HEEGNER CYCLES AND DERIVATIVES OF p-ADIC L-FUNCTIONS MARCO ADAMO SEVESO

Abstract. Let f be an even weight k ≥ 2 modular form on a p-adically uniformizable Shimura curve for a suitable Γ0 -type level structure. Let K/Q be an imaginary quadratic field, satisfying Heegner conditions assuring that the sign appearing in the functional equation of the complex L-function of f /K is negative. We may attach to f , or rather a deformation of it, a p-adic L-function of the weight variable κ, also depending on K. Our main result is a formula relating the derivative of this p-adic L-function at κ = k to the Abel-Jacobi images of so called Heegner cycles. Classification: 11F67 (Primary) 14F30, 11F80 (Secondary). Keywords: Modular forms, Heegner cycles, p-adic L-functions.

Contents 1. Introduction 2. Preliminaries and notations 3. p-adic families of modular forms and rigid analytic modular forms 3.1. Automorphic forms on definite quaternion algebras and p-adic families of automorphic forms 3.2. Automorphic forms on definite quaternion algebras, Jacquet-Langlands correspondence and rigid analytic modular forms 3.3. Specialization of p-adic families of automorphic forms 4. Cohomology of Mumford curves 4.1. de Rham cohomology of local systems 4.2. Definition of the symbol Φf (τ 1 , ..., τ r , P1 , ..., Pr ) 5. p-adic integration theory 5.1. Definite integrals 5.2. Semidefinite integrals 6. p-adic L-functions and the p-adic Abel-Jacobi map 7. Heegner cycles and p-adic Abel-Jacobi map 8. Proof of the main result 9. Final remarks 9.1. Relation with the Bertolini-Darmon-Iovita-Spiess p-adic L-function and interpolation properties 9.2. A restatement of the main result References

1 3 5 7 9 10 12 13 15 16 16 18 20 22 23 24 24 26 26

1. Introduction Let f∞ be a newform of even weight k > 2 on Γ0 (N ), which is an eigenform for the Hecke operators in TN , the Hecke algebra acting on the modular curve X0 (N ). Assume that the level of f∞ can be written as a product of three pairwise relatively prime integers N = pN + N − , where N − is the squarefree product of an odd number of prime factors and p is a rational prime. In this case the modular form f∞ corresponds, via the Jacquet-Langlands correspondence, to a modular form f on a certain Shimura curve XN + ,pN − uniformized by the p-adic upper halfplane Hp . More precisely let B be the indefinite quaternion algebra ramified at the primes dividing pN − , let OB be a fixed maximal order in B and let R = RN + ,pN − be an Eichler order of level N + in OB . The Shimura 1

curve X = XN + ,pN − over Q coarsely represents the functor associating to every Q-scheme S the set of isomorphism classes of triples (A, ι, C) where A/S is an abelian surface, ι : OB → EndS (A) is an inclusion of unitary rings defining an OB -module structure on A/S and C ⊂ A is a subscheme locally isomorphic to Z/N + Z, stable and locally cyclic under the action of R. Let B be the definite quaternion algebra ramified at the primes dividing N − ∞, let OB be a fixed maximal order in B and let R = RN + p,N − be an Eichler b and set Σ := R b× . Fix a local isomorphism order of level N + p in OB . Consider the profinite completion R ' ιp : Bp → M2 (Qp ) and let     e := ιp OB [1/p] ∩ Q Σl = ιp R [1/p]× Γ l6=p

e The Cerednik-Drinfeld Theorem asserts that Let Γ be the subgroup of the elements of reduced norm 1 in Γ. there is a rigid analytic isomorphism, defined over the quadratic unramified extension Qp2 of Qp Γ\Hp ' X an , where X an is the rigid analytification of X (see [BC]). The above identification allows us to identify f with a rigid analytic modular form, that we call again f ∈ Mk (Γ, E). This space denotes the set of rigid analytic functions on Hp defined over a complete field extension E/Qp2 such that   a b −k f (z) = (cz + d) f (γz) , for all γ = ∈ Γ. c d In [IS, Appendix 10.1] a construction of a Chow motive Mn is given, attached to the weight k modular forms on X, where n := k − 2. This motive is defined over Q and, for any field extension H/Q, we let Mn,H −

pN −new − of TN acts on it and, after taking its p-adic be its base change to  H. The pN -new quotient TN realization V := Hp Mn,Q , Qp , where Hp is the p-adic ´etale cohomology, we can consider the idempotent

component associated to f . It is shown in [IS, Lemma 5.8] that there is a GQ -module isomorphism between its p-adic realization and the p-adic representation attached by Deligne to the modular form f∞ (see [D]). In other words the modular form f can be used to study the p-adic representation attached to f∞ . Thanks to [IS, Lemma 10.1] we have CH m+1 (Mn ) = CH0m+1 (Mn ), where m := n2 . Thus, the p-adic Abel-Jacobi map takes the following form cl0m+1 = cl0 : CH m+1 (Mn ) → Ext1GQ (Qp , V (m + 1)) . Thanks to the work of Faltings (see [F89] and [F97]) there is a comparison isomorphism between the p-adic ´etale and the de Rham cohomology of Mn,E . This interpretation is made explicit in [CI], where Faltings’ definition is compared with the more explicit definition of Coleman (see [C2]). More precisely, there is a functor Dst from the category of p-adic representations of GE to the category M F := M FE (φ, N ) of filtered (φ, N )-modules over this comparison. an identification in M F between the de Rham  E, realizing    One has cohomology HdR Mn,Q , Qp and D := Dst Hp Mn,Q , Qp

. The above ext group is explicitly computed

in [IS]. After a base change from Q to E, there is an identification (let E 0 be the maximal unramified subextension of E/Qp ):
DE IS : Ext1GQ (Qp , V (m + 1)) = Ext1M F E 0 , D (m + 1) = m+1 . F DE Furthermore, since Mk (Γ, E) is identified with F m+1 DE , by Serre’s duality we have the E-dual. Summarizing, the p-adic ´etale Abel-Jacobi map takes the form:

DE F m+1 DE

∗

= Mk (Γ, E) ,

∗

log cl0 : CH m+1 (Mn,E ) → Mk (Γ, E) . Here we have to remark that, since the construction of IS depends (a priori) on the choice of a p-adic logarithm, we have chosen to include the above logarithm log cl0 in the notation. Perhaps it would be better to specify the choice of the branch, however, as explained in section 8, Lemma 8.1 implies the independence from such a choice. We refer the reader to the discussion at the beginning of section 7 for a clarifying explanation about the dependence of IS and log cl0 from the choice of a branch of the p-adic logarithm. Let K/Q be a quadratic imaginary field extension, of discriminant DK prime to pN , such that N + is a product of primes that are split in K, while pN − is a product of primes that are inert in K. In [BD96, Lemma 2.1] it is shown that the set of Heegner points associated to an order of K of conductor c prime to 2

N DK , that are naturally defined in X, is not empty. Let H/K be the ring class field associated to this order and fix an embedding σ p : H ,→ E. The p-adic Cerednik-Drinfeld uniformization yields a p-adic description of Heegner points. One may consider a higher weight analogue of Heegner points, namely Heegner cycles (n) yΨ ∈ CH m+1 (Mn,H ) defined as in [IS, Section
8], indexed by certain embeddings Ψ. × On the other hand let W := Homcont Z× p , Qp be the weight space. There is a natural inclusion Z ⊂ W, obtained by sending r ∈ Z to the function t 7→ tr−2 . The work of [BDI] can be used, following [BD07] in the weight k = 2 case, to define a (partial) two variable p-adic L-function L (f /K, Ψ, −, −) : U × Zp → Cp (κ, s) 7→ L (f /K, Ψ, κ, s) , defined on a small enough affinoid disk U ⊂ W such that k ∈ U . This is done in subsection 6. In the weight k = 2 case this is the (partial) two variable p-adic L-function studied in [BD07]. The restriction of it to the line κ = k, is the one variable p-adic L-function introduced in [BDIS] (see (29)). The paper [IS] studies the first derivative of this one variable p-adic L-function. One of its main results is the formula     d (n) (n) m+1 log cl0 yΨ (f ) . (L (f /K, Ψ, k, s))s= k = log cl0 yΨ (f ) + (−1) 2 ds Here Ψ := Ψ ◦ τ , where τ is the non-trivial element of GK/Q . Our main result computes the first derivative along the central critical line s = κ2 (see also Corollary 9.2). Theorem 1.1. The first derivative of L (f /K, Ψ, κ, s) in the weight direction is given by the following formula:      d   κ  1 (n) (n) m L f /K, Ψ, κ, = log cl0 yΨ (f ) + (−1) log cl0 yΨ (f ) . dκ 2 κ=k 2 Note that, the computation of the derivative in the direction κ = 2 in the weight k = 2 case is essentially the main result of [BD98] (see [BDIS, Introduction]), while the computation of derivative in the direction s = κ2 in the weight k = 2 case is performed in [BD07]. It should be noticed that these p-adic L-functions satisfy interpolation properties: for κ = k they have been investigated in [BDIS]; following [BD07] in the weight 2 case, they are investigated in [Se1] along the line s = κ2 (see also Section 9.1 for a brief discussion). An application of this formula is given in [Se1] in order to prove an analogue of [BD07, Theorem 1] and the rationality of Darmon cycles, a higher weight counterpart of the notion of Stark-Heegner points. The theory of Darmon cycles is generalized in [RS] in order to cover the compact case and the rationality result in [Se1] is generalized in [GSS] to the compact case. Acknowledgements. The author would like to thank Professor M. Bertolini for helpful suggestions. It is also a pleasure to thank Professors M. Bertolini, H. Darmon, A. Iovita and M. Spiess, since their papers were an indispensable inspiration. 2. Preliminaries and notations We will denote by E a field of characteristic 0, which is complete with respect to a discrete valuation and has a perfect residue field of characteristic p > 0. We denote by Qnr p the completion of the maximal unramified extension of Qp , and then we set E 0 := E ∩ Qnr p , i.e. the maximal unramified subextension of E. We will write Hp to denote the p-adic upper halfplane, viewed as a rigid analytic space, on which GL2 (Qp ) acts by fractional linear transformations. We write Hp (E) to denote the E-points. We will use the following notation for the branches of p-adic logarithm. We let log0 be the branch of the p-adic logarithm such that log0 (p) = 0 and, for every λ ∈ E, we let logλ := log0 −λ ordp : E × → E be the branch of the p-adic logarithm such that logλ (p) = −λ. When we do not make any specific choice of the p-adic logarithm we will simply write log. n k−2 As in the introduction, we let k > 2 be an even
integer, n := k − 2 and m := 2 = 2 . × × The weight space W := Homcont Zp , Qp is viewed as a rigid analytic space. The integers Z are embedded in W by sending the integer r to the function t 7→ tr−2 . Note that this normalization follows 3

[BD07], but not [BDI], where the integer r is sent to the function t 7→ tr . We always assume, for simplicity, that p 6= 2. In particular, we may consider the canonical decomposition Z× p = µp−1 × (1 + pZp ), where µp−1 is the group of p − 1-roots of unity, and write t ∈ Z× as t = [t] hti according to this decomposition. p If U ⊂ W is an open affinoid defined over E, every κ ∈ U (E) can be uniquely written as the product s × × κ (t) = ε (t) χ (t) hti , where ε : Z× is a character of order p − 1, χ : Z× is a character of order p → E p → E k−2 k−2 p and s ∈ OE . With our normalization the integer k ∈ U corresponds to the character k (t) = [t] hti , k−2 i.e. ε (t) = [t] , χ = 1 and s = k − 2. In general, up to shrinking U in a neighborhood of k ∈ Z, we can k−2 assume ε (t) = [t] and χ = 1 for every κ ∈ U (E). Then we define, for every α ∈ Qnr p , hαi (1)

hαi

κ−k κ−k 2

: :

s−k+2

= hαi = hαi

k−2 s 2− 2

= exp ((s − k + 2) log0 (α)) , = exp ((s − k + 2) /2 log0 (α)) .

This makes sense for every κ ∈ U , since hαi ∈ 1 + pOQnr and log0 (α) ∈ pOQnr . p p We let W := Q2p − {0} be the set of non-zero vectors in Q2p and consider the natural projection: π : W → P1 (Qp ) π ((x, y)) := x/y. For any Zp -lattice L in Q2p , we denote by L0 := L − pL the set of primitive vectors of L. We let L∗ := Z2p be the standard Zp -lattice in Q2p and set L∞ := Zp × pZp . The set of Zp -lattices in Q2p is denoted by L. p

p

We write L0 ⊂ L2 to denote the subset of those (L1 , L2 ) such that L2 ⊂ L1 , where ⊂ means inclusion with index p. If (L1 , L2 ) ∈ L0 , we define n o p TL1 ,L2 := L0 : L0 ⊂ L2 , L0 6= pL1 . If (L1 , L2 ), we set WL1 ,L2 := L01 ∩ L02 (then WL∗ ,L∞ = Z× p × pZp ) and we note that F (2) WL1 ,L2 = L0 ∈TL ,L WL2 ,L0 . 1

2

We let GL2 (Qp ) act from the left on Q2p by viewing the elements of Q2p as column vectors. In particular, GL2 (Qp ) acts on L and L0 , gTL1 ,L2 = TgL1 ,gL2 and gWL1 ,L2 = WgL1 ,gL2 . We remark that the GL2 (Qp )action is transitive on both L and L0 and that the stabilizer of L∗ (resp. (L∗ , L∞ )) is GL2 (Zp ) (resp. the group Γ0 (pZp ) of those matrices in GL2 (Zp ) whose lower left entry is c ∈ pZp ). If L ∈ L (resp. (L1 , L2 ) ∈ L0 ) we write gL ∈ GL2 (Qp ) (resp. gL1 ,L2 ∈ GL2 (Qp )) to denote any element such that gL L∗ = L (resp. gL1 ,L2 (L∗ , L∞ ) = (L1 , L2 )). In particular we may take gL1 ,L2 = gL1 and gL ∈ GL2 (Qp ) /GL2 (Zp ) and gL1 ,L2 ∈ GL2 (Qp ) /Γ0 (pZp ) are well defined. It follows that, if L ∈ L (resp. (L1 , L2 ) ∈ L0 ), then |L| := pordp (det gL ) (resp. |(L1 , L2 )| := pordp (det gL1 ,L2 ) = |L1 |) is a well defined quantity, called the generalized index. Recall the Bruhat-Tits tree T , whose vertices V are the homothety classes of Zp -lattices in Q2p , v = [L]   with L ∈ L. We set v∗ = Z2p . The ordered edges E are couples (v1 , v2 ) = ([L1 ] , [L2 ]) and we may always choose (L1 , L2 ) ∈ L0 . Set e∞ := ([L∗ ] , [L∞ ]). We abusively write e = [L1 , L2 ] when e = ([L1 ] , [L2 ]) and (L1 , L2 ) ∈ L0 . If we have [L1 , L2 ] = [L01 , L02 ], then (L01 , L02 ) = pr (L1 , L2 ) holds for some r ∈ Z: in other words pZ \L = V and pZ \L0 = E. We define the source s (e) and the target t (e) of e by the rule e = (s (e) , t (e)), while e := (t (e) , s (e)) denotes the opposite edge We write d (v1 , v2 ) to denote the distance between v1 , v2 ∈ V. Choose the following orientation E = E + t E − : write V + (resp. V − ) to denote the set of those vertices v which are at even (resp. odd) distance from v∗ ; define E + (resp. E − ) to be the set of those edges e such that the source s (e) ∈ V + (resp. s (e) ∈ V − ). If e ∈ E set Te := {e0 : s (e0 ) = t (e) : e0 6= e}. For any e ∈ E, let Ue ⊂ P1 (Qp ) be the open compact subset obtained by viewing the elements of P1 (Qp ) as ends of E and considering the subset of those ends originating from e. We define We := π −1 (Ue ) and note that, if e = [L1 , L2 ], F F F (3) We = r∈Z pr WL1 ,L2 = r∈Z Wpr L1 ,pr L2 and We = e0 ∈Te We0 . Of course we have gWe = Wge . Finally, we recall that there is a reduction map
red : Hp Qur →V p 4

uniquely characterized by the GL2 (Qp )-equivariance property and the fact that red (τ ) = v∗ if and only if |τ − i| ≥ 1 for i ∈ {0, ..., p − 1} and |τ | ≤ 1 (see [Dar, Proposition 5.1]). 3. p-adic families of modular forms and rigid analytic modular forms This section reviews the notion of p-adic families of modular forms on definite quaternion algebras, as presented in [BDI] in the split case. We refer the reader to [BD07], where the non-split weight 2 case is treated, and to [Se2] for the formalism employed. Having fixed E/Qp , for every open compact subset X ⊂ Q2p or X ⊂ P1 (Qp ), we write A (X) to denote the E-space of E-valued locally analytic functions on X, as defined for example in [BDI, Section 2] or [BDIS, Section 1.3] (see also [ST, Section 9] for an intrinsic definition working for any paracompact topological space X). Denote by D (X) := HomE-cts (A (X) , E) the continuous E-dual space, called the space of locally analytic distributions on X. Let A (U ) be the E-affinoid algebra of an open affinoid disk U ⊂ W (defined over E). Fix any κ ∈ U (E) and define, for any open compact (or paracompact) X ⊂ Q2p which is Z× p -stable:  Aκ (X) := F ∈ A (X) : F (tx, ty) = κ (t) F (x, y) for all t ∈ Z× p . Similarly, when X is Q× p -stable, we let Ak (X) ⊂ A (X) be the subspace of those F ∈ A (W ) such that n F (px, py) = p F (x, y) (recall n := k − 2), and we set Aκk (X) := Aκ (X) ∩ Ak (X). We let Dκ (X), Dk (X) and Dkκ (X) be the continuous E-dual spaces of Aκ (X), Ak (X) and Aκk (X). Suppose that X is Z× p -stable
× (resp. Q× p -stable). There is a natural R := D Zp -module structure on D (X) (resp. Dk (X)), defined by the formula  R R R F (tx, ty) dµ (x, y) dr (t) . F (x, y) d (rµ) (x, y) := × L0 Zp L0 ∗

∗

It extends the Z× p -action: tµ = δ t µ if δ t is the Dirac distribution at t. We can define the structure of an R-algebra on the E-affinoid algebra A (U ) of U by means of the formula h i R r 7→ κ 7→ Z× κ (t) dr (t) . p Up to the Amice-Velu Theorem establishing R ' A (W), it corresponds to the restriction map A (W) → b R D (X), the completed tensor product, and DkU (X) := A (U ) ⊗ b R Dk (X) A (U ). We set DU (X) := A (U ) ⊗ × when X is Qp -stable. × Since GL2 (Qp ) acts on L (resp. L0 ) through its quotient GL2 (Qp ) /Z× p , we have that Zp acts on L0 (resp. WL1 ,L2 ) for every L ∈ L (resp. (L1 , L2 ) ∈ L0 ). Similarly Q× p acts on W (resp. We ). It 0 follows that A (L ) and A (WL1 ,L2 ) (resp. Ak (W ) and Ak (We )) are endowed with a Z× p -action and an 0 0 R-module structure as explained above. Since g (L ) = (gL) and gW = W (resp. gWe = Wge ), L ,L gL ,gL 1 2 1 2 0
0 there are Z× -equivariant maps g : A (gL) → A (L ) and g : A (W ) → A (W ) (resp. a Q× gL1 ,gL2 L1 ,L2 p p0
0 equivariant map g : A (Wge ) → A (We )), defined by the rule F g := F ◦ g. Dually g : D (L ) → D (gL) , g : D (WL1 ,L2 ) → D (WgL1 ,gL2 ) and g : D (We ) → D (Wge ), defined by the rule (gµ) (F ) := µ (F g). The b R gives A (U )-linear operators g on DU and same applies to Aκ , Ak , Aκk and their duals. Applying A (U ) ⊗ U Dk , too.



For ? = U, κ, let C L0 , D? (W∗ ) (resp. C E, Dk? (W∗ ) , C L, D? (∗) or C V, Dk? (W ) ) be the space F F F of maps µ∗ : L0 → (L1 ,L2 )∈L0 D? (WL1 ,L2 ) (resp. µ∗ : E → e∈E Dk? (We ), µ∗ : L → L∈L D? (L0 ) or µ∗ : V → Dk? (W )) such that µL1 ,L2 ∈ D? (WL1 ,L2 ) (resp. µe ∈ Dk? (We ) or µL ∈ D? (L0 )). In view of the above discussion (see [Se2, Section 7] for more details), the rule (gµ∗ )L1 ,L2 := gµg−1 L1 ,g−1 L2 (resp. (gµ∗ )e := gµg−1 e , (gµ∗ )L := gµg−1 L or (gµ∗ )v := gµg−1 v ) defines left GL2 (Qp )-actions on these spaces.

We define Up -operators on C L0 , D? (W∗ ) and C E, Dk? (W∗ ) , using the decompositions (2) and (3), by the rule P L (Up µ∗ )L1 ,L2 := L0 ∈TL ,L µL2 ,L0 ∈ L0 ∈TL ,L D? (WL2 ,L0 ) = D? (WL1 ,L2 ) , 1 2 1 2 P L (Up µ∗ )e := pn/2 e0 ∈Te µe0 ∈ e0 ∈Te Dk? (We0 ) = Dk? (We ) .
The Up -operator on C V, Dk? (W ) is defined by the rule
P L (Up µ∗ )v := pn/2 d(v0 ,v)=1 µv0 |W v,v0 ∈ d(v0 ,v)=1 Dk? W(v,v0 ) = Dk? (W ) . (

)

5

The Up -operators are GL2 (Qp )-equivariant. We remark that there is a natural GL2 (Qp )-equivariant identification
'
(4) ds : C V, Dk? (W ) → C E, Dk? (W∗ ) , defined by the rule ds (µ∗ )e := µs(e)|We , under which the Up -operators correspond. The reader may check that the inverse is given by the rule P L δ s (µ∗ )v := s(e)=v µe ∈ s(e)=v D? (We ) = Dk? (W ) . In [BDI, Section 3] is explained how to define a continuous R-bilinear map R (5) : Aκ (X) × DU (X) → E, X R κ denoted by X F (x, y) dµU (when X is Z× p -stable). The same construction gives an R-bilinear map Ak (X)× U × U U U Dk (X) → E (when X is Qp -stable, see [Se2, Lemma 8]). If µ ∈ D (X) (or Dk (X)), we let µκ :=
R η κ µU ∈ Dκ (X) (resp. µκ ∈ Dkκ (X)) be the element defined by the rule µκ (F ) := X F (x, y) dµU ,
κ for F ∈ Aκ (X) (or Aκk (X)). The R-bilinearity implies that αµU = α (κ) µκ . These specialization maps induce GL2 (Qp ) [Up ]-equivariant maps η κ∗ from the spaces
above
of cocycles with ? = U to the corresponding κ U spaces obtained with ? = κ, by the rule η κ∗ µU := η µ , with s and element of L0 , E, L or V. Again ∗ s s

κ U κ . = α (κ) µ and we have αµ we write µκ∗ := η κ∗ µU ∗ ∗ ∗ Let Pn (E) be the E-vector space of polynomials of degree ≤ n over E with right GL2 -action (defined over E):     aX + b a b n (6) P (X) M := (cX + d) P for P ∈ Pn (E) , M = ∈ GL2 . c d cX + d Denote by Vn (E) := HomE (Pn (E) , E) the E-dual vector space, with its natural left GL2 -action. We recall that Pn := Pn (E) and Vn := Vn (E) carry a non-degenerate SL2 -invariant bilinear form (see for example [BDIS, Sec. 1.2] or [IS, (33)]): (7) 1

h−, −iPn

: Pn ⊗ Pn → E,

h−, −iVn

: Vn ⊗ Vn → E.




Let Ak P (Qp ) be the space of locally meromorphic functions on P1 (Qp ) with a pole of order at most

n at ∞, and let Dk P1 (Qp ) be its
continuous E-dual. We let GL2 (Qp ) act from the right on Ak P1 (Qp ) (and from the left on Dk P1 (Qp ) )
by the formula (6). The rules F 7→ y) := y n F (x/y) and F 7→
F (x, 1 k 1 k F (t) := F (t, 1) identify Ak P (Qp ) = Ak (W ) and then Dk P (Qp ) = Dk (W ) as GL2 (Qp )-modules. Under this identification Pn (E) corresponds to the space of homogeneous polynomials
of degree n in two variables, which we again denote by Pn (E). We will abusively identify Ak P1 (Qp ) = Akk (W ), as well as their duals. In particular, Pn (E) ⊂ Akk (W ) ⊂ Ak (W ). We write C 1 (Vn ) = C (E, Vn ) (resp. C 0 (Vn ) = C (V, Vn )) to denote
the space of maps c : E → Vn (resp. c : V → Vn ), endowed with the GL2 (Qp )-action (γc) (e) := γc γ −1 e . The Up -operator on C (E, Vn ) is defined by the rule P (8) (Up c) (e) := pn/2 e0 ∈Te c (e0 ) . We note that, restricting the integration to polynomials yields ν k : Dkk (We ) → Vn and then a GL2 (Qp ) [Up ]
k k equivariant map ν ∗ : C E, Dk (W∗ ) → C (E, Vn ).
For any P ∈ Ak (W ) (resp. Akk (W )) and τ , τ i ∈ Hp E 0 , the functions κ−k

κ FP,τ (x, y) := P (x, y) hx − τ yi

, κ−k

κ−k

κ FP,τ (x, y) := P (x, y) hx − τ 1 yi 2 hx − τ 2 yi 2 , 1 ,τ 2 R belong to Aκ (W ) (resp. Aκk (W )). We may therefore apply X (·) dµU with µU ∈ DU (X) (resp. µU ∈ × DkU (X)) to any of these functions restricted to X, that we assume to be Z× p -stable (resp. Qp -stable). 6

Proposition 3.1. The functions κ 7→

R

κ 7→

R

are locally analytic. Furthermore  d R (κ) U 0 FP,τ 1 ,τ 2 (x, y) dµ L dκ κ=k

L0

κ FP,τ (x, y) dµU

L0

κ FP,τ (x, y) dµU 1 ,τ 2

=


1 d R κ U + 0 FP,τ 1 (x, y) (x, y) dµ L κ=k 2 dκ
1 d R κ U . + 0 FP,τ 2 (x, y) (x, y) dµ L κ=k 2 dκ

b R µ, with α ∈ A (U ) and µ ∈ D (X). As explained in Section 2, near k ∈ Z Proof. We may assume µU = α⊗ k−2 s we may write κ (t) = [t] hti with s ∈ OE and then κ (x, y) = P + (s − n) log (hx − τ i yi) P + FP,τ i

∞ r X (s − n) r=2

κ FP,τ 1 ,τ 2

(x, y) =

∞ r X (s − n) r=0

=P +

2r r!

! log (hx − τ 1 yi)

r

r!

∞ r X (s − n) r=0

r

log (hx − τ i yi) P ,

2r r!

! r

log (hx − τ 2 yi)

P

∞ X s−n r (log (hx − τ 1 yi) P + log (hx − τ 2 yi) P ) + (s − n) br , 2 r=2

where br is a locally analytic function on X, to which we may apply µ. By definition R

F κ (x, y) dµU = αµ (P ) + α (s − n) µ (log (hx − τ i yi) P ) + α X P,τ i

∞ r X (s − n) r=2

r!

r

µ (log (hx − τ i yi) P ) ,

∞ X s−n r (s − n) µ (br ) , (x, y) dµ = αµ (P ) + α µ (log (hx − τ 1 yi) P + log (hx − τ 2 yi) P ) + α 2 r=2 R R κ U κ U thus showing that L0 FP,τ i (x, y) dµ and L0 FP,τ 1 ,τ 2 (x, y) dµ are analytic near k (see also [BDI, d d Lemma 4.5]). The operator dκ (resp. dκ (·)κ=k ) corresponds, in our local coordinates, to the operator d d (resp. (·) ). It follows that s=n ds ds

R

Fκ X P,τ 1 ,τ 2

U


d R κ U = α0 (n) µ (P ) + α (n) µ (log (hx − τ i yi) P ) , 0 FP,τ i (x, y) (x, y) dµ L κ=k dκ
d R α (n) κ U = α0 (n) µ (P ) + µ (log (hx − τ 1 yi) P + log (hx − τ 2 yi) P ) . 0 FP,τ 1 ,τ 2 (x, y) dµ L κ=k dκ 2 Comparing these expressions yields the claimed formula.



3.1. Automorphic forms on definite quaternion algebras and p-adic families of automorphic forms. Let B denote the rational quaternion algebra ramified at N − ∞ and let OB be a maximal order in b to denote the profinite completion of Z and set B b Let Σ = Q Σl be a decomposable b := B ⊗ Z. B. Write Z l b×. open compact subgroup of B For every prime l, let Hl be a standard choice of the unique division algebra over Ql . We can choose ' Ql -algebra isomorphisms ιl : B ⊗ Ql → M2 (Ql ) sending OB ⊗ Zl isomorphically onto M2 (Zl ) for every ' − l - N ∞, as well as Ql -algebra isomorphisms ιl : B ⊗ Ql → Hl for every l | N − ∞ (and then ιl (OB ⊗ Zl ) is the unique maximal order OHl of Hl ). Set   e Σ := ιp OB [1/p] ∩ Q Σl ⊂ GL2 (Qp ) Γ l6=p e Σ of elements of determinant 1. If V is an E-vector space, equipped with a and let ΓΣ be the subgroup of Γ left action by ιp (Σp ), let S (V ) be the space of functions ϕ : GL2 (Qp ) → V . We note that S (V ) is endowed with two commuting actions of GL2 (Qp ) and ιp (Σp ), namely for γ ∈ GL2 (Qp ) and u ∈ ιp (Σp ),
(γϕ) (g) := ϕ γ −1 g and (uϕ) (g) := u−1 ϕ (gu) . 7

Following [BDI, Section 1] and [BD07, Definition 2.1 and Remark 2.2], we give the following definition of a V -valued p-adic automorphic form on B of level Σ. Definition 3.2. A V -valued p-adic automorphic form on B of level Σ is an element of S (Σ, V ) := e ,ι (Σ ) Γ eΣ , S (V ) Σ p p . More explicitly it is a V -valued function on S (V ) such that ϕ (γgu) = u−1 ϕ (g), for γ ∈ Γ g ∈ GL2 (Qp ) and u ∈ ιp (Σp ). We will always assume ιp (Σp ) = Γ0 (pZp ). We write Σ∞ to denote the open compact obtained from Σ by replacing the local condition at p with the local condition ιp (Σ∞,p ) = GL2 (Zp ). Of special interest is the action of the following Up -operator. Write Fp−1 Γ0 (pZp ) σ 0 Γ0 (pZp ) = i=0 σ i Γ0 (pZp ) ,   1 0 ι (Σ ) where σ 0 := . The usual formalism of double cosets gives a well defined Up -operator on S (V ) p p 0 p (see [BD07, (40)]): Pp−1 Pp−1 Up ϕ := i=0 σ i ϕ, i.e. (Up ϕ) (g) := i=0 σ i ϕ (gσ i ) . Since the GL2 (Qp )-action on S (V ) commutes with the ιp (Σp )-action, it induces a well defined operator on S (Σ, V ). Definition 3.3. The space of p-adic families of
automorphic forms on B of level Σ parametrized by weights in U is by definition SU (Σ) := S Σ∞ , DU (L0∗ ) . The space of weight k automorphic forms on B of level Σ is by definition Sk (Σ) := S (Σ, Vn ). Example 3.4. The following level structure will be of particular interest for us. The open compact group Σ is defined by the following local conditions:  × (OB ⊗ Zl ) if l - N + p Σl = −1 ιl (Γ0 (N Zl )) if l | N + p Recall the Eichler order R of the introduction. Up to choosing the local isomorphisms in such a way that Rl corresponds to the standard Eichler order of level N Zl for every l | N + p, the group ΓΣ = Γ is the one considered there. e Σ , e.g. as in Example 3.4. For ? = U, κ, there are natural Up Proposition 3.5. Suppose that p ∈ Γ equivariant identifications (we define the Up -operator by transport, in the second row)
'
ΓeΣ '
ΓeΣ ·n S Σ, D? (WL∗ ,L∞ ) → C L0 , D? (W∗ ) → C E, Dk? (W∗ ) ,
'
ΓeΣ '
ΓeΣ ·n S Σ∞ , D? (L0∗ ) → C L, D? (∗) → C V, Dk? (W ) defined by the rules

µϕ,∗ L ,L := gL1 ,L2 ϕ (gL1 ,L2 ) , µϕ,∗ L := gL ϕ (gL ) (first identifications in each row) 1 2

−n/2 −n/2 µϕ,∗ [L ,L ] := |L1 | gL1 ,L2 ϕ (gL1 ,L2 ) , µϕ,∗ [L] := |L| gL ϕ (gL ) (composition in each row). 1

2

e Σ ·n means that we take invariants with respect to the twisted action γ ·n µ∗ := p−n/2 ordp (det γ) γµ∗ . Here Γ Proof. We first consider the map S D? (WL∗ ,L∞ )


Γ0 (pZp )


→ C L0 , D? (W∗ )

defined by the rule ϕ 7→ µϕ,∗ , where µϕ,L1 ,L2 := gL1 ,L2 ϕ (gL1 ,L2 ). Since we assume that ιp (Σp ) = Γ0 (pZp ), the stabilizer of (L∗ , L∞ ) in GL2 (Qp ), and that the equality gL1 ,L2 (L∗ , L∞ ) = (L1 , L2 ) holds, the map is easily checked to be well defined. It is a bijection, the inverse being given by the rule µ∗ 7→ ϕµ∗ , where ϕµ∗ (g) := g −1 µgL∗ ,gL∞ . A routine computation shows that it is GL2 (Qp )-equivariant. We compute Pp−1 µUp ϕ,L1 ,L2 = gL1 ,L2 (Up ϕ) (gL1 ,L2 ) = i=0 gL1 ,L2 σ i ϕ (gL1 ,L2 σ i ) . As remarked in [BD07, (38)], TL∗ ,L∞ = {σ i L∞ : i = 0, ..., p − 1} and σ i L∗ = L∞ . It follows that gL1 ,L2 σ i (L∗ , L∞ ) = (L2 , gL1 ,L2 σ i L∞ ) and gL2 ,gL1 ,L2 σi L∞ = gL1 ,L2 σ i . Hence gL1 ,L2 σ i ϕ (gL1 ,L2 σ i ) = µϕ,L2 ,gL1 ,L2 σi L∞ . The equality µUp ϕ,∗ = Up µϕ,∗ now follows from TL1 ,L2 = TgL1 ,L2 (L∗ ,L∞ ) = gL1 ,L2 TL∗ ,L∞ . 8


pZ


⊂ C L0 , D? (W∗ ) of those µ∗ such that µpr L1 ,pr L2 = pr µL1 ,L2 ,

pZ i.e. the pZ -invariant elements in C L0 , D(κ) (W∗ ) . Note that C L0 , D(κ) (W∗ ) is a GL2 (Qp )-submodule because pZ is in the center of GL2 (Qp ). Let us first assume for simplicity that ? = κ and define the map Consider the subspace C L0 , D? (W∗ )


pZ
C L0 , D? (W∗ ) → C E, Dk? (W∗ )   −n/2 µL1 ,L2 ·|WL1 ,L2 . The homogeneity property, together by the rule µ∗ 7→ µ∗ , where µ[L1 ,L2 ] := |L1 | −n/2

with the |L1 | -factor, make this map well defined. The disjoint decomposition (3) shows that, if F ∈ Aκ (WL1 ,L2 ), then FL1 ,L2 (x, y) := pnr F (p−r x, p−r y) for (x, y) ∈ W[L1 ,L2 ] and r such that (p−r x, p−r y) ∈ WL1 ,L2 gives a well defined element FL1 ,L2 ∈ Aκk (We ). Consider the rule µ∗ 7→ µ e∗ , where µ eL1 ,L2 (F ) :=   n/2 κ |L1 | µ[L1 ,L2 ] (FL1 ,L2 ). As we have FL1 ,L2 |WL1 ,L2 = F (for F ∈ A (WL1 ,L2 )) as well as F|WL1 ,L2 = L1 ,L2

F (for F ∈ Aκk (We )), it is easily checked that it gives the inverse of µ∗ 7→ µ∗ . Again a routine computation shows that it is GL2 (Qp )-equivariant. We leave it to the reader to check that the Up -operators correspond, once again exploiting the equality gL2 ,gL1 ,L2 σi L∞ = gL1 ,L2 σ i . When ? = U , [Se2, Lemma 8] joint with [GSS,
Lemma 163] implies that DU (WL1 ,L2 ) (resp. DkU W[L1 ,L2 ] ) is identified with the continuous A (U )-dual

b (WL1 ,L2 ) (resp. AU b of the A (U )-module AU (WL1 ,L2 ) ⊂ A (U ) ⊗A k W[L1 ,L2 ] ⊂ A (U ) ⊗Ak W[L1 ,L2 ] ) of those E-valued functions F on U (E) × X with X = WL1 ,L2 (resp. X = W[L1 ,L2 ] ) that are rigid analytic in the U variable and locally analytic in the X-variable and such that F (κ, tx) = κ (t) F (κ, x). Then a simialr argument applies.
e
Z e Σ , we have C L0 , D? (W∗ ) ΓΣ ⊂ C L0 , D? (W∗ ) p . To complete the proof we remark that, since p ∈ Γ e Σ -invariants, we may compose these two maps. Hence, after taking the Γ The claim when Σ, L0 and E are replaced by Σ∞ , L and V follows by similar arguments (or exploiting the identification (4) and the subsequent diagram (14)). 

A similar but simpler argument as the one exploited in the proof of Proposition 3.5 shows that the formulae cϕ ([L1 , L2 ]) := |L1 |

−n/2

−n/2

gL1 ,L2 ϕ (gL1 ,L2 ) , cϕ ([L]) := |L|

gL ϕ (gL )

yield identifications (9)

'

eΣ ·n Γ

S (Σ, Vn ) → C (E, Vn )

eΣ ·n Γ

= C 1 (Vn )

'

eΣ ·n Γ

, S (Σ∞ , Vn ) → C (V, Vn )

eΣ ·n Γ

= C 0 (Vn )

,

the first being Up -equivariant. 3.2. Automorphic forms on definite quaternion algebras, Jacquet-Langlands correspondence and rigid analytic modular forms. Let Σ be as in Example 3.4 until the end of this Subsection. As explained in [BD07, (37)], the space S (Σ, V ) (resp. S (Σ∞ , V )) is further endowed with an action of Tl -operators, say for simplicity for the primes l - N (resp. N/p). We note that S (Σ, V ) is further endowed with an action of the Wp -operator defined by the coset ω p Γ0 (pZp ). There are two natural maps GL2 (Zp ) GL2 (Zp ) S (Σ, V ) ⇒ S (Σ∞ , V ), namely δ s := coresΓ0 (pZ and δ t := p−n/2 coresΓ0 (pZ ◦Wp . We suppose now that p) p) V = Vn . Up to the identification (9), Wp is given by the rule (Wp c) (e) = pn/2 c (e) and the two maps δ s P P p−new and δ t correspond to δ s (c) (v) := s(e)=v c (e) and δ t (c) (v) := t(e)=v c (e). In particular, Sk (Σ) := e · ,p−new Γ

ker (δ s ) ∩ ker (δ t ) is identified with C 1 (Vn ) Σ n := ker (δ s ) ∩ ker (δ t ). The Jacquet-Langlands correN − −new spondence [BD07, Theorem 2.4] realizes Hecke equivariant identifications Sk (Σ) ' Sk (Γ0 (N )) and N − −new S (Σ∞ , Vn ) ' Sk (Γ0 (N/p)) . The degeneracy operators [BD07, After (48)] are identified with the duals of the operators δ s and δ t obtained by means of the Petersson scalar products discussed in [BD07, End of Subsection 2.2]. In particular, the Jacquet-Langlands correspondence induces a Hecke equivariant eΣ ·n ,p−new Γ

identification C 1 (Vn )

' Sk (Γ0 (N ))

pN − −new 9

.

As explained in [BDIS, After Proposition 2.16], the rule  e · ,p−new e c (e) if e ∈ E + Γ for e c ∈ C 1 (Vn ) Σ n (10) c (e) := −e c (e) if e ∈ E − e · ,p−new Γ

Γ

1 1 (Vn ) ⊂ C 1 (Vn ) is realizes a Hecke equivariant identification C 1 (Vn ) Σ n ' Char (Vn ) Σ , where Char the subspace of harmonic cocycles, defined by the conditions c ∈ ker (δ s ) and c (e) = −c (e) (then c ∈ ker (δ t )). Let Ω· (Hp ) ⊗ Vn be the (rigid analytic) de Rham complex, whose differential we denote by d. The residue map Γ

(11)

I:

(Ω (Hp ) ⊗ Vn ) Σ Γ 1  → Char  (Vn ) Σ ΓΣ d (O (Hp ) ⊗ Vn )

I (ω) (e) := Rese (ω) ∈ Vn . is a Hecke equivariant isomorphism (see [S, pag. 225] or [dS1, Section 3]). Given f ∈ Mk (ΓΣ , E), we write
ωf Pn to denote the associated Vn -valued differential form, i.e. ω f (z) := i=0 z i f (z) dz ⊗ ∂ i where ∂ i X j = δ ij . The rule f 7→ ω f gives a further Hecke equivariant isomorphism (see [S, pag. 225] or [dS1, Section 3]) '

Mk (ΓΣ , E) →

(12)

Γ

(Ω (Hp ) ⊗ Vn ) Σ   Γ d (O (Hp ) ⊗ Vn ) Σ

Summarizing, combining the identification induced by (9) on p-new parts, (10), (11) and (12) yields (13)

'

Mk (ΓΣ , E) →

Γ

e ,p−new ' (Ω (Hp ) ⊗ Vn ) Σ ' Γ ' Γ p−new 1  → Char (Vn ) Σ ← C 1 (Vn ) Σ ← Sk (Σ) ΓΣ d (O (Hp ) ⊗ Vn )

new

Recall now the modular form f∞ ∈ Sk (Γ0 (N )) that was considered in the introduction. By the Jacquet-Langlands correspondence discussed above, it gives rise to newforms (uniquely determined up to non-zero scalar factors) in the spaces appearing in (13). From now on we write f , ω f , cf , e cf and ϕf to denote a coherent choice, with respect to the identifications appearing in (13), of such newforms, ordered from left to right, i.e. f ∈ Mk (ΓΣ , E) and so on until ϕf ∈ Sk (Σ). Let wp ∈ {±1} be the eigenvalue of the Atkin-Lehner involution at p acting on f∞ , that may be also defined by the rule Wp e cf = wp pn/2 e cf . Definition 3.6. We say that f∞ (or f , ω f , cf or e cf ) is split (resp. non-split) multiplicative if wp = −1 (resp. wp = 1). We remark that f∞ is split (resp. non-split) multiplicative if and only if e cf = cf (resp. e cf = cf on E + − + − and e cf = −cf on E ). In general, we have e cf = cf on E and e cf = −wp cf on E . 3.3. Specialization of p-adic families of automorphic forms. As usual let ? = U, κ. The map

L0 S Σ∞ , D? (L0∗ ) → S Σ, D? (WL∗ ,L∞ ) given by the rule dW∗L∗ ,L∞ (ϕ) (g) := ϕ (g)|WL ,L is easily checked ∗ ∞ to be an isomorphism (this is indeed an application of Shapiro’s Lemma, see [BD07, Lemma 2.9]). We may therefore define an Up -operator on SU (Σ) by transport of structure. Proposition 3.5, joint with (4), yields a combinatorical description of this operator, since the commutativity of the following diagram is easily checked, where the vertical identifications are obtained from Proposition 3.5:
ΓeΣ ·n '
ΓeΣ ·n ds : C V, Dk? (W ) → C E, Dk? (W∗ ) (14) k k

' L0∗ ? 0 U dWL∗ ,L∞ S Σ∞ , D (L∗ ) → S Σ, D (WL∗ ,L∞ ) . Our interest in the space SU (Σ) of p-adic families of automorphic forms is justified by the fact that it comes equipped with weight r ∈ Z≥2 specialization maps for every such integer r ∈ U . In particular, we 10

have ρk : SU (Σ) → Sk (Σ) R (ρk (Φ) (g)) (P ) := WL

∗ ,L∞

P (x, y) dΦ (g) ,

where P ∈ Pn and P (x, y) := y n P (x/y) is the corresponding degree n = k − 2 homogeneous polynomial. L0 More explicitly ρk = ν k∗ ◦ η k∗ ◦ dW∗L∗ ,L∞ , where η k∗ (ϕ) (g) := η k (ϕ (g)), ν k∗ (ϕ) (g) := ν k (ϕ (g)) and η k and ν k have been defined after (5). Up to the following vertical identifications coming from Proposition 3.5 and (9), the specialization map ρk appearing in the lower row of the following commutative diagram admits the combinatorical description which is given by the upper row: (15)
ΓeΣ ·n ν k∗
ΓeΣ ·n ηk∗
ΓeΣ ·n e · d Γ → C (E, Vn ) Σ n → C E, Dkk (W∗ ) →s C E, DkU (W∗ ) C V, DkU (W ) k k k L0

SU (Σ)

dW∗

L∗ ,L∞

→


S Σ, DU (WL∗ ,L∞ )

ηk

→∗


S Σ, Dk (WL∗ ,L∞ )

νk

→∗

Sk (Σ) .

Here the map η k∗ (resp. ν k∗ ) in the first row has been defined after (5) (resp. (8)). The following theorem, due to [Ch], will play a crucial role in the definition of the p-adic L-functions (see [BDI, Theorem 3.10] for the formulation employed there). Let Σ be as defined in Example 3.4 and let eΣ = Γ e and ΓΣ = Γ). f ∈ Mk (Γ, E) be given (thus Γ Theorem 3.7. There exists U such that k ∈ U ⊂ W and a p-adic family of automorphic forms Φ ∈ SU (Σ) on B of level Σ parametrized by weights in U such that ρk (Φ) = ϕf and such that Φ is an eigenvector for the action of the Hecke operators. Furthermore, if Up Φ = ap Φ, then ordp (ap ) = n/2. If Φ0 is another such p-adic family of automorphic forms, then Φ − Φ0 ∈ Ik SU (Σ), where Ik ⊂ A (U ) is the ideal of functions that vanish at k.
Given µL ∈ Dk (L0 ) the projection π induces by push forward a measure π ∗ (µL )
∈ Dk P1 (Qp ) by the rule π ∗ (µL ) (F (t)) := µL (F (x, y)), where F (x, y) := y n F (x/y). Let Dk0 P1 (Qp ) be the subspace of
those distributions which are zero on Pn . Sitting inside of it, there is a space Dk0,b P1 (Qp ) of bounded distributions (see [T]). We recall that the methods of Amice-Velu and Vishik can be used to define an
Γ Γ ' 1 isomorphism (see [T, Proposition 9]) Char (Vn ) → Dk0,b P1 (Qp ) . The measure µc associated to c ∈ Γ 1 Char (Vn ) is uniquely determined by the equation R R (16) P (x, y) dµc (x, y) = Ue P (t) dµc (t) = c (e) (P ) , for all P ∈ Pn , We
where in the first equality we have used the identification Dk P1 (Qp ) = Dkk (W ) explained after formula (7), and P (x, y) = y n P (x/y) as usual. When c = cf we write µf to denote the corresponding measure.
V From now on we let Φ be as in Theorem 3.7 and we let µ∗ = (µL )L∈L and µV ∗ = µv v∈V (resp.


Γ· en µk∗ = µkL L∈L and µV,k = µV,k ) be the p-adic family of automorphic forms in C L, DU (∗) and ∗ v v∈V
Γ· en U k C V, Dk (W ) that corresponds to Φ via Proposition 3.5 (resp. their specializations via η ∗ ). Note that

−n/2 V,k µv = |L| π ∗ µkL , up to the identification Dk P1 (Qp ) = Dkk (W ), implicit in the following Proposition. Proposition 3.8. For every v = [L] ∈ V, −n/2

µV,k = |L| v


π ∗ µkL =



11

µf −wp µf

if v ∈ V + . if v ∈ V −

Proof. This is proved in [BDI, Proposition 4.4], when f is of split multiplicative type, and [BD07, Lemma 2.12] in the weight 2 case. We now generalize the result to our higher weight setting. We first claim that |v| + µV,k = (−wp ) µV,k (resp. v ∈ V − ). Indeed, thanks to (14), the v v∗ , where |v| = 0 (resp. |v| = 1) if v ∈ V fact that Up Φ = ap Φ implies
V n/2 P V ap µV . v = Up µ∗ v = p d(v 0 ,v)=1 µv 0 |W 0 (v,v )

k = µV v 0 |W(v,v0 ) . Applying η (v,v0 ) V,k −wp µV,k v|W(v,v0 ) = µv 0 |W(v,v0 ) . Reversing the roles of V,k −wp µV,k v|W(v0 ,v) = µv 0 |W(v0 ,v) . The claimed equality

In particular, for every v, v 0 such that d (v 0 , v) = 1, we have ap p−n/2 µV v|W and the well known equality ap (k) = −wp pn/2 yields v and v 0 we also have −wp µV,k v 0 |W

(v0 ,v)

= µV,k v|W

, i.e.

(v0 ,v) that Dkk

µV,k v0



= −wp µV,k now follows from the fact (W ) = Dkk W(v,v0 ) ⊕ Dkk W(v0 ,v) . v We now view en
Γ
Γ
Γ
Γ· U ⊂ C V, DkU (W ) = C V + , DkU (W ) ⊕ C V − , DkU (W ) µV ∗ ∈ C V, Dk (W )
Γ U V + V − and define µ bV by the rule µ bV bV ∗ ∈ C V, Dk (W ) v = µv , if v ∈ V , and µ v = −wp µv , if v ∈ V . By 0 0 construction, for every v, v such that d (v , v) = 1, we have −wp ap p−n/2 µ bV v|W

(17) 

and the specialization µ bV,k := η ∗k µ bV ∗ ∗



(v,v0 )

=µ bV v 0 |W

(v,v0 )

satisfies µ bV,k = µ bV,k v v∗ for every v ∈ V. To complete the proof of

the proposition we have to show that µ bV,k v∗ = µf . By construction and the definition of ds , the cocycle Γ V,k cf = e cf on E + and b cf = −wp e cf on E − , where we recall that e cf b cf ∈ C (E, Vn ) associated to µ b∗ satisfies b V,k + − is the cocycle associated to µ∗ . Since we already remarked that cf = e cf on E and cf = −wp e cf on E , we deduce b cf = cf . In light of (16) this means that, for every e ∈ E and every P ∈ Pn , R R P (x, y) dµV,k s(e) (x, y) = We P (x, y) dµf (x, y) . We
Γ 0,b V,k Since µV,k bV,k bV,k P1 (Qp ) . The v∗ = µf follows once we remark that µ v∗ ∈ Dk s(e) = µv∗ , the claimed equality µ
Γ
Γ V,k 0 1 fact that µ bV,k follows from the fact that µV,k = µV,k ∈ C V, Dkk (W ) v∗ ∈ Dk P (Qp ) v v∗ for every v ∈ V, µ∗ and b cf (e) = −b cf (e) for every e ∈ E. The boundedness (implicit and not proved in [BDI, Proposition 4.4] and [BD07, Lemma 2.12]), follows from the fact that b cf = cf is bounded (in the sense of [DT, Section 2.3]).  4. Cohomology of Mumford curves We let X = Γ\Hp be the Shimura curve already described in the introduction (see Example 3.4). We fix an open subscheme j : U
⊂ X (defined over E 0 ) such that the complement D is a finite number of distinct points P1 , ..., Pr ∈ X E 0 that reduce to vertices of the tree Γ\T
(in particular these points specialize to 0 smooth points of the special fiber). We write τ 1 , ..., τ r ∈ Hp E to denote fixed preimages of the points Pi under the projection p : Hp → Γ\Hp (assume for simplicity the stabilizer of the τ i s in Γ to be trivial, just to make (22) simpler). In this section we first review some of the results of [IS]. Then we prove Proposition 4.4. As explained in the proof of Theorem 7.1, joint with results of [IS] the proposition will allow us to relate the Abel-Jacobi image of Heegner cycles to the symbol Φf (τ 1 , ..., τ r , P1 , ..., Pr ) defined in subsection 4.2 when r = 1. As the proof of Theorem 7.1 will show, our Proposition 4.4 will play the role of [IS, Corollary 10.7] in the proof of [IS, Theorem 9.2]. We can not  directly apply [IS, Corollary 10.7] because this result gives a formula for the cup product I (ω f ) , PUλ (χ) Γ only when r = 2. The price to pay is to assume that a Γ-invariant Coleman primitive of ω f exists (see Remark 4.1). This amounts to choosing a suitable branch of the p-adic logarithm, as we will see in subsection 5.2. We will remove the dependence of our computations on this choice in Lemma 8.1. Finally, we relate Φf (τ , P ) evaluated at appropriate (τ , P ) to the derivatives of the p-adic L-functions to be defined in subsection 6. 12

4.1. de Rham cohomology of local systems. We write Vn to denote the locally free OX -module obtained an from the locally free OHp -module E (Vn ) := OHp ⊗E Vn by descending it to the quotient X. By definition i i HdR,c (U, Vn ) (resp. HdR (U, Vn )) is the hypercohomology of the complex L (−D) ⊗OX Vn → Ω1X ⊗OX Vn (resp. Ω·X (log D) ⊗OX Vn ). Here L (−D) is the sheaf of (rigid analytic) functions vanishing at D, while Ω·X (log D) is the de Rham complex of (rigid analytic) differentials with the possible exception of a log
arithmic pole at D. The pull-back of L (−D) (resp. Ω1X or Ω·X (log D)) is L −p−1 (D) (resp. Ω1Hp or
Ω·Hp log p−1 (D) ). Exploiting the fact that X is a rigid analytic quotient of Hp , one may compute the above cohomology groups by means of group cohomology (see [M, Ch. I, Appendix to Sec. 2]). This computation is carried out in [IS, Section 10]. One finds i HdR,c (U, Vn ) = H i (Γ, Cone (α) [−1]) and  
i HdR (U, Vn ) = H i Γ, Ω·Hp (Hp ) log p−1 (D) ⊗E Vn ,

(18) (19) where

α : Ω·Hp (Hp ) ⊗E Vn → IndΓ (Vn ) is the morphism concentrated in degree 0 defined by the formula

r

α (F ) := (α1 (F ) , ..., αr (F )) , with αi (F ) (γ) := F (γτ i ) .
Here Ind (Vn ) = M ap (Γ, Vn ) with Γ-action (γg) (δ) := γg γ −1 δ . 1 Exploiting (18) for i = 1 one finds that the group HdR,c (U, Vn ) is described as the set
r 1 (c, ω, f1 , ..., fr ) ∈ Z (Γ, O (Hp ) ⊗ Vn ) ⊕ Ω1 (Hp ) ⊗ Vn ⊕ IndΓ (Vn ) Γ

d (c (γ)) = ω − γ (ω) , αi (c (γ)) = fi − γ (fi ) for every γ ∈ Γ modulo the subgroup (∂F, dF, α1 (F ) , ..., αr (F )) : F ∈ O (Hp ) ⊗ Vn .
Here d is the differential of Ω (Hp ) ⊗ Vn , while (∂F ) (γ) := γ (F ) − F for (γ (F )) (z) := γF γ −1 z . Let K · (Vn ) := Cone (adr ) [−1], where ·

r

adr : Vn → IndΓ (Vn )

is the product of the natural inclusions ad regarding Vn ⊂ IndΓ (Vn ) as the sub Γ-module of constant functions. Exploiting (19) for i = 1 one finds that the group H 1 (Γ, K · (Vn )) is described as the set r

(c, f1 , ..., fr ) ∈ Z 1 (Γ, Vn ) ⊕ IndΓ (Vn ) ad (c (γ)) = fi − γ (fi ) for every γ ∈ Γ modulo the subgroup (∂v, ad (v) , ..., ad (v)) : v ∈ Vn .

Again (∂v) (γ) := v − γv. Let C01 (Vn ) (resp. C 0 (Vn )) be the set of maps c : V → Vn (resp. c : E → Vn such that c (e) = −c (e)). 1 The space CU1 (Vn ) (write Char (Vn ) when U = X) is defined by the following exact sequence: 0

r

→ CU1 (V ) → C 1 (V ) ⊕ IndΓ (V ) (c, f1 , ..., fr )

→ 7→

0 CP (V ) → 0 r δ (c) − i=1 χi (fi )

P s(e)=v f (e) and  fi (γ) if v = γred (τ i ) for some γ ∈ Γ χi (fi ) (v) = . 0 otherwise

where: δ (c) (v) =

As explained in [IS, pag. 379-80], there are distinguished triangles: IU,c

1 K · (Vn ) → Cone (α) [−1] → Char (Vn ) [−1] → K · (Vn ) [1] ,
IU Vn → Ω· (HΓ ) log p−1 (D) ⊗E Vn → CU1 (Vn ) [−1] → Vn [1] ,

where IU,c and IU are defined as follows. First of all note that   (d,α) r Cone (α) [−1] = O (HΓ ) ⊗E V → Ω1 (HΓ ) ⊗ V ⊕ IndΓ (V ) 13

in degree [0, 1]. Then IU,c and IU are the morphisms concentrated in degree 1 defined by the formulae: IU,c (ω, f1 , ..., fr ) (e) = Rese ω,
IU (ω) (e, γ 1 , ..., γ r ) = Rese ω, Resγ 1 τ 1 ω, ..., Resγ r τ r ω .

(20)

Here Resτ ω, for τ ∈ Hp , is the morphism appearing in the Gysin sequence [IS, Theorem 5.13]. These distinguished triangles give the following exact sequences: 0 0

→ H 1 (Γ, K · (Vn )) →

1

H (Γ, Vn )

ιU,c

→ ιU

→

1 HdR,c (U, Vn ) 1 HdR

(U, Vn )

Note that IX = I, the residue map (11), up to restriction to

IU,c

→ IU

→

Γ

1 Char (Vn )

→ 0,

Γ

→ 0.

CU1

(Vn )

(Ω(Hp )⊗Vn )ΓΣ d((O(Hp )⊗Vn )ΓΣ )

1 ⊂ HdR (X, Vn ).

λ Choosing a branch of the p-adic logarithm logλ allows us to define left inverses PU,c (resp. PUλ ) of ιU,c (resp. ιU ) as follows. Let F (Hp ) ⊗ Vn be the space of Vn -valued locally analytic functions on Hp that are primitives of elements of Ω1 (Hp ) ⊗ Vn . We refer the reader to [dS1, (2.3)] for the definition of Coleman primitive and we point out that, as explained in [dS1, (2.3)] and [IS, end of pag. 379], the definition of F (Hp ) ⊗ Vn depends on the choice of a branch of the p-adic logarithm, essentially because this amounts to the choice of a primitive of 1/z, z ∈ Hp . The existence of a Coleman primitive Fωλ of ω ∈ Ω1 (Hp ) ⊗ Vn is established in [C1]. Exploiting the definition which is given in [IS, pag. 380], one finds 

 (21) PU,c ([c, ω, f1 , ..., fr ]) = cλω − c, α1 Fωλ − f1 , ..., αr Fωλ − fr ,

where Fωλ ∈ F
(Hp ) ⊗ Vn is a (locally analytic) function such that dFωλ = ω (a Coleman primitive) and cλω (γ) := γ Fωλ − Fωλ ∈ F (H
p ) ⊗ Vn . One can easily check that cλω − c takes value in Vn and that, setting c0 := cλω − c and fi0 := αi Fωλ − fi , the relation ad (c0 (γ)) = fi0 − γ (fi0 ) is satisfied. 1 (U, Vn ) with the space of Γ-invariant Vn -valued meroThe morphism PUλ can be defined identifying HdR morphic differential forms on H that are holomorphic on p−1 (U ), modulo exact differentials, and then p
λ λ λ λ setting PU (ω) = γ Fω − Fω , where Fω ∈ F (Hp ) ⊗ Vn is a primitive of ω. The non-degenerate Γ-invariant bilinear form h−, −iVn (see (7)) induces by cup product a pairing between 1 1 HdR,c (U, Vn ) and HdR (U, Vn ). Since Im ιU,c and Im ιU are orthogonal (see [IS, Lemma 10.4]), the latter gives rise to pairings Γ

1 h−, −iΓ : Char (Vn ) ⊗ H 1 (Γ, Vn ) → E, Γ

h−, −iΓ,U : H 1 (Γ, K · (Vn )) ⊗ CU1 (Vn ) → E. The pairing h−, −iΓ,U can be made explicit as follows.(see [IS, (85)]): Pg Pr (22) hc, biΓ,U := i=1 hc (γ i ) , b (ci )iVn + j=1 hfj (1) , gj (1)iVn for c = (c, f1 , ..., fr ) and b = (b, g1 , ..., gr ) .

We end this subsection by recalling the so called hidden structures on the de Rham cohomology of X. Let σ be the absolute Frobenius of E 0 . Recall the category M FE (φ, N ) of filtered (φ, N )-modules that was considered in the introduction. Its objects are E 0 -vector spaces D such that D ⊗E 0 E is endowed with an exhaustive and separated decreasing filtration, there is a σ-linear operator φ : D → D (called the Frobenius) and there is a E 0 -linear endomorphism N : D → D such that N φ = pφN (called the monodromy operator). i i As explained in [IS, Section 3 and Section 10], the cohomology groups HdR,c (U, Vn ) and HdR (U, Vn ) belong 0 to M FE (φ, N ). Without going into details, let us simply remark that the underlying E -vector space D i i is obtained working with X/E 0 from HdR,c (U, Vn ) (resp. HdR (U, Vn )) and that the filtration is defined i i on D ⊗E 0 E, which is HdR,c (U, Vn ) (resp. HdR (U, Vn )) obtained working with X/E, by taking the Hodge filtration. 14

4.2. Definition of the symbol Φf (τ 1 , ..., τ r , P1 , ..., Pr ). In this subsection we assume that f ∈ Mk (Γ, E) is Γ λ 1 a modular form such that cλf = cλωf := PX (ω f ) ∈ H 1 (Γ, Vn ) is zero (do not confuse cλf with cf ∈ Char (Vn ) introduced after (13)). In subsection 5.1 we will see that this is always the case, once a suitable choice logλ of λ the branch of the p-adic logarithm has been made (recall that PX depends on this choice). More precisely we will choose λ in such a way that the corresponding L-invariant, as defined in Definition 5.3 (which depends on the choice of the logarithm), will be zero. When f is the modular form associated to an elliptic curve Z E (Cp ) = C× p /q , this choice corresponds to the one such that logλ (q) = 0, i.e. the one for which the logarithm logλ : C× p → Cp induces the p-adic logarithm logE (Cp ) → Cp . As we already remarked, a Coleman primitive Fωλf of ω f always exists, uniquely determined up to a constant factor in Vn , logλ being fixed. λ Remark 4.1. PX (ω f ) = 0 if and only if there exists a Coleman primitive Fωλf which is Γ-invariant, i.e.    
γ Fωλf = Fωλf for every γ ∈ Γ, where γFωλf (z) := γFωλf γ −1 z . If a Γ-invariant Coleman primitive exists, then it is uniquely determined.
Proof. By definition cλf (γ) = γ Fωλ − Fωλ in Z 1 (Γ, Vn ), so that ⇐ is clear. Conversely, suppose that  λ 
λ λ γ Fωλ −Fωλ = γv −v for some v ∈ Vn . Then F ω := Fωλ −v is a Coleman primitive such that γ F ωf = F ωf . λ

If Fωλ is another Γ-invariant Coleman primitive, their difference F ωf −Fωλ is a constant function which belongs to VnΓ = 0 (indeed Vn is an irreducible Γ-module as we assume n > 0).  Until the end of this section we fix once and for all a Γ-invariant Coleman primitive Fωλf , as granted by Remark 4.1 and our assumption cλf = 0 in H 1 (Γ, Vn ). Note that actually cλf = 0 in Z 1 (Γ, Vn ), if we agree to define cλf using Fωλf which is Γ-invariant (as already noticed in the proof of Remark 4.1).
Definition 4.2. The symbol Φf (τ 1 , ..., τ r , P1 , ..., Pr ) attached to the elements τ 1 , ..., τ r ∈ Hp E 0 and P1 , ..., Pr ∈ Pn = Pn (E) is defined by the formula E Pr D Pr Φf (τ 1 , ..., τ r , P1 , ..., Pr ) := i=1 Fωλf (τ i ) , ΛPi = i=1 Fωλf (τ i ) (Pi ) , Vn

where ΛPi = h−, Pi iPn . The following properties are trivial by definition and the Γ-invariance of Fωλf : • Φf is additive in (P1 , ..., Pr ) (for fixed (τ 1 , ..., τ r )); • for every γ 1 , ..., γ r ∈ Γ Φf (γ 1 τ 1 , ..., γ r τ r , P1 , ..., Pr ) = Φf (τ 1 , ..., τ r , P1 γ 1 , ..., Pr γ r ) . The edge morphism 1 Ω1 (X) ⊗ Vn → HdR (X, Vn ) in the Hodge spectral sequence corresponds, via the identifications (18) or (19) with U = X, to the map
Γ Ω1 (Hp ) ⊗ Vn → H 1 (Γ, Ω· (Hp ))

ω

7→ [0, ω] .


Γ

1 1 Moreover, the image of Ω1 (Hp ) ⊗ Vn in H 1 (Γ, Ω· (Hp )) = HdR (X, Vn ) is the subspace F m+1 HdR (X, Vn ) m+1 (see [dS2, Theorem 1.6] and [IS, Proposition 6.1]). Here F is the m + 1-step in the Hodge filtration of 1 HdR (X, Vn ). The inclusion j induces an epimorphism in M FE (φ, N ): 1 1 j∗ : HdR,c (U, Vn ) → HdR (X, Vn ) .


Γ 1 Let ω e f ∈ Ω1 (Hp ) ⊗ Vn be an element such that ω f = [0, ω e f ] ∈ F m+1 HdR (X, Vn ) (see [IS, Propo1 sition 6.1]). By definition of the exactness in the category M FE (φ, N ), there is ψ f ∈ F m+1 HdR,c (U, Vn ) such that j∗ ψ f = ω f . We need to make an explicit choice. Using the above group cohomological description, the morphism j∗ is given by the formula j∗ ([c, ω, f1 , ..., fr ]) = [c, ω]. Given the class ω f = [0, ω ef ] ∈ 15

1 F m+1 HdR (X, Vn ), say represented by (0, ω e f ), we define the class ψ f := [0, ω e f , 0, ..., 0]. It is an element of m+1 1 F HdR,c (U, Vn ) mapping to [0, ω e f ], i.e. j∗ ψ f = ω f . In particular (21) gives:    i
h λ (23) PU,c ψ f = cλf , α1 Fωλf , ..., αr Fωλf .


Lemma 4.3. If τ 1 , ..., τ r ∈ Hp E 0 , for every finite family ΛP1 , ..., ΛPr ∈ Vn , there exists an element 1 χ ∈ HdR (U, Vn ) such that Resτ i (χ) = ΛPi . For any such element

λ
 Φf (τ 1 , ..., τ r , P1 , ..., Pr ) = PU,c ψ f , IU (χ) Γ,U , Γ

1 where IU : HdR (U, Vn ) → CU1 (Vn ) is the residue map appearing in (20).

Proof. The exact sequence of [IS, Theorem 5.13] (Gysin sequence) 1 1 0 → HdR (X, Vn ) → HdR (U, Vn )

⊕i Resτ i

→

L

i

(Vn )τ i (−1) → 0

shows the existence of such an element χ. Choosing the Γ-invariant primitive Fωλf of ω f we find that cλf = 0. Now (23) and (22) give the claim: E

λ
 Pr D  λ  PU,c ψ f , IU (χ) Γ,U = = α F (1) , Res (χ) i τi ωf i=1 Vn D E Pr λ = . i=1 Fω f (τ i ) , ΛPi Vn

 1 Proposition 4.4. For every element χ ∈ F m+1 HdR (U, Vn ) such that Resτ i (χ) = ΛPi

 Φf (τ 1 , ..., τ r , P1 , ..., Pr ) = I (ω f ) , PUλ (χ) Γ ,

where we already remarked I = IX . 1 Proof. The structure of a filtered (φ, N )-module is defined in such a way that F m+1 HdR,c (U, Vn ) and m+1 1 F HdR (U, Vn ) are orthogonal (for the cup product h−, −iU ). The Reciprocity Law [IS, Theorem 10.6] 1 for the open subscheme U implies that, for every χ ∈ F m+1 HdR (U, Vn )

λ





 0 = ψ f , χ U = PU,c ψ f , IU (χ) Γ,U − IU,c ψ f , PUλ (χ) Γ .

Since IU,c = I ◦ j∗ and j∗ ψ f = ω f , we have



λ
 I (ω f ) , PUλ (χ) Γ = PU,c ψ f , IU (χ) Γ,U . The claim follows from Lemma 4.3.



5. p-adic integration theory 5.1. Definite integrals.
R τ Depending on the choice of a branch logλ of a p-adic logarithm, we may define an E-valued symbol τ 21 P ω λf having the properties suggested by the notation. Here τ 1 , τ 2 ∈ Hp E 0 , P ∈ Pn = Pn (E) and f ∈ Mk (Γ, E). We may define it by means of Coleman or Teitelbaum p-adic integration theory. Proposition 5.1. Let Fωλf be a Coleman primitive of ω f , obtained from the choice logλ of a p-adic logarithm, whose existence follows from [C1]. Then   i h R R τ2 t − τ2 λ λ λ P (t) dµf (t) . P ω f := Fωf (τ 2 ) − Fωf (τ 1 ) (P ) = P1 (Qp ) logλ τ1 t − τ1 Proof. This is essentially a restatement of [T, Theorem 4]. More precisely, recall that we defined ω f (z) :=
Pn i i i i j z f (z) dz⊗∂ , where ∂ X = δ . (H )⊗Vn j By definition, the Vn -valued Coleman primitive Fω f ∈ F i=0 Pn p of ω f is computed taking E-valued primitives Fzλi f ∈ F (Hp ) of the functions z i f and then Fωλf = i=0 Fzλi f ⊗ 16

∂ i . The proof of [T, Theorem 4], which applies more generally to every couple (τ 2 , τ 1 ) not necessarily of the form (γτ , τ ), gives the equality   R t − τ2 i t dµf (t) Fzλi f (τ 2 ) − Fzλi f (τ 1 ) = P1 (Qp ) logλ t − τ1
for i = 0, ..., n. Since ∂ i X j = δ ij , we have
Pn
Fωλf (τ ) X j = i=0 Fzλi f (τ ) ⊗ ∂ i X j = Fzλj f (τ ) . It follows that, for i = 0, ..., n,   h i
R t − τ2 i Fωλf (τ 2 ) − Fωλf (τ 1 ) X i = P1 (Qp ) logλ t dµf (t) . t − τ1 Now the claim follows by linearity in P ∈ Pn .



Proposition 5.2. For every λ ∈ E, setting vi := red (τ i ), Rτ R τ2 P P ω λf = τ 21 P ω 0f − λ τ1

[I (ω f ) (e)] (P ) .

e:v1 →v2

Proof. The following equality holds by definition (thanks to Proposition 5.1 we may use Teitelbaum p-adic integration theory):     R R t − τ2 t − τ2 P (t) log dµ (t) = dµf (t) + P (t) log λ f 0 P1 (Qp ) P1 (Qp ) t − τ1 t − τ1   R t − τ2 −λ P1 (Qp ) P (t) ordp dµf (t) . t − τ1     2 2 = 0 if v1 = v2 and ordp t−τ = χUe , The proof of [BDG, Lemma 2.5] shows that we have ordp t−τ t−τ 1 t−τ 1 the characteristic function on Ue , if e = (v1 , v2 ). It follows that (see equation (16))   R t − τ2 P (t) ord dµf (t) = cf (e) (P ) = [I (ω f ) (e)] (P ) . p P1 (Qp ) t − τ1 t−τ 0

t−τ 0

t−τ 2 t−τ 2 r = t−τ ... t−τ 11 in such a way that, setting vi0 := red (τ 0i ), e1 = More generally we may write t−τ 0 t−τ 0 1 r r−1
0 (v1 , v10 ), ei = vi−1 , vi0 for i = 2, ...., r and er+1 = (vr0 , v2 ). Then we find   R Pr+1 P t − τ2 P (t) ordp dµf (t) = i=1 [I (ω f ) (ei )] (P ) = [I (ω f ) (e)] (P ) . P1 (Qp ) t − τ1 e:v1 →v2

 We briefly recall the definition of the Teitelbaum L-invariant. We may define two Hecke equivariant morphisms Θλ , Φ : Mk (Γ, E) → H 1 (Γ, Vn ) , λ namely the composition Θλ of the identification (12) with the morphism PX recalled in subsection 4.1, and the composition Φ of the first two identifications considered in (13), i.e. (12) followed by I, with the boundary map δ arising from the exact sequence d

0 1 0 → Vn → Char (Vn ) → Char (Vn ) → 0

after taking Γ-invariants. Since Φ is an isomorphism (see [dS1, Theorem 3.9 and Corollary 6.6]), the following definition makes sense. Definition 5.3. The Teitelbaum L-invariant attached to the space of weight k modular forms is by definition Lλ := Θλ ◦ Φ−1 (and we simply write L = L0 to denote the genuine L-invariant). Given a newform f with eigenvalues in E, we also write Lλf ∈ E (resp. Lf = L0f ) to denote the projection onto the f -component. 17

Here we have to remark that, by multiplicity one and the Hecke equivariance of Θλ and Φ, Θλ (f ) = (ω f ) and Φ (f ) = δ (I (ω f )) differ by a scalar factor, and indeed by definition Θλ (f ) = Lλf Φ (f ). As it follows from [dS1, (4.4.5)], Lλ belongs to the Hecke Qp -algebra acting on modular forms and Lλf to the Hecke algebra generated by the eigenvalues of f . λ PX

new

Lemma 5.4. For every f ∈ Mk (X, E)

and for every λ ∈ E Lλf = Lf − λ.

R γτ λ Proof. By definition Θλ (f ) = PX (ω f ) is represented by the cocycle cλf,τ (γ) = τ P ω λf , while Φ (f ) = P δ (I (ω f )) is represented by the cocycle cf,v (γ) = cf (e) (P ) (where we recall that cf (e) = I (ω f ) (e)). e:v→γv

Thanks to Proposition 5.2 (taking v = red (τ ) ∈ V), cλf,τ = c0f,τ − λcf,v in Z 1 (Γ, Vn ). In particular,   Θλ (f ) = Θ0 (f ) − λΦ (f ) in H 1 (Γ, Vn ). Since Θ0 (f ) = L0f Φ (f ), we deduce Θλ (f ) = L0f − λ Φ (f ) and then Lλf = L0f − λ.


0


0

5.2. Semidefinite integrals. We write Hp+ E ⊂ Hp E to denote the subset of those τ such that


red (τ ) ∈ V + . In the rest of the paper we let Hpf E 0 be Hp E 0 (resp. Hp+ E 0 ), according to whenever new f ∈ Mk (Γ, E) is of split multiplicative
type (resp. non-split multiplicative type). Note that, in any case, SL2 (Qp ) (and hence Γ) acts on Hpf E 0 . Definition 5.5. A semidefinite integral for the choice logλ of a branch of the p-adic logarithm attached to new f ∈ Mk (Γ, E) is a function Rτ (τ , P ) 7→ P ωf ∈ E
0 f for every τ ∈ Hp E and P ∈ Pn = Pn (E) such that: R γτ Rτ • R P ω f = R (P γ) ω f Rfor every γ ∈ Γ; τ2 τ τ • P ω f − 1 P ω f = τ 21 P ω λf . Lemma 5.6. The above semidefinite integral is unique, if it exists (for the same choice logλ ). Proof. Let Λiτ ,f with i = 1, 2 be two such symbols and note that, by the property relating the semidefinite integral with the definite integral, the difference ∆f := Λ2τ ,f − Λ1τ ,f ∈ Vn is a well defined quantity, independent of τ . The first property, which reads Λiγ −1 τ ,f (P γ) = Λiτ ,f (P ), implies then ∆f (P γ) = Λ2γ −1 τ ,f (P γ) − Λ1γ −1 τ ,f (P γ) = Λ2τ ,f (P ) − Λ1τ ,f (P ) = ∆f (P ) i.e ∆f ∈ VnΓ = 0.



As we are going to show, a semidefinite integral attached to f always exists, but for the specific choice logf := logLf of the branch of the p-adic logarithm. In particular, we need to assume that the eigen
Γ· en values of f belong to E. Recall the p-adic families of automorphic forms µ∗ ∈ C L, DU (∗) and
en Γ· U µV introduced before Proposition 3.8. ∗ ∈ C V, Dk (W )
Proposition 5.7. Let vτ = [Lτ ] ∈ V be the reduction of τ ∈ Hpf E 0 . We have, for the choice logf of a branch of the p-adic logarithm, R   Rτ d  V  κ−k κ−k −n/2 d µ P (x, y) hx − τ yi P (x, y) hx − τ yi dµ (x, y) = . P ω f = |Lτ | 0 L v τ τ Lτ dκ dκ κ=k κ=k 0

−1

Moreover Lf = −2 (log ap ) (k) = −2ap (k)

a0p (k).


Γ U Proof. Recall the modified family µ bV that has been introduced in the proof of Proposition ∗ ∈ C V, Dk (W ) 3.8. Set  d  V  κ−k Jτ ,P := µ bvτ P (x, y) hx − τ yi , dκ κ=k 18

  
κ−k d where now τ is allowed to be an arbitrary element of Hp E 0 and note that Jτ ,P = dκ µV vτ P (x, y) hx − τ yi κ=k
when τ ∈ Hpf E 0 by definition of µ bV ∗ . The Γ-invariance property of Jτ ,P follows as in [BDI, Proposition 4.9], because µ bV bV,k = µf is zero on polynomials. In order to prove the relation with the ∗ is Γ-invariant and µ v definite integral, we claim that Rτ P (24) Jτ 2 ,P − Jτ 1 ,P = τ 21 P ω 0f − Lf [I (ω f ) (e)] (P ) . e:v1 →v2

Once this equality has been established, Proposition 5.2 with λ = Lf gives the claimed equality Jτ 2 ,P − Rτ L Jτ 1 ,P = τ 21 P ω f f . We need a generalization of [BDI, Proposition 4.10] to the non-split case. In order to
prove (24), as in the proof of [BDI, Proposition 4.10], we may assume that τ 1 , τ 2 ∈ Hp E 0 are such that e = (v1 , v2 ) where vi = red (τ i ). Since µ bV,k = µf , as in [BDI, Proposition 4.10] we find v d d (Ie )κ=k + (Ie )κ=k , dκ dκ      κ−k κ−k V − µV P (x, y) hx − τ 1 yi χWe and Ie = µV P (x, y) hx − τ 1 yi χWe v1 |We v2 |We − µv1 |We Jτ 2 ,P − Jτ 1 ,P =

 where Ie = µV v2 |We

R τ2 τ1

P ω 0f +

(here χY is the characteristic function of Y ). The relation (17) implies −wp ap p−n/2 µ bV bV v1 |We = µ v2 |We and V V −wp ap p−n/2 µ bv2 |We = µ bv1 |We . It follows that      κ−k Ie = − 1 + wp p−n/2 ap µV P (x, y) hx − τ 1 yi χWe , v1 |We     κ−k 1 + wp p−n/2 ap µ bV P (x, y) hx − τ yi χ Ie = 1 We . v2 |We
Note that α = ± 1 + wp ap p−n/2 vanishes at k, in light of the equality ap (k) = −wp pn/2 . Hence [BDI, Γ is b cf = cf , implies that Lemma 4.10], joint with the fact that the cocycle in C (E, Vn ) associated to µ bV,k ∗ d d (Ie )κ=k = −wp p−n/2 a0p (k) cf (e) (P ) and (Ie )κ=k = wp p−n/2 a0p (k) cf (e) (P ) . dκ dκ −1

Since cf (e) = −cf (e) and ap (k) Jτ 2 ,P

= −wp p−n/2 , Rτ −1 − Jτ 1 ,P = τ 21 P ω 0f + 2ap (k) a0p (k) cf (e) (P ) . −1

As in the proof of [BDI, Theorem 4.12], this implies that Lf = −2ap (k) −1

a0p (k) and (24) follows.



Remark 5.8. Specializing to k = 2, we see that Lf = −2ap (2) a0p (2) = 2wp a0p (2). If f is split (resp. non-split), Lf = −2a0p (2) (resp. Lf = 2a0p (2)). This formula agrees with [BD07, Theorem 2.25] when f is split. However, in the non-split case, [BD07, Theorem 2.25] reads again Lf = −2a0p (2), which differs from our formula. It seems to the author that there is a mistake in the proof of [BD07, Proposition 2.19], on which [BD07, Theorem 2.25] is based. More precisely the proof of [BD07, Proposition 2.19] requires [BD07, Lemma 2.23], where the non-split case is left to the reader and it seems that there is a mistake in the final 0 result in this case. This is the reason we preferred to give a complete proof of Lf = −2 (log ap ) (k), rather than refer the reader to a possible adaptation of [BD07, Proposition 2.19]. P To support our formula we argue as follows. Suppose that f∞ is non-split at p, let f = n≥1 an q n ∈ P A (U ) [[q]] be a Coleman family of level Γ0 (N ) (tame level N/p) such that fk = f∞ and let f χ := n≥1 aχn q n , where aχn := an χ (n) and χ is a quadratic Dirichlet character such that χ (p) = −1, i.e. the Dirichlet character χ χ of a quadratic field
Qχ such that p is inert. Then fk = f∞ , the χ-twist of f∞ , which is a weight k modular 2 form on Γ0 N c for c the conductor of χ that we may assume to be prime to N . The Fontaine-Mazur M L-invariant LF of a modular form g is a local invariant of the restriction of the Deligne representation Vg g attached to g at p. Furthermore, it is insensitive to base changes. Since over Qp2 we have Vf = Vfkχ and the Fontaine-Mazur L-invariant equals the Teitelbaum L-invariant (see [IS, Theorem 6.4]), we conclude M M Lf = LF = LF f f χ = Lf χ . 19

On the other hand f χ is a split modular form because aχp (k) = ap (k) χ (p) = −pn/2 and [BDI] gives (we
0 have aχp (k) = ap (k) χ (p) and aχp = a0p χ (p))
0 −1 −1 Lf χ = −2aχp (k) aχp (k) = −2ap (k) a0p (k) . L

L

The meaning of Lemma 5.4 is that Lf f = 0 and hence PX f (ω f ) = 0. As it is shown in [dS1, Corollary λ (4.2.2)], PX is an isomorphism for all but a finite number of choices of logarithm; the cases where this is not true are called exceptional and we are precisely in one of these cases. This means that there exists L a Γ-invariant Coleman primitive Fωff (see Remark 4.1) and the symbol Φf considered in subsection 4.2 is defined. Using the right hand side of Proposition 5.1 as a definition for the definite integral, we see that Rτ L Φf (τ 2 , P ) − Φf (τ 1 , P ) = τ 21 P ω f f , while the property Φf (γτ , P ) = Φf (τ , P γ) was already noticed in L

subsection 4.2 (here we have to use the fact that Fωff is Γ-invariant). Hence we find: Proposition 5.9. We have, for the choice logf of a branch of the p-adic logarithm, Rτ P ω f = Φf (τ , P ) These propositions and Lemma 5.6 imply the following corollary.
new Corollary 5.10. For every f ∈ Mk (Γ, E) , τ ∈ Hpf E 0 and P ∈ Pn = Pn (E):  R κ−k −n/2 d P (x, y) (x − τ y) dµ (x, y) , Φf (τ , P ) = |Lτ | Lτ L0τ dκ κ=k where vτ = [Lτ ] ∈ V is the reduction of τ .
Rτ Remark 5.11. As it follows from the proof of Proposition 5.7, P ω f may be extended to τ ∈ Hp E 0 R τ using the family µ bV P ω f defined using Φf (τ , P ) clearly extends to ∗ . This is consistent with the fact that
0 τ ∈ Hp E . 6. p-adic L-functions and the p-adic Abel-Jacobi map Let K/Q be an imaginary quadratic number field and assume that the following conditions hold: • the discriminant DK of K is prime to N = pN + N − ; • the prime divisors of pN − are inert in K; • the prime divisors of N + are split in K. Let O be the Z [1/p]-order in K of conductor c prime to DK N . An embedding Ψ : K → B is called optimal of O into the Eichler Z [1/p]-order R, if Ψ−1 (R) = O. The group Γ naturally acts by conjugation on the set of optimal embeddings. We denote by Γ\Emb (O, R) the set of conjugacy classes, which is non-empty under our assumptions (see [BD96, Lemma 2.1]). Let H = HO be the ring class field of conductor c and fix an embedding σ p : H ,→ Kp , the p-adic completion Kp of K (this is possible since p is inert in K and hence it splits completely in H). The p-adic uniformization provided by the Cerednik-Drinfeld Theorem allows us to view Γ\Emb (O, R) as a subset of X (Kp ) (see [BD98, After Lemma 1.1] for the last identification): Γ\Emb (O, R) ,→ Γ\Emb (Kp , Bp ) = Γ\Emb (Kp , M2 (Qp )) = X (Kp ) . Let B0 := ker (T r) be the kernel of the trace map T r : B → Q. Define a map P− (X) : B0 ⊗ Kp → P2 setting:    X −X 2 Pb (X) := T r b , 1 −X where we view b ∈ B as a matrix via the identification Bp = M2 (Qp ). The map is an isomorphism of right GL2 -modules, if we let GL2 act on Bp = M2 (Qp ) by the rule M · b := MbM, where (·) is the main involution of M2 (Qp ). Fix an embedding Ψ ∈ Emb (O, R) and define PΨ := PΨ(√−DK ) (X) ∈ Pn (Qp ) , 20

where we extend Ψ to an embedding Emb (Kp , M2 (Qp )). Let a 7→ a be the non-trivial automorphism of Kp /Q. Via the identification Γ\Emb (Kp , M2 (Qp )) = X (Kp ), the action of the non-trivial automorphism of Kp /Q is given by Ψ 7→ Ψ, where Ψ is the composition of the restriction of a 7→ a to K with Ψ. It follows that PΨ = −PΨ . Following [BD07, Definition 3.5 and Remark 3.6] in the weight k = 2 case, we may attach to the data of (f, K, Ψ) a two variable p-adic L-function: L (f /K, Ψ, −, −) : U × Zp → Cp (κ, s) 7→ L (f /K, Ψ, κ, s) as follows. Let τ Ψ be the image in Hp (Kp ) = Emb (Kp , M2 (Qp )) of Ψ and let LΨ be a lattice such that [LΨ ] = vΨ , where vΨ := red (τ Ψ ) ∈ V is the reduction of τ Ψ . Then τ Ψ = τ Ψ . Introduce the factorization PΨ (X) = A (X − τ Ψ ) (X − τ Ψ ) . Definition 6.1. The partial two variable p-adic L-function attached to (f, K, Ψ) is by definition κ−k R s− k −n κ−s− k 2 P m (x, y) hx − τ Ψ yi 2 hx − τ Ψ yi dµLΨ L (f /K, Ψ, κ, s) : = |LΨ | 2 A 2 L0Ψ Ψ   k k κ−k κ−s− 2 s− 2 m = A 2 µV hx − τ Ψ yi . vΨ PΨ (x, y) hx − τ Ψ yi We remark that the standard choice log0 of a branch of the p-adic logarithm has been done in the definition of the above p-adic L-function, in light of the defining formulae (1). However, working in a similar way as in the proof of [BD09, Theorem 2.5] shows that we may have chosen any branch of the p-adic logarithm, i.e. s we may have chosen to define the quantities hαi by means of a different choice of a branch of the p-adic logarithm without changing the resulting p-adic L-function. The first part of the following proposition is a direct consequence of the definition: Proposition 6.2. Restricting to the line s = κ2 we find   κ κ L f /K, Ψ, κ, = L f /K, Ψ, κ, κ − = 2 2 κ−k κ−k κ−k R −n P m (x, y) hx − τ Ψ yi 2 hx − τ Ψ yi 2 dµLΨ . |LΨ | 2 A 2 L0Ψ Ψ
The value at k, k2 is   k L f /K, Ψ, k, = 0. 2 The value at k of the derivative along the line s = κ/2 is d   κ  L f /K, Ψ, κ, = dκ 2 κ=k R  1 −n d κ−k = |LΨ | 2 dµLΨ (x, y) + 0 PΨ (x, y) (x − τ Ψ y) L Ψ 2 dκ κ=k m R  (−1) −n d κ−k |LΨ | 2 + P (x, y) (x − τ y) dµ (x, y) . 0 LΨ Ψ Ψ L 2 dκ Ψ κ=k
Proof. By Proposition 3.8, the value at k, k2 is R −n R |LΨ | 2 L0 PΨm (x, y) dµLΨ = P1 (Qp ) PΨm (t) dµf (t) = 0, Ψ
since µf ∈ Dn0 P1 (Qp ) and PΨm (t) ∈ Pn . It follows that taking derivatives yields d   κ  L f /K, Ψ, κ, = dκ 2 κ=k  R κ−k κ−k −n d m |LΨ | 2 hx − τ Ψ yi 2 hx − τ Ψ yi 2 dµLΨ . 0 PΨ (x, y) L Ψ dκ κ=k m Now the claim follows from Proposition 3.1. The sign (−1) is justified by PΨ = −PΨ . 21



7. Heegner cycles and p-adic Abel-Jacobi map Here we have to notice that the identification IS = ISλ described in the introduction depends a priori on the choice of a branch of the p-adic logarithm logλ . As a consequence the resulting p-adic Abel-Jacobi   map log cl0 = logλ cl0 will also depend on this choice. More precisely, recall that V = Hp Mn,Q , Qp and note that Dλ := Dλst (V ) depends on this choice because the Fontaine functor depends on it. On the other  λ hand, the (φ, N )-module structure of the de Rham realization HdR Mn,Q , Qp depends on the choice of   λ a branch of the p-adic logarithm too (see [IS, end of pag. 342]) and we have Dλst (V ) ' HdR Mn,Q , Qp in M F := M FE (φ, N ), the isomorphism being Hecke equivariant for the Hecke Qp -algebra and unique up λ to multiplication  by invertible elements in this algebra (see [IS, Theorem 5.9]). We normalize Dst (V ) '  λ HdR Mn,Q , Qp as in [IS, Remark 8.5]. We recall that, by definition, (25)   λ HdR Mn,Q , E DλE   ' Mk (Γ, E)∗ . ' λ m+1 λ F DE m+1 F HdR Mn,Q , E     λ 0 1 and HdR (X, Vn ) (really equalities) as Mn,Q , E = HdR Mn,Q , E = HdR IS

cl

1 (E, V (m + 1)) →λ logλ cl0 : CH m+1 (Mn,E ) →0 Hst

Note also that DλE = D0E

filtered E-vector spaces, since Dλst (V )E = DdR (V ) and thanks to [IS, pag. 342-343]. In particular, the last 1 identification in (25), which is given by the identification Mk (Γ, E) = F m+1 HdR (X, Vn ) and Serre’s duality, only depends on the underlying filtered E-vector spaces and for this reason does not depend on the choice of logλ . Finally, we remark that the penultimate equality only depends on the underlying filtered E-vector spaces too. Summarizing the only point where the choice of a branch of the p-adic logarithm comes into the definition of logλ cl0 is ISλ . We will come back to this point in the subsequent section. (n) We refer the reader to [Be] and [IS, section 8] for the definition of the Heegner cycles yΨ . The proof of the subsequent theorem is taken from the proof of [IS, Theorem 9.2]; as already mentioned at the beginning of section 4 we have to replace [IS, Corollary 10.7] with Proposition 4.4. As a consequence we need to assume λ = Lf and we will denote by logf cl0 the resulting p-adic Abel-Jacobi map. We briefly sketch the arguments for the convenience of the reader and also because the proof will clarify how the explicit definition of the p-adic Abel-Jacobi map works. Theorem 7.1. For every f ∈ Mk (X, E)

new

the following equality holds:   (n) logf cl0 yΨ (f ) = Φf (τ Ψ , PΨm )

Proof. Consider the Gysin sequence j∗

Resτ

1 1 E : 0 → HdR (X, Vn ) (m + 1) → HdR (U, Vn ) (m + 1) →Ψ (Vn )τ Ψ (m) → 0

As in [IS, proof of Theorem 9.2, page 373, first lines], it follows from [IS, Proposition 7.3] and [IS, Lemma ∗ 8.4] that the image in Mk (Γ, E) of the extension ϕ∗v (E) obtained from
v = h−, PΨm iVn = ϕv ∈ HomM F E 0 , (Vn )τ Ψ (m)   (n) computes the element logf cl0 yΨ , up to the identification
Ext1M F E 0 , Dλ (m + 1) =

1 DλE HdR (X, Vn ) ∗ = 1 (X, V ) = Mk (Γ, E) . λ m+1 m+1 F HdR F DE n

As it follows from the proof of [IS, Lemma 2.1], the image ϕ∗v (E) is computed by the class [d] ∈ of an element d ∈ ker N . Moreover there is a uniquely determined element e ∈ F m+1 EE ∩ ker N such that Resτ Ψ (e) = v. Then e1 := e+i (d) ∈ Em+1 ∩ker N (here Em+1 denotes the slope m+1-part). H 1 (X,Vn ) ∗ The identification F m+1dRH 1 (X,V = Mk (Γ, E) comes from the identification Mk (Γ, E) = F m+1 DE (see the n) dR proof of [IS, Proposition 6.1]) together with Serre’s duality, which is realized by the cup product. It follows 1 HdR (X,Vn ) 1 (X,V ) F m+1 HdR n

22

  (n) ∗ 1 that we must consider h−, diX ∈ F m+1 HdR (X, Vn ) in order to compute logf cl0 yΨ . The Reciprocity Law for the open subscheme X (see [IS, Theorem 10.3]) gives the first of the following equalities (the second follows from the fact that d ∈ ker N = ker I, see [IS, pag. 348] and use [dS1, Corollary 6.6]):



λ    λ λ h−, diX = PX (−) , I (d) Γ − I (−) , PX (d) Γ = − I (−) , PX (d) Γ . λ 1 We have PX (d) = PUλ (j ∗ d) = PUλ (e1 − e) (recall j : U ⊂ X). Furthermore e1 ∈ HdR (U, Vn )m+1 = ker PU (see [IS, proof of Theorem 10.6]), so that



 h−, diX = − I (−) , PUλ (e1 − e) Γ = I (−) , PUλ (e) Γ . 1 Since Resτ Ψ (e) = h−, PΨm iVn and e ∈ F m+1 HdR (U, Vn ), Corollary 4.4 gives the claim:  

 (n) logf cl0 yΨ (f ) = hω f , di = I (ω f ) , PUλ (e) Γ = Φf (τ Ψ , PΨm ) .

 8. Proof of the main result We are now ready to prove the main formula of the paper. Combining Corollary 5.10, Proposition 3.8 and Theorem 7.1 we find      d   κ  1 (n) (n) m L f /K, Ψ, κ, = logf cl0 yΨ (f ) + (−1) logf cl0 yΨ (f ) . dκ 2 κ=k 2 Our main result Theorem 1.1 is now a consequence of the following lemma, which removes from ISλ (and hence from logλ cl0) the dependence on the choice of a p-adic logarithm we made so far. Recall the  representation V := Hp Mn,Q , Qp ; we apply Lemma 8.1 to the representation V (m + 1), which is known

to satisfy the assumpion of the lemma, thanks to [IS]. As we already remarked, DdR (V ) F m+1 DdR (V )

Dλ E (m+1) F 0 DE (m+1)

=

Dλ E F m+1 Dλ E

=

(really equality) does not depend on this choice λ.

Lemma 8.1. Suppose that V is a semistable representation of GE such that N : Dλst (V ) → Dλst (V ) induces φ=1 = 0. an isomorphism between Dλst (V )0 and Dλst (V )−1 (the slope 0 and −1 parts respectively) and Dcrys (V ) The identification
DdR (V ) 1 ISλ : Hst (E, V ) = Ext1M F E 0 , Dλst (V ) = 0 F DdR (V ) does not depend on the choice of a branch of the p-adic logarithm. ·,λ Proof. Given a semistable representation V of GQp , let us denote by Cst (V ) the complex defined in [N, Section 1.19]. There is a natural morphism   DdR (V ) ·,λ ιλ : 0 → H 1 Cst (V ) , F DdR (V )   which maps x mod F 0 to 0, 0, x mod F 0 . In [N, Sections 1.19-20], Nekov´aˇr defines the morphisms ρV,st , τ V,st and β V,st . We set αλV,st := ρV,st ◦ τ V,st and β λV,st := β V,st , because they depend on λ. We have  
·,λ 1 αλV,st : Hst (E, V ) = Ext1M F E 0 , Dλst (V ) ,→ H 1 Cst (V ) ,   ' ·,λ 1 β λV,st : H 1 Cst (V ) → Hst (E, V ) . λ λ It is proved in [N, Proposition 1.21] that αλV,st ◦ β λV,st = 1. Set β 0λ V,st := β V,st ◦ ι . The fact that N : φ=1

Dλst (V )0 → Dλst (V )−1 is surjective, together with the assumption Dcrys (V ) = 0, implies the following facts. 1 (a) He1 (E, V ) = Hst (E, V ). Indeed He1 (E, V ) = Hf1 (E, V ) thanks to [N, Corollary 1.16], together with dim

Dcrys (V ) φ=1 = dim Dcrys (V ) = 0. (φ − 1) Dcrys (V ) 23

1 It is also true that Hf1 (E, V ) = Hst (E, V ) thanks to [N, Corollary 1.18], together with the remark φ=1  λ Dst (V (−1)) ∗
φ=1 = Dcrys (V ∗ (1)) , as explained in [N, proof of Corollary 1.18], the fact that N Dλ st (V ) φ=1  λ   λ Dst (V (−1)) Dst (V (−1)) ⊂ N and the fact that that N Dλ (V ) Dλ (V ) st

0

st



Dλst

(V (−1)) N Dλst (V )

(b) the Bloch-Kato exponential

 = 0

DdR,E (V ) F 0 DdR,E (V )

Dλst (V )−1 Dλst (V (−1))0 = = 0. N Dλst (V )0 N Dλst (V )0 1 → Hst (E, V ) is an isomorphism. This is a direct consequence φ=1

of (a), [N, Theorem 1.15] and our assumption Dcrys (V ) = 0. 0λ (c) β V,st = exp. This fact follows exploiting the diagrams [N, Theorem 1.15 and Theorem 1.17] and the definition of β λV,st . (d) ιλ is an isomorphism. Indeed β λV,st is an isomorphism and exp is an isomorphism by (b). Under the further assumption that N : Dλst (V )0 → Dλst (V )−1 is bijective, the morphism ISλ is defined. Following the proof of [IS, Lemma 2.1] and comparing it with the definition of αλV,st which is given in [N, 1.20], one checks that ιλ ◦ ISλ = αλV,st . Since ιλ is an isomorphism in our setting, we find λ ISλ ◦ β 0λ V,st = ι


−1

◦ αλV,st ◦ β λV,st ◦ ιλ = 1.

Hence ISλ is the inverse of β 0λ V,st = exp, which is an isomorphism, and does not depend on the choice of a branch of the p-adic logarithm.  9. Final remarks 9.1. Relation with the Bertolini-Darmon-Iovita-Spiess p-adic L-function and interpolation prop
erties. Recall the identification Dkk (W ) = Dk P1 (Qp ) obtained from the map on functions which sends
F (x, y) ∈ Akk (W ) to F (t) := F (t, 1). According to Proposition 3.8, up to Dkk (W ) = Dk P1 (Qp ) , we have = µf for every v ∈ V + . In particular, specializing our partial two variable p-adic L-function to κ = k µV,k v we find (recall τ Ψ = τ Ψ ):   k−s− k s− k 2 L (f /K, Ψ, k, s) = µV,k PΨm (x, y) hx − τ Ψ yi 2 hx − τ Ψ yi vΨ  s− k2 !  s− k2 R t − τΨ t − τΨ m m (26) = µf PΨ (t) =: P1 PΨ (t) dµf (t) t − τΨ t − τΨ Let K∞ /K be the maximal extension of K which is unramified outside p and anticyclotomic, i.e. the involution c in GK/Q acts as −1 on GK∞ /K . We specialize ourselves to the setting of [BDIS] and therefore we assume that H/K is the Hilbert ring class field. We have: 0 → GK∞ /H → GK∞ /K → GH/K → 0. √
× Assuming for simplicity that OK = {±1} (which excludes K = Q (i) and K = Q −3 ), class field × × theory identifies r : GK∞ /H ' Kp× /Q× p (see [IS, (63)]). Let Kp,1 ⊂ Kp be the subgroup of norm one × elements, so that s : Kp× /Q× p ' Kp,1 by the rule s (x) := x/x. Let Ψ be an optimal embedding of O into R and let Ψ : Kp× → Bp× = GL2 (Qp ) be (by abuse of notation) its extension to the p-adic completions, so that Kp× acts on P1 (Qp ). As explained in [BDIS, Section 2.4], the Kp× -action on P1 (Qp ) is transitive and × × 1 the stabilizer of any point is Q× p . Choosing ∞ as a base point we find η Ψ,∞ : Kp /Qp ' P (Qp ). Setting ι := s ◦ r and η Ψ := η Ψ,∞ ◦ s−1 , we find the following commutative diagram GK∞ /H (27)

r

→ Kp× /Q× p ι& ↓s × Kp,1 24

η Ψ,∞

→ % ηΨ

P1 (Qp )

Writing ξ Ψ := η Ψ,∞ ◦ r, the Teitelbaum distribution µf induces a distribution µf,Ψ on G := GK∞ /H by
the rule µf,Ψ (FG ) := µf PΨm · FG ◦ ξ −1 Ψ , where FG is a function on GK∞ /H . In particular, suppose that × t−τ Ψ F : Kp,1 → Cp and define FG := F ◦ ι. Then (27) and the equality η −1 Ψ (t) = t−τ Ψ (see [BDIS, (15)]) give  
R R R t − τΨ m (28) GK /H FG (g) dµf,Ψ (g) := P1 (Qp ) PΨm (t) FG ξ −1 (t) dµ (t) = P (t) F dµf (t) . f Ψ P1 (Qp ) Ψ ∞ t − τΨ s

s

s

s

× For every s ∈ Zp let h·i : Kp,1 → Kp be defined by the rule hαi := exp (s log0 (α)), and let h·iG := h·i ◦ ι. The Bertolini-Darmon-Iovita-Spiess (partial) p-adic L-function is by definition R s− k LBDIS (f /K, Ψ, s) = GK /H hgiG 2 dµf,Ψ (g) . ∞

Using (28) and (26) we see that (29)

LBDIS (f /K, Ψ, s) =

R

P m (t) P1 (Qp ) Ψ



t − τΨ t − τΨ

s− k2 dµf (t) = L (f /K, Ψ, k, s) .

As explained in [BDIS, Section 2.5], for a suitable χ-twist LBDIS (f /K, χ, s) of LBDIS (f /K, Ψ, s) obtained from continuous characters χ : GK∞ /K → C× of finite order, the values of
LBDIS (f /K, χ, s) := LBDIS (f /K, χ, s) LBDIS f /K, χ−1 , s at s = k/2 are expected to interpolate the special values of the complex L-function L (f∞ /K, χ, k/2) at s = k/2 as χ varies (see [BDIS, (18)]). We now specialize ourselves to unramified characters, i.e. those factoring through GH/K . Suppose now that χ : GH/K → C× is a character. Exploiting the Γ-invariance property of µV ∗ it can be proved that L (f /K, Ψ, k, s) only depends on the image of Ψ in Γ\Emb (O, R). As explained in [BDIS, Proposition 2.15], GH/K acts freely on Γ\Emb (O, R). We define P L (f /K, χ, κ, s) := σ∈GH/K χ−1 (σ) L (f /K, σΨ, k, s) ,
L (f /K, χ, κ, s) := L (f /K, χ, κ, s) L f /K, χ−1 , κ, s . Note that (29) and a slight generalization of [IS, (65)] (where χ is assumed to be 1) implies that, in this case, the χ-twist LBDIS (f /K, χ, s) touched on above, is given by LBDIS (f /K, χ, s) = L (f /K, χ, k, s) and LBDIS (f /K, χ, s) = L (f /K, χ, k, s) . Hence, L (f /K, χ, k, k/2) = LBDIS (f /K, χ, k/2) is related to the special values L (f∞ /K, χ, k/2) at s = k/2. Indeed, in this case, the interpolation property is a trivial consequence of the fact that both L (f /K, χ, k, k/2) and L (f∞ /K, χ, k/2) vanish. For this reason, we are led to consider the special values L (f /K, χ, k 0 , k 0 /2) at even integers k 0 6= k. We refer the reader to [BD07] and [Se1] for details about the following discussion. Recall the family Φ introduced before Proposition 3.8. Its specialization ρk0 (Φ) ∈ Sk0 (Σ) to even weight k 0 ∈ Z>k ∩ U is a non-zero modular form (see [Se1, Theorem 4.16]). Since ordp (ap ) = n/2 (see Theorem 3.7), the Atkin-Lehner theory implies that ρk0 (Φ) is old at p, while it can be proved to be N/p-new (for example as a consequence of the Chenevier p-adic Jacquet-Langlands correspondence and well-known results on Coleman families). In particular, if fk0 ,∞ ∈ Sk0 (Γ0 (N )) corresponds to ρk0 (Φ) via the Jacquet-Langlands correspondence, 0 pk −1 # # f 0 (q p ) fk0 ,∞ (q) = fk0 ,∞ (q) − ap (k) k ,∞ is the p-stabilization of a newform fk#0 ,∞ ∈ Sk0 (Γ0 (N/p)), that we may assume to be normalized. The   signs of the complex L-functions L fk#0 ,∞ /K, χ, s at the central critical points s = k 0 /2 are 1 for all   k 0 ∈ Z>k ∩ U , and L fk#0 ,∞ /K, χ, k/2 is expected to be frequently non-zero. The following Theorem can be proved adapting the arguments of [BD07, Theorem 3.8] to our higher weight setting (see [Se1, Theorem 5.17] for details). 25

0 >k Theorem 9.1. There exist p-adic periods λ (k) ∈ C× ∩ U, p such that, for all k ∈ Z     2 0 2 2 −2 L (f /K, χ, k 0 , k 0 /2) = λ (k 0 ) ap (k 0 ) 1 − pk −2 ap (k 0 ) L∗ fk#0 ,∞ /K, χ, k 0 /2 ,

where  L∗



 fk#0 ,∞ /K, χ, k 0 /2 :=

k0 −2 2

k0 −2

(2π)



k0 −1

!2 DK 2

D E L∗ (fk0 ,∞ /K, χ, k 0 /2) . fk#0 ,∞ , fk#0 ,∞

9.2. A restatement of the main result. We already remarked that L (f /K, Ψ, κ, κ/2) vanishes at κ = k. Since the p-adic family Φ such that ρk (Φ) = ϕf is determined up to elements of Ik SU (Σ) (see Theorem 3.7), a similar argument as the one exploited in [GSS, Remark 80] shows that the association f 7→ L0 (f /K, Ψ, k, k/2) does not depend on the choice of Φ such that ρk (Φ) = ϕf and that, furthermore, it gives rise to a Cp -linear functional L0 (Ψ, k, k/2) on Mk (Γ, Cp ) which descends to a Kp -linear functional L0 (Ψ, k, k/2) : Mk (Γ, Kp ) → Kp , where we recall that Kp is the p-adic completion of K. If χ : GH/K → C× is a character and M is a vector space, let M ⊗ χ be the vector space obtained by formally adding the values of χ (for example M ⊗ χ = M if χ2 = 1). Similarly, since L (f /K, χ, κ, κ/2) (resp. L (f /K, χ, κ, κ)) vanishes at k (resp. vanishes to order at least 2), f 7→ L0 (f /K, χ, k, k/2) (resp. f 7→ L00 (f /K, χ, k, k/2)) gives rise to a Cp -linear functional L0 (χ, k, k/2) (resp. a quadratic form L00 (χ, k, k/2)) on Mk (Γ, Cp ) ⊗ χ which descends to L0 (χ, k, k/2) , L00 (χ, k, k/2) : Mk (Γ, Kp ) ⊗ χ → Kp ⊗ χ. Define yχ(n) :=

P

σ∈GH/K

(n)

χ−1 (σ) yσΨ , y (n) χ :=

P

σ∈GH/K

(n)

χ−1 (σ) yσΨ


χ
(n) (n) Let Kχ /K be the extension cut out by the
character χ, so that yχ , y χ ∈ CH m+1 Mn,Kχ ⊂ CH m+1 Mn,Kχ ⊗ χ, the χ-component of CH m+1 Mn,Kχ ⊗ χ for the Galois action of GH/K (factoring through GKχ /K ) on
CH m+1 Mn,Kχ ⊗ χ. Writing log cl0 = log cl0 ⊗ χ by abuse of notation, Theorem 1.1 immediately implies the following Corollary. ∗

∗

Corollary 9.2. The following equalities hold in Mk (Γ, Kp ) , Mk (Γ, Kp ) ⊗ χ and in the space of Kp -valued quadratic forms on Mk (Γ, Kp ) ⊗ χ respectively:     1 (n) (n) m L0 (Ψ, k, k/2) = log cl0 yΨ + (−1) log cl0 yΨ , 2      1 m L0 (χ, k, k/2) = log cl0 yχ(n) + (−1) log cl0 y (n) , χ 2           1 (n) (n) m m L00 (χ, k, k/2) = log cl0 yχ(n) + (−1) log cl0 y (n) log cl0 yχ−1 + (−1) log cl0 y χ−1 . χ 2 We remark that crucial to our reformulation is the fact that our computation does not depend on the choice logf of the p-adic logarithm that was done in a first step towards proving our main result when computing the f -component L0 (f /K, Ψ, k, k/2) of L0 (Ψ, k, k/2) (after a base change to a large enough field E in order to let it exist). Corollary 9.2, specialized to genus characters, is the main ingredient in the proof of the generalization [Se1, Theorem 1.4] of [BD07, Theorem 1], the further ingredients being a study of the p-adic L-functions in the p split case and the analogue [Se1, Theorem 5.21] of the functional equation [BD07, Proposition 5.1]. References [BC] [BD96] [BD98] [BD07] [BD09]

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