Anticyclotomic p-adic l-functions and the exceptional zero phenomenon

Let A be a modular elliptic curve over a totally real field F, and let E/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p-adic L-function attached to (A,E), in terms of the Hasse-Weil L-function and certain p-adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p-adic L-function by means of the corresponding automorphic representation.

If A has either ordinary good or bad multiplicative reduction at all places above p, we obtain a better understanding of the arithmetic of A/F if we replace the Canalysis of L(A, s) by the a C p -analysis of its p-adic avatar L p (A, s), the (cyclotomic) p-adic L-function of A. This is a C p -valued function that interpolates the critical values L(A, ψ, 1), for any finite character ψ of the Galois group G p ≃ Z p of the cyclotomic Z p -extension F cyc p of F unramified outside p and ∞. The function L p (A, s) is defined as where ℓ : G p → Z p is a canonical isomorphism and µ p is a certain (cyclotomic) p-adic measure attached to A. By L p (A, s) interpolates L(A, ψ, 1), we mean that the measure µ p satisfies Gp ψ(γ)dµ p (γ) = e p (A, ψ)L(A, ψ, 1), for all finite characters ψ : G p → C × , where e p (A, ψ) is some Euler factor which is non-zero for almost all ψ. Observe that the p-adic L-function is univocally characterized by the C p -valued measure µ p . A p-adic analog of the Birch and Swinnerton-Dyer conjecture was stated in [11]. Let E/F be a totally imaginary quadratic extension. Some remarkable achievements towards the Birch and Swinnerton-Dyer conjecture have been obtained by means of the rich theory of Heegner points associated with E/F . This encourages us to consider A/E as an elliptic curve defined over E. Note that in this setting we can consider anticyclotomic abelian extensions of E which are linearly disjoint from F cyc p E. Indeed, for any prime ideal P of F dividing p, let E a P be the maximal abelian extension of E which is unramified outside P and ∞ and so that the complex conjugation τ ∈ Gal(E/F ) acts on G E,P := Gal(E a P /E) by −1. Up to torsion, G E,P is isomorphic to Z r p , where r = [F P : Q p ]. Motivated by the cyclotomic theory, one may ask if there is an analogous construction of p-adic L-functions attached to such anticyclotomic Z r p -extensions. The behavior of the local functional equation outside P provides a dichotomy in our scenario: on the one hand the definite case, where the number of finite places v outside P with the sign of the local functional equation (i.e. the local root number) ω v (A/E) = −1 is even; on the other hand the indefinite case, where that number of finite places is odd.
Assume that A has either ordinary good or multiplicative reduction modulo P. Our starting point is the construction of C p -valued measures of G E,P attached to A, with good interpolation properties. In the literature, we can find constructions of such measures in some particular cases: for F = Q and E/Q not ramified at p, we have the work of Bertolini-Darmon (see [5] and [6]); for arbitrary F , we have the work of Van Order also under certain restrictions for the ramification of E/F (see [18] and [17]). In this paper, we provide alternative constructions valid for arbitrary totally imaginary quadratic extensions E/F . Note the novelty of these constructions, even for F = Q, when E/Q ramifies at p. We denote by µ I E,P and µ II E,P the corresponding measures in the definite and indefinite situations, respectively. In analogy with the cyclotomic setting, our definite p-adic measure µ I E,P interpolates the critical values L(A/E, χ, 1) for any finite character χ of G E,P (Theorem 5.9). In the indefinite case, the corresponding p-adic measure µ II E,P interpolates p-adic logarithms of certain Heegner points whose height is given by the derivative L ′ (A/E, χ, 1) (Theorem 5.16). In both scenarios (and also including the cyclotomic setting), the interpolation property involves an Euler factor e P (A/E, χ) that is not identically zero.
The set of C p -valued measures Meas(G E,P , C p ) of G E,P has a natural structure of C p -algebra called Iwasawa algebra. The morphism deg : Meas(G E,P , C p ) → C p , that maps any measure to the image of the constant map 1, is indeed a C p -algebra morphism. The ideal I = ker(deg) is called augmentation ideal and the maximum power I r in which a measure µ lies is called order of vanishing of µ. Observe that, in the cyclotomic setting, L p (A, 0) = deg(µ p ) and the order of vanishing of L p (A, s) at s = 0 coincides with the order of vanishing of µ p .
In this paper, we mean by anticyclotomic p-adic L-functions the elements L I P (A, E) and L II P (A, E) of the Iwasawa algebra of G E,P defined by µ I E,P and µ II E,P , respectively. In the cyclotomic and definite anticyclotomic setting, the interpolation property relates the image deg(L I P (A, E)) with the critical value of the classical L-functions at s = 1 (analogously, in the indefinite anticyclotomic case the classical L-function is replaced by its derivative). This suggests that the order of vanishing of such p-adic L-functions coincides with the order of vanishing of the classical L-functions at s = 1 (resp. their derivatives in the indefinite anticyclotomic case). Nevertheless, a surprising phenomenon appears if A has split multiplicative reduction: the Euler factor e P (A/E, 1) vanishes and we observe a zero of the p-adic L-function, even when the classical L-function (or its derivative) is not zero. These extra zeros coming from the vanishing of the Euler factors are called exceptional zeros.
A first approach to understand this exceptional zero phenomenon is to compute the derivatives of the corresponding p-adic L-functions or, analogously in terms of Iwasawa algebras, their classes in the respective C p -vector spaces I r /I r+1 . Many authors have contributed to this research line: [4] an analogous formula involving the p-adic logarithm of a Heegner point (see also [6]). The case E/Q ramified at p is still an open problem even for F = Q. • In the indefinite anticyclotomic setting, the only results to date were for the case F = Q and p inert at E. Bertolini and Darmon proved in [3] that the image of L II p (A, E) in I/I 2 ≃ C p can be expressed in terms of the critical value L(A/E, 1) (see also [6]). The cases E/Q split or ramified at p have never been considered previously.
The main result of this paper (Theorem 7.5) establishes a formula for the classes of the p-adic L-functions L I P (A, E) and L II P (A, E) in I/I 2 , assuming that P splits in E. We remark that these results are valid for arbitrary totally real fields and, in turn, resolve the indefinite anticyclotomic case for F = Q.
In order to explain our result, let us introduce π, the automorphic representation of GL 2 (A F ) associated with A. The (twisted) Hasse-Weil L-function L(A/E, χ, s) coincides with the Rankin-Selberg L-function L(s − 1/2, π E , χ), where π E is the extension of the representation π to GL 2 (A E ). Actually, we will define our anticyclotomic measures by means of a certain Jacquet-Langlands lift π JL of π. For this reason we denote them by µ π,I E,P and µ π,II E,P , respectively L I P (π, E) and L II P (π, E) when considered as elements of the Iwasawa algebra. In §7 we will introduce a new type of L-invariants, the automorphic L-invariant vectors L P (π) ∈ I/I 2 . They are defined in terms of the cohomology of (S p -)arithmetic groups associated with Jacquet-Langlands lifts of π (see §6). If we denote by ∇L I P (π, E), ∇L II P (π, E) ∈ I/I 2 the classes of the corresponding p-adic L-functions, we obtain that where C E is a non-zero constant, C(π P ) is a non-zero Euler factor, and P T is a Heegner point with explicit height depending on L ′ (1/2, π E , 1). In the definite setting, this is a generalization of the result by Bertolini and Darmon in [5], for F = Q. With the spirit of the p-adic BSD conjecture introduced in [11], Bertolini and Darmon also conjectured in [2,Conjecture 4.6] such a result in the indefinite setting for F = Q and A(E) of rank 1, but replacing the automorphic L-invariant vector L P (π) by the corresponding geometric L-invariant.
Bearing in mind the case F = Q, we expect the L-invariant vector L P (π) to be related to the geometry of A/F P . Actually, L P (π) is defined differently in the definite and indefinite settings. In §6.3, we show in the definite setting that L P (π) is given by the geometric L-invariants. It is a work in progress to study the relation between indefinite L-invariant vectors and geometric L-invariants. Such relation, predicted by Conjecture 6.6, holds for F = Q thanks to the work by Greenberg-Stevens [9] and Longo-Rotger-Vigni [10].
The techniques used to prove these results are generalizations of the work by Spieß in [15], where the cyclotomic measure is defined by means of a certain group cohomology cocycle. In our setting, we construct our anticyclotomic measures by means of certain (S P )-arithmetic cocycles associated with the automorphic representations π JL and the torus attached to the extension E/F . In fact, our construction is given in such generality that we recover the Spieß p-adic measure if we allow E/F to be the trivial extension E = F × F . In this direction, our work helps us to understand the link between the cyclotomic and anticyclotomic framework.
Finally, let us remark that the techniques and results described in this paper could also contribute to research on the exceptional zero phenomenon whenever P does not split in E. A future research line is to compute the classes ∇L I P (π, E) and ∇L II P (π, E) in case P is inert in E, generalizing the results of Bertolini and Darmon for F = Q, or P ramified in E, which is an open problem even for F = Q.
1.1. Notation. For any field F , denoted by O F its ring of integers.
We denote the action of GL 2 (F ) on the projective line P 1 (F ) by: If F is a local field, we consider the Haar measure d × x of F × introduced in [21]. If F is a global number field, we denote by d × x v the corresponding Haar measure of the localization are the groups of finite ideles of F and E, respectively. Finally, we fix embeddingsQ ֒→ C andQ ֒→ C p .

Measures and Iwasawa algebras
2.1. Distributions and measures. Let X be a totally disconected locally compact topological space. For a topological Hausdorff ring R, we denote by C(X , R) the ring of continuous maps from X to R, and by C c (X , R) ⊆ C(X , R) the subring of compactly supported continuous maps. We denote by C(X , R) 0 and C c (X , R) 0 the corresponding rings if we consider R with the discrete topology. We denote by C ⋄ (X , R) ⊆ C(X , R) the set of continuous functions f with f (g)→0 as g→∞, more precisely, f can be extended to a continuous function on the one-point compactification of X by setting f (∞) = 0. We have We shall denote the R-module of M -valued distributions by Dist(X , M ).
Assume that R = K is a p-adic field, namely, a field of characteristic 0, complete with respect to a nonarchimedean absolute value | | : K → R, whose restriction to Q is the usual p-adic value. We denote the corresponding valuation ring by O K .
A norm on a K-vector space V is a function : V → R satisfying: Two norms in V are equivalent if they define the same topology on V . A normed K-vector space (V, ) is a (K−)Banach space if V is complete with respect to . Recall that any finite dimensional K-vector space admits a norm, any two norms are equivalent, and it is complete. The K-vector space C ⋄ (X , K) with the supremums norm f ∞ = sup γ∈X |f (γ)| is a K-Banach space.
) be a Banach space. An element µ ∈ Dist(X , V ) is a measure if µ is continuous with respect to the supremums norm, i.e. there exists C ∈ R, C > 0, such that X f dµ ≤ C f ∞ for all f ∈ C c (X , K) 0 . We will denote the space of V -valued measures on X by Meas(X , V ).
An element µ ∈ Meas(X , V ) can be integrated, not only against locally constant functions, but against any f ∈ C ⋄ (X , K), since C c (X , K) 0 is dense in the Banach space (C ⋄ (X , K), ∞ ). We obtain a continous functional Dist(X , L) ⊗ OK K −→ Dist(X , V ).

Iwasawa algebras.
Let G be a commutative pro-p group and let K be a p-adic field with valuation ring O K . Since O K is a lattice in the Banach space (K, | |), the K-vector space Meas(G, K) coincides with the tensor product Dist(G, O K ) ⊗ OK K. Let us consider the convolution product * : One checks that such convolution product endows Dist(G, O K ) with structure of O K -algebra, where the unit is given by the Dirac measure G f d1 := f (1) and the structural ring homomorphism corresponds to Observe that the natural map is a group homomorphism and the O K -module homomorphism is a Z p -module homomorphism. Indeed, The following result describes I/I 2 as the tensor product G ⊗ Zp O K .

Proposition 2.3 (Hurewicz Theorem).
Assume that G is a Z p -module of finite rank, then the map ϕ defines an isomorphism of O K -modules Proof. Let us consider the dual group Thus we have defined a O K -module morphism ψ : Since G is a Z p -module of finite rank, ι is an isomorphism. Thus, in order to prove our result, it is enough to show that ϕ is surjective.
Given the description of O K [[G]] as an inverse limit of group algebras, we observe that at each finite level This implies that the corresponding morphism ϕ H : G/H ⊗ Zp O K → I H /I 2 H is surjective. We conclude that ϕ is surjective an the result follows.
Corollary 2.4. Assume that G is a (free) Z p -module of finite rank and let G ∨ := Hom Zp (G, Z p ). Then the map Proof. Follows directly from the proof of Proposition 2.3.

3.2.
Universal unramified principal series. Let G = GL 2 (F ), and let us consider the following subgroups of G Hence Z is the center of G, and K is a maximal compact subgroup. For any ideal c ⊂ O write K 0 (c) ⊂ K for the subgroup of triangular matrices modulo c. Let R be a topological Hausdorff ring and assume α ∈ R × . Let us consider the unramified character µ α : . This provides a character on B and the induced R-representation If α = ±1, the representation Ind G B (µ −1 α ⊗µ α ) is reducible, since there is an invariant subspace of rank 1 over R generated by φ 0 (g) = α ν(det g) . We denote by (π R α , V R α ) the quotient representation of Ind G B (µ −1 α ⊗ µ α ) by this rank 1 subrepresentation. In the case α = ±1, we write (π R α , V R α ) for the induced representation Ind G Let us consider Let g ̟ = ̟ 1 . We consider the element of the Hecke algebra T ∈ H R K = (Ind G UZ 1 R ) U attached to 1 UZgω U , for U = K or K 0 (̟). Again by Frobenius reciprocity, elements φ i ∈ (V R α ) U provide G-module homomorphisms Theorem 20] Assume R is a ring endowed with the discrete topology. Then ϕ 0 is injective and ϕ 1 is an isomorphism. Moreover, if R is a field then ϕ 0 is also bijective.
Remark 3.2. We emphasize that ϕ 0 is not an isomorphism in general. If R is a domain and L its fraction field, we notice that Local distributions attached to a torus in GL 2 . Let T 2 be a 2-dimensional torus in G = GL 2 (F ). Note that T 2 acts on F 2 by means of the embedding T 2 ֒→ G. Definition 3.3. We say that T 2 is split if there is an element in F 2 that is an eigenvector for the action of all elements of T 2 . We say that T 2 is non-split otherwise.
Let R be a topological Haussdorf ring, and let us consider the action of T := T 2 /Z on C(T, R), given by t * f (x) := f (t −1 x). Assume that T 2 ⊂ B, thus B ∩ T 2 = Z and the natural map T → G/Z → B\G ≃ P 1 (F ) is injective. Let µ : B → R × be any continuous character trivial at Z. We can construct the following morphism which is well-defined, since B ∩ T = Z. Let α ∈ R × be as above, and let us consider the character µ = µ −1 α ⊗ µ α : B → R × . The composition ofδ µ T with the natural projection gives rise to a morphism (3.2) δ T : Since both Ind G B (µ) and π R α have trivial central character, restriction to T 2 provides a natural action of the group T = T 2 /Z. It is clear that both morphismsδ µ T and δ T are T -equivariant.
Lemma 3.5. Assume that R is a domain endowed with the discrete topology and α = ±1. Then there exists λ ∈ R such that where 1 H is the characteristic function of H, then the result will automatically follow. Indeed, we can choose λ ∈ R to be such that λδ T (1 H ) ∈ Ind G ZK 1 R /(T − a)Ind G ZK 1 R . Notice that the continuous equivariant map provides an injection of T in P 1 that sends 1 to ∞. The open subsets U n = {x ∈ P 1 , ord(x) < −n} (n ≥ 0) form a neighborhood basis for ∞ and their preimages give rise to neighborhood basis {H n } n of 1 ∈ T . We check that Let n 0 ∈ N be big enough that, for any n ≥ n 0 , This implies that δ T (1 Hn+1 ) = µ(g ̟ )π α (g −1 ̟ )δ T (1 Hn ) ∈ R[G]δ T (1 Hn ). By induction, we deduce that δ T (1 Hn ) ∈ R[G]δ T (1 Hn 0 ), for any n > 0. Since 1 Hn generate the T -module C c (T, R) and δ T is T -equivariant, the results follows.

3.4.
Torus and inner products. For this section, let F be a local field (either archimedean or nonarchimedean). Let D be the quaternion division algebra over F , and let T 2 be a two-dimensional torus as above. We fix an embedding T 2 ֒→ D × whenever it exists. Let χ : T = T 2 /Z → C be any continuous character. Proposition 3.6 (Saito-Tunnel [13], [16]). Let π be a representation of GL 2 (F ) with central character.
where π JL is the Jacquet-Langlands correspondence of π on D × .

Steinberg representations and intertwining operators.
In this section, we assume that our representation is Steinberg, namely, α = ±1. We denote by α s := q s α and µ s := µ −1 αs ⊗ µ αs , for any s ∈ C. We observe that the representation Ind G B (µ 1 ) admits an infinite dimensional subrepresentation (V C α1 , π C α1 ) by Corollary 8.3 with M = K, g 0 = 1 and h(g) = α ν(det(g)) φ(g) (in this case BM = BK = G). Let T 2 be a two-dimensional torus as above, and let T = T 2 /Z. Throughout this section, we assume that T 2 ⊂ B as above, hence T 2 ∩ B = Z. By Corollary 8.4 (comparison) we have that here the expression α ν(det(t)) is well-defined for t ∈ T , since ν(det(z)) ∈ 2Z, for z ∈ Z.
If t ∈ T and n ∈ N , let y ∈ T such that nω ∈ By −1 (this can be done for all n but maybe finitely many). Thus, by Comparison (Corollary 8.4).
. Thus, Due to the previous computation with the particular torus T 0 , we conclude Λ(φ 0 )(t) = I(φ 0 , 0, t) = 0 and (iii) follows. Finally, part (iv) follows from the fact that V C α is irreducible and Λ is nonzero.
3.6. Local pairings. In the previous sections, we have defined a T -equivariant morphism δ T : C ⋄ (T, C) −→ (π, V ), for any representation π := π C α associated with α ∈ C × . Moreover, we have defined the pairing β π,χ : V × V → C. The aim of this section is to compute β π, We assume that either α = ±1 or |α| 2 = q, the cardinal of the residue field of F . We know that the representation is unitarizable, namely, there is an invariant hermitian inner product ·, · : V × V → C. First, let us fix the inner product: Definition 3.11. If either α = ±1 or |α| 2 = q, we denote by ·, · the hermitian inner product on (V, π) := (V C α , π C α ) given by by Proposition 3.10 (ii).
The following result computes the pairing ·, · in terms of the torus T .
Proposition 3.13. Assume that T 2 ⊂ B is a 2-dimensional torus, T = T 2 /Z and ϑ 1 , ϑ 2 ∈ V . Then, there is a constant c T satisfying that The following proposition computes the period β π, Then, there is a non-zero constant c T , depending on T , such that and the result follows (we are allowed to exchange the order of the integrals, since f i ∈ C ⋄ (T, C)).
Let us assume for the rest of the section that (π, V ) is Steinberg, hence α = ±1. Proposition 3.14 implies that, in order to compute β π,χ (δ T (f 1 ), δ T (f 2 )), one has to describe the integral T Λ(δ T (f ))(t)χ(t)dt in terms of f ∈ C ⋄ (T, C). By Equation Again, the integral I T (χ, s) := T θ T (s)(t)χ(t)dt converges absolutely for Re(s) > 1/2 and admits analytic continuation to all s ∈ C. By abuse of notation, denote also by I T (χ, s) its analytic continuation. We conclude by Proposition 3.10 (ii). This implies that I T (χ, 0) = 0 in this case. We call this the Excepcional Zero Phenomenon.
Let E/F be the quadratic extension, such that T 2 = E × . Then T 2 is split if, and only if, E = F × F . If T 2 is non-split, we say that T 2 is inert if the extension E/F is inert, and we say that T 2 ramifies if E/F ramifies. Let O E be the integer ring of E. Let η be the quadratic character associated with T . Hence Note that the maximal compact subgroup K 0 of T admits a natural composition series K 0 ⊇ K 1 ⊇ · · · ⊇ K n ⊇ · · · such that [K n : K n+1 ] = q, for all n > 0. Given a character χ of T , its conductor n χ is the integer such that χ | Kn χ = 1 and χ | Kn χ−1 = 1.
There is an integer n T , depending on the embedding T ֒→ GL 2 (F ), such that the analytic continuation I T (χ, s) is given by where n χ is the conductor of χ and, for an unramified character χ, Proof. Let us assume that the embedding T 2 ֒→ GL 2 (F ) is given by Hence we compute Since any split torus is conjugated to the group of diagonal matrices, we compute that det(x) = x and c(x) = C(x − 1), for some C ∈ F . Since T 2 ⊂ B, we have C = 0. Hence we write n T = ord(C) > −∞. We compute Two of these integrals can be calculated easily: For any n ∈ N, write O × n := 1 + ̟ n O. Let n χ be the conductor of χ, namely, We conclude If we now consider the subrings O n := O + ̟ n αO, for every n ∈ N, we compute On the other side, if n = n χ − 1, With all these computations we obtain the value of I T (χ, s): We conclude that We compute Again we denote by n χ ≥ 1 the conductor of χ, namely, χ factorizes through Some computations analogous to the previous cases yield the formula Proof. This follows directly from the above result, observing that ζ(1) = 0, ζ(−2) −1 = 0, L(−1, π, χ) = 0 and L(0, π, χ) −1 = 0, if, and only if, χ(t) = α ν(det(t)) .
Then, there exist non-zero constants k T and k ′ T , depending on T , such that where n χ is the conductor of χ.

Cohomology of automorphic forms and Shimura curves
Let F be a totally real number field. Let G be the algebraic group associated with the multiplicative group of a quaternion algebra over F , that can be either totally definite or split at a single archimedean place σ. Write G(F ) + ⊂ G(F ) for the subgroup of elements of positive norm.
Let S be a finite set of nonarchimedean places, let F S = v∈S F v and F S = F ∩ v ∈S F v . Usually we shall consider G(F ) in G(F S ) by means of the natural monomorphism. Given any ring R, let N, M be a R[G(F )]-module and a R-module respectively. We define A S f (N, M ) to be the module of functions f : we have an isomorphism of (G(F ), G(F S ))-representations: Assume that M is an R-module and R → R ′ a flat ring homomorphism. Then the canonical map Proof. This result can be found essentially in [15,Corollary 4.7], let us reproduce its proof in our setting. It is enough to prove that Hence it is enough to prove that the N ) commutes with direct limits (any module is the direct limit of free modules of finite rank). By the Strong Approximation Theorem, there are only finitely many double cosets Since the group Γ gi is S-arithmetic, hence of type (VFL), the functor N → H q (Γ gi , N ) commutes with direct limits (see [14], p.101) and the result follows.

4.1.
Abel-Jacobi map on Shimura curves. In this section, we assume that G is the multiplicative group of a quaternion algebra that splits at a single place σ | ∞. Let us consider in this case the C-vector space A(C) of functions f : we assume that any f ∈ A(C) is K σ -finite, namely, its right translates by elements of K σ span a finite-dimensional vector space.
• We assume that any f ∈ A(C) must be Z-finite, where Z is the centre of the universal enveloping algebra of G(F σ ).
Write ρ for the action of G(F σ ) × G(F ) given by right translation, (A(C), ρ) defines a smooth G(F )-representation and a (G σ , K σ )-module, where G σ is the Lie algebra of G(F σ ). Moreover, A(C) is also equipped with the G(F )-action: In analogy with Remark 4.1, we define It is well known that the space of holomorphic forms where D is the discrete series representation of weight 2, f 2 ∈ D is a generator (see Apendix 2 §9), g f ∈ G(F ), g ∞ ∈ G(F σ ) + , and τ = g ∞ i ∈ h, where h is the Poincaré upper plane. Note that φ(f 2 )f −1 2 is a function on G(F σ ) + /SO(2)R + × G(F ) = H × G(F ). It defines a differential form on Y U because φ is G(F )-invariant. Moreover, ω φ is holomorphic since Lf 2 = 0 (see (9.37)), and φ is a morphism of (G σ , K σ )-modules.
The exact sequences (4.7) and (4.8) provide morphisms in cohomology We can identify H 1 (G(F ) + , A f (C) U ) with the singular cohomology of X U with coefficients in C. Moreover, once we interpret H 0 (G(F ), A σ (D, C) U ) as the holomorphic differentials of Y U , the compositions correspond to the morphisms by (4.9).
is the subspace where complex conjugation acts by ±1, respectively. Then it is clear makes the following diagram commutative Hence, deg * = ev ± • ι ± . The existence of ev ± 0 follows from a diagram chasing.
Hence, ev − 0 and ev + 0 define the real and imaginary part, respectively, of the image of (m, gU ) ∈ Div 0 (Y U ) under the Abel-Jacobi map.

4.2.
Multiplicity one. Let Π be an automorphic weight 2 representation of G(A F ) with trivial central character, let Π S be its restriction to G(F S ), and let L be its field of definition. Thus, there is a smooth irreducible L-representation V of G(F S ), such that Π S ≃ V ⊗ L C. For a field C containing L and a smooth semi-simple C- Let W be a G(F S )-representation over C of automorphic type. By strong multiplicity one, W Π is independent of the set S in the following sense: if S ′ ⊃ S and Proposition 4.6. Let G be a quaternion algebra over F that is either definite or splitting at a single archimedean place. Then, for any field C containing the field of definition of Π, the G(F )-representation H q (G(F ), A f (C)) is of automorphic type for all q ∈ Z. Moreover, Proof. Let U ⊂ G(F ) be an open compact subgroup.
In the definite case, the double coset space X U = G(F )\G(F )/UF × is a finite set of points. The group H k (G(F ), A f (C)) U can be identified with the singular cohomology of X U with coefficients in C. Thus, we deduce that H k (G(F ), A f (C)) = 0 for k > 0. Moreover, H 0 (G(F ), A f (C)) is in correspondence with the set of modular forms φ : G(F )\G(A F )/U G(F ∞ )F × → C, for some open compact subgroup U ⊂ G(F ). This proves that H k (G(F ), A f (C)) is of automorphic type and, by multiplicity one, H 0 (G(F ), A f (C)) Π = C.
In case G splits at a single archimedean place σ, the groups H k (G(F ) + , A f (C) U ) can be identified with the singular cohomology of the Shimura curve X U with coefficients in C. Thus, H k (G(F ) + , A f (C)) = 0, for k > 2, H 0 (G(F ) + , A f (C)) and H 2 (G(F ) + , A f (C)) contain only one-dimensional irreducible subrepresentations and, by Eichler-Shimura, H 1 (G(F ) + , A f (C)) Π = C⊕C. Moreover, the groups H k (G(F ) + , A f (C)) come equipped with an action of G(F )/G(F ) + ≃ Z/2Z given by conjugation. We can identify H k (G(F ), A f (C)) as the elements of H k (G(F ) + , A f (C)) fixed by G(F )/G(F ) + . This implies that H k (G(F ), A f (C)) is of automorphic type and H 1 (G(F ), A f (C)) Π ≃ C.

p-adic global distributions and measures
As in the previous section, F will be a totally real number field. Fix a prime P with uniformizer ̟ and residue characteristic q. We will denote by O P the integer ring of F P . 5.1. Definite anticyclotomic distributions. Throughout this section, E/F will be either a totally imaginary quadratic extension or the trivial extension E = F ×F . We will denote by T the torus E × /F × , likewise,T : Let P be a prime ideal of F above p, and denote by Σ ∞ the set of infinite places of F . Fixing an open subgroup Γ ⊆ T P , we shall consider the p-adic abelian extension of E associated with Γ, namely, the maximal abelian p-adic Galois group G Γ E,P such that the Artin map ρ A : A T /T → G Γ E,P factors through Γ.
Remark 5.1. Note that here we are considering either cyclotomic extensions of F , whenever E = F × F and therefore G Γ E,P is a quotient of A × F /F × , or anticyclotomic extensions of E, whenever E/F is totally imaginary and G Γ E,P is a quotient of Let G be the algebraic group associated with the multiplicative group of a quaternion algebra A over F that splits at P. Let (Π, V Π ) be an automorphic representation of G(A F ) with a trivial central character, and let us identify V Π with a subrepresentation of H 0 (G(F ), A(C)). Let also assume that ( Again by the Tensor Product Theorem, we have a G(F P )-equivariant morphism Remark 5.2. Note that the above morphism is not unique in general. Nevertheless, since dim C ( ′ v∤∞ V Πv ) U = 1, this morphism is unique up to constant if V Πv is trivial for all v | ∞. This will be the case when the quaternion algebra A is totally definite and the automorphic representation Π is of parallel weight 2.
Assume that there is a fixed embedding ı : E ֒→ A. Hence, we can consider E × , E × v and A × E inside G(F ), G(F v ) and G(A F ), respectively, by means of ı. Let us also assume that (E × P ∩ U P )/(F × P ∩ U P ) = Γ. We choose B P ⊂ G(F P ), some conjugate of the group of upper triangular matrices, in such a way that E × P ⊂ B P . Let R ⊂ C be any ring (endowed with the discrete topology) such that α ∈ R × . The map (3.2) provides a T P -equivariant R-module morphism and it induces the T P -equivariant morphism Finally, we define the distributionμ Π E,P on G Γ E,P as follows: where ρ A : A T /T → G Γ E,P is the Artin map, H ⊂ T P is an open subgroup small enough that g • ρ A is H-invariant, 1 H ∈ C c (T P , Z) ⊂ C c (T P , C) is the characteristic function of H, and H P is the maximal open subgroup of T P . Note that the above definition does not depend on the choice of H. Indeed, for any other H ′ ⊆ H, we where the second equality is obtained from the T P -equivariance of δ.
In the cyclotomic setting (E = F × F and T = F × ), the maximal abelian extension of F unramified outside ∞ and p is isomorphic to Z p . The following lemma describes the free part of the Galois group G Γ E,P , if E/F is a totally imaginary quadratic extension, and establishes a big difference between the cyclotomic and the anticyclotomic setting:

Lemma 5.3. Assume that E/F is a totally imaginary quadratic extension and let
Remark 5.4. Observe that G E,P does not depend on Γ.
Proof. First note that, in this case, the Artin map ρ A : A T /T Γ → G Γ E,P factors throughT /ΓT . Since Γ ⊆ T P is open,T /Γ = T P × T P /Γ where T P /Γ is discrete. Moreover, Hence T is a discrete subgroup ofT . We deduce that the Z p -rank of G Γ E,P coincides with the Z p -rank of O × E,P /O × F,P , which is clearly [F P : Q p ]. From now on, we shall consider the distributionsμ Π E,P restricted to functions supported on G E,P . 5.1.1. Waldspurger formula and interpolation properties. Let E be a quadratic extension of F , as above. Again, we denote by T the torus E × /F × . Let us consider (π, V π ), an irreducible cuspidal automorphic representation in L 2 (GL 2 (F )\GL 2 (A F )) with trivial central character and parallel weight (2, 2, · · · , 2), and let χ be a finite character of A T /T . Write L(s, π E , χ) for the Rankin-Selberg L-series associated with π, χ and E. We also consider the finite sets of places of F Let A be a quaternion algebra with ramification set Σ A and let G be the algebraic group associated with the multiplicative group of A. For any place v of F , denote by (π JL v , V π JL v ) the Jacquet-Langlands lift of the local representation . We also consider the global Jacquet-Langlands lift (π JL , V π JL ) of π on G(A F ). Invoking again the Tensor Product Theorem, . We define the following pairing on π JL is the quadratic character associated with the extension E/F (η is trivial if E = F × F ) and α π JL v are the local pairings defined in §3.4. The pairing β π JL ,χ is well-defined by Proposition 3.7. The following result is due to Waldspurger [19]: Moreover, the above expression is 0 unless Σ A = Σ χ π . By Saito-Tunnel (Proposition 3.6), if T 2 does not split at some place σ | ∞, then Σ ∞ ⊆ Σ χ π for every finite character χ. Since the C-vector space in C(G E,P , C) is generated by the set of finite characters with trivial component outside P, we deduce the following direct consequence of the Waldspurger formula: Corollary 5.6. Let E/F be a totally imaginary quadratic extension. Let G be the algebraic group associated with the multiplicative group of a quaternion algebra A. Then Σ 1 π \ {P} = Σ A is a necessary condition for the Jacquet-Langlands lift π JL and the distributionμ π JL E,P to be non-zero. In particular, A is totally definite and #(Σ 1 π \ {P}) + [F : Q] is even. Remark 5.7. The above result implies that whenever #(Σ 1 π \ {P}) + [F : Q] is odd we will not be able to construct non-zero anticyclotomic p-adic distributions of G E,P using the previous procedure. In §5.4, we will explain a different procedure to deal with this situation.
From now on, we assume that E/F is a totally imaginary quadratic extension, thus, we deal with the anticyclotomic situation. For a complete description using similar techniques to the cyclotomic case (interpolation property and exceptional zero phenomenon) see [15].

5.2.
Cohomological interpretation of µ π,I E,P . We showed in the proof of Lemma 5.3 that T is a discrete subgroup inT . The Artin map provides an isomorphism G E,P ≃T /T Γ, for some open compact subgroup Γ ⊂ T P . Hence, for any Hausdorff topological ring R, we have an isomorphism Moreover, the natural map is an isomorphism and, since T P /Γ is discrete, any f P ∈ C c (T P /Γ, R) 0 can be written as a finite combination of characteristic functions of T P /Γ. Since the representation (π JL , V π JL ) satisfies (π JL P , V π JL P ) ≃ (π C α , V C α ), equation (5.14) provides an element φ ∈ Hom G(FP ) (V C α , A f (C) U P ) = A P f (V C α , C) U P , which is unique up to constant by Remark 5.2. Since π JL is an automorphic representation Lemma 5.10. We have a natural isomorphism  a) (C = C orQ) by Lemma 3.1, the above result implies that we can assume that φ ∈ H 0 (G(F ), We deduce that where res : Q)) is the restriction map. 5.3. Heegner points and Gross-Zagier-Zhang formula. As in previous sections, let G be the multiplicative group of a quaternion algebra A/F that splits at a single archimedean place σ. Let E/F be a totally imaginary quadratic extension admitting an embedding E ֒→ A.
Proof. Note that A 0 π (C) is a free L-module of infinite rank, hence the first claim follows from the fact that H 0 (G(F ), A f (L)) π JL = 0 (Proposition 4.6).
On the other side, we know that ρ π is generated by ω φ ∈ ρ U π . Thus, if we prove that ∂(ϕ(ω φ )) = 0 the assertion will follow. By Remark 4.3, we have that under the identifications given in §4.1.1. For all c ∈ H 1 (X U , Z), Hence, the result follows.
The above Lemma implies that there exists a uniqueφ ∈ H 0 (G(F ), A f (∆, A 0 π (C))) π JL extending ϕ. Let us describeφ(ω φ ) geometrically in two different ways: On the one hand,φ can be described as the extension of the composition of the F -morphism ι : X U → Jac(X U ), given by a suitable multiple of the Hodge class (resp., of the cusp at infinity if A = M 2 (Q)), with the modular parametrization Jac(X U ) → A π . Indeed, on the degree zero divisors, by (4.13). On the other hand, let H U = (Ind UF × 1 Z ) U be the Hecke algebra of compactly supported and U -bi-invariant functions. By Frobenius reciprocity, H U acts on the 1-dimensional space ρ U π , providing a morphism of algebras We write where N : G(F ) →F × is the norm map. Notice that both H U and H 0 U act on Div(X U ).
Lemma 5.12. For any T ∈ H 0 U and any (z, gU ) ∈ Div(X U ), Proof. Since A is a quaternion algebra that splits at a single archimedean place, the norm map induces an isomorphism G(F )\G(F )/U ≃ F × \F × /N (U ). Write G(F )g a U for the class corresponding to F × aN (U ). We compute, has degree zero, hence T * (z, gU ) ∈ Div 0 (X U ).
Let T ∈ H 0 U such that λ(T ) = 0. Then, where AJ is the Abel-Jacobi map of (5.22). Moreover, the above description does not depend on T .
where ω q ∈ O FQ is an uniformizer and q = #(O FQ /ω q ). It is easy to check that T ∈ H 0 U . Moreover, λ(T ) = a Q − q − 1, where |a Q | ≤ q 1/2 , hence λ(T ) = 0. Let τ E ∈ h ⊂ C \ R be the unique point in the Poincaré hyperplane such that tτ E = τ E , for all t ∈ E × . Such τ E defines a G(F )-equivariant monomorphism Ind ). Shimura's reciprocity law asserts that the image of Φ T (ω φ ) lies in A 0 π (E ab ), where E ab is the maximal abelian extension of E, and the Galois action of Gal(E ab /E) is given by In analogy with §5.1.1, we consider the pairing on and let us also consider the Neron-Tate pairing on A π : , : A π (Q) × A π (Q) −→ R.
In contrast with Definition 5.8, we assume that #(Σ 1 π \ {P}) + [F : Q] is odd. Let A be the quaternion algebra with ramification set (Σ 1 π \ {P}) ∪ (Σ ∞ \ σ), for a fixed archimedean place σ. Let π JL be the corresponding Jacquet-Langlands lift to G, the algebraic group associated with the multiplicative group of A.
The previous conditions imply that A admits an embedding ı : E ֒→ A, that we fix once for all. Let us denote by T the torus associated with E × /F × . As above, let U = U P ×U P ⊂ G(F ) be an open compact subgroup such that dim C ′ v∤∞ π JL v U = 1 and U P = GL 2 (O P ) or K 0 (̟). We have constructed a morphism of G(F )representations Let R ⊂ C be any ring (endowed with the discrete topology) such that α ∈ R × . In analogy with the definite situation, we consider the T P -equivariant morphism Similarly as above, we decompose G Γ E,P = G Γ E,P × G E,P , where G Γ E,P is its torsion subgroup. By Lemma 5.3, G E,P ≃ Z does not depend on Γ. We will consider the anticyclotomic distributions of functions supported on G E,P . We know that the Artin map ρ A :T /T → G Γ E,P factors through Γ. For g ∈ C(G E,P , C), we define G Γ E,P g(γ)dµ π,II E,P (γ) := [H P : H] where the second equality is obtained from the T P -equivariance of Φ T •δ.
Remark 5.15. By Theorem 5.14, a necessary condition for the Jacquet-Langlands lift π JL and the above expression to be non-zero is precisely that the ramification

Theorem 5.16 (Interpolation Property).
There is a non-zero constant C E depending on E/F such that, for any continuous character χ : G E,P → C × , and n χ is the conductor of χ P .
Proof. The proof is completely analogous to the proof of Theorem 5.9, if the Waldspurger formula (Theorem 5.5) is replaced by the Gross-Zagier-Zhang formula (Theorem 5.14).

5.5.
Cohomological interpretation of µ π,II E,P . Recall that Notice that the composition π (E ab ))), we deduce that the image of (5.24) . Given the fixed embeddings O L ֒→Q ֒→ C p , let log p = log ω φ : A 0 π (E ab ) ⊗ LQ −→ C p . be the formal group logarithm attached to the differential ω φ ∈ Ω 1 XU . Composing with such formal logarithm, we obtain a morphism . From now on, we denote by µ π,II E,P ∈ Dist(G E,P , C p ) (again denoted the same way by abuse of notation) the p-adic distribution Thus µ π,II E,P ∈ Dist(G E,P , C p ) can be interpreted as the logarithm of the corresponding (A 0 π (E ab ) ⊗ L C)-valued distribution. Analogously to §5.2, equation (5.25) provides an element log φ ∈ H 0 (T, A P f (VQ α , C p ) U P ). If we recall the T P -equivariant morphism (5.21): κ : A P f (VQ α , C p ) U P −→ C c (T /Γ, C p ) ∨ 0 , we deduce that GE,P g(γ)dµ π,II E,P (γ) = κ(log φ) ∩ ∂g.
5.6. p-adic measures and p-adic L-functions. Let E/F be a totally imaginary quadratic extension. Let π be a cuspidal automorphic representation of GL 2 with parallel weight 2 and a trivial central character. Let Σ 1 π be the finite set of places defined in previous sections.
In the definite case (#(Σ 1 π \ {P}) + [F : Q] even), we have constructed the distribution µ π,I E,P defined by is some generator of the Jacquet-Langlands lift to a totally definite quaternion algebra.
In the indefinite case (#(Σ 1 π \ {P}) + [F : Q] odd), we have constructed the distribution µ π,II E,P defined by where log φ ∈ H 0 (T, A P f (VQ α , C p ) U P ) has been obtained from the p-adic formal logarithm of Heegner points associated with E on the Shimura curve attached to a quaternion algebra that splits at a single place.
Definition 5.17. We say that π is P-ordinary if π P is of the form Proposition 5.18. If π is P-ordinary then both µ π,I E,P and µ π,II E,P are p-adic measures.
We claim that, if ψ is in the image of the natural monomorphism then the distribution g → κ(ψ) ∩ ∂(g) is a p-adic measure. Indeed, by (5.19), the restriction of ∂ provides an isomorphism Moreover, by (5.20), Since α ∈Ō × (ordinary), by Lemma 3.5 there exists a fixed λ ∈Ō such that the restriction of the morphism κ of (5.21) factors through We conclude that, for any ψ ∈ H 0 (T, V), the map g → κ(ψ)∩∂(g) is a distribution in Dist(G E,P , λ −1 O Cp ). The claim follows from (2.1).
In the definite case, is an automorphic form on a totally definite quaternion algebra, hence its image is a finite set of values. This implies that, after scaling, we can assume that φ(v 0 ) has values inŌ. Since VŌ =Ō[G(F P )]v 0 , we deduce that φ(f ) has values inŌ, for any f ∈ VŌ. This implies that resφ ∈ H 0 (T, V), hence µ π,I E,P ∈ Meas(G E,P , C p ). In the indefinite case, ψ = log φ. By (5.23), where g ∈ G(F P ), g P ∈ G(F P ) ((g P , g) ∈ G(F )), T is some auxiliary Hecke operator in H U 0 , and AJ is the Abel-Jacobi map (5.22). By Shimura's reciprocity law, Thus, the Galois action on λ( P ⊂ C p be the local field extension generated by E Γ , let P Γ above P be its maximal ideal and let k be its residue field. Since G Γ E,P = G E,P × G Γ E,P , where G Γ E,P is finite and G E,P is the Galois group of an extension totally ramified at P, k is finite. Note that the formal logarithm (after normalization) is given by a formal series inŌ [[t]]. We have the exact sequence where G Aπ is the formal group of A π and A π (k) is finite. This implies that the set log p (A π (E Γ P )) has bounded denominators. We conclude that there exists λ ′ ∈ C p such that log Thus, log φ ∈ H 0 (T, V) ⊗ O Cp C p and µ π,II E,P ∈ Meas(G E,P , C p ).
Hence, in the P-ordinary case, µ π,I E,P , µ π,II E,P ∈ Meas(G E,P , C p ) define elements L I P (π, E) and L II P (π, E) in the Iwasawa algebra O Cp [[G E,P ]] ⊗ O Cp C p , called (anticyclotomic) p-adic L-functions. Interpolation properties (Theorem 5.9 and Theorem 5.16) and the vanishing of the local factor L(0, π P , 1) −1 when π P is the Steinberg representation with α = 1, show the appearance of what are known as Exceptional Zeros. In terms of Iwasawa algebras, this means that both L I P (π, E) and L II P (π, E) lie in the augmentation ideal I := ker(deg), where Corollary 5.19 (Exceptional Zero). Assume that π P is Steinberg with α = α(π P ) = 1. Then the p-adic L-functions satisfy in both definite and indefinite cases.

Automorphic L-invariants
Let G be the algebraic group associated with the multiplicative group of a quaternion algebra over F that can be either definite or split at a single archimedean place. In addition, assume that G(F P ) = GL 2 (F P ), where P is a prime ideal of F dividing p. Let (Π, V Π ) be an automorphic representation of G(A F ) of weight 2 with a trivial central character, and assume that (Π P , V ΠP ) is the Steinberg representation (π C α , V C α ) with α = 1. Let U ⊂ G(F ) be an open compact subgroup such that dim C (Π | G(F ) ) U = 1. In this section, we will assume that T 2 is a torus in G(F ) associated with a totally imaginary extension that splits at P. Write T = T 2 /Z as usual.
6.1. Extensions of the Steinberg representation. The main references in this section are [15, §2.7] and [8]. Let G := G(F P ) and B is its Borel subgroup. Recall that the Steinberg representation V C := V C 1 is defined by means of the exact sequence 1 for all g ∈ G(F P ). For any topological ring R, we defined in §3.2 its R-valued analogue (π R , V R ) := (π R 1 , V R 1 ). Since T 2 P splits, there are two eigenvectors v 1 , v 2 ∈ F 2 P and two eigenvalue morphisms λ 1 , λ 2 : T 2 P −→ F × P , such that tv i = λ i (t)v i and det(t) = λ 1 (t)λ 2 (t), for all t ∈ T 2 P . We fix the isomorphism ψ : T P ≃ → F × P provided by t → λ 1 (t)/λ 2 (t). Notice that we have the map that identifies B\G with P 1 (F P ). If T 2 P ⊂ B, the restriction of ϕ provides an injection where x 1 and x 2 ∈ P 1 (F P ) correspond to the spaces generated by v 1 and v 2 , respectively. Since ϕ(1) = ∞ ∈ ϕ(T P ), the points x i = ∞ can be seen as values in F P . If It is easy to check that Λ i (gt) = Λ i (g)λ i (t), for all t ∈ T 2 P . Let H ⊂ T P be the maximal open subgroup. Thus T P /H ≃ ̟ Z for some ̟ ∈ T P . Assume that x 1 = lim xn∈̟ n H x n in P 1 (F P ) and let U = x 1 ∪ n∈N ̟ n H.
6.2. L-invariants. As in §4.2, let C be a field endowed with the discrete topology and containing the field of definition of Π. Remark 4.1 implies that be the restriction of the class associated with the continuous morphism ord : F × P → Z (here the ring Z is considered with the discrete topology). Proposition 6.3. The cup product by c ord provides an isomorphism of 1-dimensional C-vector spaces Proof. Let us describe the extension E (ℓ) in the particular case ℓ = ord : This element is clearly K-invariant, thus, by Frobenius reciprocity, (φ K , 1) ∈ E (ord) KZ = Hom KZ (1 C , E (ord)) = Hom G (Ind G KZ 1 C , E (ord)). Let T ∈ End G (Ind G KZ 1 C ) = (Ind G KZ 1 C ) K be the Hecke operator corresponding to 1 Kg̟KZ . It is easy to compute that T (φ K , 1) = (q + 1)(φ K , 1), where q = O P /P, thus (φ K , 1) provides an homomorphism of C[G]-modules On the other side, the identity in Hom KZ (C, C) = Hom G (Ind G KZ 1 C , C) is compatible with T − a (a = q + 1) and provides a surjective homomorphism deg : Write K 0 := K 0 (̟), and let U ∈ End G (Ind G K0Z 1 C ) ≃ (Ind G K0Z 1 C ) K0 be the Hecke operator corresponding to 1 K0g̟ K0Z . The two natural forgetful morphism v 1 , v 2 : Ind G K0Z 1 C → Ind G KZ 1 C provides a surjective intertwining operator v 2 − v 1 : Ind G K0Z 1 C → ker(deg). It is easy to check that av 2 • (U − 1) = T v 1 • (U − 1), hence the above morphism gives rise to an isomorphism Conjecture 6.6. Let G 1 and G 2 be the multiplicative groups of two quaternion algebras over F that are either definite or split at a single archimedean place, but both split at P. If Π 1 and Π 2 are two automorphic representations of G 1 and G 2 , respectively, related by the Jacquet-Langlands correspondence, then the L-invariants coincide, namely, L P (Π 1 , ℓ) = L P (Π 2 , ℓ), for any continuous morphism ℓ : F × P → Z p . 6.3. Geometric L-invariants. Let A Π be the abelian variety of GL 2 -type associated with Π, and let B Π be its dual abelian variety. Let L be the field of definition of Π. Since Π is Steinberg with α = 1 at P, the abelian varieties A Π and B Π have purely multiplicative reduction at P, hence they admit the following analytic description: There is a pairing (determined up to canonical isomorphism) where X and Y are free abelian groups of rank d = [L : Q], and j is a bimultiplicative mapping such that the composition ord P • j tensored with Q gives a perfect duality of Q-vector spaces moreover, such that there is a pair of exact sequences of Gal(Q p /F P )-modules (X and Y endowed with trivial Galois action) where the morphisms on the left are induced by j.
Let O L ⊂ L be the endomorphism ring of A Π (and B Π ). Thus, X and Y are O L -modules and j(αx, y) = j(x, αy) for all x ∈ X, y ∈ Y and α ∈ O L . The non-degenerate pairing (6.30) provides an isomorphism of 1-dimensional L-vector spaces α : X ⊗ Q −→ Hom(Y ⊗ Q, Q) Given a continuous morphism ℓ : F × P → Z p , we consider the corresponding bilinear pairing and the corresponding homomorphism of L ⊗ Q p modules (free of rank 1) Hence there exist L P (A Π , ℓ) ′ ∈ L ⊗ Q p such that . We define L P (A Π , ℓ) ∈ C p to be the image of L P (A Π , ℓ) ′ under the homomorphism L ⊗ Q p → C p given by the fixed embeddingQ ֒→ C p .
Assume we are in the definite case, hence Π is generated by an automorphic form φ ∈ H 0 (G(F ), A f (O L ) U ), where G is the multiplicative group of a totally definite quaternion algebra. Since O L ≃ Z d , φ can be seen as . Fix g ∈ G(F P ). Since V R is the quotient of the space C(P 1 (F P ), R) modulo the subspace generated by 1 P 1 (FP ) , we can interpret f i (g) ∈ Hom(V Z , Z) as a distribution of P 1 (F P ) with integral values and such that P 1 (FP ) df i (g) = 0. Hence, it makes sense to consider the corresponding multiplicative integral. Let H p (Q p ) := P 1 (Q p ) \ P 1 (F P ) be the p-adic Poincaré hyperplane, write ∆ P = Z[H P ], is analogous to d 2 .

Exceptional zero phenomenon in the split case
The aim of the rest of the paper is to compute the class of L I P (π, E) and L II P (π, E) in I/I 2 in the presence of the Exceptional Zero phenomenon and in case T P splits. We will invoke Corollary 2.4 and so such class will be characterized by the integrals for all ℓ ∈ Hom Zp (G E,P , Z p ) = G ∨ E,P . Let H be the maximal open compact subgroup of T P ≃ F × P . We have the exact sequence Then we can consider the real manifold M = R × T P /Γ with the following natural action of T : Since we can identify H 0 (M, Z) with C c (T P /Γ, Z), we can consider the fundamental class ϑ of the oriented compact manifold M/T as an element of H 1 (T, C c (T P /Γ, Z)) by means of the identifications Let ℓ : G E,P → Z p be a continuous group homomorphism (ℓ ∈ C(G E,P , Z p ) not necessarily locally constant). It corresponds to a continuous homomorphism ℓ :T → Z p that factors through ΓT . Let ℓ P : T P → Z p and ℓ P : T P → Z p be its corresponding restriction to T P and T P . By topological reasons, ℓ P = 0. Let F be a fundamental compact domain forT /Γ under the action of T . Hence ∂ℓ = [l1 F ], where [l1 F ] ∈ H 0 (T, C c (T /Γ, Z p )) is the image ofl1 F and 1 F ∈ C c (T /Γ, Z) is the characteristic function of F . As above, we consider the natural injections ι P : T ֒→ T P and ι P : T ֒→ T P .
Let us consider again the open compact subset U = P 1 ∪ n∈N ̟ n H of P 1 . Since ℓ P = 0, the morphism ℓ depends only on ℓ P . Hence it makes sense to consider the cocycle z ℓP ∈ H 1 (T, C c (T P , Z p )) defined by z ℓP (t) := (1 − ι P (t))(ℓ P 1 U ).
Proposition 7.2. We have that Proof. Let F ⊂T /Γ be an open compact fundamental domain for the action of T such that HF = F . By definition, ∂(ℓ) is the image of the compactly supported functionl1 F in H 0 (T, C c (T /Γ, Z p )).
Let us consider a finite index subgroup of the form T = t Z × T P ⊆ T , such that ι P (T P ) ⊂ H and ι P (t) ∈ Γ. Let G ⊆ T P /H be the subgroup generated by t and F P a fundamental domain for T P /Γ under the action of T P . Hence, as above, U G × F P is a fundamental domain forT /Γ under the action of T . Note that, in this situation, by Shapiro's Lemma, H 1 (T , C c (T P /Γ, Z)) = H 1 (t Z , C(F P , Z)). We describe ϑ as the co-restriction of the class in H 1 (t Z , C(F P , Q)) defined by the cocycle Therefore, we compute where the second equality has been obtained from Remark 7.1. Since 1 UG×F P = g∈T /T g1 F andl is trivial on T , we deduce that the class ofl1 UG×F P = g∈T /T g(l1 F ) coincides with the class of [T : T ]l1 F in H 0 (T, C c (T /Γ, Z p )), hence the result follows.
Let us consider the subset T 1 = {t ∈ T : ι P (t) ∈ H} ⊂ T and let F 1 be a fundamental domain for T P /Γ under the action of T 1 . We can consider the subgroup X := U × F 1 ⊂T /Γ. As in previous results, let us identify V Z and C c (P 1 , Z). Hence ord : Proposition 7.3. The class of the characteristic function 1 X satisfies Proof. Since ι P (T ) is dense in T P , we have the exact sequence where, by abuse of notation, we also denote by ord the composition ord • ι P . Note that z ord is the image through the inflation map of an elementz ord ∈ H 1 (T /T 1 , (V Z ) T1 ).
On the other hand, let us consider ϑ 1 ∈ H 1 (T /T 1 , Z) ≃ H 1 (R/T, Z) given by the fundamental class of R/T (t ∈ T acts on R by tx = x + ord(ι P (t)) as above).
Then it is clear that [1 F1 ] ⊗ ϑ 1 is mapped to ϑ by means of the composition Thus, the result follows if we show thatz ord ∩ ϑ Indeed, if t ∈ T /T 1 is a generator then For any continous Z p -module homomorphism ℓ : G E,P → Z p , letl :T → Z p be the composition ℓ•ρ A , whereρ A :T → G E,P is the Artin map, and let ℓ P : T P → Z p be its restriction to T P (recall that ℓ P = 0). We will identify T P with F × P by means of the isomorphism ψ of §6.1. Recall the automorphic L-invariant L P (π JL , ℓ P ) introduced in Definition 6.5. Definition 7.4. By Corollary 2.4, the morphism in Hom Zp (G ∨ E,P , C p ) that maps ℓ ∈ G ∨ E,P to L P (π JL , ℓ P ) ∈ C p defines a class L P (π) ∈ I/I 2 , where I is the augmentation map of O Cp [[G E,P ]] ⊗ O Cp C p . Such class L P (π) is called automorphic L-invariant vector attached to π.
We have shown that L i P (π, E) ∈ I whenever π P Steinberg with α = 1. The main result of this section computes the class of L i P (π, E) in I/I 2 in terms of the automorphic L-invariant vector.
The (G, H)-module structure of I can be described as follows: Let L, R ∈ G be the Maass differential operators defined in [