Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture

Massimo Bertolini Henri Darmon Adrian Iovita January 23, 2010

Contents

1 Automorphic forms on quaternion algebras 7 2 Teitelbaum’s L-invariant 11 3 Families of automorphic forms on B 14 4 A geometric interpretation of p-adic

families of automorphic forms 21 5 Orton’s L-invariant 32 6 Distribution-valued modular symbols 37

Introduction

Let f = ∑ a_{n}q^{n} be a newform of even weight k_{0} + 2 ≥ 2 on Γ_{0}(Np), where N ≥ 4 is a positive integer and p is a prime which does not divide N. We denote by L(f, s) the complex L-function attached to f, and by L(f, χ, s) its twist by a Dirichlet character χ. A theorem of Shimura asserts the existence of a complex period Ω_{f} such that the special values

L(f, χ, j)/Ω_{f} with 1 ≤ j ≤ k_{0} + 1

belong to the subfield K_{f} of ℂ generated by the Fourier coefficients of f, and even to its ring of integers. These special values (when χ ranges over the Dirichlet characters of p-power conductor) can be interpolated p-adically, yielding the Mazur-Swinnerton-Dyer p-adic L-function L_{p}(f, s), a p-adic analytic function whose definition depends on the choice of Ω_{f}. Denote by

L^{∗}(f, χ, 1 + k_{0}/2) := L(f, χ, 1 + k_{0}/2)/Ω_{f},

the algebraic part of L(f, χ, s) at the central critical point s = 1 + k_{0}/2. The modular form f is said to be split multiplicative if

f|U_{p} = p^{k_{0}/2}f.

In that case, L_{p}(f, s) has a so-called exceptional zero at s = 1 + k_{0}/2 arising from the p-adic interpolation process. In fact, like its classical counterpart, the p-adic L-function L_{p}(f, s) has a functional equation of the form

L_{p}(f, k_{0} + 2 − s) = ϵ_{p}(f)⟨N⟩^{s−1−k_{0}/2}L_{p}(f, s),  (1)

and the sign ϵ_{p}(f) = ±1 that appears in this equation is related to the the sign ϵ_{∞}(f) in the classical functional equation for L(f, s) by the rule

ϵ_{p}(f) = \{ \begin{matrix}−ϵ_{∞}(f) & if f is split multiplicative; \\ ϵ_{∞}(f) & otherwise. \end{matrix}

In the case where f is a split multiplicative newform, Mazur, Tate and Teitelbaum made the following conjecture in [MTT]:

Conjecture 1. There exists a constant L(f) ∈ ℂ_{p}, which depends only on the restriction of the Galois representation attached to f to a decomposition group at p, and such that

L′_{p}(f, χ, 1 + k_{0}/2) = L(f)L^{∗}(f, χ, 1 + k_{0}/2),  (2)

for all χ with χ(−1) = χ(p) = 1.

The constant L(f), which Mazur, Tate and Teitelbaum called the Linvariant, was only defined in [MTT] in the weight two case k_{0} = 0. In the higher weight case k_{0} > 0, several a priori inequivalent definitions of L(f) were subsequently proposed.

1. In [T], Teitelbaum offered the first definition for L(f). This invariant, denoted L_{T} (f), is based on the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves and is only defined for modular forms which are the Jacquet-Langlands lift of a modular form on a Shimura curve uniformized by Drinfeld’s p-adic upper half plane. This occurs, for example, when the conductor of f can be written as a product of three pairwise relatively prime integers of the form

pN = pN^{+}N^{−},

where N^{−} is the square-free product of an odd number of prime factors. A modular form which satisfies this condition will be said to be padically uniformisable.

2. Coleman [Co2] then proposed an analogous but more general invariant L_{C}(f) by working directly with p-adic integration on the modular curve attached to the group Γ_{0}(p) ∩ Γ_{1}(N).

3. Fontaine and Mazur [Ma] gave a definition for the so-called Fontaine-Mazur L-invariant L_{F M}(f) in terms of the filtered, Frobenius monodromy module of the p-adic Galois representation attached to f.

4. In [Or], Orton has introduced yet a further L-invariant L_{O}(f), based on the group cohomology of arithmetic subgroups of GL_{2}(ℤ[1/p]), extending to forms of higher weight the approach taken in [Da1] for k_{0} = 0.

5. Finally Breuil defined in [Br] the L-invariant L_{Br}(f) in terms of the p-adic representation of GL_{2}(ℚ_{p}) attached by him to f.

We now know that all the above L-invariants are equal (when they are defined) as result of work of many people, which we briefly list below (see [Cz4] for a more detailed account of these various articles and preprints).

The equality of the L-invariants L_{C}(f) and L_{F M}(f) was proved in [CI] by making explicit the comparison isomorphism between the p-adic étale cohomology and log-crystalline cohomology of the modular curve X_{0}(Np) with respective coefficients. The equality of L_{T} (f) and L_{C}(f) (when they are both defined) was proved in [IS] by interpreting L_{T} (f) as the L-invariant of a filtred, Frobenius monodromy module. Breuil proved in [Br] the equality L_{Br}(f) = L_{O}(f), which is a manifestation of the local-global compatibility for the p-adic Langlands correspondence.

It was first observed by Greenberg and Stevens for weight two (in [GS]) and by Stevens in general (in [St]) that p-adic deformations of f, i.e. p-adic families of modular eigenforms are relevant for conjecture (1). To describe these objects precisely, let

W := Hom_{cont}(ℤ_{p}^{×}, ℚ_{p}^{×})

denote the weight space, viewed as the ℚ_{p}-points of a rigid analytic space. There is a natural inclusion ℤ ⊂ W by sending k to the function x ↦ x^{k}. Write A(U) for the ring of rigid analytic functions on U, for any affinoid disk U ⊂ W.

A p-adic family of eigenforms interpolating f is the data of a disk U with k_{0} ∈ U, and of a formal q-expansion

f_{∞} = ∑_{n=1}^{∞} a_{n}q^{n},  (3)

with coefficients in A(U) satisfying:

1. For every k ∈ U ∩ ℤ^{≥0},

f_{k} := ∑_{n=1}^{∞} a_{n}(k)q^{n}

is the q-expansion of a normalized eigenform of weight k + 2 on the congruence group Γ_{1}(p) ∩ Γ_{0}(N);

2. f_{k_{0}} = f.

The existence and essential uniqueness of the family f_{∞} interpolating f is proved in [Co1].

Greenberg and Stevens for weight two and Stevens in general first proved that L_{C}(f) = −2(dloga_{p})_{κ=k_{0}}. Colmez generalized the Galois cohomology calculations in [GS] by working inside Fontaine’s rings and proved the equality L_{F M}(f) = −2(dloga_{p})_{κ=k_{0}} in [Cz2]. He also proved the equality L_{Br}(f) = −2(dloga_{p})_{κ=k_{0}} in [Cz3] by using the p-adic local Langlands correspondence for trianguline representations. Let us remark that in fact the quantity L(f)_{NoName} := −2(dloga_{p})_{κ=k_{0}} behaves like an L-invariant: it satisfies the equation (2) of conjecture 1 (see [St]) and it is a local invariant of f in the sense that it is invariant to twists of f by Dirichlet characters trivial at p (in fact it is invariant to all twists by Dirichlet characters.)

Conjecture (1) was first proved in [GS] for weight two, and several different proofs have been announced in the higher weight case:

1. By Kato-Kurihara-Tsuji, working with the invariant L_{F M}(f);

2. By Glenn Stevens, working with L_{C}(f);

3. By Orton, working with L_{O}(f) in [Or];

4. By Emerton working with L_{Br}(f) in [E].

The first two proofs are still unpublished but an account of the approach of Kato-Kurihara-Tsuji can be found in [Cz1] while Stevens gave a series of lectures on his theory during the Automorphic Forms semester in Paris, 1998. Notes to these lectures, to which we will refer as [St], although not yet published circulated widely in the mathematical community and greatly influenced articles like [Bu], [Ch] and the present note. As these notes have not been published we will sketch proofs of all the results quoted from them.

The main goal of this note is to describe a new proof of Conjecture 1 which applies to forms which are p-adically uniformisable.

Theorem 2. Assume that f is p-adically uniformisable. Then

L′_{p}(f, χ, 1 + k_{0}/2) = L_{T} (f)L^{∗}(f, χ, 1 + k_{0}/2),  (4)

for all Dirichlet characters χ satisfying χ(−1) = χ(p) = 1.

Our proof of Theorem 2 is based on Teitelbaum’s definition of the Linvariant: this is why it needs to be assumed that f is p-adically uniformisable. Thus the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves and the Jacquet-Langlands correspondence, which play no role in the earlier proofs of Stevens and Kato-Kurihara-Tsuji, are key ingredients in our approach. Section 1 supplies the necessary definitions concerning automorphic forms on definite quaternion algebras, and Section 2 recalls a few basic facts concerning p-adic integration on Shimura curves, including Teitelbaum’s theory of the “p-adic Poisson kernel” and his definition of the invariant L_{T} (f).

Guided by the Jacquet-Langlands correspondence between classical modular forms and automorphic forms on quaternion algebras, Section 3 describes a theory of p-adic families of automorphic forms on definite quaternion algebras, based on ideas of Stevens, Buzzard and Chenevier. The resulting structures are used to prove the following theorem in Section 4, which relates Teitelbaum’s L-invariant to the derivative of the Fourier coefficient a_{p}(k) with respect to k.

Theorem 3. Suppose that f is p-adically uniformisable. Then

L_{T} (f) = −2dlog(a_{p})_{κ=k_{0}}.  (5)

The ideas of Orton in [Or], which are recalled in Section 5, make it apparent that the definition of the invariants L_{T} (f) and L_{O}(f) are very similar in flavour. The calculations of Sections 1 to 4, when transposed to the context of a modular form on GL_{2}(ℚ), with the “integration on H_{p} × H” defined in terms of modular symbols playing the role of the p-adic line integrals on Drinfeld’s upper half-plane, leads to the proof of the following analogue of Theorem 3, which is described in Section 6:

Theorem 4. Let f be a modular form of weight k on Γ_{0}(N) which is split multiplicative at p. Then

L_{O}(f) = −2dlog(a_{p})_{κ=k_{0}}.  (6)

Theorem 2 now follows directly from Theorems 3 and 4, in light of Orton’s proof of Conjecture (1).

The remainder of the text will focus on explaining the proofs of Theorems 3 and 4, which are independent (both in their statement, and their formulation) of the existence and basic properties of either the p-adic L-function or the p-adic Galois representation attached to f and the Coleman family interpolating it.

We emphasize that the proof of Theorem 2 owes much to the ideas that are already present in the earlier (although still unpublished) approaches of Stevens and Kato-Kurihara-Tsuji. The main virtues (and drawbacks) of our method are inherently the same as those in Teitelbaum’s approach to defining the L-invariant: a gain in simplicity (because the method involves p-adic integration on a Mumford curve rather than a modular curve, and requires no information about Galois representations) offset by a certain loss of generality (since the method only applies to automorphic forms that can be obtained as the Jacquet-Langlands lift of a modular form on a p-adically uniformized Shimura curve). A second, less immediately apparent advantage of our approach lies in the insights arising from the connection that is drawn between the two-variable p-adic L-function L_{p}(k, s) attached to f_{∞} and the p-adic uniformisation of Shimura curves. In particular, the new ideas introduced in this article form the basis for the proof of the main result of [BD], which, in the case where f corresponds to a modular elliptic curve E over ℚ and ϵ_{∞}(f) = −ϵ_{p}(f) = −1, relates the leading term of L_{p}(k, s) at the central critical point (k, s) = (2, 1) to the formal group logarithm of a global point on E(ℚ).

1 Automorphic forms on quaternion algebras

Suppose from now on that f is p-adically uniformisable, so that its level pN can be factored as

pN = pN^{+}N^{−}, where gcd(N^{+}, N^{−}) = 1,  (7)

and where N^{−} is square-free and has an odd number of prime factors. Let B denote the quaternion algebra over ℚ ramified exactly at N^{−}∞, and let R denote a maximal order in B. For each ℓ not dividing N^{−} we fix an isomorphism

ι_{ℓ} : B ⊗ ℚ_{ℓ} ≅ M_{2}(ℚ_{ℓ}), with ι_{ℓ}(R ⊗ ℤ_{ℓ}) = M_{2}(ℤ_{ℓ}).

Let ℤ̂ denote the profinite completion of ℤ and let B̂ := B ⊗_{ℤ} ℤ̂.

Let Σ = ∏_{ℓ} Σ_{ℓ} be any compact open subgroup of B̂^{×}, and let V be any ℚ_{p}-vector space equipped with a right action by Σ_{p}. The following definition is taken from Section 4 of [Bu].

Definition 1.1. A V -valued automorphic form on B of level Σ is a function

φ : B̂^{×} ⟶ V satisfying φ(bsσ) = φ(s)σ_{p},  (8)

for all b ∈ B^{×}, s ∈ B̂^{×}, and σ ∈ Σ, where σ_{p} denotes the component of σ at p.

The space of all V -valued automorphic forms on B of level Σ will be denoted S(Σ, V). It is equipped with the action of Hecke operators T_{ℓ} with ℓ |̸N as well as the operator U_{p}, defined as in [Bu], section 4.

Let

Γ̃ = ι_{p} (R[1/p]^{×} ∩ ∏_{ℓ≠p} Σ_{ℓ}) ,

and let Γ denote the subgroup of Γ̃ of elements of determinant 1. The strong approximation theorem for B asserts that

B̂^{×} = B^{×}GL_{2}(ℚ_{p})Σ,

so that we may write

S(Σ, V) = {φ : GL_{2}(ℚ_{p}) ⟶ V | φ(γgu) = φ(g)u}  (9)

for all γ ∈ Γ̃, g ∈ GL_{2}(ℚ_{p}), and u ∈ Σ_{p}.

We will be mostly interested in a specific choice of level structure Σ. Let Σ(N, p) := ∏_{ℓ} Σ_{ℓ} ⊂ B̂^{×} be the compact open subgroup defined by

• Σ_{p} = ι_{p}^{−1}(Γ_{0}(pℤ_{p}));

• Σ_{ℓ} = (R ⊗ ℤ_{ℓ})^{×}, if ℓ divides N^{−};

• Σ_{ℓ} = ι_{ℓ}^{−1}(Γ_{1}(Nℤ_{ℓ})) if ℓ divides N^{+};

• Σ_{ℓ} = (R ⊗ ℤ_{ℓ})^{×}, otherwise.

The group Σ(N, 1) is defined in a similar way, with Γ_{0}(pℤ_{p}) replaced by GL_{2}(ℤ_{p}) in the definition of Σ_{p}. Weights. If k is a positive integer, let P_{k} := ℚ_{p}[z]^{deg≤k} be the space of polynomials of degree ≤ k, equipped with the right action of GL_{2}(ℚ_{p}) given by

(P β)(z) = (cz + d)^{k}P (\frac{az + b}{cz + d}) , for β = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℚ_{p}).

Let V_{k} = Hom_{ℚ_{p}}(P_{k}, ℚ_{p}) denote its ℚ_{p}-dual, equipped with the left action given by

(βh)(P) = h(P β) for P ∈ P_{k} and h ∈ V_{k}.

We may also make V_{k} into a right GL_{2}(ℚ_{p})-module by the rule

hβ = β^{−1}h, for h ∈ V_{k} and β ∈ GL_{2}(ℚ_{p}).

The module V_{k} is isomorphic to P_{k} as a GL_{2}(ℚ_{p})-module, and hence the following definition of the space of (classical) automorphic forms on B of weight k + 2 and level Σ(N, p) is equivalent to the one given in Section 4 of [Bu]:

S_{k+2}(N, p) := S(Σ(N, p), V_{k}).

Of crucial importance for our arguments is the Hecke operator U_{p} acting on the space S_{k+2}(N, p), whose precise definition we now describe. Let α_{1}

be the matrix ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) and decompose the double coset space Σ_{p}α_{1}Σ_{p} as a

disjoint union of left cosets:

Σ_{p}α_{1}Σ_{p} = ∪_{j=1}^{p}α_{j}Σ_{p}.

Then

(U_{p}φ)(g) = det(α_{1})^{k/2} ∑_{j} φ(gα_{j})α_{j}^{−1}.

It is useful to have a geometric interpretation of automorphic forms in terms of certain functions on the edges of the Bruhat-Tits tree T of PGL_{2}(ℚ_{p}). Recall that T denotes the tree whose vertices are in bijection with the homothety classes of ℤ_{p}-lattices in ℚ_{p}^{2}, two vertices being joined by an (unordered) edge if they admit representatives which are contained one in the other with index p. Let T_{0} and T_{1} denote the set of vertices and edges of T respectively, and let E(T ) denote the set of ordered edges of T , i.e., the set of ordered pairs of adjacent vertices. If e = (v_{s}, v_{t}) is such an ordered edge, we will call the vertex s(e) := v_{s} the source of e, and t(e) := v_{t} its target. The edge ē:= (v_{t}, v_{s}) obtained from e by interchanging its source and target is called the edge opposite to e.

Let v_{∗} be the vertex associated to the homothety class of the standard lattice ℤ_{p}^{2}. The index p sublattices of ℤ_{p}^{2} are naturally in bijection with ℙ^{1}(𝔽_{p}) by setting

L_{j} := {(x, y) ∈ ℤ_{p}^{2} such that [x : y] ≡ j (mod p)}, j = 0, 1, …, p − 1, ∞.

Let v_{j} be the vertex associated to the homothety class of L_{j}, and let

e_{j} = (v_{∗}, v_{j}) ∈ E(T ).

A vertex in T_{0} is said to be even or odd if its distance from v_{∗} is even or odd. Likewise, an ordered edge in E(T ) is even (resp. odd) if its source is even (resp. odd).

The groups GL_{2}(ℚ_{p}) and PGL_{2}(ℚ_{p}) act naturally on T via their left action on ℚ_{p}^{2}, viewed as column vectors. The resulting actions of these groups on T_{0}, T_{1}, and E(T ) are transitive, while the subgroup PSL_{2}(ℚ_{p}) preserves the even and odd elements in T_{0} and E(T ). The stabiliser of v_{∗} in PGL_{2}(ℚ_{p}) is the group PGL_{2}(ℤ_{p}), while the stabiliser of the ordered edge e_{∞} is the projective image of the group Γ_{0}(pℤ_{p}). Hence the assignment g ↦ ge_{∞} identifies the quotient PGL_{2}(ℚ_{p})/Γ_{0}(pℤ_{p}) with E(T ).

If g ∈ GL_{2}(ℚ_{p}) we denote by |g| := p^{ord_{p}(det(g))}. To each η ∈ S_{k+2}(N, p), viewed as a function on GL_{2}(ℚ_{p}) via the description (9), is attached a V_{k}- valued, Γ-invariant function c_{η} on E(T ) by setting, for all e = ge_{∞} with g ∈ GL_{2}(ℚ_{p}), and for all P ∈ P_{k},

c_{η}(e)(P) := |g|^{−k/2}(gη(g))(P).  (10)

It is easy to see that the expression on the right of equation (10) depends only on the class of g in PGL_{2}(ℚ_{p})/Γ_{0}(pℤ_{p}), so that the value of c_{η} is welldefined. Moreover, if γ is any element of Γ, and e = ge_{∞} is any edge in E(T ), we have

c_{η}(γe)(P) = |γg|^{−k/2}(γgη(γg))(P) = γ(|g|^{−k/2}gη(g))(P)
= (γc_{η}(e))(P) = c_{η}(e)(P γ).

Since η can be recovered from the datum of c_{η}, the assignment η ↦ c_{η} identifies η ∈ S_{k+2}(N, p) with an element c_{η} in the space C(E, V_{k})^{Γ} of Γinvariant V_{k}-valued functions on E(T ). Let us spell out the action of the Hecke operator U_{p} which is deduced from this identification.

Lemma 1.2. For all η ∈ S_{k+2}(N, p), we have

(c_{U_{p}η})(e) = p^{k/2} ∑_{e′≠\overline{e}^{s(e′)=t(e)}} c_{η}(e′).

Proof. This follows from a direct calculation.

2 Teitelbaum’s L-invariant

Let f be the normalised eigenform of weight k_{0} + 2 on Γ_{0}(N) that was discussed in the introduction. The definition of Teitelbaum’s invariant L_{T} (f) rests crucially on the Jacquet-Langlands correspondence which associates to f an automorphic form on a definite quaternion algebra in the sense of the previous section.

Theorem 2.1. There exists an automorphic form ϕ ∈ S_{k_{0}+2}(N, p) which is an eigenform for the Hecke operators and satisfies

ϕ|T_{ℓ} = a_{ℓ}(k_{0})ϕ, for all ℓ |̸Np, ϕ|U_{p} = p^{k_{0}/2}ϕ.

This ϕ is unique up to multiplication by a non-zero scalar in ℂ_{p}^{×}.

Let ϕ ∈ S_{k_{0}+2}(N, p) be the modular form obtained from f via Theorem 2.1, and recall the Γ-equivariant V_{k}-valued function c_{ϕ} on E(T ) that was associated to it in the previous section. A function c on E is called a harmonic cocycle if

c(ē) = −c(e), ∑_{s(e)=v} c(e) = 0, for all v ∈ T_{0}.

Lemma 2.2. The function c_{ϕ} attached to ϕ is a V_{k_{0}}-valued harmonic cocycle on T .

Proof. The fact that c_{ϕ}(ē) = −c_{ϕ}(e) follows directly from the fact that the Atkin-Lehner involution W_{p} acts as multiplication by −1 on c_{ϕ}. Let v be any vertex of E and let e be an ordered edge of T satisfying t(e) = v. Since c_{ϕ}|U_{p} = p^{k_{0}/2}c_{ϕ}, it follows from the description of U_{p} given in Lemma 1.2 that

p^{k_{0}/2}c_{ϕ}(e) = (c_{ϕ}|U_{p})(e) = p^{k_{0}/2} ∑_{e′≠\overline{e}^{s(e′)=v}} c_{ϕ}(e′),

so that

∑_{s(e′)=v} c_{ϕ}(e′) = 0

for all e ∈ E(T ) and v = t(e).

We now explain how the cocycle c_{ϕ} gives rise to a locally analytic distribution on ℙ^{1}(ℚ_{p}), denoted μ_{ϕ}. To do this, let W := ℚ_{p}^{2} − {0}, equipped with its natural p-adic topology. There is a natural continuous projection

π : W ⟶ ℙ^{1}(ℚ_{p}), π((x, y)) = x/y.

If L is any ℤ_{p}-lattice in ℚ_{p}^{2}, let L′ := L − pL be the compact open subset of W consisting of the primitive vectors in L. If e = (s, t) ∈ E(T ) is an ordered edge of T , let L_{s} and L_{t} denote ℤ_{p}-lattices whose homothety classes correspond to the source and the target of e respectively, chosen in such a way that L_{s} contains L_{t} with index p. To the edge e are associated the subset W_{e} ⊂ W and the compact open subset U_{e} ⊂ ℙ^{1}(ℚ_{p}) by the rules

W_{e} = L′_{s} ∩ L′_{t}, U_{e} = π(W_{e}).

Note that the set W_{e} depends on the choice of L_{s} and L_{t}, so that W_{e} is only well-defined (as a function of e) up to multiplication by elements of ℚ_{p}^{×}. The subset U_{e}, on the other hand, depends only on e and not on the choices of representative lattices L_{s} and L_{t} that were made to define it.

Let us now briefly recall some of the theory of locally analytic distributions. Let X be a compact open subset of W ⊂ ℚ_{p}^{2}. For each integer n ≥ 0, denote by B[X, p^{−n}] the affinoid subdomain of ℂ_{p}^{2} given by

B[X, p^{−n}] := {z ∈ ℂ_{p}^{2} | there exists x ∈ X with |z − x| ≤ p^{−n}}.

The region B[X, p^{−n}] is a finite disjoint union of closed polydisks of radius p^{−n} defined over ℚ_{p}. Therefore B[X, p^{−n}] is also defined over ℚ_{p}. Let A_{n}(X) denote the ℚ_{p}-affinoid algebra of B[X, p^{−n}]. It is a Banach algebra over ℚ_{p} under the spectral norm,

||h||_{A_{n}(X)} := sup_{z∈B[X,p^{−n}]} |h(z)|.

If m ≥ n ≥ 0, restriction defines a continuous map

A_{n}(X) ⟶ A_{m}(X).

The direct limit

A(X, ℚ_{p}) := lim_{→,n} A_{n}(X)

is called the space of locally analytic functions on X. It is endowed with the inductive limit of the Banach topologies on each of the A_{n}(X)’s. Let

D_{n}(X) := Hom_{cont}(A_{n}(X), ℚ_{p})

denote the ℚ_{p}-Banach-dual to A_{n}(X) and let

D(X, ℚ_{p}) := lim_{←,n} D_{n}(X) = Hom_{cont}(A(X, ℚ_{p}), ℚ_{p}).

This space, endowed with the projective limit of the Banach topologies of the D_{n}(X)’s, is called the space of locally analytic distributions on X. It is a Fréchet space over ℚ_{p}.

These definitions can be extended without difficulty to the case where X is a compact open subset of the projective space ℙ^{1}(ℚ_{p}). (see [St].)

Following the approach described in [T], the harmonic cocycle c_{ϕ} can be used to define a locally analytic distribution μ_{ϕ} on ℙ^{1}(ℚ_{p}), determined by the property:

∫_{U_{e}} P (t)μ_{ϕ}(t) = c_{ϕ}(e)(P),  (11)

for all e ∈ E(T ) and P ∈ P_{k_{0}}.

Let H_{p} := ℙ^{1}(ℂ_{p}) − ℙ^{1}(ℚ_{p}) denote the p-adic upper half-plane. In [T], the distribution μ_{ϕ} is used to define a rigid analytic function

ψ = ψ_{f} : H_{p} ⟶ ℂ_{p}

by the rule

ψ(z) = ∫_{ℙ^{1}(ℚ_{p})} (\frac{1}{t − z} ) dμ_{ϕ}(t).  (12)

By Theorem 3 of [T], the function ψ is a rigid analytic modular form on Γ\H_{p} of weight k_{0} + 2, i.e., it satisfies the relation

ψ(γz) = (cz + d)^{k_{0}+2}ψ(z), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

The p-adic Coleman line integral attached to ψ, a polynomial P ∈ V_{k_{0}}, and two endpoints τ_{1} and τ_{2} ∈ H_{p} is defined in terms of the distribution μ_{ϕ} by the rule

∫_{τ_{1}}^{τ_{2}} ψ(z)P (z)dz := ∫_{ℙ^{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} ) P (t)μ_{ϕ}(t).  (13)

This formula can be used as a definition for the Coleman line integral in this setting, in light of Teitelbaum’s theory of the “p-adic Poisson kernel”. (See [T] for a more complete discussion.) In particular, it satisfies the additivity properties suggested by the line integral notation. Let us now fix base points v_{0} ∈ T_{0} and z_{0} ∈ H_{p}. (For example, one could take v_{0} = v_{∗}, but this is not necessary.) The harmonic cocycle c_{ϕ} gives rise (after extending scalars from V_{k_{0}} to V_{k_{0}} ⊗ ℂ_{p}) to a V_{k_{0}} ⊗ ℂ_{p}-valued one-cocycle on Γ (where V_{k_{0}} is viewed as a left Γ-module) defined by the rule:

κ_{ϕ}^{ord}(γ)(P) = ∑_{e:v_{0}→γv_{0}} c_{ϕ}(e)(P),  (14)

where the sum is taken over the ordered edges in the path joining v_{0} to γv_{0}. Likewise, the associated rigid analytic modular form ψ gives rise to the V_{k_{0}} ⊗ ℂ_{p}-valued one-cocycle on Γ defined by

κ_{ϕ}^{log}(γ)(P) = ∫_{z_{0}}^{γz_{0}} ψ(z)P (z)dz.  (15)

The images [κ_{ϕ}^{ord}] and [κ_{ϕ}^{log}] of κ_{ϕ}^{ord} and κ_{ϕ}^{log} in H^{1}(Γ, V_{k_{0}}⊗ℂ_{p}) are independent of the choices of v_{0} and z_{0} that were made to define them. These classes lie in the one-dimensional f-isotypic component of H^{1}(Γ, V_{k_{0}} ⊗ ℂ_{p}) for the action of the Hecke operators. Furthermore, Theorem 1 of [T] shows that the class of κ_{ϕ}^{ord} is non-zero. We are now in a position to recall the definition of L_{T} (f) given in [T].

Definition 2.3. The Teitelbaum L-invariant attached to f is the unique scalar L_{T} (f) ∈ ℂ_{p} such that

[κ_{ϕ}^{log}] = L_{T} (f)[κ_{ϕ}^{ord}].

Note that multiplying ϕ, and the resulting cocycle and locally analytic distribution, by a non-zero scalar multiplies both κ_{ord} and κ_{log} by that same scalar and hence does not affect the value of L_{T} (f), which is therefore a genuine invariant of f (once the factorisation (7) has been fixed) in light of the uniqueness of ϕ described in Theorem 2.1.

3 Families of automorphic forms on B

The group GL_{2}(ℚ_{p}) acts naturally on W := ℚ_{p}^{2} − {0} on the left, by viewing elements of W as non-zero column vectors. Of considerable importance is the resulting action of the scalar matrices in ℤ_{p}^{×}, which commutes with the GL_{2}(ℚ_{p}) action, and preserves L′ for any ℤ_{p}-lattice L ⊂ ℚ_{p}^{2}. This latter action is denoted by

λ · (x, y) := (λx, λy).

Recall the standard lattice L_{∗} = ℤ_{p}^{2}, and let A(L′_{∗}, ℚ_{p}) denote as above the space of locally analytic ℚ_{p}-valued functions on L′_{∗}. It is equipped with a right action by GL_{2}(ℤ_{p}) given by:

(f|u)(x, y) = f(ax + by, cx + dy) for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ_{p}).

Let

𝔻 := D(L′_{∗}, ℚ_{p})

be the space of locally analytic distributions on L′_{∗}. The natural, continuous left action of GL_{2}(ℤ_{p}) can be turned into a right action by the rule:

μ · u := u^{−1}μ, for u ∈ GL_{2}(ℤ_{p}), μ ∈ 𝔻.

Let R := D(ℤ_{p}^{×}, ℚ_{p}) be the ℚ_{p}-algebra of locally analytic distributions on ℤ_{p}^{×}.

The ℤ_{p}^{×}-action on L′_{∗} equips 𝔻 with a natural R-module structure

R × 𝔻 ⟶ 𝔻 sending (α, μ) to α · μ,

where α · μ is defined by the rule:

∫_{L′_{∗}} F (x, y)(α · μ)(x, y) := ∫_{ℤ_{p}^{×}×L′_{∗}} F (tx, ty)α(t)μ(x, y),

where F (x, y) belongs to A(L′_{∗}, ℚ_{p}) and the variables of integration t and (x, y) range over ℤ_{p}^{×} and L′_{∗} respectively.

Let us now fix an integer k_{0} ≥ 0 and let U be an affinoid disk defined over the finite extension K of ℚ_{p} such that k_{0} ∈ U ⊂ W. Let A(U) denote the K-affinoid algebra of U. Then we have a natural ℚ_{p}-algebra homomorphism R ⟶ A(U) defined by rule

α → (κ → ∫_{ℤ_{p}^{×}} κ(t)α(t)), for all α ∈ R, κ ∈ U.  (16)

Remark 3.1. Let κ ∈ U(K), then κ can be uniquely written κ = ϵ(t)χ(t)⟨t⟩^{c} for ϵ : ℤ_{p}^{×} ⟶ K^{×} a character of order dividing p − 1, χ : ℤ_{p}^{×} ⟶ K^{×} a character of order a power of p and c ∈ O_{K}. So we may think of κ as determined by the pair (ϵχ, c). Let us remark that if K is fixed and the radius r of U is small enough the associated pair (ϵχ, c) is characterized by:

ϵ(t) = (\frac{t}{⟨t⟩})^{k_{0}}, χ(t) = 1 and |c − k_{0}| ≤ r. In other words κ is entirely

determined by c.

Denote by 𝔻_{U} := A(U) ˆ⊗_{R}𝔻 and let GL_{2}(ℚ_{p}) act on the right on 𝔻_{U} via its action on 𝔻.

A natural R-module structure on 𝕊(N, p) is obtained by setting

(α · Φ)(g) := α · Φ(g), for α ∈ R, Φ ∈ 𝕊(N, p), and g ∈ GL_{2}(ℚ_{p}).

Definition 3.2. Fix k_{0} and U as above. The space

𝕊_{U}(N, p) := S(Σ(N, 1), 𝔻_{U})

is called the space of p-adic families of automorphic forms on B of level Σ(N, p) parametrized by weights in U.

Remark 3.3. Note that the space 𝕊_{U}(N, p) is defined using a level structure Σ(N, 1) in which the prime p has been removed. In other words, these functions satisfy an equivariance property, on the right, by the full group GL_{2}(ℤ_{p}) and not just Γ_{0}(pℤ_{p}).

The terminology introduced in Definition 3.2 is justified by the fact that 𝕊_{U}(N, p) is equipped with natural Hecke-equivariant specialization maps ρ_{k} to S_{k+2}(N, p) for every even integer k ≥ 0 in U. In order to define ρ_{k}, it is convenient to introduce

W_{∞} := L′_{∗} ∩ L′_{∞} = ℤ_{p}^{×} ⊕ pℤ_{p} ⊂ W,

where L_{∞} = ℤ_{p} ⊕ pℤ_{p}, as before. If P ∈ P_{k} is a polynomial (of degree ≤ k), let

P̃(x, y) = y^{k}P (x/y)

denote the corresponding homogeneous polynomial in x and y of degree k. More generally let κ ∈ U and let k ≥ 0 be an integer and define for X = L′_{∗} or W:

A^{(κ)}(X) := {f : L′_{∗} ⟶ K locally analytic | f(tx, ty) = κ(t)f(x, y)
for all t ∈ ℤ_{p}^{×}, (x, y) ∈ X}

and

A_{k}^{(κ)}(W) := {f : W ⟶ K locally analytic | f(tx, ty) = κ(t)f(x, y)
and f(px, py) = p^{k}f(x, y) for all t ∈ ℤ_{p}^{×}, (x, y) ∈ W }

Let us fix κ ∈ U(K) and define

B_{κ} : A(U) × 𝔻 ⟶ Hom_{cont,ℚ_{p}}(A^{κ}(L′_{∗}), K)

by

B_{κ}(α, μ)(f) := α(κ) ∫_{L′_{∗}} f(x, y)μ(x, y),

where f ∈ A^{(κ)}(L′_{∗}), α ∈ A(U), and μ ∈ 𝔻. Moreover we have

|B_{κ}(α, μ)(f)|_{K} = |α(κ)|_{K} · | ∫_{L′_{∗}} fμ|_{ℚ_{p}} ≤ ||κ|| · ||α||_{A(U)} · ||f|| · ||μ||_{𝔻}
= ||κ|| · ||f|| · ||(α, μ)||.

Therefore, for every (α, μ) ∈ A(U)×𝔻, B_{κ}(α, μ) is continuous and ℚ_{p}-linear, therefore an element of Hom_{cont,ℚ_{p}}(A^{(κ)}(L′_{∗}), K), and B_{κ} is a continuous, ℚ_{p}- bilinear map. Moreover if r ∈ R, α ∈ A(U), μ ∈ 𝔻 we have

B_{κ}(α, rμ)(f) = α(κ ∫_{L′_{∗}} f(rμ) = ∫_{ℤ_{p}^{×}×L′_{∗}} f(tx, ty)r(t)μ(x, y)
= α(κ)r(κ) ∫_{L′_{∗}} f(x, y)μ(x, y) = B_{κ}(rα, μ)(f).

By the universal property of completed tensor product, there is a unique continuous, ℚ_{p}-linear map L_{κ} : A(U) ˆ⊗_{R}𝔻 ⟶ Hom_{cont,ℚ_{p}}(A^{κ}(L′_{∗}), K) such that the following diagram is commutative

A(U) × 𝔻 ⟶^{B_{κ}} Hom_{cont,ℚ_{p}}(A^{(κ)}(L′_{∗}), K)
↓  ||
A(U) ˆ⊗_{R}𝔻 ⟶^{L_{κ}} Hom_{cont,ℚ_{p}}(A^{(κ)}(L′_{∗}), K)

Finally, if μ ∈ 𝔻_{U} = A(U) ˆ⊗_{R}𝔻 and f ∈ A^{(κ)}(L′_{∗}) we denote

L_{κ}(μ)(f) = ∫_{L′_{∗}} fμ.

Let now k ≥ 0 be an integer such that k ∈ U and P ∈ P_{k}. Let us remark that as the ℤ_{p}^{×}-action on L′_{∗} preserves W_{∞}, the function P̃χ_{W_{∞}} ∈ A^{(k)}(L′_{∗}) where χ_{W_{∞}} is the characteristic function of W_{∞} in L′_{∗}. Let Φ ∈ 𝕊_{U}(N, p).

Definition 3.4. The specialisation map in weight k + 2 is the map

ρ_{k} : 𝕊_{U}(N, p) ⟶ S_{k+2}(N, p)

defined by

ρ_{k}(Φ)(g)(P) = ∫_{W_{∞}} P̃(x, y)Φ(g)(x, y),

for g ∈ GL_{2}(ℚ_{p}) and P ∈ P_{k}.

Remark 3.5. Note that the stabiliser of the ordered edge e_{∞} := ([L_{∗}], [L_{∞}]) in PGL_{2}(ℤ_{p}), and therefore of W_{∞}, is the image of Γ_{0}(pℤ_{p}). This is why the prime p arises in the level of the specialisation ρ_{k}(Φ), even though Φ was taken to be equivariant under the larger group Σ(N, 1).

The group B̂^{×} can be written as a finite disjoint union of double cosets

B̂^{×} = ∪_{i=1}^{q}B^{×}d_{i}Σ(N, p),

for elements d_{i}, i = 1, ..., q in B̂^{×}. The condition N ≥ 4 insures that the groups d_{i}^{−1}B^{×}d_{i} ∩ Σ(N, p) are trivial, so that there is a natural identification

𝕊(N, p) ⟶ 𝔻^{q}, given by φ ⟶ (φ(d_{i}))_{1≤i≤q}.

The spaces R and 𝔻 with their natural topologies are Fréchet spaces. Thus 𝕊(N, p) inherits from 𝔻 a topology under which it becomes a Fréchet space (just like R and 𝔻). Moreover 𝕊_{U}(N, p) = S(Σ(N, 1), 𝔻_{U}) ≅ 𝕊(N, p) ˆ⊗_{R}A(U). See Section 4 of [Bu] for more details.

Definition 3.6. Let M be a Fréchet space which is an R-module. We’ll say that M is an orthonormalizable R-module if, for each n ≥ 0 there are orthonormalizable R_{n} = D_{n}(ℤ_{p}^{×})-Banach modules M_{n} such that M ≅ lim_{←,n} M_{n}

as R-modules.

Theorem 3.7. The Fréchet spaces 𝔻 and 𝕊(N, p) are orthonormalizable Rmodules.

Proof. We have a natural projection π : W_{∞} ⟶ ℤ_{p} given by (x, y) ⟶ y/x, whose fibers are isomorphic to ℤ_{p}^{×}. Moreover π is equipped with a natural continuous section s defined as follows. For each i = 0, 1, ..., p−1, ∞ ∈ ℙ^{1}(𝔽_{p}) let C_{i} ⊂ ℙ^{1}(ℚ_{p}) denote the residue class of i. The we can write L′_{∗} as the disjoint union (ℤ_{p}^{×} ⊕ pℤ_{p}) ∪ (ℤ_{p} ⊕ ℤ_{p}^{×}) such that π(ℤ_{p}^{×} ⊕ pℤ_{p}) = C_{∞} and π(ℤ_{p} ⊕ ℤ_{p}^{×}) = ℙ^{1}(ℚ_{p}) − C_{∞} = ∪_{i=0}^{p−1}C_{i}. Define s : ℙ^{1}(ℚ_{p}) ⟶ L′_{∗} by s(z) = (1, 1/z) if z ∈ C_{∞} and s(z) = (z, 1) else. Then both π, s are locally analytic functions and they induce locally analytic isomorphisms:

u : L′_{∗} ⟶ ℤ_{p}^{×} ⊕ ℙ^{1}(ℚ_{p}) and v : ℤ_{p}^{×} ⊕ ℙ^{1}(ℚ_{p}) ⟶ L′_{∗}

by: (u(x, y) = (x, π(x, y)) and v(a, z) = as(z).

Moreover we have actions of L′_{∗} and ℤ_{p}^{×} ⊕ ℙ^{1}(ℚ_{p}) as follows: if α ∈ ℤ_{p}^{×}, (x, y) ∈ L′_{∗}, (a, z) ∈ ℤ_{p}^{×} ⊕ ℙ^{1}(ℚ_{p}) then α(x, y) = (αx, αy) and α(a, z) = (αa, z). Then both u, v are equivariant with respect to these actions and they induce, for each n ≥ 1 natural isomorphisms as Banach spaces

A_{n}(L′_{∗}) ≅ A_{n}(ℤ_{p}^{×}) ˆ⊗A_{n}(ℙ^{1}(ℚ_{p})).

By duality they induce D_{n}(ℤ_{p}^{×})-linear isomorphisms

D_{n}(L′_{∗}) = Hom_{cont,ℚ_{p}}(A_{n}(L′_{∗}), ℚ_{p})
≅ Hom_{cont}(A_{n}(ℤ_{p}^{×}) ˆ⊗A_{n}(ℙ^{1}(ℚ_{p})), ℚ_{p})
≅ Hom_{cont}(A_{n}(ℙ^{1}(ℚ_{p})), D_{n}(ℤ_{p}^{×})).

The last term in the sequence naturally contains D_{n}(ℙ^{1}(ℚ_{p})) ˆ⊗D_{n}(ℤ_{p}^{×}) as the subspace of completely continuous (or compact) ℚ_{p}-linear maps from A_{n}(ℙ^{1}(ℚ_{p})) to D_{n}(ℤ_{p}^{×}). (See [Se1] section 4.) Since D_{n}(ℙ^{1}(ℚ_{p})) is a Banach space over ℚ_{p}, it is orthonormalizable and therefore D_{n}(ℙ^{1}(ℚ_{p})) ˆ⊗D_{n}(ℤ_{p}^{×}) is an orthonormalizable Banach module over D_{n}(ℤ_{p}^{×}). Now we claim that the natural inclusions above induce isomorphisms

lim_{←,n} D_{n}(ℙ^{1}(ℚ_{p})) ˆ⊗D_{n}(ℤ_{p}^{×}) ⟶ lim_{←,n} Hom_{cont,ℚ_{p}}(A_{n}(ℙ^{1}(ℚ_{p})), D_{n}(ℤ_{p}^{×})).

The map above is clearly injective. Let us show that it is surjective. Let

(f_{n})_{n} ∈ lim_{←,n} Hom_{cont}(A_{n}(ℙ^{1}(ℚ_{p})), D_{n}(ℤ_{p}^{×})).

We have the following commutative diagram:

A_{n}(ℙ^{1}(ℚ_{p})) ⟶^{f_{n}} D_{n}(ℤ_{p}^{×})
↓ r_{n}  ↑ s_{n}
A_{n+1}(ℙ^{1}(ℚ_{p})) ⟶^{f_{n+1}} D_{n+1}(ℤ_{p}^{×}),

where r_{n} is the restriction and s_{n} is dual to restriction. Therefore,

f_{n} = s_{n}f_{n+1}r_{n},

and because r_{n} is the restriction induced by the inclusion

B[ℙ^{1}(ℚ_{p}), p^{−n−1}] ⊂ B[ℙ^{1}(ℚ_{p}), p^{−n}],

it is completely continuous. (See [Se1], Section 8.) Therefore f_{n} is completely continuous for all n ≥ 0. So we have an isomorphism as R-modules 𝔻 ≅ lim_{→,n} D_{n}(ℙ^{1}(ℚ_{p}) ˆ⊗D_{n}(ℤ_{p}^{×}) which implies that 𝔻 is an orthonormalizable R-

module. As 𝕊(N, p) ≅ 𝔻^{q} it is an orthonormalizable R-module as well.

Corollary 3.8. Let U be an affinoid disk contained in the weight space W. Then 𝔻_{U} and 𝕊_{U}(N, p) are orthonormalizable A(U)-modules.

Theorem 3.7 can be used to define actions of Hecke operators T_{ℓ} for ℓ not dividing Np and U_{p}, as in Sections 6 and 8 of [Bu]. The following theorem now follows from a standard argument.

Theorem 3.9. Let U be an affinoid disk contained in the weight space W. The operator U_{p} : 𝕊_{U}(N, p) ⟶ 𝕊_{U}(N, p) is a compact A(U)-linear operator.

Proof. See [St] and [Bu].

Recall the Coleman family f_{∞} of eigenforms on Γ_{1}(N) ∩ Γ_{0}(p) interpolating f that is given in equation (3) of the introduction. The Fourier coefficients a_{n}(k) of f_{∞} correspond to elements of A(U) for some rigid analytic disk U containing k_{0} and contained in the weight space W. We will be making crucial use of the following “Jacquet-Langlands correspondence” applied to the family f_{∞}.

Theorem 3.10 (G.Chenevier, [Ch]). To the expense of possibly shrinking U, there exists an eigenfamily Φ ∈ 𝕊_{U}(N, p) such that

Φ|T_{ℓ} = a_{ℓ}Φ for (ℓ, Np) = 1  and  Φ|U_{p} = a_{p}Φ.

4 A geometric interpretation of p-adic families of automorphic forms

In this section, we attach to any family Φ ∈ 𝕊_{U}(N, p) a collection of locally analytic distributions (μ_{L})_{L⊂ℚ_{p}^{2}} on W, indexed by the ℤ_{p}-lattices in ℚ_{p}^{2}.

Definition 4.1. Let us first fix a weight κ ∈ U. Let L = gL_{∗} be a ℤ_{p}-lattice in ℚ_{p}^{2}, for some g ∈ GL_{2}(ℚ_{p}). The distribution μ_{L} on A^{(κ)}(W) is defined by

∫_{W} F (z)μ_{L}(z) = ∫_{L′} F (z)μ_{L}(z) := ∫_{L′_{∗}}(F |g)(z)Φ(g) = ∫_{L′_{∗}} F (gz)Φ(g),

where F : W ⟶ ℚ_{p} is any function in A^{(κ)}(W).

Note that if F ∈ A^{(κ)}(W) then (F |g) ∈ A^{(κ)}(W) for any g ∈ GL_{2}(ℚ_{p}) and that μ_{L} is supported, by definition, on the compact subset L′ of W.

Here are some elementary properties of the collection {μ_{L}}.

1. The distribution μ_{L} is well defined, i.e. it does not depend on the choice of g. Indeed, let g_{1}, g_{2} ∈ GL_{2}(ℚ_{p}) be such that g_{1}L_{∗} = g_{2}L_{∗} = L. Then g_{1} = g_{2}u with u ∈ GL_{2}(ℤ_{p}) and we have

∫_{L′_{∗}}(F |g_{1})(z)Φ(g_{1}) = ∫_{L′_{∗}}(F |g_{2}u)(z)Φ(g_{2}u) = ∫_{L′_{∗}}(F |g_{2}u)(z)(u^{−1}Φ(g_{2}))
= ∫_{L′_{∗}}(F |g_{2}uu^{−1})(z)Φ(g_{2}) = ∫_{L′_{∗}}(F |g_{2})(z)Φ(g_{2}),

for all functions F ∈ A^{(κ)}(W).

2. Let γ be any element of Γ̃. Then

∫_{(γL)′} F (z)μ_{γL}(z) = ∫_{L′}(F |γ)(z)μ_{L}(z)

for all locally analytic functions F in the space A^{(κ)}(W). In particular for

γ = ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) ∈ Γ̃ we have

∫_{(pL)′} F (z)μ_{pL}(z) = ∫_{L′} F (pz)μ_{L}(z).

3. For any α ∈ A(U) and any lattice L ⊂ ℚ_{p}^{2}, there is a natural multiplication α · μ_{L}, such that α · μ_{L} is a locally analytic distribution on L′, and the family (α · μ_{L})_{L⊂ℚ_{p}^{2}} is associated to αΦ ∈ 𝕊(N, p) by the procedure described above. The specialization map

ρ_{k} : 𝕊_{U}(N, p) ⟶ S_{k+2}(N, p)

can be reinterpreted geometrically as a map assigning a V_{k}-valued cocycle on T to a family of distributions (μ_{L})_{L} indexed by lattices in ℚ_{p}^{2} and satisfying properties 1 to 3 above. More precisely, for all P ∈ P_{k}, let

P̃(x, y) := y^{k}P (x/y)

denote the homogeneous polynomial in x and y, satisfying P̃(z, 1) = P (z). Let us also denote |L| := p^{ord_{p}(det(B))}, for B any ℤ_{p}-basis of L.

Lemma 4.2. For each even integer k ≥ 0, the Γ-invariant cocycle on T attached to the specialisation ρ_{k}(Φ),

c_{Φ,k} : E(T ) ⟶ V_{k}

is expressed in terms of the system of distributions (μ_{L})_{L} associated to Φ by the rule:

c_{Φ,k}(e)(P) = |L|^{−k/2} ∫_{W_{e}} P̃(x, y)μ_{L}(x, y).

where the lattice L above is any representative of the origin of e.

Proof. The proof is a direct consequence of the definitions.

Let k ≥ 0 be an even integer and let κ ∈ U. Let us recall that we have defined the space A_{k}^{(κ)}(W) of locally analytic functions on W, homogeneous of degree κ for the action of ℤ_{p}^{×} and homogeneous of degree k for the action of p on W. Let us remark that if P is a locally meromorphic function on ℙ^{1}(ℚ_{p}) with at worst a pole of order k at ∞ then P̃(x, y) := y^{k}P (x/y) ∈ A_{k}^{(k)}(W). In particular if P ∈ P_{k} then P̃ ∈ A_{k}^{(k)}(W).

Suppose now that Φ ∈ 𝕊_{U}(N, p) is an eigenvector for the operator U_{p}, so that

Φ|U_{p} = a_{p}Φ, with a_{p} ∈ A(U).

Lemma 4.3. Suppose that L_{2} ⊂ L_{1} are ℤ_{p}-lattices in ℚ_{p}^{2} with [L_{1} : L_{2}] = p. Let ϵ = ([L_{1}], [L_{2}]) ∈ E(T ) be the corresponding edge. Then

∫_{W_{ϵ}} F (x, y)μ_{L_{2}}(x, y) = ∫_{W_{ϵ}} F (x, y)(a_{p}μ_{L_{1}})(x, y),

for every locally analytic function F in A_{k}^{(κ)}(W), where k ≥ 0 is an even integer and κ ∈ U.

Proof. Let D_{k}^{(κ)}(W) be the continuous dual of A_{k}^{(κ)}(W). We will extend the definition in 4.2 and will attach to Φ a D_{k}^{(κ)}(W)-valued cocycle on T as follows: let C_{Φ,κ,k} : E(T) ⟶ D_{k}^{(κ)}(W) be defined by

C_{Φ,κ,k}(e)(F) := |L|^{−k/2} ∫_{W_{e}} F (x, y)μ_{L}(x, y),

where e = [L, L′] with L, L′ lattices in ℚ_{p}^{2} such that L′ ⊂ L has index p and F ∈ A_{k}^{(κ)}(W). Let us remark that due to the homogeneity of F with respect to the action of p, the definition is independent of the choice of L, L′. Then C_{Φ,κ,k} enjoys the same formal properties as c_{Φ,k}, in particular we have

C_{U_{p}Φ,κ,k}(e) = p^{k} ∑_{s(e′)=t(e),e′≠\overline{e}} C_{Φ,κ,k}(e′),

for all e ∈ E(T ).

Let us now prove the lemma. We have

|L_{1}|^{−k/2} ∫_{W_{ϵ}} F (x, y)(a_{p}μ_{L})(x, y) = C_{U_{p}Φ,κ,k}(ϵ)(F)
= p^{k} ∑_{s(ϵ′)=t(ϵ),ϵ′≠\overline{ϵ}} C_{Φ,κ,k}(ϵ′)(F).

For every ϵ′ in the above sum let us choose lattices ϵ′ = ([L_{2}], [L_{ϵ′}]), then we have

|L_{1}|^{−k/2} ∫_{W_{ϵ}} F (x, y)(a_{p}μ_{L_{1}})(x, y) = |L_{1}|^{−k/2}p^{k} ∑_{ϵ′} ∫_{W_{ϵ′}} F (x, y)μ_{L_{2}}(x, y) =
= |L_{1}|^{−k/2} ∫_{W_{ϵ}} F (x, y)μ_{L_{2}}(x, y).  (17)

For every ℤ_{p}-lattice L ⊂ ℚ_{p}^{2} we define a distribution π_{∗}(μ_{L}) on ℙ^{1}(ℚ_{p}) by the formula

∫_{ℙ^{1}(ℚ_{p})} P (t)π_{∗}(μ_{L})(t) := |L|^{−k_{0}/2} ∫_{W} P̃(x, y)μ_{L}(x, y),

where P is any locally meromorphic function on ℙ^{1}(ℚ_{p}), with at worst a pole of order k_{0} at ∞.

Assume now that ρ_{k_{0}}(Φ) = ϕ, where ϕ ∈ S_{k_{0}+2}(N, p) is the automorphic form on B attached to f via Theorem 2.1. In particular, Φ is an eigenvector for U_{p} whose associated eigenvalue a_{p}(k) satisfies

a_{p}(k_{0}) = p^{k_{0}/2}.

Recall the distribution μ_{ϕ} attached to ϕ that was defined in Section 2.

Proposition 4.4. For all ℤ_{p}-lattices L in ℚ_{p}^{2},

π_{∗}(μ_{L}) = μ_{ϕ}.

Proof. First note that the function (x, y) ⟶ P̃(x, y) is a locally analytic homogeneous function of degree k_{0} with respect to the action of ℚ_{p}^{×} on W, in particular P̃ ∈ A_{k_{0}}^{(k_{0})}(W). The relationship between μ_{pL} and μ_{L} described after Definition 4.1 implies that π_{∗}(μ_{L}) only depends on the homothety class of L. Moreover, let L_{1} and L_{2} be any two ℤ_{p}-lattices in ℚ_{p}^{2}. Suppose without loss of generality that L_{2} is contained in L_{1} with index p, and that |L_{1}| = 1, and |L_{2}| = p. Let e = ([L_{1}], [L_{2}]) be the corresponding edge. Using Lemma 4.3 we have

∫_{U_{e}} P (t)π_{∗}(μ_{L_{2}})(t) = |L_{2}|^{−k_{0}/2} ∫_{W_{e}} P̃(x, y)μ_{L_{2}}(x, y)
= p^{−k_{0}/2} ∫_{W_{e}} P̃(x, y)(a_{p}μ_{L_{1}})(x, y)
= p^{−k_{0}/2}a_{p}(k_{0}) ∫_{W_{e}} P̃(x, y)μ_{L_{1}}(x, y)
= ∫_{U_{e}} P (t)π_{∗}(μ_{L_{1}})(t).

Arguing in the same way for \overline{e} = ([(1/p)L_{2}], [L_{1}]), one finds that

∫_{U\overline{e}} P (t)π_{∗}(μ_{L_{2}})(t) = ∫_{U\overline{e}} P (t)π_{∗}(μ_{L_{1}})(t),

for all locally meromorphic functions P on ℙ^{1}(ℚ_{p}) with at worst a pole of order k_{0} at ∞. Because ℙ^{1}(ℚ_{p}) = U_{e} ∪ U\overline{e}, we conclude that the distribution π_{∗}(μ_{L}) is independent of the lattice L.

On the other hand, for all P as above we have

∫_{U_{e}} P (t)μ_{ϕ}(t) = C_{Φ,k_{0},k_{0}}(e)(P̃) = |L_{1}|^{−k_{0}/2} ∫_{W_{e}} P̃(x, y)μ_{L_{1}}(x, y)
= ∫_{U_{e}} P (t)π_{∗}(μ_{L_{1}})(t).

Similarly, it follows that

∫_{U\overline{e}} P (t)μ_{ϕ}(t) = ∫_{U\overline{e}} P (t)π_{∗}(μ_{L_{2}})(t),

which allows us to conclude.

Given τ ∈ ℙ^{1}(ℂ_{p}), let τ̄ denote the natural image of τ in ℙ^{1}(𝔽̄_{p}) obtained by reducing τ modulo the maximal ideal of the ring of integers of ℂ_{p}. Let H_{p}(ℚ_{p}^{ur}) denote the unramified p-adic upper half-plane, consisting of elements in ℚ_{p}^{ur} − ℚ_{p}. Finally, let

r : H_{p}(ℚ_{p}^{ur}) ⟶ T_{0}

denote the so-called reduction map which is determined by the rules

1. r(τ) = v_{∗} if and only if τ̄ ∉ ℙ^{1}(𝔽_{p});

2. r(γτ) = γr(τ) for all γ ∈ PGL_{2}(ℚ_{p}).

(See Chapter 5 of [Da2], for example, for more details.)

We will now extend the definition to a more general class of functions. Let us fix τ ∈ H_{p}(ℚ_{p}^{ur}), k_{0} ≥ 0 an integer and let U be an affinoid disk containing k_{0} and contained in the weight space W. Let P ∈ A^{(k_{0})}(L′_{∗}) and μ ∈ 𝔻_{U}. We’d like to define

∫_{L′_{∗}} log(x − τy)P (x, y)μ,

where the branch of log in the above formula and to the end of this article is such that log(p) = 0.

Let F : U(K) × L′_{∗} ⟶ ℂ_{p} be defined by:

F (κ, (x, y)) = P (x, y)(x − τy)^{κ−k_{0}}.

By the above expression we mean the following. Suppose first that the radius r of U is small enough and let κ be determined by the pair (ϵ, c) as in remark

3.1. Here ϵ is the character t ⟶ (\frac{t}{⟨t⟩})^{k_{0}} and c ∈ O_{K} such that |c − k_{0}| ≤ r.

Then by (x − τy)^{κ−k_{0}} we mean (x − τy)^{c−k_{0}} = exp((c − k_{0}) log(x − τy)).

Let us remark that if t ∈ ℤ_{p}^{×}, we have F (κ, (tx, ty)) = κ(t)F (κ, (x, y)), i.e. F (κ, −) ∈ A^{(κ)}(L′_{∗}).

Lemma 4.5. Let μ ∈ 𝔻_{U}. The function U(K) ⟶ ℂ_{p} defined by

κ ⟶ ∫_{L′_{∗}} F (κ, (x, y))μ(x, y),

is analytic near k_{0}.

Proof. Let us remark that we have the following expansion

F (κ, (x, y)) = P (x, y) ∑_{n=0}^{∞} \frac{(κ − k_{0})^{n}}{n!} log^{n}(x − τy),

which converges for all (x, y) ∈ L′_{∗} as log(x − τy) ∈ pO_{ℚ_{p}^{ur}}. Moreover, for all n ≥ 0 the function (x, y) → P (x, y) log^{n}(x − τy) is locally analytic, more precisely P (x, y) log^{n}(x − τy) ∈ A_{m}(L′_{∗}) with m depending only on τ and P. Let us fix an orthonormal basis {μ_{i}}_{i=0}^{∞} of D_{m}(L′_{∗}), so that the m-th component of μ in A(U) ˆ⊗_{R_{m}}D_{m}(L′_{∗}), μ^{(m)} can be uniquely written

μ^{(m)} = ∑_{i=0}^{∞} α_{i} ⊗ μ_{i}, where α_{i} ∈ A(U) with ||α_{i}|| ⟶ 0.

We have, according to our definition

∫_{L′_{∗}} F (κ, (x, y))μ = ∫_{L′_{∗}} F (κ, (x, y))μ^{(m)} = ∑_{i=0}^{∞} α_{i}(κ) ∫_{L′_{∗}} F (κ, (x, y))μ_{i}

= ∑_{i=0}^{∞} α_{i}(κ) ∑_{n=0}^{∞} \frac{(κ − k_{0})^{n}}{n!} ∫_{L′_{∗}} P (x, y) log^{n}(x − τy)μ_{i}(x, y).

The lemma now follows from the fact that α_{i}(κ) is analytic around k_{0} for all i ≥ 0.

Let notations be as above, i.e. let μ ∈ 𝔻_{U} and P ∈ A^{(k_{0})}(L′_{∗}).

Definition 4.6. We define ∫_{L′_{∗}} P (x, y) log(x − τy)μ(x, y) to be

(\frac{d}{dκ} ∫_{L′_{∗}} F (κ, (x, y))μ)_{κ=k_{0}} .

Remark 4.7. Let us give an explicit formula for ∫_{L′_{∗}} P (x, y) log(x−τy)μ(x, y).

Let us suppose that P (x, y) log^{n}(x − τy) ∈ A_{m}(L′_{∗}) for some m independent of n and let us fix an orthonormal basis {μ_{i}}_{i=0}^{∞} as in the proof of lemma 4.5.

We write μ = ∑_{i=0}^{∞} α_{i} ⊗ μ_{i}, with α_{i} ∈ A(U) such that ||α_{i}|| ⟶ 0. Then we

have

∫_{L′_{∗}} P (x, y) log(x − τy)μ(x, y) = ∑_{i=0}^{∞}(\frac{d}{dκ}α_{i})_{κ=k_{0}} ∫_{L′_{∗}} P (x, y)μ_{i}(x, y) +

+ ∑_{i=0}^{∞} α_{i}(k_{0}) ∫_{L′_{∗}} P (x, y) log(x − τy)μ_{i}(x, y).

Let now Φ ∈ 𝕊_{U}(N, p) and let {μ_{L}}_{L⊂ℚ_{p}^{2}} be the family of distributions attached to it. Let as above τ ∈ H_{p}(ℚ_{p}^{ur}), P ∈ A^{(k_{0})}(W) and define f(x, y) := P (x, y) log(x − τy). Let L ⊂ ℚ_{p}^{2} be a lattice and let g ∈ GL_{2}(ℚ_{p}) be such that L = gL_{∗}. For z = (x, y) ∈ W, (f|g)(z) can be written

(f|g)(z) = f(gz) = C(g, τ)(P |g)(x, y) + (P |g)(x, y) log(x − τ′y),

where C(g, τ) is independent of (x, y) and τ′ ∈ H_{p}(ℚ_{p}^{ur}). Therefore it makes sense to define

∫_{W} P (x, y) log(x − τy)μ_{L} := C(g, τ) ∫_{L′_{∗}}(P |g)(x, y)μ_{L_{∗}}(x, y) +
+ ∫_{L′_{∗}}(P |g)(x, y) log(x − τ′y)μ_{L_{∗}}(x, y).

We are now ready to define the main object of this section. Given τ ∈ H_{p}(ℚ_{p}^{ur}), let v_{τ} = r(τ) ∈ T_{0} and let L_{τ} be any ℤ_{p}-lattice in the homothety class of v_{τ}. Recall the rigid analytic modular form ψ defined in equation (12) of Section 2.

Definition 4.8. For all P ∈ P_{k_{0}}, the indefinite integral attached to τ and ψ is defined by the formula

∫ ^{τ} ψ(z)P (z)dz := |L_{τ}|^{−k_{0}/2} ∫_{W} log(x − τy)P̃(x, y)μ_{L_{τ}} (x, y), (18)

where the branch of the p-adic log used above is the one satisfying log(p) = 0.

Note that because

log(px − pτy)P̃(px, py) = p^{k_{0}} log(x − τy)P̃(x, y),

formula (18) only depends on the homothety class of L_{τ}, so that the indefinite integral is well-defined.

The main properties of the indefinite integral of Definition 4.8 are summarized in the following two propositions.

Proposition 4.9. For all γ ∈ Γ and P ∈ P_{k_{0}},

∫ ^{γτ} ψ(z)P (z)dz = ∫ ^{τ} ψ(z)(P γ)(z)dz.

Proof. Let

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

Then

∫ ^{γτ} ψ(z)P (z)dz = |γL_{τ}|^{−k_{0}/2} ∫_{W} log(x − (γτ)y)P̃(x, y)μ_{γL_{τ}} (x, y).

Performing the change of variables

( \begin{matrix}u \\ v \end{matrix} ) = γ^{−1} ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}dx − by \\ −cx + ay \end{matrix} ) ,

we obtain

∫ ^{γτ} ψ(z)P (z)dz = |L_{τ}|^{−k_{0}/2} ∫_{W} log (\frac{u − τv}{cτ + d} ) (P̃ γ)(u, v)μ_{L_{τ}} (u, v)
= ∫ ^{τ} ψ(z)(P γ)(z)dz
− log(cτ + d)|L_{τ}|^{−k_{0}/2} ∫_{W} (P̃ γ)(u, v)μ_{L_{τ}} (u, v).

On the other hand by proposition 4.4 we have

|L_{τ}|^{−k_{0}/2} ∫_{W} (P̃ γ)(u, v)μ_{L_{τ}} (u, v) = ∫_{ℙ^{1}(ℚ_{p})}(P γ)(t)μ_{ϕ}(t) = 0.

Proposition 4.9 follows.

The next proposition relates the indefinite integral to the p-adic line integral of equation (13).

Proposition 4.10. Let τ_{1}, τ_{2} ∈ H_{p}(ℚ_{p}^{ur}) and let v_{i} = r(τ_{i}) = [L_{i}] ∈ T_{0} be the corresponding vertices. For all P ∈ P_{k_{0}},

∫ ^{τ_{2}} ψ(z)P (z)dz − ∫ ^{τ_{1}} ψ(z)P (z)dz

= ∫_{τ_{1}}^{τ_{2}} ψ(z)P (z)dz + 2p^{−k_{0}/2}a′_{p}(k_{0}) ∑_{e:v_{1}→v_{2}} c_{ϕ}(e)(P).

Proof. Suppose without loss of generality that L_{2} ⊂ L_{1} and [L_{1} : L_{2}] = p. Let e = ([L_{1}], [L_{2}]) ∈ E(T ). Then

∫ ^{τ_{2}} ψ(z)P (z)dz − ∫ ^{τ_{1}} ψ(z)P (z)dz  (19)
= |L_{2}|^{−k_{0}/2} ∫_{W} log(x − τ_{2}y)P̃(x, y)μ_{L_{2}}(x, y)  (20)
− |L_{1}|^{−k_{0}/2} ∫_{W} log(x − τ_{1}y)P̃(x, y)μ_{L_{1}}(x, y)  (21)
= |L_{2}|^{−k_{0}/2} ∫_{W} log (\frac{x − τ_{2}y}{x − τ_{1}y} ) P̃(x, y)μ_{L_{2}}(x, y)  (22)
+ ∫_{W} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}μ_{L_{2}} − |L_{1}|^{−k_{0}/2}μ_{L_{1}}) . (23)

By proposition 4.4, the first integral (22) appearing in the last expression is equal to

∫_{ℙ^{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} ) P (t)μ_{ϕ}(t) = ∫_{τ_{1}}^{τ_{2}} ψ(z)P (z)dz.

In order to calculate the second integral (23), we will need the following describing the distribution α · μ_{L} for α ∈ A(U).

Lemma 4.11. Let α be an element of A(U). For all e ∈ E(T ), τ ∈ H_{p}, and P ∈ P_{k_{0}},

∫W_{e}log(x − τy)P̃(x, y)(αμ_{L})(x, y) = α′(k_{0})|L|^{k_{0}/2}c_{ϕ}(e)(P)
+α(k_{0}) ∫_{W_{e}} log(x − τy)P̃(x, y)μ_{L}(x, y).

Proof of Lemma 4.11: This proof is a consequence of the following calculation:

∫W_{e} log(x − τy)P̃(x, y)(αμ_{L})(x, y)
= \frac{d}{dκ} ((α(κ)(∫_{W_{e}} P̃(x, y)(x − τy)^{κ−k_{0}}μ_{L}(x, y))_{κ=k_{0}}
= α′(k_{0}) ∫_{W_{e}} P̃(x, y)μ_{L}(x, y)
+α(k_{0}) ∫_{W_{e}} log(x − τy)P̃(x, y)μ_{L}(x, y)
= α′(k_{0})|L|^{k_{0}/2} ∫_{U_{e}} P (z)μ_{ϕ}(z) + α(k_{0}) ∫_{W_{e}} log(x − τy)P̃(x, y)μ_{L}(x, y).

This proves the lemma.

End of proof of Proposition 4.10: We return to the evaluation of the integral

J := ∫_{W} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}μ_{L_{2}} − |L_{1}|^{−k_{0}/2}μ_{L_{1}})

appearing in (23). It is useful to express J as a sum of two contributions J_{e} and Jē obtained by integrating over the disjoint subsets W_{e} and Wē of W associated to the ordered edge e = ([L_{1}], [L_{2}]) of T . By Lemma 4.3,

J_{e} = ∫_{W_{e}} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}a_{p} − |L_{1}|^{−k_{0}/2}) μ_{L_{1}}(x, y)
= |L_{1}|^{−k_{0}/2} ∫_{W_{e}} log(x − τ_{1}y)P̃(x, y) ((p^{−k_{0}/2}a_{p} − 1)μ_{L_{1}}) (x, y).

Now applying Lemma 4.11 with α = p^{−k_{0}/2}a_{p} −1, and noting that α(k_{0}) = 0, we find

J_{e} = p^{−k_{0}/2}a′_{p}(k_{0})c_{ϕ}(e)(P).  (24)

On the other hand, \overline{e} = ([(1/p)L_{2}], [L_{1}]) and [(1/p)L_{2} : L_{1}] = p so we have μ_{L_{1}}|_{W\overline{e}} = a_{p}μ_{(1/p)L_{2}}|_{W\overline{e}} and the same computation gives:

J\overline{e} := ∫_{W\overline{e}} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}μ_{L_{2}} − |L_{1}|^{−k_{0}/2}μ_{L_{1}})
= −p^{−k_{0}/2}a′_{p}(k_{0})c_{ϕ}(\overline{e})(P) = p^{−k_{0}/2}a′_{p}(k_{0})c_{ϕ}(e)(P).

Therefore

J = J_{e} + J\overline{e} = 2p^{−k_{0}/2}a′_{p}(k_{0})c_{ϕ}(e)(P).

This concludes the proof of Proposition 4.10.

We are now able to prove Theorem 3 of the introduction:

Theorem 4.12. Let L_{T} (f) denote Teitelbaum’s L-invariant attached to f. Then

−2p^{−k_{0}/2}a′_{p}(k_{0}) = L_{T} (f).

Proof. Let Φ be the family of automorphic forms associated to f_{∞} by Theorem 3.10. Fix τ ∈ H_{p}(ℚ_{p}^{ur}) and let v_{τ} = [L_{τ}] ∈ T_{0} be the corresponding vertex. Let h_{τ} ∈ V_{k_{0}} ⊗ ℂ_{p} be the map sending P ∈ P_{k_{0}} to

h_{τ}(P) := ∫ ^{τ} ψ(z)P (z)dz.

For all γ ∈ Γ and P ∈ P_{k_{0}}, Proposition 4.10 gives

h_{γτ}(P) − h_{τ}(P) = ∫_{τ}^{γτ} ψ(z)P (z)dz + 2p^{−k_{0}/2}a′_{p}(k_{0}) ∑_{e:v_{τ} →γ(v_{τ})} c_{ϕ}(e)(P).

In the notations of Section 2 this formula can be rewritten as

h_{γτ} − h_{τ} = κ_{ϕ}^{log}(γ) + 2p^{−k_{0}/2}a′_{p}(k_{0})κ_{ϕ}^{ord}(γ).

On the other hand, Proposition 4.9 implies that

h_{γτ} − h_{τ} = γh_{τ} − h_{τ}

is a V_{k_{0}} ⊗ ℂ_{p}-valued coboundary for Γ. It follows that

[κ_{ϕ}^{log}] = −2p^{−k_{0}/2}a′_{p}(k_{0})[κ_{ϕ}^{ord}].

Theorem 4.12 now follows from Definition 2.3 of L_{T} (f).

5 Orton’s L-invariant

This section recalls the definition of Orton’s L-invariant, which involves the theory of modular symbols. The reader is referred to [Or] for more details.

Write Δ for the group Div^{0}(ℙ^{1}(ℚ)) of degree zero divisors supported on the rational cusps of the Poincaré upper half plane. For any unitary commutative ring A of ℂ, let P_{k}(A) denote the A-algebra of polynomials of degree ≤ k with coefficients in A, and let V_{k}(A) be the A-dual of P_{k}(A). When A is a subfield of ℂ, the group GL_{2}(ℚ) acts on the right on P_{k}(A) by the rule

(P γ)(z) = (cz + d)^{k}P (γz), γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

This induces a right action of GL_{2}(ℚ) on V_{k}(A) by setting

(ϕγ)(P) = ϕ(P γ^{−1}).

A modular symbol with values in a GL_{2}(ℚ)-module V is a homomorphism from Δ to V. The space of all such modular symbols is denoted

MS(V) := hom(Δ, V).

It is equipped with a right GL_{2}(ℚ)-action by the rule

(mγ)(δ) = m(γδ)γ,

for m ∈ MS(V), δ ∈ Δ, and γ ∈ GL_{2}(ℚ). If the divisor δ is of the form (s) − (r), write m{r → s} for m(δ).

A modular eigenform g of weight k + 2 on Γ_{0}(Np) gives rise to a Γ_{0}(Np)- invariant V_{k}(ℂ)-valued modular symbol

Ψ_{g} : Δ ⟶ V_{k}(ℂ)

by the rule

Ψ_{g}(δ)(P) = 2πi ∫_{δ} g(z)P (z)dz,

with δ ∈ Δ and P ∈ P_{k}(ℂ). Write Ψ_{g}^{±} for the projection of Ψ_{g} to the ±-eigenspace of Hom(Δ, V_{k}(ℂ)) for the action of the involution

c = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) .

Let K_{g} be the extension of ℚ generated by the Hecke eigenvalues of g. By a result of Shimura, there exist complex periods Ω_{g}^{±} such that

Φ_{g}^{±} = Ψ_{g}^{±}/Ω_{g}^{±}

takes values in V_{k}(K_{g}). Note that the modular symbols Ψ_{g} and Φ_{g}^{±} are all Γ_{0}(Np)-invariant.

Let f be the newform on Γ_{0}(Np) considered in the introduction. Fix a choice of sign w_{∞} ∈ {−1, 1} and let

Φ_{f} = \{ \begin{matrix}Φ_{f}^{+} & if w_{∞} = 1; \\ Φ_{f}^{−} & if w_{∞} = −1. \end{matrix}

be the modular symbol in MS(V_{k_{0}}(K_{f})) attached to f. Define

Γ̃ = \{γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ[1/p]) : N | c and det(γ) = p^{2h}, for h ∈ ℤ\}.

Write Γ for the group of elements in Γ̃ having determinant one. For (s)−(r) ∈ Δ, define a harmonic cocycle

c_{f}{r → s} : E(T ) ⟶ V_{k_{0}}(ℚ̄_{p})

by the rule

c_{f}{r → s}(e)(P) = Φ_{f}((γs) − (γr))(P γ^{−1}),

where γ ∈ Γ is such that γe = e_{∞}. This definition is independent of the choice of γ such that γe = e_{∞}: for if γ′ is another such element, the element γ′γ^{−1} belongs to Γ_{0}(Np), the stabiliser of e_{∞}. The claim then follows from the Γ_{0}(Np)-invariance of Φ_{f}.

The cocycle c_{f}{r → s} gives rise to a locally analytic distribution on ℙ^{1}(ℚ_{p}), denoted μ_{f}{r → s}, and determined by setting

∫_{U_{e}} P (t)μ_{f}{r → s}(t) = c_{f}{r → s}(e)(P)

for all e ∈ E(T ) and P ∈ P_{k_{0}}(ℚ̄_{p}), and extending to functions on ℙ^{1}(ℚ_{p}) which are locally analytic on ℚ_{p} and have a pole of order at most k_{0} at infinity. Note the analogy between this definition and the definition of the locally analytic distribution μ_{ϕ} in equation (11) of Section 2.

The following definition is modelled on the description of the Coleman line integral given in equation (13) of Section 2.

Definition 5.1. For τ_{1}, τ_{2} ∈ H_{p} and r, s ∈ ℙ^{1}(ℚ), the definite double integral is defined by

∫_{τ_{1}}^{τ_{2}}∫_{r}^{s}ω_{f}P = ∫_{ℙ^{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} )P (t)μ_{f}\{r → s\}(t).

The notation ω_{f} in Definition 5.1 is meant to suggest that the definite double integral should be thought of as the integration of a form of parallel weight (k_{0} + 2, k_{0} + 2) on H_{p} × H associated to f. This point of view is explained in detail in [Da2], Chapter 9 and [Or], Chapter 2.

Set P_{k} = P_{k}(ℚ̄_{p}), V_{k} = V_{k}(ℚ̄_{p}) and write

M_{k} := MS(V_{k}) = Hom(Δ, V_{k}).

The following definitions are motivated by the definition of the 1-cocycles κ_{ϕ}^{ord} and κ_{ϕ}^{log} given in equations (14) and (15) respectively.

Definition 5.2.

1. The 1-cocycle κ_{f}^{ord} ∈ Z^{1}(Γ, M_{k_{0}}) is defined by choosing v ∈ T_{0} and

setting

κ_{f}^{ord}(γ){r → s}(P) = ∑_{e:v→γv} c_{f}{r → s}(P)(e).

2. The 1-cocycle κ_{f}^{log} ∈ Z^{1}(Γ, M_{k_{0}}) is defined by choosing τ ∈ H_{p} and setting

κ_{f}^{log}(γ){r → s}(P) = ∫_{τ}^{γτ}∫_{r}^{s}ω_{f}P.

Lemma 5.3.

1. The image [κ_{f}^{ord}] of κ_{f}^{ord} in H^{1}(Γ, M_{k_{0}}) is independent of the choice of base vertex v.

2. The image [κ_{f}^{log}] of κ_{f}^{log} in H^{1}(Γ, M_{k_{0}}) is independent of the choice of base point τ ∈ H_{p}.

Proof. See [Or], Lemma 5.1 and 5.2. Note that the one-cocycles κ_{f}^{ord} and κ_{f}^{log} are denoted õc_{f,v} and l̃c_{f,τ} respectively in [Or].

Proposition 5.4. The class [κ_{f}^{ord}] is non-zero.

Proof. Proposition 13 of Section II.2.8 of [Se2], applied to the case M = M_{k_{0}} and G = Γ acting on T , yields a linear transformation

θ : H^{0}(Γ_{0}(Np), M_{k_{0}}) ⟶^{θ} H^{1}(Γ, M_{k_{0}})

whose kernel is identified with the p-old subspace of the space of modular symbols on Γ_{0}(Np). The map θ is described explicitly in Section 3.1 of [Da1], where it is shown that θ(Φ_{f}) = [κ_{f}^{ord}]. (Although [Da1] assumes k_{0} = 0, the treatment of the general case is no different.) Proposition 5.4 follows from the fact that the form f is new at p.

Let

H^{1}(Γ, M_{k_{0}})^{f}, H^{1}(Γ, M_{k_{0}})^{f,w_{∞}} ⊂ H^{1}(Γ, M_{k_{0}})

denote, respectively, the f-isotypic subspace and its w_{∞}-eigenspace for the action of the involution c defined at the beginning of this section. The classes [κ_{f}^{ord}] and [κ_{f}^{log}] both belong to H^{1}(Γ, M_{k_{0}})^{f,w_{∞}}. In [Or], Proposition 7.1, it is shown that this space is one-dimensional over ℂ_{p}. This makes it possible to define Orton’s L-invariant L_{O}(f) in a way which parallels closely Definition 2.3 of Teitelbaum’s L-invariant.

Definition 5.5. The Orton L-invariant attached to f is the unique scalar L_{O}(f) ∈ ℂ_{p} such that

[κ_{f}^{log}] = L_{O}(f)[κ_{f}^{ord}].

Remark 5.6. Note that L_{O}(f) depends a priori on the choice of sign w_{∞} which determines whether Φ_{f} is taken to be the even or odd modular symbol attached to f. Hence there are two a priori distinct Orton L-invariants attached to f, which could be denoted L_{O}^{+}(f) and L_{O}^{−}(f). A by-product of our study of L_{O}(f) is a direct proof that these two invariants are in fact equal. (Cf. Theorem 6.8.)

Recall the Shimura period Ω_{f} (depending on the choice of sign w_{∞}) that was used to define the modular symbol attached to f. Let L_{p}(f, χ, s) denote the Mazur-Tate-Teitelbaum p-adic L-function attached to f and χ, which is constructed in terms of the modular symbol of f and hence depends on the choice of Ω_{f}. Set w = 1 if f is split multiplicative at p, and w = −1 if f is non-split multiplicative at p. The following theorem of Orton is crucial for the proof of Theorem 2 of the Introduction.

Theorem 5.7 (Orton). For all Dirichlet characters χ satisfying χ(p) = w and χ(−1) = w_{∞},

L′_{p}(f, χ, 1 + k_{0}/2) = L_{O}(f)L^{∗}(f, χ, 1 + k_{0}/2).

It is worth noting that it is at this stage, and this stage only, that a connection is made between the cohomologically-defined L-invariants and special values of L-series.

Let us briefly recall some of the ideas that go in Orton’s proof of Theorem 5.7. Fix a positive integer c prime to Np. For any positive integer ν prime to c, define an embedding Ψ_{ν} : ℚ × ℚ → M_{2}(ℚ) by setting

Ψ_{ν}(a, a) = ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) , Ψ_{ν}(c, 0) = ( \begin{matrix}c & ν \\ 0 & 0 \end{matrix} ) .

When ν varies in a full set of representatives for (ℤ/cℤ)^{×}, Ψ_{ν} describes the set of all Γ-conjugacy classes of oriented optimal embeddings of conductor c: see [Da1], Section 2. Set

r_{Ψ_{ν}} = ∞, s_{Ψ_{ν}} = −ν/c, γ_{Ψ_{ν}} = Ψ_{ν}(p^{u}, p^{−u}) = ( \begin{matrix}p^{u} & (p^{u} − p^{−u})ν/c \\ 0 & p^{−u} \end{matrix} ) ,

where u denotes the order of p^{2} in (ℤ/cℤ)^{×}. The element γ_{Ψ_{ν}} is a generator for the image of Ψ(ℚ^{×} × ℚ^{×}) ∩ Γ in PGL_{2}(ℚ). Moreover, r_{Ψ_{ν}}, resp., s_{Ψ_{ν}} is the repulsive, resp., attractive fixed point for the action of Ψ(γ_{Ψ_{ν}}) on ℙ^{1}(ℚ). Define the polynomial

P_{Ψ_{ν}}(z) = (cz + ν)^{k_{0}/2} ∈ P_{k_{0}}.

Note that P_{Ψ_{ν}} is invariant under the weight k_{0} + 2 action of γ_{Ψ_{ν}}.

The one-cocycles κ_{f}^{ord} and κ_{f}^{log} can be used to associate to the embedding Ψ_{ν} the following numerical invariants:

J_{Ψ_{ν}}^{ord} = κ_{f}^{ord}(γ_{Ψ_{ν}}){r_{Ψ_{ν}} → s_{Ψ_{ν}}}(P_{Ψ_{ν}}) = ∑_{e:v→γ_{Ψν} v} c_{f}{r_{Ψ_{ν}} → s_{Ψ_{ν}}}(P_{Ψ_{ν}})(e); (25)

J_{Ψ_{ν}}^{log} = κ_{f}^{log}(γ_{Ψ_{ν}}){r_{Ψ_{ν}} → s_{Ψ_{ν}}}(P_{Ψ_{ν}}) = ∫_{τ_{Ψν}}^{γτ_{Ψν}}∫_{r_{Ψν}}^{s_{Ψν}}ω_{f}P_{Ψ_{ν}}.

(26)

Remark 5.8. Note that

b(γ_{Ψ_{ν}}){r_{Ψ_{ν}} → s_{Ψ_{ν}}}(P_{Ψ_{ν}}) = 0

for any coboundary b ∈ B^{1}(Γ, M_{k_{0}}). This implies that the quantities J_{Ψ_{ν}}^{ord} and J_{Ψ_{ν}}^{log} do not depend on the choice of cocycles representing the cohomology classes [κ_{f}^{ord}] and [κ_{f}^{log}] respectively, and hence, by the definitiono f L_{O}(f), that

J_{Ψ_{ν}}^{log} = L_{O}(f)J_{Ψ_{ν}}^{ord}.  (27)

Let χ be a Dirichlet character of conductor c, such that χ(p) = w and χ(−1) = w_{∞}. The following formula of Orton relates the numerical invariants J_{Ψ_{ν}}^{ord} and J_{Ψ_{ν}}^{log} defined in (25) and (26) to special values of L-series, and derivatives of the corresponding p-adic L-functions, respectively:

∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)J_{Ψ_{ν}}^{ord} = (2u)L^{∗}(f, χ, 1 + k_{0}/2);  (28)
∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)J_{Ψ_{ν}}^{log} = (2u)L′_{p}(f, χ, 1 + k_{0}/2).  (29)

The first formula is Corollary 6.1 of [Or], while the second formula is Corollary 6.2 of [Or].

Theorem 5.7 now follows directly from these formulae and equation (27).

6 Distribution-valued modular symbols

Recall from the introduction the p-adic family of eigenforms

f_{∞} = ∑_{n=1}^{∞} a_{n}q^{n}, with a_{n} ∈ A(U)

interpolating the given newform f = ∑_{n=1}^{∞} a_{n}q^{n} of weight k_{0} + 2 on Γ_{0}(Np). This means that

f_{k} = ∑_{n=1}^{∞} a(k)q^{n}

is a normalised eigenform of weight k+2 on Γ_{1}(N)∩Γ_{0}(p), for all k ∈ U ∩ℤ_{≥0}, and that f_{k_{0}} = f. Let Φ_{f_{k}} be the modular symbol in MS(V_{k}) defined in section 5, associated to the choice of sign w_{∞}. Note that Φ_{f_{k}} also depends on a choice of complex period Ω_{f_{k}}, and thus is only really well-defined up to multiplication by a non-zero scalar. Two modular symbols m_{1} and m_{2} in MS(V_{k}) are said to be equivalent if there exists a non-zero scalar λ ∈ ℂ_{p}^{×} such that m_{1} = λm_{2}; one then writes m_{1} ∼ m_{2}.

Assume throughout this section that f = f_{k_{0}} is split multiplicative at p, so that

w = +1, a_{p}(k_{0}) = p^{k_{0}/2}.

As in section 3, let 𝔻 be the space of locally analytic distributions on L′_{∗}, with L_{∗} = ℤ_{p}^{2}. Recall that the ℚ_{p}-algebra R of locally analytic distributions on ℤ_{p}^{×} acts on 𝔻. The space MS_{Γ_{0}(N)}(𝔻) of Γ_{0}(N)-invariant 𝔻-valued modular symbols is equipped with a natural action of the Hecke operators T_{n} with p |̸n, as well as an action of R arising from the R-module structure on 𝔻. Let us fix as in the previous sections an affinoid disk U, defined over K, containing k_{0} and contained in the weight space W and let 𝔻_{U} := 𝔻 ˆ⊗_{R}A(U).

Proposition 6.1. There exists a distribution-valued modular symbol Φ_{f_{∞}} ∈ MS(𝔻_{U}) satisfying the following properties:

1. (Γ_{1}(N)-invariance) Φ_{f_{∞}} is Γ_{0}(N)-invariant, that is,

Φ_{f_{∞}}(γδ) · γ = Φ_{f_{∞}}(δ)

for all γ ∈ Γ_{1}(N).

2. (Weight specialisation) Following the notations of Definition 3.4, for k ∈ U(K) ∩ ℤ_{≥0} and P ∈ P_{k}, define a V_{k}-valued modular symbol

ρ_{k}(Φ_{f_{∞}}) : Δ ⟶ V_{k}

by the rule

ρ_{k}(Φ_{f_{∞}})(δ)(P) = ∫_{W_{∞}} P̃(x, y)Φ_{f_{∞}}(δ)(x, y).

Then,

ρ_{k}(Φ_{f_{∞}}) ∼ Φ_{f_{k}}, and ρ_{k_{0}}(Φ_{f_{∞}}) = Φ_{f}.

Proof. When f has weight 2 (i.e., k_{0} = 0), the existence of a modular symbol with values in the module of bounded distributions on L_{∗} is proved in [GS]. In general, it follows from results of Stevens in [St].

For a divisor δ = (s) − (r) in Δ, write μ_{L_{∗}}{r → s} for the locally analytic distribution Φ_{f_{∞}}(δ). It will be viewed as a distribution on W, supported on L′_{∗}.

Definition 6.2. For any lattice L in ℚ_{p}^{2}, the locally analytic distribution μ_{L}{r → s} on W is defined by the rule

∫_{W} F (x, y)μ_{L}{r → s}(x, y) = ∫_{L′_{∗}} F (g^{−1}(x, y))μ_{L_{∗}}{gr → gs}(x, y),

where F : W ⟶ ℚ_{p} is any locally analytic function, and g ∈ Γ̃ is any element such that gL = L_{∗}.

Note that the above definition does not depend on the choice of g ∈ Γ̃ such that gL = L_{∗}: if g′ is another element of Γ̃ such that g′L = L_{∗}, it follows that g′g^{−1} belongs to the stabiliser of L_{∗} in Γ̃, which is the group Γ_{0}(N). The claim then follows from the Γ_{0}(N)-invariance of Φ_{f_{∞}}, stated in part 1 of Proposition 6.1.

The system of distributions μ_{L}{r → s} satisfies similar properties to those of the system μ_{L} introduced in section 4. Since the proofs of these new properties are analogous to those presented in section 4, details are usually omitted.

Lemma 6.3. Let κ ∈ U(K), L_{2} ⊂ L_{1} be ℤ_{p}-lattices in ℚ_{p}^{2} with [L_{1} : L_{2}] = p, and let e = ([L_{1}], [L_{2}]) ∈ E(T ) be the corresponding edge. Then

μ_{L_{2}}{r → s}|_{W_{e}} = a_{p}μ_{L_{1}}{r → s}|_{W_{e}},

so that

∫_{W_{e}} F (x, y)μ_{L_{2}}{r → s}(x, y) = ∫_{W_{e}} F (x, y)(a_{p}μ_{L_{1}}{r → s})(x, y),

for every locally analytic function F ∈ A^{(κ)}(W).

Proof. The proof is identical to that of lemma 4.3.

Let π be as before the projection of W onto ℙ^{1}(ℚ_{p}). For every ℤ_{p}-lattice L ⊂ ℚ_{p}^{2}, define a locally analytic distribution π_{∗}(μ_{L}{r → s}) on ℙ^{1}(ℚ_{p}) by the formula

∫_{ℙ^{1}(ℚ_{p})} P (t)π_{∗}(μ_{L}{r → s})(t) = |L|^{−k_{0}/2} ∫_{W} η^{k_{0}}P (x/y)μ_{L}{r → s}(x, y), where P is any locally meromorphic function on ℙ^{1}(ℚ_{p}) with a pole of order at most k_{0} at ∞.

Proposition 6.4. For all ℤ_{p}-lattices L in ℚ_{p}^{2},

π_{∗}(μ_{L}{r → s}) = μ_{f}{r → s},

where μ_{f} is the locally analytic distribution on ℙ^{1}(ℚ_{p}) defined in section 5.

Proof. The proof is similar to that of Proposition 4.4. It uses lemma 6.3 instead of lemma 4.3, and part 3 of Proposition 6.1, which guarantees that the specialisation at k_{0} of Φ_{f_{∞}} is the modular symbol attached to f.

Let τ ∈ H_{p}(ℚ_{p}^{ur}) and let v_{τ} = [L_{τ}] ∈ T_{0} be the vertex corresponding to τ under the reduction map. The following definition is modelled on that of the indefinite integral of Definition 4.8:

Definition 6.5. For all P ∈ P_{k_{0}}, the indefinite integral attached to τ ∈ H_{p}, to r, s ∈ ℙ^{1}(ℚ), and to f is defined by the formula

∫ ^{τ}∫_{r}^{s}ω_{f}P = |L_{τ}|^{−k_{0}/2} ∫_{W} log(x − τy)P̃(x, y)μ_{L_{τ}} {r → s}(x, y).  (30)

Since

log(px − pτy)P̃(px, py) = p^{k_{0}} log(x − τy)P̃(x, y),

formula (30) depends only on the homothety class of L_{τ}, and hence only on τ. The main properties of the indefinite double integral of Definition 6.5 are summarized in the following two propositions.

Proposition 6.6. For all γ ∈ Γ and P ∈ P_{k_{0}},

∫ ^{γτ}∫_{γr}^{γs}ω_{f}P = ∫ ^{τ}∫_{r}^{s}ω_{f}P γ.

Proof. It is identical to the proof of Proposition 4.9.

Proposition 6.7. Let τ_{1}, τ_{2} ∈ H_{p}(ℚ_{p}^{ur}), and let v_{1} = [L_{1}], v_{2} = [L_{2}] ∈ T_{0} be the corresponding vertices under the reduction map. For all P ∈ P_{k_{0}},

∫ ^{τ_{2}}∫_{r}^{s}ω_{f}P − ∫ ^{τ_{1}}∫_{r}^{s}ω_{f}P

= ∫_{τ_{1}}^{τ_{2}}∫_{r}^{s}ω_{f}P + 2p^{−k_{0}/2}a′_{p}(k_{0}) ∑_{e:v_{1}→v_{2}} c_{f}{r → s}(e)(P).

Proof. Assume without loss of generality that L_{2} ⊂ L_{1} and [L_{1} : L_{2}] = p. Set e = ([L_{1}], [L_{2}]) ∈ E(T ). Then

∫ ^{τ_{2}}∫_{r}^{s}ω_{f}P − ∫ ^{τ_{1}}∫_{r}^{s}ω_{f}P

= |L_{2}|^{−k_{0}/2} ∫_{W} log(x − τ_{2}y)P̃(x, y)μ_{L_{2}}{r → s}(x, y)
− |L_{1}|^{−k_{0}/2} ∫_{W} log(x − τ_{1}y)P̃(x, y)μ_{L_{1}}{r → s}(x, y)
= I_{log} + I_{ord},

where

I_{log} = |L_{2}|^{−k_{0}/2} ∫_{W} log (\frac{x − τ_{2}y}{x − τ_{1}y} ) P̃(x, y)μ_{L_{2}}\{r → s\}(x, y),

and

I_{ord} = ∫_{W} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}μ_{L_{2}}{r → s} − |L_{1}|^{−k_{0}/2}μ_{L_{1}}{r → s}) .

Using Proposition 6.4, and the fact that the function involved in the integral defining I_{log} is constant along the fibers of π, one finds that

I_{log} = ∫_{ℙ^{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} ) P (t)μ_{f}\{r → s\}(t) = ∫_{τ_{1}}^{τ_{2}}∫_{r}^{s}ω_{f}P.

Now, write the integral defining I_{ord} as the sum of two contributions J_{e} and Jē, obtained by integrating over the disjoint subsets W_{e} and Wē. By Lemma 6.3,

J_{e} = ∫_{W_{e}} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}μ_{L_{2}} − |L_{1}|^{−k_{0}/2}μ_{L_{1}}) {r → s}(x, y)
= ∫_{W_{e}} log(x − τ_{1}y)P̃(x, y) (|L_{2}|^{−k_{0}/2}a_{p} − |L_{1}|^{−k_{0}/2}) μ_{L_{1}}{r → s}(x, y)
= |L_{1}|^{−k_{0}/2} ∫_{W_{e}} log(x − τ_{1}y)P̃(x, y) ((p^{−k_{0}/2}a_{p} − 1)μ_{L_{1}}{r → s}) (x, y).

The formula

∫_{W_{e}}log(x − τy)P̃(x, y)(αμ_{L}{r → s})(x, y) =  (31)

α′(k_{0})|L|^{k_{0}/2}c_{f}{r → s}(e)(P) + α(k_{0}) ∫_{W_{e}} log(x − τy)P̃(x, y)μ_{L}{r → s}(x, y)

(e ∈ E(T ), α ∈ A(U), τ ∈ H_{p}, and P ∈ P_{k_{0}}) is obtained by adapting the approach that is followed in the proof of lemma 4.11. By applying (31) with α = p^{−k_{0}/2}a_{p} − 1, the above expression for J_{e} becomes

J_{e} = p^{−k_{0}/2}a′_{p}(k_{0})c_{f}{r → s}(e)(P).

Moreover, a similar argument proves that

Jē = −p^{−k_{0}/2}a′_{p}(k_{0})c_{f}{r → s}(ē)(P) = p^{−k_{0}/2}a′_{p}(k_{0})c_{f}{r → s}(e)(P).

Hence,

J_{ord} = J_{e} + Jē = 2p^{−k_{0}/2}a′_{p}(k_{0})c_{f}{r → s}(e)(P),

as was to be shown.

We are now ready to prove the main result of this section.

Theorem 6.8. The equality

−2p^{−k_{0}/2}a′_{p}(k_{0}) = L_{O}(f)

holds. In particular, Orton’s L-invariant L_{O}(f) is independent of the choice of sign w_{∞} that was made in defining it.

Proof. Let τ be a point in H_{p}(ℚ_{p}^{ur}), and let v_{τ} = [L_{τ}] ∈ T_{0} be the corresponding vertex. Fix a divisor (s) − (r) in Δ. Given τ ∈ H_{p}, one defines an P_{k_{0}}^{∨}-valued modular symbol h_{τ} by the rule

h_{τ}{r → s}(P) = ∫ ^{τ}∫_{r}^{s}ω_{f}P.

Proposition 6.7 gives

h_{τ}{γ^{−1}r → γ^{−1}s}(γ^{−1}P) − h_{τ}{r → s}(P) = h_{γτ}{r → s}(P) − h_{τ}{r → s}(P)

= ∫_{τ}^{γτ}∫_{r}^{s}ω_{f}P + 2p^{−k_{0}/2}a′_{p}(k_{0}) ∑_{e:v_{τ} →γ(v_{τ})} c_{f}{r → s}(e)(P),  (32)

for all γ ∈ Γ. In the notations of Definition 5.2 of Section 5, this relation can be rewritten as

h_{τ}{γ^{−1}r → γ^{−1}s}(γ^{−1}P) − h_{τ}{r → s}(P)  (33)

= κ_{f}^{log}(γ){r → s} + 2p^{−k_{0}/2}a′_{p}(k_{0})κ_{f}^{ord}(γ){r → s}.

Since the expression on the left of (33) is a M_{k_{0}}-valued one-coboundary, it follows upon projecting this equation to H^{1}(Γ, M_{k_{0}}) that

[κ_{f}^{log}] = −2p^{−k_{0}/2}a′_{p}(k_{0})[κ_{f}^{ord}].

Theorem 6.8 is now a direct consequence of Definition 5.5 of L_{O}(f).

Corollary 6.9.

1. The equality L_{T} (f) = L_{0}(f) holds.

2. For all Dirichlet characters χ satisfying χ(p) = 1,

L′_{p}(f, χ, 1 + k_{0}/2) = L_{T} (f)L^{∗}(f, χ, 1 + k_{0}/2).

Proof. Part 1 follows by combining theorem 6.8 with theorem 4.12. Part 2 follows by combining part 1 of this corollary with theorem 5.7.

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