$P$-adic $L$-functions of Bianchi modular forms

The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the $p$-adic $L$-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular symbol attached to a Bianchi modular form and critical values of its $L$-function, which then allows us to construct $p$-adic $L$-functions of Bianchi modular forms.


Introduction
In recent years, p-adic L-functions have been a subject of considerable study. As an example, the 'main conjecture' of Iwasawa theory says that p-adic L-functions control the size of cohomology groups of Galois representations. They have also been used in the construction of Stark-Heegner points, which are conjecturally global points on non-CM elliptic curves over number fields. Via the theory of overconvergent modular symbols, Pollack and Stevens (see [19] and [20], or for an exposition, [18]) gave a beautiful and computationally effective method of constructing the p-adic L-function of a suitable rational modular form. They defined the space of overconvergent modular symbols to be a space of modular symbols taking values in some p-adic distribution space D k (L), exhibiting a surjective Hecke-equivariant specialisation map from this space to the space of classical modular symbols of weight k; the crux of their work is Stevens' control theorem, which says that this map is an isomorphism on the small slope eigenspaces of the U p operator. This draws comparison with the work of Coleman on overconvergent modular forms, and in particular his result that a small slope overconvergent modular form is in fact classical.
In this paper, we give an analogue of their theory for the case of Bianchi modular forms, that is, the case of modular forms over imaginary quadratic fields. The theory of modular symbols over the rationals generalises to arbitrary number fields; however, the definitions can be made far more explicit over Q or imaginary quadratic fields, motivating the study of the imaginary quadratic case as a stand-alone topic. We give a brief description of the completely general case as motivation for the sequel. To this end, let F be a number field with adele ring A F (with finite adeles A f F ) and degree d F = r + 2s, where r (respectively, s) is the number of real (respectively, pairs of complex) embeddings of F . Let Ω be an open compact subgroup of GL 2 (A f F ), which takes the (adelic) role of a congruence subgroup. The analogue of the modular curve is then the locally symmetric space where K ∞ is the standard maximal compact subgroup of GL 2 (F ⊗ Q R). Let q = r + s. For a suitable local system V on Y Ω , we have a canonical link between spaces of automorphic forms of fixed weight and level Ω and elements of the cohomology group H q c (Y Ω , V). We call this latter space the space of modular symbols of level Ω and weight V. When det(Ω) = (O K ⊗ Z) × , and via a result called strong approximation, Y Ω breaks down into components indexed by the class group of F ; explicitly, we choose representatives for the class group, which then allows us to define a collection (Γ α ) α∈Cl(F ) of discrete subgroups of SL 2 (F ), and a collection of spaces . We see that Y Ω = α∈Cl(F ) Y α , and this leads to a decomposition In the Bianchi case, where q = 1, we can study the geometry of H 3 to find that each of these components is isomorphic to a space with a much simpler description, namely the natural analogue of that used in [19], where they define modular symbols to be functions on divisors on cusps into some polynomial space. This isomorphism-described in [1]-only exists for q = 1, showing that this more explicit description of modular symbols only applies in the rational and imaginary quadratic cases. In this paper, we give a complete account of this explicit description in the latter setting. The literature regarding Bianchi modular forms is widespread; in particular, an account of the general theory over arbitrary number fields is given in Weil's book [26], whilst accounts in the imaginary quadratic case for weight 2 are given by Cremona and two of his students in [2], [4] and [5]. However, there seem to be a large number of different conventions pertaining to the theory surrounding these objects, and as such, we start with a brief introduction to the theory-largely following the account of Ghate in [7]-fixing as we do so the conventions that we will use in the sequel. Broadly speaking, a Bianchi modular form is an adelic automorphic form for GL 2 over an imaginary quadratic field K; such a form corresponds to a collection of automorphic functions F 1 , . . . , F h : GL 2 (C) → V 2k+2 (C), where h is the class number of K and V k (C) is the space of homogeneous polynomials in two variables of degree k over C, and then further to functions F 1 , . . . , F h on the upper half-space H 3 := C × R >0 , the analogue of the upper half-plane in this setting. Such a Bianchi modular form has a Fourier expansion, and accordingly an L-function that converges absolutely on some right half-plane. As in the case of classical modular forms, one can give a formula for the L-function in terms of integrals of the automorphic form over certain paths in H 3 . In Theorem 1.8, we prove a precise version of: Theorem. Let Φ be a cuspidal Bianchi modular form of weight (k, k) and level Ω 1 (n). Let ψ be a Grossencharacter with a prescribed infinity type. Then there is an integral formula for L(Φ, ψ, s) for s ∈ C.
The heart of the paper is in the study of Bianchi modular symbols. To a cuspidal Bianchi modular form Φ of weight (k, k) and level Ω 1 (n), where (p)|n, we associate a collection F 1 , . . . , F h of functions on H 3 , as above, each satisfying an automorphy condition for some discrete subgroup Γ i of SL 2 (K). To each of these F i , we associate a classical V k (C) ⊗ C V k (C)valued modular symbol φ F i for Γ i , and exhibit a link between values of this symbol and critical values of the part of the L-function corresponding to F i . In particular, we have the following (see Theorem 2.11 for a precise statement): Theorem. Let Φ be a cuspidal Bianchi modular form of parallel weight (k, k) and level Ω 1 (n) with renormalised L-function Λ(Φ, ψ). For a Grossencharacter ψ of K of conductor f and infinity type 0 (q, r) (k, k), we can express Λ(Φ, ψ) explicitly in terms of the coefficients of φ F i ({α} − {∞}), for 1 i h and α ranging over a set of representatives of f −1 /O K with αf coprime to f. Here the coefficients are taken in a suitable basis of V k (C) ⊗ C V k (C).
We then define the space of overconvergent Bianchi modular symbols to be the space of modular symbols taking values in some p-adic distribution space; precisely, we fix a finite extension L/Q p , and denoting by A k (L) the space of rigid analytic functions on the unit disc defined over L, our distribution space is D k,k (L) := Hom(A k (L)⊗ L A k (L), L). After renormalising the values of φ F i , we may consider it to have values in V k (L) ⊗ L V k (L), and we then have a specialisation map from overconvergent to classical modular symbols by dualising the inclusion V k (L) ⊗ L V k (L) → A k (L)⊗ L A k (L), much like in the rational case. We prove the following analogue of Stevens' control theorem; in the case of p inert, it is proved in Corollary 4.3, in the case of p ramified it is proved in Corollary 4.3 combined with Lemma 6.9, and in the case p split it is proved in Theorem 6.10.
Theorem (The control theorem for Bianchi modular symbols). Let p be a rational prime with pO K = p|p p ep . For each prime p|p, let λ p ∈ L. Then, when v p (λ p ) < (k + 1)/e p for all p|p, the restriction of the map to the simultaneous λ p -eigenspaces of the U p operators is an isomorphism.
The proof draws from work of Greenberg in [8], in that we define a series of finite approximation modules, and lift compatibly through this system to obtain a overconvergent symbol from a classical one.
It is worth remarking that whilst in the rational case, the control theorem gives an analogue of Coleman's small slope classicality theorem, no such theory of 'overconvergent Bianchi modular forms' yet exists.
In the remainder, the values of an overconvergent eigenlift Ψ F i are studied; namely, we prove that such a symbol takes values in some space of locally analytic distributions, and that it is admissible (or tempered).
Suppose we are in the setup of the control theorem, and let Φ be a small slope cuspidal Bianchi eigenform of weight (k, k) and level Ω 1 (n). Then to Φ we can associate a small slope eigensymbol (φ 1 , . . . , φ h ) in a direct sum of symbol spaces, which we can lift uniquely to an overconvergent symbol (Ψ 1 , . . . , Ψ h ) using the control theorem. Then there is a way of patching together the distributions Ψ i ({0} − {∞}) to a locally analytic distribution μ p on the ray class group Cl(K, p ∞ ). We then have the following (see Theorem 7.4): Theorem. Let Φ be a cuspidal Bianchi eigenform of weight (k, k) (for k 0) and level Ω 1 (n), where (p)|n, with U p -eigenvalues a p , where v(a p ) < (k + 1)/e p for all p|p. Then there exists a locally analytic distribution μ p on Cl(K, p ∞ ) such that for any Grossencharacter of conductor f|(p ∞ ) and infinity type 0 (q, r) (k, k), we have where ψ p-fin is a character on Cl(K, p ∞ ) corresponding to ψ, ( * ) is an explicit factor depending on ψ and given in Theorem 7.4 , and hence is unique.
We call μ p the p-adic L-function of Φ. Such a result is the natural analogue of the results of Pollack and Stevens in the rational case.
Comparison to relevant literature. There are a number of people who have worked on similar things in the recent past. Perhaps of most relevance is Trifković, who in [25] performed computations with overconvergent Bianchi modular symbols. He proved a lifting theorem in the case of weight 2 ordinary eigenforms over an imaginary quadratic field of class number 1, using similar explicit methods to [8]. The lifting results in this paper are a significant generalisation of his theorem, though the author has not made any efforts to repeat the computational aspects of Trifković's work in this more general setting. Looking in a different direction, in his PhD thesis [21] Barrera Salazar has generalised the results of Pollack and Stevens to the case of Hilbert modular forms (that is, to modular symbols over totally real fields), though the methods he uses are vastly different to those in the Bianchi case, as such symbols live in higher compactly supported cohomology groups and do not lend themselves to such explicit study.
It is also worth mentioning the work of Haran. In [9], he constructed p-adic L-functions for weight 2 ordinary eigenforms over any number field of class number 1 (in particular, not just over imaginary quadratic fields).

Bianchi modular forms
A Bianchi modular form is an automorphic form for GL 2 over an imaginary quadratic field. Here, we give only a very brief description of the theory; more detailed descriptions are given by Bygott in [2] (focusing on weight 2) and Ghate in [7] (for higher weights). The basic definitions are given in Section 1.1, whilst Section 1.2 looks at the L-function of a Bianchi modular form, giving an integral formula for the twisted L-function.
Notation 1. Throughout this paper, we will take K to be an imaginary quadratic field with ring of integers O K , different D and discriminant −D, p a rational prime, n an ideal of O K divisible by (p). At each prime p of K, denote by K p the completion of K with respect to p and O p the ring of integers of K p . Denote the adele ring of K by where K ∞ are the infinite adeles and A f K are the finite adeles. Furthermore, define O K := O K ⊗ Z Z to be the finite integral adeles. Denote the class group of K by Cl(K) and the class number of K by h, and-once and for all-fix a set of representatives I 1 , . . . , I h for Cl(K), with I 1 = O K and each I j for 2 j h integral and prime, with each I i coprime to n and D. (We will assume that for any Grossencharacter of conductor f considered in the sequel, these representatives are also coprime to f.) Let k −1 be an integer, and for any ring R, let V 2k+2 (R) denote the ring of homogeneous polynomials over R in two variables of degree 2k + 2. Note that V 2k+2 (C) is an irreducible complex right representation of SU 2 (C), and denote the corresponding antihomomorphism by ρ : SU 2 (C) → GL(V 2k+2 (C)). Finally, define an antihomomorphism ρ : For a general Grossencharacter ψ of K, for each prime p of K we denote by ψ p the restriction of ψ to K × p ; then we also write ψ ∞ for the restriction of ψ to the infinite ideles, and ψ f for the restriction to the finite ideles.

Definitions
In the spirit of Weil, we define Bianchi modular forms adelically. First, we fix a level.
Henceforth, we will always take k 0.
Definition 1.2. We say a function Φ : GL 2 (A K ) → V 2k+2 (C) is a cusp form of weight (k, k) and level Ω 1 (n) if it satisfies: (i) Φ(zgu) = Φ(g)ρ(u, z) for u ∈ SU 2 (C) and z ∈ Z(GL 2 (C)) ∼ = C × ; (ii) Φ is right-invariant under the group Ω 1 (n); (iii) Φ is left-invariant under GL 2 (K); (iv) Φ is an eigenfunction of the operator ∂, where ∂/4 denotes a component of the Casimir operator in the Lie algebra sl 2 (C) ⊗ R C (see [13,Section 1.3]), and where we consider Φ(g ∞ g f ) as a function of g ∞ ∈ GL 2 (C); and (v) Φ satisfies the cuspidal condition that for all g ∈ GL 2 (A K ), we have where we consider A k to be embedded inside GL 2 (A K ) by the map sending u to ( 1 0 u 1 ), and du is the Lebesgue measure on A K . Remarks 1. (i) The cuspidal condition is a natural one; the value of the integral for a fixed g corresponds to a constant Fourier coefficient.
(ii) The general definitions given above are already slightly tailored to work with cusp forms, that appear only at parallel weights (see [10]); more generally, it is possible to define general automorphic forms of weight (k 1 , k 2 ), for distinct integers k 1 and k 2 , over K.
A cusp form Φ of weight (k, k) and level Ω 1 (n) descends to give a collection of h functions where h is the class number. Such a collection is non-canonical, depending on choices of representatives for the class group of K. To obtain this decomposition, we note that GL 2 (A K ) decomposes as the disjoint union of h sets. To describe this, recall that we took I 1 , . . . , I h to be a complete set of (prime or trivial) representatives for the class group; Set t 1 = 1 and for each i 2, define Then we have: There is a decomposition Proof. See [2, Section 5.2].
It is now clear that Φ descends in the way claimed above via F i (g) := Φ(g i g). Furthermore, we can descend further using condition (i) to obtain h functions F i : is hyperbolic 3-space. Such functions are automorphic forms on H 3 of weight (k, k) and level This process is described in [2]. Note here that Γ 1 is nothing other than the usual Γ 1 (n) SL 2 (O K ).

L-functions
be a cusp form of weight (k, k) and level Ω 1 (n). Then, from [14], Φ has a Fourier expansion of the form where: (iii) the Fourier coefficient c(·, Φ) may be considered as a function on the fractional ideals of K, with c(I, Φ) = 0 for I non-integral; (iv) e K is an additive character of K\A K defined by where K n (x) is a (modified Bessel function that is) a solution to the differential equation with asymptotic behaviour If our adelic cusp form Φ corresponds to a collection of h cusp forms F 1 , . . . , F h on H 3 , where h is the class number of K, then the Fourier expansion stated above descends to give a Fourier expansion for each F j , which can be worked out to be (see [7]) Let F 1 , . . . , F h be the h automorphic forms corresponding to Φ with respect to I 1 , . . . , I h . Then to each one, we assign a 'part' of this L-function, in the following sense: where here we scale by w −1 as when we sum over elements of K × , we include each ideal w times (once for each unit).
The following lemma (see [26, Chapter II]) shows that each component of the L-function is a holomorphic function on a right half-plane: Lemma 1.6. For each i, L i (Φ, s) converges absolutely in the right half-plane Re(s) > C, for some constant C.
Let ψ be a Grossencharacter of conductor f. We want to define the twist of the L-function by ψ. For each ideal m coprime to f, define ψ(m) := p n ||m ψ p (π p ) n ; then define where, for ideals I and J, the notation (I, J) = 1 means that I and J are coprime. It is more convenient in terms of our future work to define the twist in terms of the L i (Φ, s). Recall that D = (δ) is the different of K and the definition of L i (Φ, s) above, and set where here |αδ| is the usual norm of αδ ∈ K × and |t i | f = v ∞ |t i | v is the finite idelic norm.
Remark 1. Note that in [26,Chapter II], it is proved that the twisted L-function still converges absolutely in some suitable right half-plane.
There is one more useful alternative method of writing the twist. Define Then note that as ψ is trivial on the diagonal embedding of K × in A × K , we have, for (α) coprime to f, Accordingly, we can also write the twist as 3. An integral formula. In the rational case, the twist of an L-function attached to a modular form can be written in terms of an integral formula, which is crucial to its study via modular symbols. We now obtain such an integral formula for L i (Φ, ψ, s). To do so, we will need some results about Gauss sums. Hecke wrote down a suitable analogue of Gauss sums over general number fields in [11] and [12], and Deligne wrote down local analogues (via the theory of local ε-factors) in [6, Section 3]; his work is translated into English in [24,Section 3]. In [16, Proposition 6.14(ii)], it is shown that Hecke's Gauss sums satisfy a product law, and moreover that the local factors agree with Deligne's local ε-factors. The version we will use is one obtained from Hecke's definitions by Nemchenok in [17,Section 6]. For a Grossencharacter ψ of conductor f, the Gauss sum is defined to be where as before δ is a generator of the different D of K. The key proposition we require is the following, which follows easily from [16, Proposition 6.14(i)], and allows us to deal with the terms of the L-series which correspond to ideals not coprime to the conductor.
(ii) By replacing ψ with ψ −1 , we have From hereonin, ψ denotes a Grossencharacter of infinity type (−u, −v) and conductor f. Ultimately, the result we will prove is: is an explicit function of s.
The proof is contained in the rest of this section. Using Proposition 1.7 to replace ψ f (αδ) −1 in the expression for L i (Φ, ψ, s) we obtained above, and rearranging, we get To get our integral formula, we will use the standard integral (see [14,Section 7]) Setting λ = 4π|α|, and j = 2s, we get an expression for |α| −2s . Now, to force our expressions into the form of the Fourier expansion for F i , we fix (u, v) = ((k + 1 − n)/2, −(k + 1 − n)/2); then substituting the expression for |α| −2s and rearranging the terms, we get that this is equal to is an explicit function of s.

Bianchi modular symbols
In this section, we describe Bianchi modular symbols. Recalling the general motivation given in the introduction, we elaborate in detail on the Bianchi case, focusing entirely on the description in terms of functions rather than compactly supported cohomology groups. We will also show how to obtain a Bianchi modular symbol from a Bianchi modular form; the process is, unfortunately, significantly less intuitive than the rational case. We conclude the section by exhibiting a link between the modular symbol of a Bianchi modular form and critical values of its L-function.

Definitions
Definition 2.1. We define, for a non-negative integer k and a ring R, the space V k (R) as above to be the space of homogeneous polynomials of degree k in two variables over R. Furthermore, Note that we can identify V k,k (C) with the space of polynomials that are homogeneous of degree k in two variables X, Y and homogeneous of degree k in two further variables X, Y . Furthermore, V k (C) is an irreducible SU 2 (C)-module, with SU 2 (C) acting on the right by The following defines a different action on this space, with a view to obtaining a 'nice' action on the dual space of V k,k (C).
We then obtain a left action of Remark 2. This action, whilst appearing unconventional, is chosen so that it is compatible with an action on a space of locally analytic functions on some p-adic space; see Section 3.3 for further details. This compatibility simplifies matters considerably when considering specialisation maps from overconvergent to classical modular symbols.
The left action of SL 2 (C) described above translates into a right action on the dual space We also have a left action of SL 2 (C) on the space Δ 0 := Div 0 (P 1 (K)) by fractional linear transformations.
Definition 2.3. Let Γ SL 2 (K) be a discrete subgroup, and let V be a right Γ-module. Given a function φ : We define the space of V -valued modular symbols for Γ to be the space Symb Remark 3. Relating back to the description of modular symbols in the introduction, where we gave motivation for the explicit study of modular symbols in the Bianchi case, we see that using the arguments in [1] that there is a canonical isomorphism Recall that we set n to be an ideal of O K with (p)|n, and recall the definition of Ω 1 (n) in Definition 1.1. With our choices of explicit representatives I 1 , . . . , I h for the class group of K (see the start of Section 1), the discrete subgroups of SL 2 (K) corresponding to Ω 1 (n) are The space of Bianchi modular symbols of parallel weight (k, k) and level Γ i is defined to be the space The space of Bianchi modular symbols of parallel weight (k, k) and level Ω 1 (n) is defined to be the space

Aside on group actions
For the purposes of clarity, we take a slight detour to elaborate on the various group actions we are endowing the polynomial spaces above with. In our previous work, we stated that V 2k+2 (C) was an irreducible right-SU 2 (C)-module. In the next section, we will need to use a left action on this space, namely, the one we obtain from the right action above; that is, we define, for γ ∈ SU 2 (C), In an attempt to avoid confusion, we denote the space by V r 2k+2 (C) when we consider the right action of SU 2 (C), and by V 2k+2 (C) when we consider the corresponding left action. We also use this notation for the tensor product in the obvious manner. Note that these actions obviously extend to give actions of SL 2 (C) on the spaces involved.

Differentials on H 3
In the sequel, we will require some facts about differentials on H 3 ; these are listed here for reference.
The space Ω 1 (H 3 , C) of differential 1-forms on H 3 is a three-dimensional C ∞ (H 3 )-module, spanned by dz, dt and dz. The usual action of SL 2 (C) on H 3 induces a left action of SL 2 (C) on Ω 1 (H 3 , C) by pull-back. We also have: There is a natural map from Ω 1 (H 3 , C) to Ω 1 0 (H 3 , C) given by evaluation at (0, 1) ∈ H 3 . Combining this map and the isomorphism of Proposition 2.5 with the action of SL 2 (C) above allows us to define a left action of SL 2 (C) on V 2 (C). An explicit check, contained in [7], shows that this action can be explicitly described as

Construction
Let Φ be a Bianchi cusp form, giving rise to a collection of h functions F i , . . . , F h on GL 2 (C).
To each of these 'components' of Φ, we can associate a harmonic V k,k (C)-valued differential ω i on H 3 , which we can then integrate between cusps to obtain a modular symbol of level Γ i . These then combine to give the modular symbol of level Ω 1 (n) attached to Φ. This process is described in full in [7], and here we recap the construction. Firstly, note that the Clebsch-Gordan formula gives an injection of SU 2 (C)-modules. With this, and using Proposition 2.5, we can define a V k,k (C)-valued differential on H 3 by is invariant under translations by SU 2 (C) and thus descends to a differential on H 3 .) We can view this as a V r k,k (C)-valued differential in the manner described above. This is still not quite what we require; the definition calls for a V * k,k (C)-valued differential, where the right action on . This is dealt with via: Proof. This is a simple, but lengthy, check.
The following is another elementary check, which uses the modularity of F i and the action of SL 2 (C) on Ω 1 (H 3 , C): Definition 2.8. The modular symbol attached to Φ is

Relation to L-values
We now exhibit a link between values of this modular symbol and critical L-values of the original Bianchi modular form. To do this, we appeal to [7], in which Ghate calculates what this symbol looks like more explicitly.
Proposition 2.9. We have, for a ∈ K, Proof. The bulk of the work is completed in [7, Section 5.2]. We note that as we integrate over vertical paths, we fix z = a, and we are not interested in the dz and dz terms of the differential.
This now combines with the integral formula we obtained for the L-function L i (Φ, ψ, s) in Theorem 1.8, where ψ is a Grossencharacter with infinity type and conductor f, to give the required link. We want to set 2s − 2 = q + r, that is, Some cancellation and the explicit form for A(i, n, ψ, (q + r + 2)/2) now gives This gives us a link between modular symbols and L-values. Indeed, combining the results above for each i, we have Proposition 2.10. In the setup of above, we have

L-functions as functions on characters
It is convenient to rephrase the above; instead of seeing the L-function as a complex function of one variable, we think of it as a function on Grossencharacters by setting Let ψ := ψ| · | (q+r)/2 . This is now a Grossencharacter of conductor f and infinity type (q, r). To see that there is a relation of Gauss sums between ψ and ψ , we need some new notation. To any ideal I of K, we associate an idele x I in the following way: if (α) is principal, then define (x (α) ) v := α : v = p is finite, and p|(α), 1 : otherwise.
which is proved using the easily seen identity where here we have used that |δ| 2 = D. This simplifies further; indeed, for any Grossencharacter ψ, an explicit check shows that when so that this becomes where ψ is a Grossencharacter of conductor f and infinity type (q, r) with 0 q, r k. We make one further change with the aim of massaging this formula into something a little nicer; namely, we renormalise, using the Deligne Γ-factor at infinity. Define from which we deduce that: Theorem 2.11. Let K/Q be an imaginary quadratic field of discriminant D and w be the size of O × K , and let Φ be a cuspidal Bianchi modular form of parallel weight (k, k) and level Ω 1 (n) with renormalised L-function Λ(Φ, ψ). For a Grossencharacter ψ of K of conductor f and infinity type 0 (q, r) (k, k), we have

2)
where c i q,r (a) is as defined in Proposition 2.9.

Algebraicity results
Ideally, we would like to work with algebraic coefficients. For this, we require the following: Proposition 2.12. Let Φ be a Bianchi modular form and φ Φ ∈ Symb Ω1(n) (V * k,k (C)) its associated modular symbol. Then there exists a number field F and a non-zero complex period Proof. See [14,Chapter 8].
In particular, by renormalising, we can consider φ Φ to have coefficients in a sufficiently large extension of Q p . This also means that the critical L-values given by the integral formula above are all algebraic multiples of a single complex number.

Overconvergent Bianchi modular symbols
Overconvergent modular symbols are modular symbols that take values in a space of p-adic distributions. In a similar style to the rest of the paper, we begin this section by defining these spaces in their most natural way-in the process, directly generalising the work of Pollack and Stevens-before passing to a more workable and explicit description. We will then write down a filtration on the space of overconvergent Bianchi modular symbols of fixed weight and level, putting us in the situation of the next section and allowing us to prove a control theorem for these objects.
In the following, let F be either Q or an imaginary quadratic field with ring of integers O F . We can write down the polynomial space associated to modular symbols (either V k (C p ) over Q or V k,k (C p ) over an imaginary quadratic field) in a more intrinsic way, namely as , giving a surjective specialisation map. To see that we can define modular symbols with values in D(O F ⊗ Z Z p ), we consider the semigroup (Note that when p ramifies as p 2 in F , we can consider the larger group coming from the condition c ∈ π p O F ⊗ Z Z p , where π p is a uniformiser at p. We make no further mention of this here, however.) This has a natural action on A(O F ⊗ Z Z p ), depending on k, defined as follows: Using this we obtain an action of suitable discrete subgroups of SL 2 (F ) on the distribution space, allowing us to use it as a value space. We also get a Hecke action on the resulting modular symbols. All of this directly generalises the work of Pollack and Stevens in the case F = Q, and with suitable small adjustments would generalise further to the case of arbitrary number fields. In our setting, we can describe these distribution spaces more explicitly. Later, when p is split or inert, we will show that the space A(O F ⊗ Z Z p ) can be thought of as just being the space of rigid analytic functions on Z 2 p , or, in the weak topology, the completed tensor product of two copies of the space of rigid analytic functions on one copy of Z p . The group Σ 0 (O F ⊗ Z Z p ) naturally injects into two copies of the analogue of the semigroup Σ 0 (p) as written down by Pollack and Stevens (we will call this semigroup Σ 0 (O L )). When p is not split, we are carrying around some redundant information with this approach; in particular, the useful information given by the action of Σ 0 (O L ) 2 is entirely determined by the action of one of the components. There are significant advantages to using this more explicit approach, however. The spaces have nice descriptions that are easy to work with and allow us to generalise the filtration proof of Greenberg in [8] to the imaginary quadratic case. In the remainder of this section, we work with this explicit approach exclusively.

Notation and preliminaries
Notation 3.1. Throughout, as before, K/Q denotes an imaginary quadratic field. Let p be a rational prime with pO K = p ep and define f p to be the residue class degree of p. Note that e p f p = 2. Fix an embedding Q → Q p ; then for each prime p|p, we have e p f p embeddings K p → Q p , and combining these for each prime, we get an embedding If a ∈ K ⊗ Q Q p , we write σ 1 (a) and σ 2 (a) for the projection of σ(a) onto the first and second factors, respectively. Now, let L be a finite extension of Q p such that the image of σ lies in L 2 . We equip L with a valuation v, normalised so that v(p) = 1, and denote the ring of integers in L by O L , with uniformiser π L . Then in fact, for each integral ideal I of K coprime to (p), we have In the obvious way, we then have an embedding Note that in particular this means that the groups in the following definition-and hence the groups Γ i = Γ Ii of previous sections-embed into the right-hand side. Whilst the groups Γ I are implicitly dependent on the ideal n, since we work exclusively at level n throughout this paper, for convenience we drop n from the notation.
Definition 3.2. Let n be an ideal of O K with (p)|n, and let I be an integral ideal of K that is coprime to n. Define the twist of Γ 1 (n) by I to be The embedding above will be used in the sequel to define the action of each Γ i as well a Hecke action on suitable modular symbol spaces. We will define some monoid Σ 0 (O L ) GL 2 (O L ) and an action of Σ 0 (O L ) × Σ 0 (O L ). Every matrix whose action we study will have image in Σ 0 (O L ) 2 under the embedding above. Thus in proving facts about the action of Σ 0 (O L ) × Σ 0 (O L ), we will encapsulate everything we will later need regardless of the splitting behaviour of p in O K .
Remark 4. For our purposes, we may need to take L to be larger than this. Let Φ be a cuspidal Bianchi eigenform with Fourier coefficients c(I, Φ), normalised so that c(O K , Φ) = 1. Then the Fourier coefficients are algebraic. In particular, when studying the action of the U p operator we may need to consider eigenvalues living in the number field F := K({c(p, Φ) : p|p}).
As we can easily enlarge L to contain all possible embeddings of the completions of this field into Q p , we will henceforth assume that these eigenvalues live in L.

Overconvergent modular symbols
To define overconvergent modular symbols, Stevens used spaces from p-adic analysis. For more details on the results here, including p-adic function and distribution spaces as well as the completed tensor product, see [3]. a n x n : a n ∈ R, a n → 0 as n tends to ∞ ⎫ ⎬ ⎭ .
For R = L, this is a L-Banach space with the sup norm. We write A 2 (R) for the completed tensor product A(R)⊗ R A(R). We let D(R) = Hom cts (A(R), R), the space of rigid analytic distributions over R, be the topological dual of A(R), and analogously we let D 2 (R) = Hom cts (A 2 (R), R) be the topological dual of A 2 (R).
We have the following identification, as promised in the introduction to this section. where the coefficients lie in L as we took L large enough to contain K ⊗ Q Q p . Moreover, rigid analyticity means that the coefficients a r,s tend to zero as r + s tends to infinity. Accordingly, we can identify f with an elementf (X, Y ) = r,s a r,s X r Y s of the Tate algebra L X, Y in two variables defined over L. But this can be viewed as the space of rigid analytic functions on Z 2 p defined over L, and the result follows.
Remarks 2. (i) When p is inert, the fact that we can make this identification relies on our functions being rigid analytic. If instead we considered functions that were locally analytic of order 1, such an identification does not hold. This is because any rigid analytic function on O K ⊗ Z Z p can be extended to O Cp , whereas the same is not true of locally analytic functions.
(ii) Furthermore, this identification does not carry over immediately to the case p ramified. Indeed, consider the function x → r,s π n p ([x − x]/π p ) n 2 . This is a rigid analytic function on O K ⊗ Z Z p that cannot be written as an element of the Tate algebra over L. However, we will work with the more explicit definition in the case p ramified as well, as ultimately the construction still gives us a p-adic L-function in this case. In this case, the relevant distribution space for constructing the p-adic L-function, D(O K ⊗ Z Z p ), is naturally a subspace of D 2 (L); indeed, in Proposition 5.8, we show that the overconvergent modular symbol we construct lives in this smaller space (and, indeed, in the even smaller space of locally analytic distributions).
Having defined these spaces of distributions, our primary spaces of interest, we immediately give two alternate descriptions that are easier to work with.
Definition 3.5. (i) Let μ ∈ D be a distribution. Define the moments of μ to be the values (μ(x i )) i 0 , noting that these values totally determine the distribution since the span of the x i is dense in A.
(ii) Let μ ∈ D 2 be a two variable distribution. Define the moments of μ analogously to be the values (μ(x i y j )) i,j 0 .
The following simple proposition gives a simple description of the distribution spaces.
extends in the obvious way to give a weight (k, k) and, more generally, a weight (k, ) action of Σ 0 (O L ) 2 on A 2 (R) by where γ i = ( ai ci bi di ). This gives rise to a right weight (k, ) action of Σ 0 (O L ) 2 on D 2 (R) defined by When talking about these spaces equipped with the weight (k, ) action, we denote them by A k, (R) and D k, (R), respectively.
Remarks. (i) Importantly, for p split or inert, the isomorphism of Proposition 3.4 respects the actions of Σ 0 (O K ⊗ Z Z p ) on the left-hand side and Σ 0 (O L ) 2 on the right-hand side.
(ii) Note that the subspace V k, (R) of A 2 (R) is stable under the action of Σ 0 (O L ), and hence it inherits a left action of Σ 0 (O L ). This is the action we earlier defined in Definition 2.2.
Note that for any ideal I coprime to n, we have a right action of Γ I (as in Definition 3.2) on the space D k, (L).
We can work out the double coset operators explicitly to be given by 1 a 0 α i using the usual methods. Note that if n is an integer such that p n = (σ) is principal, then this action becomes significantly simpler; namely, we just act on each component, with no permuting, via the sum a (mod p n ) ( 1 0 a σ ). Because of this much simpler description, in the sequel we much prefer to use a principal power of p and prove results using just one component at a time.
Ideally, we would prefer to work with integral distributions.
Lemma 3.8. Let K be a number field and I be an ideal coprime to n. Then Div 0 (P 1 (K)) is a finitely generated Z[Γ I ]-module.
Proof. This follows from the fact that Γ I is a finitely generated group (see, for example, [23]) and that it has finitely many cusps. Proof. Let φ ∈ Symb ΓI (D ⊗ OL L). Using Lemma 3.8, take a finite set of generators α 1 , . . . , α n for Δ 0 as a Z[Γ I ]-module. We can find some element c ∈ O L such that cφ(α i ) ∈ D for each i. But then it follows immediately that cφ ∈ Symb ΓI (D), and the result follows.
With this structure in place, we can now work with the space D k, (O L ).

Finite approximation modules
Remark 5. From now on, we work with parallel weights, that is, we consider only k = and use the space D k,k (R). There are no classical cuspidal Bianchi modular forms at non-parallel weights, so in proving a control theorem in the spirit of Stevens' work, it suffices to exclude the case k = . We will also focus on looking at the space of Bianchi modular symbols one component at a time. Henceforth, to this end, Γ will denote one of the Γ i for i ∈ {1, . . . , h}.
In the one variable case, in [8] Greenberg gave an alternative proof of Stevens' control theorem using finite approximation modules, defining a Σ 0 (O L )-stable filtration of D k (L) and then lifting modular symbols through this filtration. We aim to mimic this. First we recap Greenberg's filtration, recasting it slightly to make it more favourable for our generalisation. He defines: To define our own filtration, we take a similar route, imposing suitable conditions on the moments of distributions.
Definition 3.11. Define: Proof. There is an obvious switching map s : D k, (O L ) → D ,k (O L ). Thus, it suffices to prove the result for elements of form (γ, I 2 ) for γ ∈ Σ 0 (O L ), as the action of a more general element can be described as To each two variable distribution μ ∈ D k,k (O L ), associate a family of distributions {μ j ∈ D k (O L )} by defining the moments of μ j to be Then note that we have where the condition must hold for all j 0. The result we require then follows from the observation that In particular, this result shows that we have a collection of Σ 0 (O L ) 2 -modules with action inherited from D k,k (O L ), and where this is well-defined since the where T is some finite product of copies of Z/p, Z/p 2 , and so on up to Z/p N −k−1 .
In particular, we also have , and we see that π 0 is in fact the map given by projection (and hence also maps ρ M,N ). The projection maps are all compatible in the obvious ways. Thus, we get an inverse system of modules, and it is straightforward to see that It follows from equation (3.2) that

Moments of functionals on polynomials
Given an element f ∈ V * k,k (O L ), define the moments of f to be the quantities f (x m y n ) for 0 m, n k, analogously to the moments of more general distributions from previously. Any such f is entirely determined by its moments.
The next result is a technical lemma describing the action of certain matrices in Σ 0 (O L ) 2 . It gives us nice properties of the U p operator.
Proof. Take some μ ∈ D k,k (O L ). Then . It follows that each term of the sum above lies in λO L , and hence the result follows. If instead i + j > v(λ), then the result follows as μ(x i y j ) ∈ O L .
(ii) Now suppose μ ∈ F N D k,k (O L ). Again considering the sum above, the terms where i, j k vanish. If i + j > k, then since λ has p-adic valuation < k + 1. As p i+j and λ are divisible by integral powers of π L , it follows that p i+j ∈ π L λO L . Hence, as μ( Remark 6. Note that we have made some choice of orbit representatives for the action of Γ on the double coset Γ( 1 0 0 p )Γ. If we consider U p as a double coset operator on set-theoretic maps, this is not well-defined up to choice of such representatives. However, as long as we are consistent in our choice, it shall not matter; hence we simply define U p in this very specific way and ignore where it comes from.

Summary
Before embarking on the proof of a suitable lifting theorem, we first take stock of the work so far. We have a monoid Σ 0 (O L ) acting on an Suppose γ is a summand of the U p operator defined above; then Lemma 3.15(i) tells us that if μ ∈ D, then μ|γ ∈ λD k,k (O L ), while Lemma 3.15(ii) This then gives us the exact situation described in the next section.

A lifting theorem
The following is an abstraction of some of the elements of Greenberg's work in [8]. Since we will use essentially the same ideas multiple times, it is presented here in a completely general setting. The notation given is suggestive, and throughout, the reader should imagine the objects of (i)-(vii) below to be the obvious analogues from Section 3.6. Suppose L/Q p is a finite extension, and that we have: Then there is a natural map ρ 0 : Symb Γ (D A ) → Symb Γ (A) whose restriction to the λ-eigenspaces of the U operator is an isomorphism.
For clarity, the proof will be broken into a series of smaller steps. We have natural Σ-equivariant projection maps (ii) The reason we do not simply just define V γ = λ −1 γ is that 'dividing by λ' is not in general a well-defined notion on D.
so that we have an equality of operators λV = U | Hom(Δ,DA) (where the right-hand side is an operator on Hom(Δ, D A ) by Remark (i) above).
It is easy to see that, for each N , V gives rise to an operator V N : Take a U -eigensymbol φ 0 ∈ Symb Γ (A) = Symb Γ (A 0 D A ) with eigenvalue λ. Suppose a lift to a U -eigensymbol φ N ∈ Symb Γ (A N D A ) exists. We can take an arbitrary lift of φ N to some Z-homomorphism Such a lift exists, as we can take some Z-basis of Δ (using countability) and define φ on this basis, extending Z-linearly (noting that this gives a well-defined lift since φ N is also a Z-homomorphism). Now, since φ N is a U -eigensymbol with eigenvalue λ, it follows that φ|V is also a lift of φ N to Hom(Δ, D A ). The maps π N , inducing the maps ρ N , can be used immediately to extend the definition of ρ N to the space of Z-homomorphisms from Δ to D A (rather than just the ones that are Γ-equivariant). Define a Z-homomorphism from Δ to A N +1 D A . Note that since the maps ρ N , ρ M,N are Σ-equivariant (and hence V -equivariant), we have compatibility relations so that the following lemma says that the family {φ N } we obtain gives an element of the inverse limit given in equation (4.1).
We prove this lemma in a series of claims.
In particular, φ N +1 is independent of the choice of φ above φ N .
Proof. To say that φ and φ are both lifts of φ N is to say that the image of φ − φ under ρ N takes the value 0 in A N D A , that is, φ − φ takes values in F N D A . Thus, by condition (b) in Theorem 4.1, As φ N is a U -eigensymbol of eigenvalue λ, the image of (φ − φ )|V γ under ρ N +1 lies in A N +1 D A , and by the above, it is 0. But this is precisely the statement that we required.
Proof. Let γ ∈ Γ. As the map ρ N +1 is Σ-equivariant, it follows that where the division by λ is purely formal and well-defined by the remarks above. By condition (vi) above, we have a double coset decomposition Since φ N is Γ-invariant, φ|γ j is a lift of φ N , and hence by Claim 4.2.1 it follows that as required.
Proof. Recall that when we consider the operator V acting on A N D A , we denote it V N , and that φ N is a U -eigensymbol with eigenvalue λ if φ N is a fixed point of V N . Note then that φ|V is also a lift of φ N to Hom(Δ, D A ). In particular, φ|V also lives in Hom(Δ, D A ), so we can apply V to it again. Thus, by Σ-equivariance and Claim 4.2.1, as required.
Proof (Theorem 4.1). Surjectivity follows from the results above; take an element Then for each N we can construct φ N ∈ Symb Γ (A N D A ), compatibly with the projection maps π M,N , and thus obtain a well-defined element of the inverse limit, which is the domain. By construction, this element has image φ 0 under ρ 0 .
To prove injectivity, take some Applying the operator V recursively gives indeed, by condition (b) in Theorem 4.1, ϕ|V N ∈ F N D, and since ϕ is a U -eigensymbol with eigenvalue λ, we can make sense of ϕ|V N = ϕ ∈ D A . Thus the image of ϕ lies in the intersection of all the F N D A , that is, ϕ = 0 (by condition (iii)) as required.
Thus the map is a bijection, and thus an isomorphism, as required.
An immediate consequence of this is the following corollary (the control theorem for Bianchi modular symbols): Corollary 4.3. (i) Let K/Q be an imaginary quadratic field, p a rational prime, and L/Q p the finite extension defined in Notation 3.1. Let Γ = Γ i be a twist of Γ 1 (n) SL 2 (O K ) with (p)|n. Let λ ∈ L × . Then, when v p (λ) < k + 1, the restriction of the specialisation map (ii) In the same setup, with Ω 1 (n) defined as in Definition 1.1, we have an isomorphism Proof. To prove (i), recall that in the setup of Section 3.6, Theorem 4.1 says that the restriction of ρ 0 to the map Up=λ is an isomorphism. The result now follows by right-exactness of tensor product and Proposition 3.9, since D ⊗ OL L ∼ = D k,k (L) and V λ k,k ⊗ OL K ∼ = V * k,k (L). Part (ii) is a trivial consequence as the U p operator acts separately on each component.

Values of overconvergent lifts
This section will examine the spaces in which overconvergent lifts take values, refining our earlier results. Recall that Γ = Γ i is a twist of Γ 1 (n).

Locally analytic distributions
We have shown that any classical Bianchi eigensymbol of suitable slope can be lifted to an overconvergent symbol that takes values in a space of rigid analytic distributions. However, the space we are truly interested in is a smaller space of distributions. A p-adic L-function should be a function on characters; but a rigid analytic distribution can take as input only functions that can be written as a convergent power series, and most finite order characters are locally constant, and thus cannot be written in this form. Instead, we want our lift to take values in the dual of locally analytic functions.
where σ 1 , σ 2 are the projections of the embedding σ : K ⊗ Q p → L 2 to the first and second factors, respectively.
When p is split and r and s are both at least 1, B(O K ⊗ Z Z p , r, s) is the cartesian product of the closed discs of radii r and s in C p ; and if r = s = 1/p, this is the cartesian product of two copies of the disjoint union of closed discs of radius 1/p with centres at 0, 1, . . . , p − 1. We endow A[L, r, s] with a weight k action of Σ 0 (O K ⊗ Z Z p ) identical to the action defined earlier on A 2 (L); it is obvious that this action extends immediately to the larger space. It is then clear that by dualising, the action we obtain on D[L, r, s] is the restriction of the action on D 2 (L). When talking about these spaces equipped with these actions, we denote them A k,k [L, r, s] and D k,k [L, r, s].
For r r and s s , we have a natural and completely continuous injection r , s ). Since the polynomials are dense in each of these spaces, the image of this injection is dense. Using this compatibility, we make the following definitions.
Proof. The natural maps in each direction are inverse to each other and Σ 0 (O K ⊗ Z Z p )equivariant.

The action of
The action of certain elements of Σ 0 (O K ⊗ Z Z p ) naturally moves us up and down the direct/inverse systems.
For such x, y, we have γ · k g(x, y) = g(σ 1 (a) + p n x, σ 2 (a) + p n y), and since g is defined on B(O K ⊗ Z Z p , r, s), the result follows.
For part (ii), note that the action of γ gives a map and hence dualising, the action gives a map This is the required result (though scaled by a factor of p n ).
Proof. Firstly, D k,k (L) = D k,k [L, 1, 1]. Note that U p acts invertibly on the U p -eigenspace, so that for each integer n, there exists some eigensymbol Ψ with Ψ = Ψ |U n p . The U n p operator can be described explicitly as Let λ ∈ L × . Then, when v(λ) < k + 1, the restriction of the natural map (ii) In the setup of above, and with Ω 1 (n) as defined in Definition 1.1, the restriction of the natural map Symb Ω1(n) (D k,k (L)) Up=λ −→ Symb Ω1(n) (V * k,k (L)) Up=λ is an isomorphism.

Admissible distributions
For each pair r, s, the space D k,k [L, r, s] admits a natural operator norm || · || r,s via ||μ|| r,s = sup where | · | p is the usual p-adic absolute value on L and | · | r,s is the sup norm on A k,k [L, r, s]. Note that if r r , s s , then ||μ|| r,s ||μ|| r ,s for μ ∈ D k,k [L, r , s ]. These norms give rise to a family of norms on the space of locally analytic functions. It is natural to classify locally analytic distributions by growth properties as we vary in this family.
Definition 5.10. Let μ ∈ D k,k (L) be a locally analytic distribution. We say μ is h-admissible if The following lemma is a useful technical result describing the family of norms of a Γ-orbit in A k,k [L, r, s], where Γ is as before. It gives universal constants that will be useful in the sequel.
Lemma 5.11. There exist positive constants C and C such that Proof. The action of γ by say, noting that C −1 is certainly positive. Thus from which the first inequality follows. The reverse direction follows from symmetry by considering the action of γ −1 .
In particular, the supremum is finite and hence gives a well-defined norm, as required.
The values of an overconvergent eigensymbol satisfy further conditions of the type above depending on their slope.

The case p split
The results above hold for the U p operator regardless of the splitting behaviour of the prime p in K. When p is inert, this is the whole story, and if p is ramified, then there are only slight modifications to make to obtain a finer result for the U p operator, where (p) = p 2 in K. In the case that p splits in K as pp, however, we can obtain more subtle results. The crux of this section is that we can prove a control theorem for a 'small slope' condition that encompasses far more possible eigensymbols. For example, suppose E/K is a modular elliptic curve with supersingular reduction at both primes above p. The associated weight (k, k) = (0, 0) eigenform gives rise to a Bianchi eigensymbol φ ∈ Symb Γ (V * k,k (L)) with slope 1/2 at p and 1/2 at p. This symbol will have slope 1 = k + 1 under the U p operator, and hence we cannot lift it using the control theorem proved above. The results below will allow us to lift even symbols such as this.
Many of the results and proofs closely mirror those of previous sections. We first prove a more refined control theorem, again by writing down a suitable filtration and using Theorem 4.1, then go on to prove an admissibility result for such lifts.
Throughout, we assume that p splits as pp. Whilst ultimately we want to prove a control theorem for the full space of modular symbols for Ω 1 (n), it is simpler to instead work with a principal power of p and look at each individual component of the direct sum separately, combining together at the end.

Lifting simultaneous eigensymbols of U p and U p
The following results will show that it is possible to lift a classical Bianchi symbol to a space of Bianchi modular symbols that are overconvergent in one variable, and then again from this space to the space of fully overconvergent Bianchi modular symbols we considered previously.
To do so, consider the space with the appropriate action of Σ 0 (O L ) 2 (where this makes sense) induced from the action on A k,k . This gives us . Now put R = O L , and recall the filtration in the one variable case from Definition 3.10. We now define new filtrations to reflect lifting by one variable at a time.
Further define Proof. This follows from the one variable case, as these filtrations are defined to be a tensor product of Σ 0 (O L )-stable spaces, and in the two variable case, Σ 0 (O L ) 2 acts as Σ 0 (O L ) separately on each component.
which again give maps ρ N 1 and ρ N 2 on the corresponding symbol spaces. We want an analogue of Lemma 3.15 for this setting. Choose n such that p n = (β) is principal (noting that this also forces p n to be principal). Then we define U p n = U n p as a (mod p n ) 1 a 0 β . since λ has p-adic valuation < n(k + 1). As β i and λ are divisible by integral powers of π L , it follows that β i ∈ π L λO L . Hence, as μ( which completes the proof.
Proof. Identical to that of Lemma 6.6-up to notation-but with j replacing i where appropriate.
We are hence in exactly the situation of Theorem 4.1, and applying it twice gives us: Lemma 6.8. Let K/Q be an imaginary quadratic field, p a rational prime that splits as pp in K, n an integer such that p n is principal, and L/Q p the finite extension defined in Notation 3.1. Let Γ = Γ i be a twist of Γ 1 (n) SL 2 (O K ) with (p)|n. Let λ ∈ L × . Then, when v(λ) < n(k + 1), we have: (i) the restriction of the specialisation map (where the superscript (U p = λ) denotes the λ-eigenspace for U p ) is an isomorphism; (ii) the restriction of the specialisation map is an isomorphism.
Proof. The only remaining details to fill in are formalities regarding tensor products, which are analogous to before and are omitted.
We require results about the U p operator, whilst the results above are for the U n p operator. We work around this using: Lemma 6.9. Suppose we have two L-vector spaces D, V with a right action of an operator U , and a U -equivariant surjection ρ : D → V such that, for some positive integer n and λ ∈ L × , the restriction of ρ to the λ n -eigenspaces of the U n operator is an isomorphism. Then the restriction of ρ to the λ-eigenspaces of the U operator is an isomorphism.
The result then follows directly from Lemma 6.9, since there are well-defined U p and U p operators on each of the spaces, and ρ 0 is equivariant with respect to these operators.

The action of
The following results are proved in an almost identical manner to those in Section 5.2.
Lemma 6.11. Let p split as pp in K, with p n = (β) principal. We also have an entirely analogous result for the U n p operator. Combining the two then gives the following.
Proposition 6.12. Suppose that Ψ ∈ Symb Γ (D k,k (L)) is simultaneously a U n p -and U n p -eigensymbol with non-zero eigenvalues. Then Ψ is an element of Symb Γ (D k,k (L)).
Corollary 6.13. Let K/Q be an imaginary quadratic field, p a rational prime that splits as pp in K, and L/Q p the finite extension defined in Notation 3.1. Let Ω 1 (n) be as defined in Definition 1.1.

Admissible distributions
In this new setting, we need a new definition of admissibility-namely one that encodes the slope at both p and p.
Definition 6.14. Let μ ∈ D k,k (L) be a locally analytic distribution. We say μ is ( uniformly in s as r → 0 + , and uniformly in r as s → 0 + . Proposition 6.15. Suppose p splits in K as pp, with p n = (β) principal, and suppose that Ψ ∈ Symb Γ (D k,k (L)) is a U n p -eigensymbol with eigenvalue λ n 1 and a U n p -eigensymbol with eigenvalue λ n 2 with slopes h i = v(λ i ). Then, for every D ∈ Δ 0 , the distribution Ψ(D) is (h 1 , h 2 )admissible.
Definition 7.1. Let (φ 1 , . . . , φ h ) be a classical Bianchi eigensymbol (respectively, Φ a classical Bianchi eigenform), with U p eigenvalue(s) a p for p|p. We can canonically see a p as living in Q p under the fixed embedding Q → Q p , and thus they have a well-defined valuation. We say that (φ 1 , . . . , φ h ) (respectively, Φ) has small slope if v(a p ) < (k + 1)/e p for all p|p, where e p is the ramification index of p in K. Note that this is precisely the condition that allows us to lift φ using one of the control theorems above. We say that the slope is (v(a p )) p|p . We define an operator U f as follows: Take some small slope classical Bianchi eigensymbol (φ 1 , . . . , φ j ) ∈ Symb Ω1(n) (V * k,k (L)) with U f -eigenvalue λ f , and lift it to an overconvergent eigensymbol (Ψ 1 , . . . , Ψ h ) using the control theorem. Evaluating (Ψ 1 , . . . , Ψ h ) at {0} − {∞}, and then evaluating the resulting distribution at this polynomial, we obtain where here the sum for U f is 'absorbed' by the indicator function; indeed, we have The ray class group fits into a useful exact sequence; we have where the map β takes an element α + f to the class (α) + K 1 f and the surjection is the natural quotient map. Now let f = (p n ). Piecing this together as we let n vary, taking the inverse limit of this family of exact sequences, we obtain an exact sequence is defined to be the inverse limit.

Grossencharacters as characters of Cl(K, p ∞ )
Let g be some ideal of O K that is coprime to (p). The following theorem is due to Weil; note here that we consider Cl(K, gp ∞ ) as a quotient of the adeles in the natural way.
Theorem 7.3 (Weil). There is a bijection between algebraic Grossencharacters of conductor dividing gp ∞ and locally algebraic characters of Cl(K, gp ∞ ) such that if ψ corresponds to ψ p-fin , then where here consider the adelic definition of the ray class group and we restrict to the adeles away from the infinite place and the primes above p.
The correspondence is simple to describe. If ψ is an algebraic Grossencharacter of conductor f|gp ∞ and infinity type (q, r), then to ψ we associate a K × -invariant function by fixing an isomorphism C ∼ = C p and defining : p splits as pp, It is simple to check that this is trivial on K × . By unravelling the definitions, we see that for α ∈ K × , and an idele x α,p defined by (x α,p ) p := α : p|(p) 1 : otherwise, we have This association now gives the construction of Theorem 7.3.

Admissible distributions on Cl(K, p ∞ )
In earlier sections, we gave definitions of admissibility for distributions on O K,p . We quickly mention how the condition translates to the space Cl(K, p ∞ ), which is covered in detail in [15]. Let f : Cl(K, p ∞ ) → L be a function, and note that it naturally gives rise to a collection of h functions f 1 , . . . , f h : O K,p → L, each supported on O × K,p and invariant under global units; indeed, we can see from the exact sequence (7.3) that Cl(K, p ∞ ) is the disjoint union of a collection of spaces each isomorphic to a quotient of O × K,p . We say that f is locally analytic of radius (r, s) if each f i is, and the spaces of such functions give rise to a family of normsindexed by (r, s)-on the space of locally analytic distributions on Cl(K, p ∞ ). The admissibility conditions are then defined identically to before.

Constructing the p-adic L-function
The p-adic L-function of a modular form over a field K is naturally a locally analytic distribution on Cl(K, p ∞ ). We have reformulated the complex L-function as a function of Grossencharacters; we will now prove that we can construct such a locally analytic distribution μ p that satisfies an interpolation property that, for any Grossencharacter of conductor f|(p ∞ ) and infinity type 0 (q, r) (k, k), we have μ p (ψ p-fin ) = ( * )Λ(Φ, ψ), where ( * ) is some completely explicit factor. If Φ has small slope, then in conjunction with the admissibility condition, this defines μ p uniquely (see, for example, [15] for this result in the weight (0,0) case). The p-adic L-function of Φ will then be defined to be the distribution μ p . To make this tie in with the work of Section 7.1, we describe the character ψ p-fin explicitly as a locally analytic function on the ray class group.
First, we describe what Cl(K, p ∞ ) actually looks like. By choosing a set of representatives for the class group, we are choosing a section of the map Cl(K, p ∞ ) → Cl(K), and thus, going back to the exact sequence above, we can identify Cl(K, p ∞ ) with a disjoint union of h copies of O × K,p /O × K , indexed by our class group representatives. On each of the h components, the character ψ p-fin gives a locally polynomial function on O × K,p . To describe this, define a function P i (z) on O K,p by where η is the character sending z → z q z r , and 1 b (mod f) is the indicator function for the open subset b + fO K,p . The ith component of ψ p-fin is given by the restriction of P i (z) to O × K,p .
Remark 7. The function P i (z) is a function on O K,p supported at elements that are units (mod f). In the case where f is divisible by every prime above p, this means it is zero outside O × K,p . If this condition does not hold-that is, there is some prime above p at which ψ is unramified-then P i (z) is actually supported on a slightly larger space. This means we have to consider the case where not all primes above p divide the conductor of ψ separately in Section 7.6.
With the above data, combined with equation (2.2), we are in a position to construct our distribution on Cl(K, p ∞ ). We recall the construction. Let n be an ideal divisible by (p), and let Φ be a cuspidal Bianchi eigenform of level Ω 1 (n) and weight (k, k). Then to such a function we associate a modular symbol (φ 1 , . . . , φ h ), where h is the class number. By scaling by the periods of the eigenform by the period Ω Φ , we can consider each φ i as an element of Hom Γi (Δ 0 , V k,k (L)) for some extension L/Q p . In the case where Φ (and hence each φ i ) has small slope (h p ) p|p , we can lift these symbols to give an overconvergent modular symbol (Ψ 1 , . . . , Ψ h ) in ⊕ h i=1 Hom Γi (Δ 0 , D k,k (L)). The values taken by such symbols are then (h p ) p|p -admissible in the sense of the previous section.
Define, for each i, a distribution Then we know that, for f|(p ∞ ), if b = 0 (mod p) for each prime p|p, and μ i (P q,r b,f (z)) = 0 if there exists a prime p|p such that b ≡ 0 (mod p). This leads us to define a distribution

Interpolation of L-values
We prove that μ p interpolates critical L-values of Φ. We split this into two cases.
(i) Firstly, we consider the case of characters that are ramified at each prime p|p. To this end, take a Grossencharacter ψ of infinity type (q, r)-where 0 q, r k-and conductor f|(p ∞ ), with f divisible by each prime above p. Then we have using equation (2.2) (noting that we have scaled everything by Ω Φ ). In this equation, recall that ψ f = p|f ψ p , the idele x f is as defined in Section 2.6, λ f is the U f -eigenvalue of Φ, −D is the discriminant of K, w is the size of the unit group of K and τ (ψ −1 ) is a Gauss sum as defined in Section 1.2.3.
(ii) The situation where ψ is unramified at one or more of the primes above p is slightly more difficult. The complication occurs since μ i is the restriction to O × K,p , meaning that μ i (P q,r b,f ) = 0 for some values of b that appear in the integral formula. As an example, suppose p is inert and that ψ is the trivial character; then The first term on the right-hand side is the one in the integral formula for Λ(Φ, 1), but we also have an extra term. In general, we can deal with this term by a careful application of the U p operator for each prime p above p for which ψ is unramified. In this case, we obtain an extra factor of Z p (ψ) := ψ p-fin (π p )(1 − λ −1 p ψ(p) −1 ) on the right-hand side of the interpolation formula for each such p. For more details on this calculation in the case of general number fields, see [22].
This interpolation property, in addition to the admissibility condition described above, means we can now prove: Theorem 7.4. Let K/Q be an imaginary quadratic field of class number h and discriminant −D, and let p be a rational prime. Let Φ be a cuspidal Bianchi eigenform of weight (k, k) and level Ω 1 (n), where (p)|n, with U p -eigenvalues a p , where v(a p ) < (k + 1)/e p for all p|p. Let Ω Φ be a complex period as in Proposition 2.12. Then there exists a locally analytic distribution μ p on Cl(K, p ∞ ) such that for any Grossencharacter of K of conductor f|(p ∞ ) and infinity type 0 (q, r) (k, k), we have where ψ p-fin is as in Theorem 7.3 and Z p (ψ) := ψ p-fin (π p )(1 − λ −1 p ψ(p) −1 ) : p f, 1 : otherwise.
The distribution μ p is (h p ) p|p -admissible in the sense of Definitions 5.10 and 6.14, where h p = v p (a p ), and hence is unique. We call μ p the p-adic L-function of Φ.
Proof. The eigenform Φ corresponds to a collection of h cusp forms F 1 , . . . , F h on H 3 ; associate to each F i a classical Bianchi eigensymbol φ F i with coefficients in a p-adic field L, and lift each to its corresponding unique overconvergent Bianchi eigensymbol Ψ i . Define and define a locally analytic distribution μ p on Cl(K, p ∞ ) by μ p := h i=1 μ i 1 i . Then by the work above, μ p satisfies the interpolation and admissibility properties.
Remark 8. As an example of where this theorem applies, suppose p splits in K and let E/K be a modular elliptic curve with supersingular reduction at both primes above p. Then to E we can associate a modular symbol which will have slope 1/2 at each of the primes above p. Accordingly, our construction will give the p-adic L-function of E.