p-adic FAMILIES OF MODULAR FORMS AND p-adic ABEL-JACOBI MAPS

Transcription

1 p-adic FAMILIES OF MODULAR FORMS AND p-adic ABEL-JACOBI MAPS MATTHEW GREENBERG AND MARCO ADAMO SEVESO Abstract We show that p-adic families of modular forms give rise to certain p-adic Abel-Jacobi maps at their p-new specializations We intruduce the concept of differentiation of distributions, using it to give a new description of the Coleman-Teitelbaum cocycle that arises in the context of the L-invariant Contents 1 Introduction 1 2 L-invariants of rigid analytic tori and formal logarithms 4 3 Definite multiplicative integrals and the Manin-Drinfeld Theorem 6 4 Definite additive integrals 8 5 Definite additive integrals: proof of the main result 11 6 The analogous results in the indefinite setting 14 7 Derivatives of distributions and the Coleman-Teitelbaum cocycle Generalities on distributions and locally trivial bundles Derivatives of distributions The relation with the Coleman-Teitelbaum cocycle 28 References 30 1 Introduction After Serre s observations involving the family of Eisentein series see [29], a theory of p-adic families of modular forms was systematically developed by Hida in the ordinary case see [23] and [24], and by Coleman in the more general case of finite slope see [10] This theory was subsequently encapsulated geomerically via the eigencurve by Coleman-Mazur see [11] Since then, other points of view on p-adic families of modular forms have been developed; one of the most flexible is the cohomological approach inspired by the theory of modular symbols and the Eichler-Shimura isomorphism This approach was developed in a sequence of papers, see [22, 42], with the development culminating in the authoritative [3] The connection between this cohomological approach and original theories of Coleman and Coleman-Mazur approach to p-adic families was spelled out in [2], the essential input being an overconvergent Eichler-Shimura isomorphism When the algebraic group underlying the p-adic families is the group of units of a definite quaternion algebra, the cohomological approach simplifies, due to the fact that the associated algebraic group is compactmod-center at infinity In this case, one can define p-adic families of modular forms very concretely as in [9] Such families are similar in spirit to those considered in the first and the third parts of the this paper, while the arguments of 6 are couched in the language of [3] There are many ways in which the arithmetic of a modular forms f is determined by the p-adic family f passing through it One such example is proof by Greenberg-Stevens [22] of the Mazur-Tate-Teitelbaum conjecture [27], in which the key point is the identity 1 L p A f = 2 d log a p 2 The quantity on the left hand side, the L-invariant attached to f S 2 Γ 0 Np new, does not involve the family f while the right hand side clearly does The aim of this paper is to highlight other manifestations Date: March 8, 2016 MG s research is supported by NSERC of Canada 1

2 of this phenomenon, whereby a quantity which depends only on on the modular form f is described by one attached to its deformation f One such manifestation in fact, a direct analogue of 1 is Proposition 714, relating the Coleman-Teitelbaum cocycle to another cocycle constructed via a differentiation of measures operation described in 7 Another phenomenon of this kind is obtained showing that the p-adic Abel-Jacobi map of an appropriate Shimura curve can be computed by means of the deformations identified with the tangent space to the the Jacobian of the Shimura curve in question The result is the formula stated in Theorem 11; its proof is based on computations of Bertolini, Darmon, and Iovita appearing in [4] and [5] As an application, one may extend their main result [4, Theorem 54] by removing the restriction that the Fourier coefficients of f lie in Q Our final example is Theorem 65, presented in 6 see also Theorem 11 Suppose that p is a prime, that N + and N are coprime integers, and that N is the squarefree product of an odd number of primes Fix a complete subfield E C p We let φ S 2 pn +, N p-new be a weight two, p-new, cuspidal eigenform on the definite quaternion algebra B over Q of discriminant N and level Γ 0 pn + Fix a maximal order O B B and an Eichler order R 0 pn +, N B of level pn + We choose a local isomorphism ι p : B p GL2 Q p and let Γ N +,N := ι p R 0 pn +, N [1/p] GL 2 Q p be the arithmetic group attached to R 0 pn +, N Its subgroup of norm one elements Γ N +,N is a Schottky group whose associated Mumford curve Γ N +,N π N +,N : H p Γ N +,N \H p is the rigid analytic space associated to the Shimura curve X N +,N p attached to the indefinite quaternion algebra over Q of discriminant N p; the isomorphism is defined over Q p 2, the quadratic unramified extension of Q p see [4, 41] Abusing notation slightly, we will frequently identify Γ N +,N \H pand X N +,N p Write J N +,N p for the Jacobian variety of X N +,N p We recall that the rigid analytic p-adic upper halfplane is a tubular neighborhood of the Bruhat-Tits tree T whose set of edges resp vertices we denote by E resp V The ends of T are identified with P 1 Q p and the combinatorics of T can be used to conveniently describe the p-adic topology on P 1 Q p as follows: A basis of compact-open subsets of the topology on the projective line is given by the subsets U e P 1 Q p, where U e consists of those points associated to ends of T emanating from e We refer the reader to [31] for basic notation and facts about the Bruhat-Tits tree We let C har E, E be the space of harmonic cocycles on T, maps c : E E such that c e = c e and se=v c e = 0 for every v V For a p-adic manifold X we write A X be the space of E-valued, locally analytic functions on X, and D X for its strong E-dual see 71 Write A P 1 Q p 0 A Qp for the space of locally analytic functions which are constant at see [28, Definition 32], for example We let D P 1 Q p 0 be its dual and write D 0 P 1 Q p for the subspace of D P 1 Q p 0 consisting of those locally analytic distributions µ such that µ 1 = 0 Because φ is p-new, when E contains the eigenvalues of φ one can associated to φ a harmonic cocyle c φ C har E, E Γ N +,N and a rigid analytic modular form f φ on X N +,N p see [34, 13] There is a unique distribution µ φ D 0 P 1 Q p Γ N +,N such that R µ φ = cφ, where 2 R : D 0 P 1 Q p Γ N +,N C har E, E Γ N +,N is defined by the rule R µ e := µ χ Ue, χue being the characteristic function of U e In order to state our results precisely, we need to introduce a few more spaces of distributions Setting W := Q 2 p {0} defines a trivial Q p -bundle π : W P 1 Q p, π x, y := [x : y] Let O be an affinoid E-algebra and letω : Z p O be a continuous character; it extends uniquely to a homomorphism of algebras ω : D Z p O Write A0 W for the space of E-valued locally analytic 2

3 functions on W such that F px, py = F x, y and let D 0 W be its strong dual Then D 0 W is naturally a D Z p -module and, regarding O as a D Z p -module via ω, we may form the completed tensor product D ω,0 W := O ω D 0 W If ϕ : O O is a morphism of E-algebras then there is a natural specialization map 3 D ω,0 W D ω,0 W, ω := ϕ ω The reader may consult 61 for the precise definition of this map Recall that continuous characters from Z p are parametrized by the weight space X := Hom cts Z p, G m Suppose that ϕ : O Ω E is the morphism corresponding to 0 Ω, for an open affinoid Ω X containing the weight 0 := [ t t 0 = 1 ] The inclusion Ω X corresponds to a character ω : Z p O and, in this case, we write D Ω,0 W := D ω,0 W ; we can also consider D 0,0 W = D ω,0 W with ω = 0 With this notation, 3 can be written in the form 4 D Ω,0 W D 0,0 W The invariance property of F A 0 W implies that F is uniquely determined by its restriction to L := Z 2 p pz 2 p, and one easily checks thatthere is a canonical isomorphism D P 1 Q p 0 = D0,0 W see [21, proof of Lemma 62] and [35, Remark 5] We use this identification to vary µ φ in a p-adic family Let C V, D Ω,0 W Γ N +,N be the space of Γ N +,N -invariant maps c : V D Ω,0 W see [34, Proposition 35] for a description in terms of lattices We write V + for the set of vertices at even distance from v = [L ] Then there exists an eigenfamily of distributions µ Ω, := { µ Ω,v }v V with the property that, writing µ 0,v for the specialization of µ Ω,v at 0 obtained via 4, we have 5 µ 0,v = µ φ for every v V + See [4, Theorem 25 and Lemma 212] and [34, Theorem 37 and subsequent discussion, Proposition 38] For an element τ Hp ur := Q ur p Q p, we write r τ V for its reduction see [13, Proposition 51] and consider the quantity d [ 6 µ dκ Ω,rτ x τy κ] E κ=0 It can be shown that it does not depend on the choice of family µ Ω, satisfying 5; the proof is similar to that of [21, Lemma 71] The Picard variety J N +,N p is endowed with a canonical morphism AJ 0 X N +,N p : Div 0 X N +,N p E := Div 0 X N +,N p C p G Qp /E J N +,N p E We write G K /K for the associated Galois group of K /K and G K for G K/K For every q pn, the Hecke operator t q has degree q + 1 and it follows that, setting t q := q + 1 T q, we have t q : Div X N +,N p Div 0 X N +,N p Let T 0 JN +,N p be the tangent space to JN +,N p at the identity, whose E-points are identified with apply [6, Ch 8, 84, Theorem 1] and Serre duality: 7 T 0 JN +,N p E = HomE H 0 X N +,N p/e, Ω 1 X N+,N p/e, E The formal logarithm is a map log JN +,N p : J N +,N p E T 0 JN +,N p E Noticing that t q is invertible on T 0 JN +,N p we may define the following composite: 8 log JN +,N p AJ := t 1 q log JN +,N p AJ 0 X N +,N p t q : Div X N +,N p T 0 JN +,N p We remark that this definition is indeed independent of q pn We recall that taking residues of the pull-back to H p of a differential form ω H 0 X N +,N p,e, Ω 1 X N +,N p,e yields an identification H 0 X N +,N p,e, Ω 1 X N +,N p,e Char E, E Γ N +,N 3

4 Let ω φ be the differential form whose residues are given by c φ Changing our point of view slightly, we assume that ω H 0 X N +,N p,e, Ω 1 X gives rise to c N +,N ω C har E, E Γ N +,N which, in turn, gives a p,e measure µ ω D 0 P 1 Q p Γ N +,N We choose a family µ ω Ω, such that µω 0,v = µ ω for every v V + The main result of the first part of this paper is the following: Theorem 11 Let τ H ur p Writing [τ] for the natural image of τ in Div X N +,N p log JN +,N p AJ [τ] ω = d dκ [ µ ω Ω,rτ x τy κ] κ=0 Q ur, we have The first part of the paper is devoted to the proof of this result Theorem 11, specialized to the quotient J N +,N p E f of J N +,N p associated to a weight 2 modular form with rational Fourier coefficients, is a key ingredient in the proof of the main result of [4] Our results can be viewed as a lift to the Jacobian of the computations appearing there, yielding generalizations of the corresponding theorems In 6 we move to the setting where N is a squarefree product of an even number of primes Here, we first lift the original construction of [12], as generalized in [19] Our approach is a bit different from related constructions of of [15] and [26], but the resulting rigid analytic varieties are isogenous Then we prove the analogue of Theorem 11 in Theorem 65 As an application, we can lift the rationality main result of [25] on rationality of Darmon points to the Jacobians This result can also be deduced from the rationality result of [21], specialized to weight 2 modular forms, using 6 to translate the results of loc cit to our setting see [21, Remarks 111 and 14] Finally, 7 introduces the notion of derivative of a family of distributions in complete generality This is employed in order to differentiate the family specializing to µ φ possibly of higher, even weight and attach to it a cocycle in H 1 Γ N +,N, V k As an application, in Proposition 714 we get a new description of the Coleman-Teitelbaum cocycle in terms of differentiation of families of distributions In particular, an a priori a new definition of the Teitelbaum L-invariant can be proposed and, possibly, generalized to other settings Let us remark that, the formula which appears in Proposition 714 is another manifestation of the phenomenon relating invariants of modular forms the Coleman cocycle to quantities defined using p-adic deformations the derivative of the family passing through it It is, thus, of the same nature as 1 and the formulas appearing in Theorem 11 and Theorem 65 Because p-adic L-functions are defined in terms of measures, we hope to apply these notions to the study of derivatives of p-adic L-functions in other settings Regarding notation for the various arithmetic group actions appearing in this paper, we follow the convention of [28] 2 L-invariants of rigid analytic tori and formal logarithms In the literature, L-invariants have been attached to a variaty of objects, eg, abelian varieties, p-adic representations, and modular forms The aim of this section to review the definition of the L-invariant L A see [22, 3, 35 definition] attached to an abelian variaty, to extend this definition to rigid analytic tori, and to relate this L-invariant to the formal logarithm map We work over our complete field E C p, assumed to be locally compact in this section We write k E for the residue field of E From now on we will write log λ : E E for the choice of the p-adic logarithm determined by log λ p = λ, where λ E; hence log 0 = log and log λ = log 0 λ ord Recall that, if A = A /E is as above, then the dual abelian variety B has the same reduction type and, hence, they both admit a p-adic uniformization see for example [37, 1] for a short account on p-adic uniformization Indeed, there are canonical free Z-modules X and Y of rank dima the characters groups of B/k 0 E and, respectively, A 0 /k E on which G E acts trivially, a Z-linear pairing and exact G E /E-equivariant sequences j : X Z Y E 0 X j Hom Y, E A E 0, 0 Y j Hom X, E B E 0 4 p

5 for every E E E A continuous group homomorphism l : E E induces a map l : Hom Y, E Hom Y, E Taking E = E, we define: j l := l j : X Z Y E and l := l j : X Hom Y, E =: T 0 A E Set X E := E Z X and Y E := E Z Y and write j l,e for the corresponding maps j l,e : X E E Y E E and j l,e : X E Hom Y E, E = T 0 A E Noticing that j ord is non-degenerate, the L-invariant L A is defined in [22, 3, 35 definition] to be the unique E-linear endomorphism L A : X E X E such that 9 j log0,e x, y = j ord,e L A x, y On the other hand, since j ord,e is non-degenerate, it induces an isomorphism j ord,e : X E T0 A E and there is a unique E-linear endomorphism L A : T 0 A E T 0 A E such that 10 j log0,e = L A j ord,e : X E T 0 A E If we transport L A to X E via j ord,e, it is easy to see that 9 and 10 are equivalent, ie L A = L A Furthermore, since X E := E Z X, 10 is equivalent to 11 j log0 = L A j ord : X T 0 A E We now assume that we are given a rigid analytic torus J = J /E defined by an exact sequence of the form 0 X j Hom Y, G an m J 0, where X and Y are free Z-modules of the same rank on which G E acts trivially By the definition of a rigid analytic torus, j factors through 12 j : X Hom Y, E, identifying X with a lattice in Hom Y, E that is discrete in Hom Y, G an m, ie such that its intersection with every affinoid is finite see [36, 42] The underlying E-analytic manifold of J is J E := Hom Y, E X and the tangent space to J and J E at the identity are identified with T 0 J E := Hom Y, E and agree with T 0 A E when J = A an is the rigid analytic manifold attached to an abelian variety A over E Let us write log J : J E T 0 J E for the formal logarithm as defined, for example, in [7, Ch III, 7, no 6, Def 2, Prop 10 and Prop 13] We can define j l := l j : X T 0 J E using 12 in this lightly more general setting We take 11 as the definition of the L-invariant L J of J, ie L J : T 0 J E T 0 J E is the unique E-linear endomorphism such that 13 j log0 = L J j ord : X T 0 J E The following simple result connects L-invariants and formal logarithms Lemma 21 The formal logarithm is explicitly given by the rule 14 log J [z] = log 0 z L J ord z, where z Hom Y, E represents [z] J E Proof The formal logarithm is characterized as the unique group homomorphism J E T 0 J E that is locally analytic at the identity with differential the identity see [7, Ch III, 7, no 6, Prop 10] Since log 0 has this property and the differential of ord is zero, we can conclude that the differential of the right hand side of 14 has this property once we have checked that it is well defined on the quotient J By definition of j l x = l j x, we see that, for every x X, log 0 j x L ord j x = j log0 x L J j ord x 13 = 0 5

6 3 Definite multiplicative integrals and the Manin-Drinfeld Theorem Let C P 1 Q p, E be the space of continuous, E -valued functions and let C har E, Z be the space of Z-valued harmonic cocyles We can define a GL 2 Q p -equivariant pairing 15, : C har E, Z Z C P 1 Q p, E /E E induced by the rule cf [19, 9] and [4, 63] 16 c, f := lim f t e,t0 ce, T 0 T e T 0 f C P 1 Q p, E Here, T 0 runs over all the net of finite connected subtrees of T, T 0 denotes the boundary edges of T 0 oriented towards T = P 1 Q p and t e,t0 is an arbitrary point in U e The limit exists, it is well defined and induces 15 For every τ 1, τ 2 H p, let us consider the function θ τ 2 τ 1 z := z τ 2, z P 1 Q p z τ 1 Write [ θ τ ] 2 τ 1 for its image in C P 1 Q p, E /E Define Then we have the following exact sequence: E E = Div H p G Q p /E and 0 E = Div 0 H p GQp /E Define a GL 2 Q p -equivariant integration pairing i E deg E 0 18 I 0 : C har E, Z Z 0 E E by linear extension of the rule I 0 c, τ 2 τ 1 := c, [ θ τ 2 τ 1 The integration pairing I 0 gives rise to a morphism 19 I 0 : 0 E Γ N +,N Hom C har E, Z Γ N +,N, E =: T ΓN +,N E On the other hand 17 induces the exact sequence 20 H 1 ΓN +,N, Z 0 E Γ N +,N ] i deg E ΓN Z 0 +,N Defining q ΓN +,N := Im I 0, it follows from that I 0 induces AJ 0 = AJ 0 Γ N +,N : 0 E Γ N +,N T Γ N +,N E q ΓN +,N =: J ΓN +,N E Proposition 31 We have that q ΓN +,N T ΓN +,N Q p is Hecke stable and the quotient J ΓN +,N E is represented by a rigid analytic torus J ΓN +,N over Q p endowed with an action of the Hecke algebra Proof The fact that q ΓN +,N is Hecke stable follows from the fact that and I 0 are Hecke equivariant The proof of the other assertions will be given after Lemma 41 Let AJ 0 X N +,N p : Div 0 X N +,N p J N +,N p be the algebraic Abel-Jacobi map 6

7 Theorem 32 Manin-Drinfeld There is a unique isomorphism J ΓN +,N J N +,N p over Q p 2 making the following diagram commutative for every complete field E C p : 0 E ΓN+,N AJ 0 Γ N +,N J ΓN +,N E Div 0 X N +,N p This isomorphism is Hecke equivariant EAJ 0 X N +,N p J N +,N p E Proof We base our computations on those of [14, 22 and 23] that we briefly recall Let us write T ΓN +,N := Γ N +,N \T for the quotient graph, whose oriented edges are E Γ N +,N := Γ N +,N \E and whose vertices are V ΓN +,N := Γ N +,N \V Note that the action of Γ N +,N is orientation preserving Define Div E ΓN +,N to be the quotient of Div E ΓN +,N by the relations e + e = 0 for every e E ΓN +,N Sending a vertex v to δ s v := se=v e induces an injective map δs 0 Div V ΓN +,N Div E ΓN +,N Div har E ΓN +,N 0, where Div har E ΓN +,N is, by definition, the cokernel of δ s It follows that 21 C har E, Z Γ N +,N = Hom Div har E ΓN +,N, Z Consider the canonical morphism 22 Div har E ΓN +,N Div har E ΓN +,N = C har E, Z Γ N +,N, where we write X := Hom X, Z By abstract nonsense, 22 is canonically the torsion free quotient of Div har E ΓN +,N As explained in [14, 23], it is the universal module considered in [14, 22] in case M = Z, and [14, 26] becomes I 0 : 0 E Γ N +,N C har E, Z Γ N +,N E = Hom C har E, Z Γ N +,N, E Our notation is justified by the fact that I 0 is indeed explicitly given by the morphism 18 On the other hand, the mapping e e := t e s e induces an exact sequence 0 Div har E ΓN +,N Div E ΓN +,N Div 0 V ΓN +,N 0, where Div har E ΓN +,N is defined to be the kernel As explained in [14, 23], there is a natural pairing, : Div E ΓN +,N Z Div E ΓN +,N Z, induced by the pairing on Div E ΓN +,N defined by declaring that e and e are orthogonal when e e or e, while e, e = e, e = 1 As is also observed in loc cit, e e, induces an identification 23 Div har E ΓN +,N Hom Div har E ΓN + free,n, Z where Div har E ΓN +,N is the torsion free quotient of Div har E ΓN + free,n There are natural spectral sequences in Γ N +,N -equivariant homology, which degenerate to a long exact sequence from which we extract the following exact sequence see [8, Ch VII, 7, 73 and 77] and [31, Ch II, 28, Proposition 13]: 24 H 1 Γ v N +,N, Z H 1 ΓN +,N, Z Div har E ΓN +,N 0 σ V ΓN +,N 7

8 Here, Γ v N +,N Γ N +,N is the finite stabilizer of v in Γ N +,N and H 1 ΓN +,N, Z = Γ ab N +,N, canonically It follows that the left-most term of 24 is finite see [30, Ch VIII, 2, Corollaries 1 and 2], implying that the epimorphism in 24 gives an isomorphism 25 Hom Div har E ΓN +,N, Z Hom Γ ab N +,N, Z Applying Z E and employing the canonical identification X Z E = Hom X, E, we see that free gives λ : Hom C har E, Z Γ N +,N, E = C har E, Z Γ N +,N E = Div har E ΓN +,N Z E Γ ab N +,N Z E = Hom Γ ab N +,N, E 26 free We deduce that there is a commutative diagram 0 E ΓN+,N I 0 Hom C har E, Z Γ N +,N, E I 0,D Hom λ Γ ab N +,N, E where I 0,D is the left hand side of [14, 213], as the reader may check using its explicit expression [14, 214] Let us set q D Γ := Im I 0,D N +,N so that Hom Γ ab JΓ D E := N +,N, E N +,N q D Γ N +,N is the rigid analytic torus considered in [14] Writing AJ 0,D : 0 E Hom Γ ab Γ N +,N, E =: J D N +,N Γ E N +,N q D Γ N +,N for the induced morphism, 26 yields the following commutative diagram: 27 0 E ΓN+,N AJ 0 J N +,N E AJ 0,D λ J D N +,N E Thanks to 27, we see that λ identifies our analytic torus with that of [14] and our AJ 0 corresponds to the map AJ 0,D induced by the multiplicative integral I 0,D, the left hand side of [14, 213] But [14, Proposition 231] expresses I 0,D in terms of theta functions and, using this result, the Manin-Drinfeld Theorem takes the form AJ 0 X N +,N p [τ 2 ] [τ 1 ] = I 0,D τ 2 τ 1 mod q D Γ = AJ0,D N +,N τ 2 τ 1, where τ 2, τ 1 H p, [τ i ] is the corresponding point in X N +,N p E, and [τ 2 ] [τ 1 ] Div 0 X N +,N p E see [14, Theorem 232] The uniqueness follows by taking E = C p 4 Definite additive integrals Let D 0,b P 1 Q p resp Char b E, E be the space of bounded distributions in D0 P 1 Q p resp C har E, E They have been considered in [44], [17, 23], [28, 3] and [35, Remark 5] They are defined so that the morphism the induced map R : D 0,b P 1 Q p C b har E, E 8

9 is an isomorphism Furthermore, it is known that D 0,b P 1 Q p Γ N +,N = D 0 P 1 Q p Γ N +,N and C b har E, E Γ N +,N = C har E, E Γ N +,N see [44, Proposition 9] and [17, 23], so that 28 D 0 P 1 Q p Γ N +,N Char E, E Γ N +,N Let 29 l : E E be a continuous group homomorphism, which is therefore locally analytic Lemma 41 Suppose that µ D 0,b P 1 Q p is such that R µ C har E, Z and that f C P 1 Q p, E is such that l f A P 1 Q p Then µ l f = l R µ, f Proof The boundedness of µ D 0,b P 1 Q p implies that the tautological pairing of a distribution µ against any f A 0 Q p given by 30 µ f := lim µ f t e,t0 χ Ue = lim R µ e f t e,t0 T 0 T T 0 T e T 0 e T 0 Here, we may choose t e,t0 for every e and T 0 We remark that we have R µ C har E, Z C b har E, E D 0,b P 1 Q p Applying 30 with f = l f, the claim follows from the fact that l is a continuous group homomorphism and from definition 16 For every τ 1, τ 2 H p, let us consider the function θ τ 2 τ 1 l z := l θ τ 2 τ 1 z = l z τ 2 z τ 1, z P 1 Q, whose equivalence class modulo E constant functions is denoted [ θ τ ] l 2 τ 1 We may then define a GL2 Q p - invariant pairing 31 Il 0 : D 0 P 1 Q p 0 E E by extending of the rule Il 0 µ, τ 2 τ 1 := µ, [ θ τ ] l 2 τ 1, well defined as µ 1 = 0 As an immediate consequence of Lemma 41 and 28, we deduce the commutativity of the following diagram: 32 0 E ΓN+,N I 0 l Hom C har E, Z Γ N +,N, E ι E 0 E ΓN+,N I 0 Hom C har E, Z Γ N +,N, E l Here, l is induced by 29 We can now complete the proof of Proposition 31 as follows First, we remark that, according to [18, 64], must prove that 33 q ΓN +,N Hom C har E, Z Γ N +,N, Q ord p Hom C har E, Z Γ N +,N, Z Hom R C har E, Z Γ N +,N, R 9

10 has finite kernel K and that its image is a lattice in the classical sense 1 Consider the following diagram, commutativeby H 1 ΓN +,N, Z 0 E I 0 Hom C ΓN+,N har E, Z Γ N +,N, E ι H 1 ΓN +,N, E ι E 0 E ΓN+,N I 0 l Hom C har E, Z Γ N +,N, E We are interested in the case l = ord and E = Q p The same formalism as in [28, proof of Theorem 311] applies in this setting with a shift by 1 in the degrees This reduces the proof of the fact that Iord 0 is an isomorphism to the proof that the natural boundary map δ : C har E, Q p Γ N +,N H 1 Γ N +,N, Q p is an isomorphism This is true thanks to [37, discussion before Theorem 39] By definition we have q ΓN +,N := Im I 0 and it follows from the commutative diagram 34 and the fact that Iord 0 is injective that K ker ι But according to the Universal Coefficients Theorem, we have 35 ι : H 1 ΓN +,N, Z H 1 ΓN +,N, Z Z Q p = H 1 ΓN +,N, Q p in 34 In particular we see that # ker ι <, being the torsion subgroup of the finitely generated abelian group H 1 ΓN +,N, Z The finiteness of K follows from the inclusion K ker ι It remains to show that ord q ΓN +,N ord q ΓN +,N = Im ord I 0 Hom C har E, Z Γ N +,N, Z is a lattice Since is finitely generated, we need only show that its rank is the R-dimension of the codomain Hom R C har E, Z Γ N +,N, R = Hom C har E, Z Γ N +,N, Z Z R of 33 or, equivalently, that it equals the Q p -dimension of Hom Qp C har E, Z Γ N +,N, Q p This is true if and only if the Q p -span of ord q ΓN +,N is Hom Qp C har E, Z Γ N +,N, Q p = Hom C har E, Z Γ N +,N, Z Z Q p But this is a consequence of the commutativity of 34, the surjectivity of Iord 0 and 35 We remark that I 0 ord, I 0 log 0 : H 1 ΓN +,N, E T 0 J ΓN +,N l E are Hecke equivariant maps Because we have proved that Ilog 0 0 is an isomorphism and we may take E = Q p, the following definition makes sense cf [28, Corollary 313] Recall that we have T 0 J ΓN +,N E = Hom C har E, Z Γ N +,N, E and we write T Qp for the Hecke algebra with Q p coefficients acting on T 0 J ΓN +,N Q p Definition 42 The Teitelbaum L-invariant is the unique endomorphism L End TQp T 0 J ΓN +,N Q p such that Ilog 0 0 = L Iord 0 1 Indeed let K be the finite kernel, so that we may write qγn +,N = K q Γ N +,N with q Γ N +,N q Γ N +,N such that ord is injective on q Γ N +,N and such that the image is a lattice Then, according to [18, 64], T Γ N +,N q Γ N +,N is an analytic torus But since K is finite, T Γ N +,N q ΓN +,N = K TΓ N +,N q exists again by [18, 64] and it is also an analytic torus Γ N +,N 10

11 By construction of J ΓN +,N, we have L = L J ΓN +,N : apply 13 with j : q ΓN +,N := Im I 0 Hom C har E, Z Γ N +,N, E and use 32 to deduce that log 0 I 0 = I 0 log 0 = L I 0 ord = L ord I 0, from which we get j log0 = L J j ord We deduce from Lemma 21 that the formal logarithm : J ΓN +,N E T 0 J ΓN +,N E log JΓN +,N is given by the rule log JΓN [z] = log +,N 0 z L ord z, where z Hom C har E, Z Γ N +,N, E represents [z] J ΓN +,N E The following result is now a consequence of 34 with l = ord and l = log 0 Proposition 43 We have: log JΓN +,N AJ 0 = I 0 log 0 ι L I 0 ord ι =: log AJ 0 5 Definite additive integrals: proof of the main result We remark that the Hecke action on the Z appearing in the long exact sequence 20 is Eisenstein: for a prime q pn, we define t q := q + 1 T q ; then t q = 0 on it It follows that we may regard t q as a morphism Noticing that t q End Qp T 0 J ΓN +,N t q : E ΓN 0 E +,N Γ N +,N is invertible because T 0 J ΓN +,N is cuspidal, we may define J ΓN +,N 36 log AJ := t 1 q log AJ 0 t q : E ΓN +,N T 0 Proposition 51 The map log AJ does not depend on the choice of q pn and is the unique Hecke equivariant morphism making the following diagram commutative: 37 0 E ΓN+,N i log AJ 0 E ΓN +,N log AJ T 0 J ΓN +,N With the notation introduced before the statement of Theorem 11, it is given the formula log AJ τ c ω = d [ µ ω Ω,rτ dκ x τy κ] κ=0 Proof The fact that log AJ is Hecke equivariant follows from the Hecke equivariance of the maps in 36 The Hecke equivariance of log AJ : τ d [ µ Ω,rτ dκ x τy κ] T 0 J ΓN + κ=0,n can be proved as in [4, Proposition 218] The fact that log AJ makes 37 commute is an easy consequence of the Hecke equivariance of log AJ 0 The fact that log AJ makes 37 commutative can be proved as in [4, Proposition 219] or, more explicitly, using [34, displayed equation after 24] It remains to check that there is a unique Hecke equivariant morphism making 37 commutative, from which independence of q and the equality log AJ τ = log AJ τ will follow Set T := T 0 J ΓN +,N and suppose that Hom T E, T ΓN is the difference of two morphisms making 37 commutative, +,N 11

12 where we write T for the Hecke algebra It suffices to consider the Hecke algebra generated by the T q for q pn for our purposes Then the restriction Imi of to Im i is zero It follows from 20 that there is an exact sequence deg 0 Im i E ΓN Z 0 +,N Taking the T -dual yields 0 Hom T Z, T Hom T E ΓN +,N, T Hom T Im i, T But we have Hom T Z, T = 0 because t q = 0 on Z while it is invertible on T Therefore, Imi is injective and = 0 as claimed Our Theorem 11 is now a consequence of Propositions 51 and 43 and Theorem 32 as follows First, Theorem 32 implies that, up to the identification J ΓN +,N J N +,N p, we have 38 AJ 0 X N +,N p [τ 2 ] [τ 1 ] = AJ 0 τ 2 τ 1 for every τ 2, τ 1 H p, where [τ i ] is the corresponding point in X N +,N p E As explained in 2, the isomorphism J ΓN +,N J N +,N p induces an identification T 0 J ΓN +,N E T 0 JN +,N p and the formal logarithms are identified by the characterization appearijg in Lemma 21 We deduce from 38 that we have 39 log JN +,N p AJ 0 X N +,N p [τ 2 ] [τ 1 ] = log JΓN AJ 0 τ 2 τ 1 +,N It now follows from Proposition 43 and 39 that we have 40 log JN +,N p AJ 0 X N +,N p [τ 2 ] [τ 1 ] = log AJ 0 τ 2 τ 1 Recalling the definitions 8 of log JN +,N p AJ and 36 of log AJ, we deduce from 40 that we have log JN +,N p AJ [τ] = log AJ τ for every τ H p Now Theorem 11 follows from the explicit expression of log AJ that appears in Proposition 51 We end this section explaining how the main result of [4], generalized to the case where the modular form may not have rational Fourier coefficients, follows from our Theorem 11 To this end, for a quadratic Dirichlet character χ, we write S 2 Γ 0 Np new χ N=ω N,χp=a p for the space of those modular forms on GL 2 that are new and such that the eigenvalues of W N and U p acting on f are χ N and χp, respectively Let Q χ /Q be the extension cut out by the character χ and write Q χ p for the p-adic completion at a prime p p determined by a fixed embedding Q C p The Fourier expansion at the -cusp determines a Q-rational structure on the space of modular forms using which we may consider S 2 Γ0 Np, Q χ new p χ N=ω N,χp=a p The conditions 41 χ N = ω N and χ p = a p cut out a quotient Jχ N=ω new N,χp=a p of the new part J new of the Picard variety J = J pn,1 of the modular curve Let f S 2 Γ 0 Np new χ N=ω N,χp= ω p and we f be the normalized Hida eigenfamily through f and write L f,χ for the Mazur-Kitagawa p-adic L-function, restricted to the central critical line Due to the sign conditions 41, L f,χ vanishes at the weight 0 := [ t t 0 = 1 ] to order at least 2 and the association f L f,χ 0 gives rise to a well defined function to L χ 0 : S 2 Γ0 Np, Q χ new p χ N=ω N,χp=a p Q χ p Let us remark that it could also be regarded as a map L χ 0 : H 0 X Np,1/Q χ p, Ω 1 X Np,1/Q χ p new 12 χ N=ω N,χp=a p Q χ p,

13 χ it being just log 2 J P for every P Q J new χ N=ω χ N=ω new N,χp=ap N,χp=a p Q χ As a final piece of notation, for a Q [ G Q /Q] χ -module V, we write V χ for the χ-component of V, we and, for a T new χ N=ω N,χp= ω p -module M and f S 2 Γ 0 Np new χ N=ω N,χp=a p with companion class [f], we write M [f] for the [f]-isotypic component We recall that the companion class of a modular form is the set of its Galois conjugates We now explain give our generalization of [4, Theorem 54] write T new χ N=ω N,χp= ω p for the rational Hecke algebra acting on S 2 Γ 0 Np new χ N=ω N,χp=a p Theorem 52 Suppose that there is a prime q with q N 1 There is an element P χ χ Q Jχ N=ω new N,χp=a p Q χ and a Hecke operator t = t χ T new, χ N=ω N,χp= ω p such that 42 L χ 0 = t log 2 J P χ new χ N=ω N,χp=ap 2 If then 0 P χ [f] Q J [f] Q χ χ, Q J[f] Q χ χ = T[f] Q P χ [f] T [f] Q, where P χ [f] is the [f]-component of P χ 3 We have 0 P χ [f] if and only if L f, χ, If t f Q f is the f-component of t, then for any quadratic Dirichlet character ɛ such that ɛ N = χ N, ɛ p = χ p, and L f, ɛ, 1 0, the following congruence holds t f L f, ɛ, 1 in Q f /Q f 2 Here Q f is the field generated by the Fourier coefficients of f Proof We sketch the required adjustments to the proof of [4, Theorem 54] We will apply our Theorem 11 with B = B q the unique definite quaternion algebra of discriminant q By the Jacquet-Langlands correspondence Jχ N=ω new N,χp=a p appears as a quotient of J N/q,qp over Q; indeed P χ will be constructed from a point on J N/q,qp In order to deduce the main formula 42, we extend scalars to a field E containing the p-adic completion of Q f We write ω f for the differential associated to f and ω φ for its pull-back to J N/q,qp We need to find an element whose image P χ satisfies Q χ Q J N/q,qp Q χ χ L χ,f 0 = t log 2 J P χ ω new f = t log 2 J χ N=ω N,χp=ap N/q,qp Q χ ω φ Applying Theorem 11 when τ = τ Ψ corresponds to the Heegner point attached to an embedding Ψ as in [4, 31] and [34, 6], the k 0 = 0 case of [34, Theorem 11] follows formally From here, the generalization of [4, Theorem 49 and Corollary 410] which is the k 0 = 0 case of [33, Corollaries 526 and 527] follows too Now the proof of Theorem 52 follows the same pattern as that of [4, Theorem 54] See also the proof of [33, Theorem 61] for the minor adjustments needed to handle the case where the Fourier coefficients need not to be rational 13

14 6 The analogous results in the indefinite setting Suppose that p, N + and N are pairwise coprime integers, with p prime and N the squarefree product of an even number of primes We write B for the indefinite quaternion algebra over Q of discriminant N, we fix a maximal order O B B and an Eichler order R 0 pn +, N B of level pn +, with group of invertible elements Γ 0 pn +, N We choose a local isomorphism ι p : B p GL2 Q p and we set Γ N +,N := ι p R 0 pn +, N [1/p] GL 2 Q p Finally, as in the definite setting, we let Γ N +,N Γ N +,N be the subgroup of norm one elements Consider the exact sequence 43 0 C har E, Z C 0 E, Z δs C V, Z 0, where δ s c v := se=v c e, which induces, taking Γ N +,N -cohomology and applying Shapiro s Lemma see [28, 25], the exact sequence where 0 E H 1 Γ N +,N, C har E, Z H 1 Γ 0 pn +, N, Z p new 0, E := coker δ s : C 0 E, Z Γ N +,N C V, Z Γ N +,N, H 1 Γ 0 pn +, N, Z p new := ker δs : H 1 Γ N +,N, C 0 E, Z H 1 Γ N +,N, C V, Z The notation is justified by the following lemma and the fact that H 1 Γ 0 pn +, N, Z p new is identified with the p-new part of H 1 Γ 0 pn +, N, Z, because δ s corresponds to the degeneracy maps under Shapiro s isomorphism The proof of the next result is easy and left to the reader For a Hecke module M, we write M c for its cuspidal part see [28, 24] Lemma 61 1 We have that t q = 0 on E Z p+1z Z for every q pn while t q : H 1 Γ 0 pn +, N, Z p new,c H 1 Γ 0 pn +, N, Z p new,c is injective on H 1 Γ 0 pn +, N, Z p new,c Z g 2 There exists E H H 1 Γ N +,N, C har E, Z such that t q : H H is injective for every q pn, is Z-free with torsion cokernel, inducing H H 1 Γ 0 pn +, N, Z p new,c Q H = H 1 Γ N +,N, C har E, Q c = Q Z H H 1 Γ 0 pn +, N, Q p new,c 3 There exists a unique H which is maximal having the property stated in 2 We fix from now on any H H 1 Γ 0 pn +, N, Z p new,c as granted by Lemma 61 2, eg, the unique maximal H of Lemma 61 3 The integration pairing I 0 gives rise, for every field E, to a Hecke equivariant morphism 44 I 0 : H 1 ΓN +,N, 0 E Hom H 1 Γ N +,N, C har E, Z, E T ΓN +,N E, where T ΓN +,N E := Hom H, E and the second map is induced by H H 1 Γ N +,N, C har E, Z Note that, since H is Z-free, T ΓN +,N is indeed a rigid analytic torus, endowed with an action of the p-new Hecke Q-algebra T acting on the weight two, p-new, level pn + cusp forms on B On the other hand 17 induces the exact sequence 45 H 2 ΓN +,N, Z H 1 ΓN +,N, 0 E i H 1 ΓN +,N, E H 1 ΓN +,N, Z Define q ΓN +,N := Im I 0 The following result is a generalization of [19, Proposition 30] specialized to F = Q 14

15 Proposition 62 We have that q ΓN +,N T ΓN +,N Q p is Hecke stable and the quotient J ΓN +,N E is represented by a rigid analytic torus J ΓN +,N over Q p endowed with an action of the Hecke algebra T Proof The fact that q ΓN +,N is Hecke stable follows from the fact that and I 0 are Hecke equivariant The proof of the other assertions follows exactly the same pattern as the proof of Proposition 31 Indeed, 28 is replaced by 46 H 1 Γ N +,N, D0,b P 1 Q p c H 1 Γ N +,N, C har E, E c = E H, which is [28, Theorem 35] the equality follows from Lemma 61 2 This allows us to define the analoge of the second line appearing in 32, namely I 0 l : H 1 ΓN +,N,E 0 E Hom E H 1 Γ N +,N, C har E, Z, E Hom E H, E where the second arrow is given by H H 1 Γ N +,N, C har E, Z Then Lemma 41 and 46 give the analogue of 32: one need only add one to the co/homology degrees With this formal adjustment, the proof proceed along the same lines, replacing [37, discussion before Theorem 39] with [28, Theorem 311] specialized to the weight 2 case See also [19, Lemma 32] specialized to F = Q, which is [28, Theorem 311] on the component corresponding to f with rational eigenvalues, in light of Iord 0 ι = ord I0 by the analogous of 32 Here is the analogue of the Manin-Drinfeld theorem in this setting: Theorem 63 There is a Hecke equivariant isogeny J ΓN +,N J p new pn +,N J p new pn +,N over Q p 2, where J p new pn +,N is the p-new quotient of the Picard variety of the indefinite Shimura curve attached to an Eichler order of level pn + in the indefinite quaternion algebra of discriminant N Proof The action of the involution at infinity W induces an isogeny J ΓN +,N J + Γ N +,N J Γ N +,N over Q p ; it follows from [16], extended to include the required lift to our abelian varieties in [35] specialized to the weight 2 case, that the L-invariant of J ± p new Γ equals the L-invariant of J N +,N pn +,N, which has split multiplicative reduction over Q 2 p 2 By a standard argument exploiting the fact that the abelian varieties under consideration are endowed with a T-action and their dimension is twice the dimension of T over Q, it follows that there is an isogeny J ± Γ J p new N +,N pn +,N over Q p 2 Remark 64 In [15] and [26], a dual construction of J Γ endowed with an isogeny J N +,N Γ J p new N +,N pn +,N J p new pn +,N is described In particular, J ΓN +,N and J Γ are isogenous N +,N By definition of q ΓN +,N, 44 induces AJ 0 : H 1 ΓN +,N, 0 E H 2 ΓN +,N, Z T Γ E N+,N =: J ΓN +,N E q ΓN +,N Since H 1 ΓN +,N,, Z is a finite group it is finitely generated and H 1 ΓN +,N,, E = E H 1 ΓN +,N,, Z = 0 by the universal coefficient theorem, in view of [28, Lemma 310] it has finite exponent h It follows from 45 that there is a unique h through which h factors: We define h : H 1 ΓN +,N, E h H 1 ΓN +,N, 0 E H 2 ΓN +,N,Z j H 1 ΓN +,N, E AJ h : H 1 ΓN +,N, E h H 1 ΓN +,N, 0 E 0 AJ J ΓN H 2 ΓN + +,N, Z,N E 2 Indeed, we have I 0 l ι = l I 0 for l = ord or log 0 by the analogue of 32 It follows that the L-invariant of [28] considered in [35] is the L-invariant of the monodromy module attached to J, ie the L-invariant of J ΓN +,N Γ Hence, the main N +,N result of [35] gives the required lift of the main result of [16] 15

16 We will need to consider the formal logarithm log JΓN +,N : J ΓN +,N E T 0 J ΓN +,N E By its construction and Lemma 61 2, the target of log JΓN is identified with +,N 47 T 0 J ΓN +,N E = Hom E H, E Hom E H 1 Γ N +,N, C har E, E c, E On the other hand, let d dκ [, ] κ=0 : H1 Γ N +,N,C V, D Ω,0 W H 1 ΓN +,N,E E E be the pairing induced by [35, Lemma 17] which is the faux Abel-Jacobi map of [21, 71], up to Shapiro s isomorphism, by [21, Proposition 81] It is essentially induced by a rule analogous to 6 There is a cohomological family c ω Ω H1 Γ N +,N,D Ω,0 W passing through every ω H 1 Γ N +,N, C har E, E c see [20, Corollary 114] The following result is the analogue f Theorem 11 in this indefinite setting Theorem 65 For every z H 1 ΓN +,N, E and ω H 1 Γ N +,N, C har E, E c, we have d log JΓN AJ h z ω = h +,N dκ [cω Ω, ι z] κ=0 Proof Writing L for the L-invariant of J ΓN +,N, we have log JΓN +,N [x] := log 0 x L ord x Since AJ 0 is induced by I, 0 we see that recall the morphisms Il 0 with l = log 0 or ord from log JΓN AJ 0 = log +,N 0 I 0 L ord I 0 = Ilog 0 0 LIord 0 ι, where the second equality follows from 32 We may form the following commutative diagram H 1 ΓN +,N, E h H 1 ΓN +,N, 0 E H 2 ΓN +,N,Z j H 1 ΓN +,N, E ι H 1 ΓN +,N,E E h ι H 1 ΓN +,N,E 0 E H 2 ΓN +,N,E j H 1 ΓN +,N,E E ι It follows that 49 log JΓN +,N AJ h = log JΓN AJ 0 h = Ilog 0 +,N 0 LIord 0 ι h = Ilog 0 0 LIord 0 h ι = Ilog 0 0 LIord 0 j 1 j h ι = Ilog 0 0 LIord 0 j 1 h ι = h Ilog 0 0 LIord 0 j 1 ι, the second equality following from 48 The claim now follows from 49 and [35, Proposition 26] or [21, Porposition 75] and Shapiro s lemma, giveing Ilog 0 0 LIord 0 j 1 ω ι z = d dκ [cω Ω, ι z] κ=0 16

17 7 Derivatives of distributions and the Coleman-Teitelbaum cocycle 71 Generalities on distributions and locally trivial bundles Let X be a locally compact, paracompact strictly paracompact by [39, Proposition 87] p-adic manifold For a Banach K-algebra O, we let A X, O be the space of O-valued locally analytic functions on X and let D X, O := L O A X, O, O L A X, O, O, the strong O-dual of A X, O Here L V, W denotes the space of continuous linear maps between the two locally convex topological spaces V and W, with its strong topology, and L O V, W is its subspace of O-linear maps, when V and W are given an O-module structure, with the induced topology If f : X Y is a morphism of p-adic manifolds as above, we have maps f O : A Y, O A X, O and f : D X, O D Y, O, the first being the pull-back and the second given by duality We note that 50 f δ O x = δ O fx, for every x X if δ O : X D X, O denotes the Dirac distribution map It can be shown that there are topological identifications 3 and such that T O DX : O D X D X, O P O1,O2 DX : D X 1,DX 2 1, O 1 ι D X 2, O 2 D X 1 X 2, O 1 O 2 51 T O DX 1O δ x = δ O x and P O1,O2 DX 1,DX 2 δ O1 x 1 ι δ O2 x 2 = δ O1 O 2 x 1,x 2 Since the K-span of {δ x : x X} is dense in D X use strict paracompactness { } in order to reduce to the compact case [40, Lemma 31], we also deduce that the O-span of δ O x : x X is dense in D X, O: { } 52 O δ O x : x X = D X, O It follows from 52 that the properties in 50 and 51 characterize the respective maps When X is compact, then A X is the inductive limit lim V n of Banach spaces V n with injective and n compact transition maps, while D X is the inverse limit lim V n of Banach spaces of V n by compact and n dense transition maps namely V n = L V n, K These well known facts can be deduced from [38, Lemma 164, Proposition 165 and Proposition 1610], noticing that lim V n V n has dense image if the transition n maps have this property It follows from loc cit that both A X and D X mutually dual, hence reflexive We call such a locally convex space V := lim V n resp V := lim V n n n a space of compact type resp a nuclear Fréchet space Because we may write X = i X i where X i is open and compact, we see that A X = i A X i by [39, Proposition 125] and, taking duals, yields D X = i D X i the reader may check that the canonical map i L V i, W L i V i, W is a topological identification, when W is normed Consider the canonical morphism of locally convex spaces 1 O DX : O ι D X O D X Then D X = i D X i and we may form the following commutative diagram: i O t ι D X i O ι i D X i = O ι D X i 1 O DXi i O D X i 1 O i DX i t O i D X i = 1 O DX O D X Since O is a Banach space, it is not difficult to see that the canonical morphisms t are topological identifications On the other hand, 1 O DXi is a topological identification because O and D X i are Fréchet spaces; therefore, we may apply [38, Proposition 176] We have proved the following remark: 3 We write V ι W resp V W to denote V K W with the inductive resp projective tensor topology Also, a morphism V W is a continuous, K-linear if V and W are locally convex spaces Completion means Hausdorff completion 17

18 Remark 71 The canonical morphism 1 O DX : O ι D X O D X is a topological O-linear identification Suppose, from now on, that a locally compact, paracompact, p-adic Lie group T acts on X by means of a locally analytic map a : T X X The left action of T on D X can be extended, with respect to δ : T D X, to a left action of D T making D X a D T -module under the convolution product: ν µ F = P DT,DX ν µ F a, where ν D T, µ D X and F A X More formally, the multiplication is given by the separately continuous morphism 53 D T ι D X P DT,DX We note the formula D T X a D X 54 δ t δ x = δ tx for t T and x X, characterizes the multiplication law, thanks to 51 Since 54 characterizes the respective maps, upon setting X = T we obtain a K-algebra structure on D T with identity element δ 1 and that, for a general X, we get a D T -module structure on D X Note that, when T and X are compact, the multiplication law is continuous and, hence, D T is a locally convex K-algebra and D X is a locally convex D T -module We write Hom A T, O for the subgroup of Hom T, O consisting ofthose group homomorphisms whose composition with the inclusion O O belongs to A T, O; give Hom A T, O A T, O the subspace topology We let Hom L D T, O L D T, O be the subspace consisting of those morphisms of locally convex spaces that are also morphisms of K-algebras 4 Lemma 72 The map C O : Hom L D T, O Hom A T, O defined by the rule C O k t := k δ t is a well defined topological identification Proof By [40, Theorem 22] we have to prove that C O k Hom A T, O if and only if k Hom L D T, O 5 We claim that, for a continuous K-linear morphism k L D T, O to belong to Hom L D T, O, it is necessary and sufficient that k δ 1 = 1 and k δ t1 δ t2 = k δ t1 k δ t2 Indeed we note that both the left and the right hand sides of the equality k ν 1 ν 2 = k ν 1 k ν 2 are linear in the variables ν 1 and ν 2 Furthermore, if we fix one of these variables, the two resulting functions are K-linear and continuous in the remaining variable thanks to 53, showing that the multiplication law is separately continuous Hence, the claimed equality k ν 1 ν 2 = k ν 1 k ν 2 follows from the identity k δ t1 δ t2 = k δ t1 k δ t2 and 52 Since k δ t1t 2 = k δ t1 δ t2 by 54, the claim follows Suppose that we are given a map k Hom L D T, O By Lemma 72 its restriction to δ : T D T gives rise to a locally analytic group homomorphism k : T O sending t to t k := k δ t We define the following closed sub O-module of A X, O 6 : A k X = A X, k = { F A X, O : F tx = t k F x } Viewing both O and D X as D T -modules by means of k and, respectively, the convolution product, we may define D k X := O k D X and D X, k := L O A k X, O We assume, further, that X is endowed with a right action by a semigroup Σ with the property that the mapping σ : X X is locally analytic for every σ Σ There are induced left resp right actions of Σ on the spaces of locally analytic functions resp distributions defined by the rule σf x := σ F x = F xσ 4 Since the multiplication law D T D T D T is only separately continuous, D T is not a locally convex algebra When T is compact the multiplication law is continuous and Hom L D T, O is the space of morphisms of locally convex algebras 5 We remark that [40, Theorem 22] can be refined to a bicontinuous isomorphism and extended to the case where the manifold is strictly paracompact 6 Suppose that Fi F is a net in A k X converging to F A X, O If t T let t : X X be the locally analytic map given by left multiplication by t, so that t F i t F Since the multiplication in O is separately continuous, G t k 2G is continuous and t k 2F i t k 2F Since t F i = t k 2F i we deduce that t F = t k 2F for every t T, meaning that F A k X 18

19 resp Λσ F := σ Λ F = Λ σf We will assume that the right Σ-action is compatible with the left T -action in the sense that txσ = txσ for all t T, x X, and σ Σ It follows that Σ acts on A k X, D k X and D X, k These spaces of distributions are related by the following lemma Lemma 73 There is a unique morphism T k DX commutative: of locally convex spaces making the following diagram 55 O D X T O DX D X, O D k X T k DX D X, k The morphism T k DX is O, Σ-equivariant Proof We first need to prove that, if α O, µ D X and ν D T, then 56 T O DX αk ν µ = T O DX α ν µ on A k X By definition F A k X if and only if δ O tx F = F tx = t k F x = k δ t δ O x F By 54 and 51 we have T O DX 1 δ t δ x = T O DX 1 δ tx = δ O tx and T O DX k δt δ x = k δt δ O x We deduce, by O-linearity, the identity T O DX αk δ t δ R x = T O DX α δ t δ x on A k X We note that both the left and right hand sides of 56 are linear in the variables µ and ν Furthermore, if we fix one of these variables, the two resulting functions are continuous in the remaining variable thanks to 53, showing that the multiplication law is separately continuous Hence, 56 follows from 52 The existence and uniqueness of T k DX follows Since O D X D k X is surjective and all the arrows other than T k DX in 55 are O, Σ-equivariant, Tk DX is equivariant as well We note that, writing 1 k δ x D k X for the image of 1 δ x O D X and δ k x D X, k for the evaluation at x morphism the image of δ O x D X, O, 51 implies that the morphism T k DX appearing in 55 can be characterized by the formula T k DX 1 k δ x = δ k x for every x X We say that π : X T \X =: X is a trivial T -bundle if the quotient X := T \X has the structure of a locally compact, paracompact, p-adic manifold 7 and there exists a locally analytic section s : X X of π such that the locally analytic mapping h = h s : T X X given by h t, x := ts x is a locally analytic isomorphism Let s : A X, O A X, O be the pull-back map on functions We let k Hom L D T, O in the following discussion Lemma 74 The map s restricts to an topological identification s = s k : A k X A X, O Proof Define r = r k : A X, O A k X by r f h t, x = t k f x We claim that r is the inverse of s If f A X, O, F A k X, x X and t T, then s r f x = r f s x = r f h 1, x = f x and r s F h t, x = t k s F x = t k F s x = F ts x = F h t, x 7 According to [32, Part II, Ch III 12 Theorem 1] there exists at most one p-adic manifold structure on X making π : X X a topological quotient in such a way that π is a submersion See [32, Part II, Ch III 12 Theorem 2] for natural conditions granting the existence of such a quotient 19