G Q,S . Let ρ x = ρ f | GK,S K , the Galois representation attached to x. Throughout, we use decomposition groups and complex conjugation c ∈ G Q,S defined by the choices of embeddings from §2. Likewise, I q ⊂ G Kq denotes an inertia subgroup; similarly, we use I q over Q.
Definition 5.7. Let A L denote the category of Artinian local L-algebras A with residue field L, and for each A ∈ A L , let X ref (A) be the set of deformations (under strict equivalence) ρ A of ρ x to A satisfying the following.
(i) If q is a prime of K dividing n but coprime to p, then ρ A | Iq is constant.
(ii) For each p | p in K, we have: (1) (null weights) for each embedding τ : K p ֒→ L, one of the τ -Hodge-Sen-Tate weights of ρ A | GK p is 0; (2) (crystalline periods/weakly refined) there exists α p ∈ A such that the K p ⊗ Qp A-module We can evaluate ρ f at complex conjugation c, and note that the operation is a functorial involution on X ref . We thank Carl Wang-Erickson for explaining the utility of this involution, and for supplying the appendix that proves the following.
Lemma 5.9. There exists a neighbourhood V of x in E bc and a Galois representation ρ V : G K,SK → GL 2 (O(V )) such that for each classical point y ∈ V , the specialisation ρ V,y of ρ V at y is the Galois representation attached to y.
Proof. By a theorem of Rouquier and Nyssen (see [Rou96] or [Nys96]), one obtains such a representation from the Galois pseudorepresentation on E bc ⊂ E constructed in [JN19a]. One can check  f (Q, ad ρ f ) = 0, he bounds the dimension of the Zariski tangent space of D, which he denotes t D , by 1. We will show there exists an isomorphism t D ∼ = t ref,bc . Indeed, by ignoring all the deformation conditions, we can view t ref,bc as a subspace of the tangent space without conditions, which we identify with H 1 (K, ad ρ x ). Using condition (iii) and Prop. 5.8 it is moreover a subspace of H 1 (K, ad ρ x ) ι ∼ = H 1 (Q, ad ρ f ).
As dim L t D ≤ 1, the result then follows from the following claim.

The Σ-smoothness condition.
We would like to conclude that E par is smooth at basechange points (or twists thereof). However, there might exist other irreducible components of E par , not contained in E bc , that meet E bc at such points.
Definition 5.12. A point x ∈ E bc is Σ-smooth if every irreducible component I ⊂ E par through x is contained in E bc (equivalently, if the natural inclusion E bc ⊂ E par is locally an isomorphism at x).
For decent f satisfying Conditions 2.2[Q] with base-change F , by Props. 5.4 and 5.10 E par is smooth at x F if and only if x F is Σ-smooth. If F is non-critical then x F is Σ-smooth by Cor. 4.8.
We conjecture that every decent classical base-change point is Σ-smooth. At non-critical points, this holds by Cor. 4.8. In general, this is implied by the following generalisation of a conjecture of Calegari-Mazur [CM09, Conj. 1.3].
Recall if y is a classical point of the Bianchi (resp. Coleman-Mazur) eigenvariety, then F y (resp. f y ) is the corresponding modular form. If y is such a point, and ϕ is a finite order Hecke character of K (resp. Q), write y ⊗ ϕ for the classical point (in the relevant eigenvariety) attached to the modular form F y ⊗ ϕ (resp. f y ⊗ ϕ). Note y ⊗ ϕ might appear in an eigenvariety of different tame level to y. Calegari and Mazur conjecture that every ordinary component of E par is either twisted basechange (as in Conj. 5.13) or is CM (so is transfer from a GL 1 -eigenvariety). Non-ordinary CM components do not exist (by slope considerations). This is a Bianchi version of a folklore conjecture, which says that automorphic representations vary in p-adic families with a Zariski-dense set of classical points if and only if they satisfy a self-duality condition [APS08], [Urb11,Intro.].
Proposition 5.14. Conj. 5.13 implies that every classical base-change point Proof. Let I ⊂ E par be an irreducible component through x F . We must prove that I is contained in E bc . If I is ordinary, then x F is ordinary and hence small slope; so E par is étale over Σ by Thm. 4.5, hence smooth, and x F is Σ-smooth.
Suppose I is non-ordinary. By the conjecture, there exists some M , an irreducible component J ⊂ C M and a Hecke character ϕ of K such that I = BC(J ) ⊗ ϕ. Thus there exists some classical modular form g such that BC(x g ) ⊗ ϕ = BC(x f ), and we have an equality of Galois representations ρ f | GK = ρ g | GK ⊗ ϕ, identifying ϕ with its associated Galois character (via class field theory). Since ρ f | GK and ρ g | GK both admit extensions to G Q , so does ϕ, and it follows that there exists a Dirichlet character ϕ Q such that ϕ = ϕ Q • N K/Q . Further, perhaps after multiplying ϕ Q by the quadratic character χ K/Q attached to K/Q, we see that Let y ∈ I be a classical point. By the conjecture, y = BC(z) ⊗ ϕ for some classical z ∈ J . We see ρ fz | GK ⊗ ϕ = (ρ fz ⊗ ϕ Q )| GK . By the same argument as Proposition 3.5, for z in a neighbourhood of x g in C M we have ρ fz ⊗ ϕ Q unramified outside N , so z ⊗ ϕ Q appears in C. For such z, we have y = BC(z) ⊗ ϕ = BC(z ⊗ ϕ Q ) ∈ E bc . The set of such y is Zariski-dense in I, as it accumulates at x F . Thus a Zariski-dense set of points in I appear in E bc , so I red ⊂ E bc by e.g. [JN19b, Thm. 3.2.1]. But by Prop. 5.2, I = I red is reduced, and we conclude.

Three-variable and critical p-adic L-functions
Throughout §6, let f ∈ S k+2 (Γ 1 (N )) be a decent eigenform satisfying Conditions 2.2[Q], and let F ∈ S λ (U 1 (n)) be its base-change. Then F satisfies Conditions 2.2[K], and by the previous section it varies in a family V = Sp T ⊂ E bc ⊂ E par over Σ = Sp Λ ⊂ W K,par . If F is critical, suppose it is Σ-smooth in the sense of Def. 5.12, whence E par is smooth at x F by Prop. 5.10. We will write x . . = x F ∈ V for the point corresponding to F , with associated maximal ideal m x ⊂ T , and let e be the ramification degree of V → Σ at x. We also write y for a general point of V , with associated maximal ideal m y ⊂ T ; if y is classical we write F y for the associated Bianchi modular form.
By Σ-smoothness, V is the unique irreducible component of Sp(T Σ,h ) ⊂ E par through x. Possibly shrinking Σ, we may take V connected and smooth, so there exists an idempotent ε on T Σ,h such that T = εT Σ,h ⊂ T Σ,h is a summand, and

Three-variable p-adic L-functions.
Recall the Mellin transform Mel : H 1 c (Y 1 (n), D Σ ) ≤h → D(Cl K (p ∞ ), Λ) is valued in a space of three-variable analytic functions -two variables coming from functions on Cl K (p ∞ ), and one variable on Σ. When F is critical, we expect that V → Σ is not étale, and we cannot identify V and Σ; then the p-adic L-function should be an element of D(Cl K (p ∞ ), T ), not D(Cl K (p ∞ ), Λ). Following Bellaïche, we define a Mellin transform over V , rather than Σ. For this, we consider the space H 1 c (Y 1 (n), D Σ ) ≤h ⊗ Λ T, which has natural T -structures on each factor (with T acting on H 1 c (Y 1 (n), D Σ ) ≤h via T ⊂ T Σ,h ). The two T -structures are not the same in general.
Definition 6.1. Let y ∈ V (L) and κ = w(y). Define This map ('specialisation in the second factor') is equivariant for the action of the Hecke operators on the cohomology in both the target and source (i.e. for the first T -structure on the source). It is not in general equivariant if we equip the source with the second T -structure and the target with the natural Hecke action.
Similarly, we can define a specialisation map at the level of distributions.
Definition 6.2. With y as above, define sp y to be the map where the last isomorphism is [Han17, Prop. 2.2.1].
Define the Mellin transform over V to be the map Mel V . .= Mel ⊗ id in the top row of (6.1). From the definitions, we see the following diagram commutes: (6.1) Recall the p-adic L-function at y is defined as the Mellin transform of a class in the generalised eigenspace H 1 c (Y 1 (n), D κ (L)) my . To use (6.1), we would like to find a class Ψ ∈ H 1 c (Y 1 (n), D Σ ) ≤h ⊗ Λ T such that sp y,2 (Ψ) lies in this generalised eigenspace. We have a criterion for this: (that is, the first and second T -structures agree on Ψ). Then sp y,2 (Ψ) ∈ H 1 c (Y 1 (n), D κ (L)) my is a Hecke eigenclass for all y ∈ V (L).
We say an affinoid Σ = Sp(Λ) ⊂ W K,par is nice if Λ is a principal ideal domain. Every classical weight has a basis of nice affinoid neighbourhoods (see after [Bel12,Defn. 3.5]); so we may shrink Σ and V so that Σ is nice. We now remove the assumption that F is non-critical in Thm. 4.5 and Cor. 4.8. H 1 c (Y 1 (n), D Σ ) ≤h ⊗ T Σ,h T is free of rank one over T . Proof. To prove part (i) we use [Bel12,Lem. 4.1]. This says that if R and T are discrete valuation rings, with T a finite free R-algebra and M a finitely generated T -module that is free as an R-module, then M is finite free over T .
As Λ is a PID, and the module H 1 c (Y 1 (n), D Σ ) ≤h is finite over Λ (by general properties of slope decompositions) and torsion-free (by Lem. 4.7), it is finite free over Λ. It follows that T Σ,h is also finite and torsion-free over Λ, and hence also a finite free Λ-module. Since W K,par and E par are rigid curves that are smooth and reduced (Prop. 5.2) at λ and x, the local rings Λ m λ and (T Σ,h ) mx are discrete valuation rings. We conclude (i) by Bellaïche's lemma.
The argument from Cor. 4.8 shows that we can shrink Σ and V = Sp T so H 1 c (Y 1 (n), D Σ ) ≤h ⊗ T Σ,h T is free of finite rank over T . This rank is preserved by localising at any point of V . Let y ∈ V (L) be a non-critical classical cuspidal point satisfying (4.1); such points are Zariski-dense. By Thm. 4.5, H 1 c (Y 1 (n), D Σ ) ≤h my is free of rank one over T my , completing the proof.
As V is smooth and reduced at x, the extension T mx /Λ m λ is a finite extension of DVRs, so T mx = Λ m λ [X]/(X e − u) for a uniformiser u ∈ Λ m λ (recalling e is the ramification degree of w at x). Possibly shrinking, this lifts to T = Λ[X]/(X e − u).
Note Ψ V depends on the choices made only up to multiplication by an element of T × (in either T -structure). We now formally have (compare [Bel12,Prop. 4.14]): Proof. This follows directly from Lems. 6.3 and 6.6.
For y as in Prop. 6.7, if Ψ Fy is the overconvergent modular symbol attached to F y in §2.3, then sp y,2 (Ψ V ) = c y Ψ Fy for some c y ∈ L × . By (6.1) we see that sp y (L p (V )) = Mel(sp y,2 (Ψ V )) = c y L p (F y ). When combined with Thm. 2.12, this proves Thm. A from the introduction. To get the precise statement of Thm. A: -for twists by a finite order Hecke character φ, -and finally, we obtain the function on V Q × X (Cl K (p ∞ )) by pulling back under BC : For h ∈ (Q ≥0 ) p|p , recall the notion of µ ∈ D(Cl K (p ∞ ), L) being h-admissible from [Wil17, Defs. 5.10,6.14, §7.4]. These definitions generalise directly to µ ∈ D(Cl K (p ∞ ), T ) using the Banach algebra structure on T . Recall α p is the U p -eigenvalue of F . Slopes at p are locally constant over V , so we may shrink so that the slope is constant over V . Then identically to [Wil17] we have: Proposition 6.9. Let h p = v p (α p ) and h = (h p ) p|p . Then L p (V ) is h-admissible. Remark 6.10: Our construction of L p (V ) required F to be Σ-smooth (which is conjecturally always the case). It seems likely that this is a necessary condition to carry out the construction of L p (V ) via overconvergent cohomology. There are two key inputs: the class Ψ V and the Mellin transform Mel V . If F is not Σ-smooth, then over V we have the Mellin transform but we have little control on the geometry of E par at F and the structure of H 1 c (Y 1 (n), D Σ ) mx , so it is hard to construct the class Ψ V . One could instead work over the (unique, smooth) base-change component V bc = Sp(T bc ) ⊂ V through F . Using smoothness, one can exhibit a canonical quotient M V bc of H 1 c (Y 1 (n), D Σ ) ⊗ T Σ,h T that is free of rank one over T bc , and construct a 'good' class Ψ V bc in this quotient. However, the Mellin transform does not descend to this quotient. Thus Σ-smoothness appears necessary to simultaneously obtain both Ψ V and Mel V . Remark 6.11: More generally, suppose we do not assume that F ∈ S λ (U 1 (n)) is base-change. Suppose F satisfies Conditions 2.2[K] and is non-critical. By Prop. 3.7 and Thm. 3.8, there exists at least one (not necessarily parallel) curve Σ = Sp(Λ) ⊂ W K such that F varies in a family V = Sp(T ) over Σ. If Σ is smooth at λ, then Cor. 4.8 says V → Σ is étale at x F and H 1 c (Y 1 (n), D Σ ) ≤h ⊗ T Σ,h T is free of rank one over Λ. Let Ψ V be a generator. The methods of this section show L p (V ) . . = Mel V (Ψ V ⊗ 1) ∈ D(Cl K (p ∞ ), T ) interpolates the p-adic L-functions of all classical non-critical y ∈ V . If V contains a Zariski-dense set of classical points, then Σ is parallel (as it then contains a Zariski-dense set of parallel weights; see also [Ser,Thm. 1.1]), hence smooth. This construction thus gives a three-variable p-adic L-function over any classical family through F .
In this generality, classical y are not always Zariski-dense in V , but one might expect that L p (V ) still carries interesting information about the arithmetic of F . For example, if one could prove a functional equation for L p (V ), then it might be possible use L p (V ) to prove [BSW19a,Conj. 11.2], giving an arithmetic description of the L-invariants attached to F (similar to [GS93,BSDJ]).
To prove the interpolation property for L p (F ) from Thm. B of the introduction, we give another description of L p (F ) using a strategy of Bellaïche.
Proof. The proof closely follows the strategy of Bellaïche; each step is proved identically to the referenced result. Firstly, there is an isomorphism and we again write m x for the maximal ideal of this space corresponding to x (see [Bel12,Cor. 4.4]).
After possibly enlarging L, there exists a uniformiser u of Λ m λ and an isomorphism of Λ m λalgebras that sends X to a uniformiser of (T Σ,h ) mx , where e is the ramification index of the weight map w : E par → W K,par at x (see [Bel12,Prop. 4.6]). This is enough to show that the generalised eigenspace H 1 c (Y 1 (n), D λ (L)) (F ) = H 1 c (Y 1 (n), D λ (L)) mx has dimension e over L and is free of rank one over the algebra If e > 1, then X acts nilpotently on this eigenspace, hence its image under ρ λ , which is thus 0. If e = 1 and ρ λ = 0, then ρ λ is a non-trivial map between 1-dimensional vector spaces, so must be an isomorphism, contradicting F being critical; so ρ λ = 0.
Theorem 6.14. The p-adic L-function L p (F ) is (v p (α p )) p|p -admissible. For Hecke characters ϕ of K of conductor f|(p ∞ ) and infinity type 0 ≤ (q, r) ≤ (k, k), we have Proof. Firstly, L p (F ) is admissible by specialising Prop. 6.9 at x. By Thm. 6.13 the eigenspace H 1 c (Y 1 (n), D λ )[F ] is one-dimensional; let Ψ F be a generator. Now Mel(Ψ F )(ϕ p−fin ) = 0 for all ϕ as in (6.3), as evaluation of Mel(Ψ F ) at ϕ p−fin depends only on ρ λ (Ψ F ) [Wil17, §7.6], and ρ λ (Ψ F ) = 0. Thus it suffices to prove that L p (F ) and Mel(Ψ F ) are equal up to rescaling. Recall H 1 c (Y 1 (n), D Σ ) ≤h ⊗ T Σ,h T is free of rank one over T . Let Ψ V be as in Def. 6.5. Identically * to [Bel12,Prop. 4.14], at x (up to rescaling) we have sp x,2 (Ψ V ) = Ψ F . Finally specialisation is compatible with Mellin transforms by (6.1), so By construction, L p (V ) is well-defined only up to the choice of Φ V in Def. 6.5, corresponding to changing the p-adic periods {c y }. Specialising, we see L p (F ) is only well-defined up to scalar multiple. However, this scalar indeterminacy is expected, arising from scaling the periods of F .
Unlike in the non-critical slope case, the admissibility and interpolation properties are not sufficient to determine L p (F ) uniquely in D(Cl K (p ∞ ), L)/L × . However, up to scaling L p (F ) is uniquely determined by interpolation over V , and does not depend further on the method of construction: Proposition 6.15. Suppose L p (F ) = 0. Let L ′ p : V ×X (Cl K (p ∞ )) → L be analytic and (v p (α p )) p|padmissible satisfying the interpolation property of Thm. A, with possibly different constants c ′ y . Then L ′ p (x) = c · L p (F ) for some c ∈ L. * There is a typo op. cit.; from the context, Symb ± Proof. Let L ′ p (F ) . .= L ′ p (x). If L ′ p (F ) = 0 we take c = 0; so assume L ′ p (F ) = 0. Let To see this is well-defined, note there exists a Zariski-dense set V ss ⊂ V of classical small slope points y in V of weight (k, k), where k > 2. For any finite order Hecke character ϕ of conductor p r > 1, the quantity L(F y , ϕ, k + 1) converges absolutely to a non-zero number; it follows that L p (F y , (ϕ| · | k ) p−fin ) = 0, since the p-adic L-function does not have an exceptional zero there. As every connected component of O(X (Cl K (p ∞ ))) contains a character of the form (ϕ| · | k ) p−fin , it follows that L p (F y ) is not a zero-divisor in O(X (Cl K (p ∞ ))) (as on every such component the only zero-divisor is 0). Now let D ∈ D(Cl K (p ∞ ), T ) such that DL p (V ) = 0. At any y ∈ V ss , we have sp y (D)sp y (L p (V )) = 0, and as sp y (L p (V )) = c y L p (F y ) is not a zero-divisor we see sp y (D) = 0.
As D vanishes at a Zariski-dense set of points, we have D = 0, so L p (V ) is not a zero-divisor and C(y, φ) is well-defined. At each classical y ∈ V ss , the p-adic L-function L p (F y ) is uniquely determined up to scalar multiple by its interpolation and admissibility properties, as in [Wil17,Thm. 7.4]. Thus L p (y) and L ′ p (y) are scalar multiples, so C(y, −) ∈ L. As such points are Zariski-dense, for all z ∈ V we see C(z, −) is constant, that is, C ∈ Frac(O(V )). Since (by assumption) neither L p (F ) nor L ′ p (F ) is zero, C does not have a zero or pole at x. Hence we may shrink V further so that C has no zeros or poles, that is, C ∈ O(V ) × . Specialising to x, we see that L p (F ) and L ′ p (F ) differ by scalar multiplication by c . .= C(x) ∈ L × , as required.
Remark 6.16: This proof shows there is a subspace W F ⊂ D(Cl K (p ∞ ), L), of dimension ≤ 1, such that for any L p : V × X (Cl K (p ∞ )) → L satisfying the interpolation property of Thm. A, we have L p (x) ∈ W F . Further, up to shrinking V any two such functions L p , L ′ p with L p (x), L ′ p (x) = 0 differ by O(V ) × , as claimed in the introduction (also cf. Prop. 7.9). We can also treat twisted p-adic L-functions L φ p by using twisted Mellin transforms Mel φ (Rem. 2.13).

Factorisation of base-change p-adic L-functions
Let f be a classical eigenform with base-change F to K, and let χ K/Q denote the quadratic Hecke character attached to K/Q. Artin formalism says that for any rational Hecke character ϕ, we have We now prove an analogue of this for p-adic L-functions (see Thm. 7.5).
7.1. p-adic L-functions attached to classical eigenforms. Let f ∈ S k+2 (Γ 1 (N )) be a decent eigenform satisfying Conditions 2.2[Q], and let Λ(f, ϕ) be its L-function, normalised to include the Euler factors at infinity. Here ϕ ranges over Hecke characters of Q. Denote the eigenvalue of f at p by α p (f ) and the periods of f by Ω ± f ∈ C × , which are well-defined up to Q × . Let h . .= v p (α p (f )).
For any Dirichlet character χ of conductor M , let τ (χ) . . = a (mod M) χ(a)e 2πia/M be its Gauss sum. Let η = η p η p be a Dirichlet character, where η p has conductor p t and η p has conductor C prime to p. The following is due to many people.
Theorem 7.1. There exists a locally analytic distribution L η p (f ) on Z × p such that, for any Hecke character ϕ = χ| · | j , where χ is finite order of conductor p n > p t and 0 ≤ j ≤ k, we have The sign of Ω ± f is given by χη(−1)(−1) j = ±1. The distribution is admissible of order h, and if h < k + 1, it is uniquely determined by this interpolation property.
If η is the trivial character, we just write L p (f ) for this distribution.
where L anti p (F ) is the restriction of L p (F ) to the anticyclotomic line (see e.g. [Geh19] for this result in the ordinary case). Suppose there exists such a two-variable function L anti p (V Q ), over a neighbourhood V Q in C, interpolating the anticyclotomic p-adic L-functions at classical weights. Then the methods of this section show that, under an analogous non-vanishing condition, and up to multiplication by an element of O(V Q ) × , we have an equality of two-variable distributions L anti p (V Q ) 2 = L anti p (V ). (7.6) If h ≥ k+1 2 , we can obtain the identity (7.5) for F by specialising (7.6) at F .

Appendix: A base-change deformation functor
by Carl Wang-Erickson † The point of this appendix is to supply the proof of Prop. 5.8, regarding deformations of Galois representations. The main idea we will apply here applies under the following running assumptions: (A) there is an index 2 subgroup H ⊂ G and a chosen element c ∈ G H of order 2. Equivalently, G is expressed as a semi-direct product H ⋊ c ; (B) char(L) = 2, for L the base coefficient field of the deformed representation.
In the first section we set up the theory of the base change deformation functor. In the second section, we verify that this theory is compatible with arithmetic conditions imposed when G is a Galois group over Q.
A.1. The base change deformation functor. We work under assumptions (A)-(B) above. Let ρ : G → GL d (L) be a representation that is absolutely irreducible after restriction to H. Let A L be the category of Artinian local L-algebras (A, m A ) with residue field L. We denote by X the deformation functor for ρ| H . This is the functor from A L to the category of sets given by where ∼ is the equivalence relation of "strict equivalence," that is, conjugation by 1 + M d (m A ) ⊂ GL d (A). We will let ρ A ∈ X(A) denote a deformation of ρ| H with coefficients in A. This is in contrast to the notationρ A , which we reserve for a lift of ρ| H to A, i.e. a homomorphismρ A ∈ ρ A as in (A.1). Let X bc denote the subfunctor of X cut out by the condition that some (equivalently, all) ρ A ∈ ρ A admits an extension to a homomorphismρ G A : G → GL d (A) such thatρ G A | H =ρ A . In this case, we say that ρ A admits an extension to an A-valued deformation ρ G A of ρ. For h ∈ H, we write h c := chc ∈ H for twisting by c. Likewise, for a group homomorphism η with domain H, let η c (h) := η(h c ). To justify this claim, we calculate that for any x ∈ GL d (A), ι(ad x ·ρ A ) = ad ρ(c) · ad x ·ρ c A = ad y · (ι(ρ A )), where y = ad ρ(c) · x. Let X ι denote the ι-fixed subfunctor of X, and let t (resp. t bc ) denote the tangent space X(L[ε]/(ε 2 )) (resp. X bc (L[ε]/(ε 2 )). (iii) There is a canonical injection t bc ֒→ t of tangent spaces. The image of this injection is the subspace t ι ⊂ t fixed by the involution ι * : t → t induced by ι.
Proof. Part (i) follows directly from Lem. A.1. For Part (ii), it is well-known that X is pro-representable; see e.g. [Maz89]. It is a brief exercise that a homomorphism R → A kills (1 − ι * )(R) if and only if the corresponding deformation of ρ| H is ι-fixed. Then the pro-representability of X bc by R bc follows from (i).
Part (iii) follows from Part (ii) and the perfect L-linear duality of m R /m 2 R and X(L[ε]/(ε 2 )).
A.2. Galois-theoretic conditions. Let G = G Q,S and H = G K,SK (see Not. 5.6). We also use the decomposition groups and complex conjugation c ∈ G given in (5.1). The data (G, H, c) satisfy assumption (A), as K/Q is imaginary quadratic. Because the level of the modular form f of Prop. 5.10 is supported by S, and because p, ∞ ∈ S, the representation ρ f of the absolute Galois group of Q factors through G Q,S . We let ρ . .= ρ f : G → GL 2 (L), as in Def. 5.5, with its critical refinement with eigenvalue α p . It is an L-linear representation, where L is a p-adic field; thus we have satisfied assumption (B).
Deformation theory as in §A.1 can be carried out for continuous representations of G and H, using the p-adic topology of L, and the arguments therein make good sense in this setting. This is standard; see e.g. [Kis03,§9]. From now on, we impose continuity without further comment.
Because (2) there exists α p ∈ A such that the A-module D crys (ρ G A | Gp ) ϕ= αp is free of rank 1 and ( α p mod m A ) = α p .
Proof. It is a straightforward exercise about representations and the corresponding Frobenius isocrystals to verify that the statements of (i)-(ii) of Lem. A.4 are equivalent to (i)-(ii) of Def. 5.7 under both extension and restriction.