SINGULAR MODULI FOR REAL QUADRATIC FIELDS: A RIGID ANALYTIC APPROACH

HENRI DARMON AND JAN VONK

Abstract. A rigid meromorphic cocycle is a class in the first cohomology of the discrete group Γ := SL_{2}(ℤ[1/p]) with values in the multiplicative group of non-zero rigid meromorphic functions on the p-adic upper half plane H_{p} := ℙ_{1}(ℂ_{p})−ℙ_{1}(ℚ_{p}). Such a class can be evaluated at the real quadratic irrationalities in H_{p}, which are referred to as “RM points”. Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lifts: their zeroes and poles are contained in a finite union of Γ-orbits of RM points, and their RM values are conjectured to lie in ring class fields of real quadratic fields. These RM values enjoy striking parallels with the CM values of modular functions on SL_{2}(ℤ)\H: in particular they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high p-adic accuracy leads to convincing numerical evidence for the algebraicity and factorisation of the resulting singular moduli for real quadratic fields.

Contents

Introduction  1
1. Meromorphic cocycles of weight two  6
2. Analytic cocycles of weight two  19
3. Multiplicative cocycles  26
4. Real quadratic singular moduli  39
5. Gross-Stark units and Stark-Heegner points  51
References  56

Introduction

Drinfeld’s p-adic upper half-plane, a rigid analytic space whose ℂ_{p}-points are identified with H_{p} := ℙ_{1}(ℂ_{p}) − ℙ_{1}(ℚ_{p}), offers an enticing framework for explicit class field theory for real quadratic fields, since it contains a large supply H_{p}^{RM} of real multiplication (RM) points belonging to real quadratic fields in which the prime p is either inert or ramified. Let M^{×} denote the multiplicative group of rigid meromorphic functions on H_{p}, consisting of ratios of non-zero rigid analytic functions. The discrete group Γ = SL_{2}(ℤ[1/p]) acts on H_{p} by Möbius transformations, inducing an action on M^{×} (written either on the right or on the left) by the rule

(1)  (f|γ)(τ) := (γ^{−1}f)(τ) := f(\frac{aτ + b}{cτ + d}) ,  where γ := ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

A naive attempt at explicit class field theory for real quadratic fields could proceed by examining the RM values of Γ-invariant functions in M^{×}. However, because the Γ-orbits in H_{p} are dense for the rigid analytic topology, any such function is constant, i.e.,

1991 Mathematics Subject Classification. 11G18, 14G35.

(2)  H^{0}(Γ, M^{×}) = ℂ_{p}^{×}.

It is then natural to consider the first cohomology group H^{1}(Γ, M^{×}) instead. A class in this group is said to be parabolic if its restriction to the subgroup Γ_{∞} ⊂ Γ of upper triangular matrices is trivial, and is said to be quasi-parabolic if this restriction lies in H^{1}(Γ_{∞}, ℂ_{p}^{×}). The groups of such classes are denoted by H_{par}^{1}(Γ, M^{×}) and H_{f}^{1}(Γ, M^{×}) respectively.

Definition 1. A class in H_{f}^{1}(Γ, M^{×}) is called a rigid meromorphic cocycle for Γ.

A rigid meromorphic cocycle is thus a function J : Γ ⟶ M^{×} satisfying

J(γ_{1}γ_{2}) = J(γ_{1}) × γ_{1}J(γ_{2}),

taken modulo 1-coboundaries, of the form ξ(γ) = γf ÷ f, with f ∈ M^{×}, and admitting a quasi-parabolic representative, whose values on Γ_{∞} consist of constant functions. This representative is even unique (up to torsion), because M contains no translation-invariant elements.

This article initiates the study of rigid meromorphic cocycles, with special emphasis on their application to the analytic construction of class fields of real quadratic fields.

The relevance of Definition 1 for explicit class field theory rests on the fact that rigid meromorphic cocycles can be meaningfully evaluated at RM points. More precisely, the RM points in H_{p} are characterised by the fact that their associated order

O_{τ} := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(ℤ[1/p]) such that aτ + b = cτ^{2} + dτ\}

is isomorphic to a ℤ[1/p]-order in the real quadratic field K = ℚ(τ), via the inclusion

ι : O_{τ} ⟶ K, ι (\begin{matrix}a & b \\ c & d \end{matrix}) = cτ + d.

The stabiliser of τ in Γ is generated up to torsion by a fundamental unit of norm one in O_{τ}. It is called the automorph of τ, and denoted γ_{τ}. The value of a rigid meromorphic cocycle J at an RM point τ is then defined to be

(3)  J[τ] := J(γ_{τ})(τ) ∈ ℂ_{p} ∪ {∞},

a numerical invariant which depends only on the class of J in cohomology (and not on the choice of cocycle representing it) and on the Γ-orbit of τ. The cocycle J thus gives rise to a function

(4)  J : Γ\H_{p}^{RM} ⟶ ℂ_{p} ∪ {∞}.

Conjecture 1 below asserts that it takes algebraic values that lie in (composita of) abelian extensions of real quadratic fields, thus behaving in many key respects like the function SL_{2}(ℤ)\H^{CM} ⟶ ℂ ∪ {∞} induced by the classical j-function, or by any other meromorphic modular function defined over ℚ̄.

Let S be the standard matrix of order 2 in Γ/⟨±1⟩ that fixes i = \sqrt{−1}.

Definition 2: A rigid meromorphic period function is the value at S of the quasi-parabolic representative of a rigid meromorphic cocycle.

The assignment J ↦ j := J(S) identifies H_{f}^{1}(Γ, M^{×}) with the multiplicative group R^{×} of rigid meromorphic period functions, and any function in R^{×} satisfies the functional relations

(5)  j(−1/z) = j(z)^{−1},  j(p^{2}z) = j(z),  \frac{j(z + 1)}{j(z)} = j (−\frac{z + 1}{z}) .

The main result of the first three chapters is

Theorem 1. The group R^{×} is of infinite rank. The zeroes and poles of any j ∈ R^{×} are contained in a finite union of Γ-orbits of RM points in H_{p}.

Theorem 1 suggests that rigid meromorphic period functions might be viewed as the real quadratic counterpart of Borcherds’ singular theta lifts of modular forms of weight 1/2, insofar as the latter are meromorphic modular functions with divisor concentrated at CM points.

Let H_{τ} denote the ring class field (in the narrow sense) associated to the order O_{τ}. It is an abelian extension of K = ℚ(τ) whose Galois group over K is identified via global class field theory with the narrow class group Pic^{+}(O_{τ}) of projective oriented O_{τ} -modules. If j is any rigid meromorphic period function and J is its associated rigid meromorphic cocycle, Theorem 1 implies that the field

H_{j} = H_{J} := Compositum_{j(τ)=∞}(H_{τ})

is a finite extension of ℚ; it is called the field of definition of j, or of J. The main conjecture of this paper, which is discussed in greater detail in Chapter 4, is

Conjecture 1. If J is a rigid meromorphic cocycle, and τ ∈ H_{p} is an RM point, then the value J[τ] is an algebraic number belonging to the compositum of H_{J} and H_{τ}.

Conjecture 1 gives ample motivation for the systematic study of rigid meromorphic cocycles. This study is carried out in Chapters 1, 2 and 3, where Theorem 1 is proved by giving a complete classification of rigid meromorphic period functions. These functions, and their additive counterparts known as rigid meromorphic period functions of weight two, are reminiscent of the “rational period functions” that are studied in [Kn], [Ash], [CZ], and can be classified along similar lines. The classification obtained in Chapter 3 is constructive and leads to explicit product expansions for rigid meromorphic period functions. To describe these, for any τ ∈ Γ\H_{p}^{RM}, let

(6)  Σ_{τ} := {w ∈ Γτ such that ww′ < 0},

where w′ is the algebraic conjugate of w ∈ K := ℚ(τ). The subset Σ_{τ} of Γτ contains only finitely many integer translates of any given w ∈ Γτ, and it can in fact be shown that it is a discrete subset of the full Γ-orbit, relative to the rigid p-adic topology on H_{p} . After fixing a real embedding of K, let δ_{∞}(w) ∈ {−1, 1} denote the sign of w ∈ Σ_{τ}.

A prime p is said to be monstrous if it divides the cardinality of the Monster sporadic simple group, or equivalently (by a famous observation of Andrew Ogg) if the quotient of the modular curve X_{0}(p) by its Atkin Lehner involution has genus zero, which occurs precisely when

p ∈ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.

One of the illustrative results of Chapter 3 is Theorem 2. Let p be a monstrous prime, and let τ be any RM point in H_{p}. The infinite product

j_{τ}^{+}(z) := ∏_{w∈Σ_{τ}} (\frac{t_{w}(z)}{t_{pw}(z)})^{δ_{∞}(w)} , where t_{w}(z) = \{ \begin{matrix}z − w & if |w| ≤ 1, \\ z/w − 1 & if |w| > 1, \end{matrix}

converges to a rigid meromorphic period function on H_{p}.

The constructions described in the first three chapters also lead to a complete classification of rigid meromorphic period functions for arbitrary p: the multiplicative group of such functions is spanned by suitable Hecke translates of simple generalisations of the example in Theorem 2.

Chapter 3 concludes by describing an efficient algorithm for computing rigid meromorphic period functions to high p-adic accuracy, in terms of their images in the Tate algebra of a suitable affinoid subset of H_{p}. This algorithm makes it feasible to compute the RM values of rigid meromorphic cocycles to hundreds of significant digits, and has been used to test Conjecture 1 numerically in a variety of situations.

For example, if p is a monstrous prime that is inert in ℚ(\sqrt{5}), i.e., if

(7)  p ∈ {2, 3, 7, 13, 17, 23, 47},

and φ := (1 + \sqrt{5})/2 is the golden ratio, the rigid meromorphic period function j_{φ}^{+} and associated rigid meromorphic cocycle J_{φ}^{+} defined by setting τ = φ in Theorem 2 can be viewed as a convincing analogue of the j-function, whose divisor is concentrated on the CM point (1 + \sqrt{−3})/2 of smallest negative discriminant. Calculations performed to 100 digits of 3-adic and 13-adic accuracy (the primes p = 3, 13 being precisely those in (7) which are also inert in ℚ(\sqrt{2})) suggest that

J_{φ}^{+}[2\sqrt{2}] =^{?} \{ \begin{matrix}(33 + 56\sqrt{−1})/(5 · 13) & in ℂ_{3}; \\ (1 + 2\sqrt{−2})/3 & in ℂ_{13}, \end{matrix}

consistent with the fact that the order ℤ[2\sqrt{2}] has narrow class number two, and that its associated narrow ring class field is equal to ℚ(\sqrt{2}, \sqrt{−1}). Likewise, the real quadratic field K := ℚ(\sqrt{223}) has narrow class number 6, and the element \sqrt{223} can be viewed as belonging to H_{7}, H_{13}, and H_{47}. The value of J_{φ}^{+} at τ = \sqrt{223}, computed to 100 digits of p-adic accuracy for p ∈ {7, 13, 47}, appears to satisfy the following sextic polynomials:

p = 7. 282525425x^{6} + 27867770x^{5} + 414793887x^{4} − 128906260x^{3} + 414793887x^{2}

+27867770x + 282525425,

p = 13. 464800x^{6} + 1275520x^{5} + 1614802x^{4} + 1596283x^{3} + 1614802x^{2}
+1275520x + 464800,
p = 47. 4x^{6} + 4x^{5} + x^{4} − 2x^{3} + x^{2} + 4x + 4.

All three of these polynomials have the Hilbert class field of K as their splitting field, confirming Conjecture 1 in a setting where H_{τ} is non-abelian over ℚ.

A number of patterns emerge from the above experiments, notably:

• The RM value J_{φ}^{+}[τ] appears to belong to H_{τ}^{σ_{p}σ_{∞}=1}, where σ_{p} and σ_{∞} denote frobenius elements at the prime p and ∞ respectively. In that sense the situation is even a bit better than predicted by Conjecture 1, which only asserts that J_{φ}^{+}[τ] should be defined over the compositum of H_{τ} with the field of definition H_{φ} = ℚ(\sqrt{5}) of J_{φ}^{+}. The experiments, which have been carried out for numerous other examples as well, suggest that j_{φ}^{+} should really be viewed as being “defined over ℚ” rather than over ℚ(\sqrt{5}).

• The primes that occur in the factorisation of J_{φ}^{+}[τ] lie above rational primes that are inert or ramified in both the real quadratic fields ℚ(\sqrt{5}) and ℚ(τ), and divide an integer of the form \frac{5Disc(τ)−m^{2}}{4} ≥ 0.

• The value of J_{φ}^{+}[τ]–and even the precise field over which it is defined–depends very much on the monstrous prime p relative to which it is computed. However, if p and q are two monstrous primes for which φ and τ belong to both H_{p} and H_{q}, the primes above q very often (but not always!) occur in the factorisation of the p-adic J_{φ}^{+}[τ] with the same multiplicity as the primes above p in the factorisation of the selfsame q-adic invariant.

In an attempt to better understand this last feature, Chapter 4 focusses on a prime p ∈ {2, 3, 5, 7, 13}, i.e., a prime for which the modular curve X_{0}(p) has genus zero. For each pair (τ_{1}, τ_{2}) of RM points in H_{p} with associated ring class fields H_{1} = H_{τ_{1}} and H_{2} = H_{τ_{2}} a p-adic arithmetic intersection number

J_{p}(τ_{1}, τ_{2}) ∈^{?} H_{12} := H_{1}H_{2}

is defined. Roughly speaking, it is the value Ĵ_{τ_{1}}[τ_{2}], where Ĵ_{τ_{1}} is a simple modification of the cocycle J_{τ_{1}}^{+} of Theorem 2, with zeroes and poles concentrated in Γτ_{1}. The quantity J_{p}(τ_{1}, τ_{2}) seems to enjoy many of the same properties as the difference

J_{∞}(τ_{1}, τ_{2}) := j(τ_{1}) − j(τ_{2}), j(q) = \frac{1}{q} + 744 + 196884q + ⋯

of “classical” singular moduli studied in [GZ1], and is conjectured to admit analogous factorisations.

The prediction made in Conjecture 4.26 of Chapter 4 can be loosely paraphrased as follows:

Conjecture 2. The p-adic intersection number J_{p}(τ_{1}, τ_{2}) ∈^{?} H_{12} is divisible only by primes of H_{12} lying above rational primes that are non-split in both of the real quadratic fields K_{1} := ℚ(τ_{1}) and K_{2} := ℚ(τ_{2}), and divide a positive integer of the form \frac{D_{1}D_{2}−x^{2}}{4} . If q is such a prime, then the valuations of J_{p}(τ_{1}, τ_{2}) at the primes above q are determined by certain qweighted topological intersection numbers of modular geodesics attached to τ_{1} and τ_{2} on the Shimura curve arising from the indefinite quaternion algebra ramified at q and p.

This prediction resonates closely with the factorisations described in [GZ1], where the valuation at q of J_{∞}(τ_{1}, τ_{2}) is determined by a similar intersection of 0-cycles attached to the CM points τ_{1} and τ_{2} on the 0-dimensional Shimura variety arising from the definite quaternion algebra ramified at q and ∞.

Chapter 5 recalls the construction of the Gross–Stark units given in [DD] and the Stark– Heegner points of [Da], which are conjecturally defined over ring class fields of real quadratic fields, and explains how these constructions can be recast in the framework of RM values of rigid analytic cocycles “modulo suitable periods”.

In closing, it is worth noting that the infinite rank group H_{f}^{1}(Γ, M^{×}) admits no non-trivial finite rank Hecke stable subspaces. Rigid meromorphic cocycles and their RM values thus bear no direct relationship to Hecke eigenforms and to special values of L-functions with Euler products, unlike the Gross-Stark units and the Stark-Heegner points of Chapter 5, which are expected to satisfy analogues of the Kronecker limit formula and the Gross-Zagier formula. In that sense, the main thesis of this paper — that rigid meromorphic cocycles play the role of meromorphic modular functions in extending the theory of complex multiplication to real quadratic fields — breaks more decisively from the tradition of the Stark conjectures than either [DD] or [Da], where the leading terms of motivic L-functions continue to play a central role.

Acknowledgements. This article builds on the ideas of a great many people: the seminal work of Marvin Knopp, Avner Ash, Youngju Choie and Don Zagier on rational modular cocycles, and of Glenn Stevens, Peter Schneider and Jeremy Teitelbaum on overconvergent modular symbols and p-adic integral transforms, are the basis for Chapters 1 and 2 respectively. One of the main conjectures of Chapter 4 concerning the factorisations of “real quadratic singular moduli” is modelled on the analogous factorisations in the CM setting explored by Benedict Gross and Don Zagier. The last chapter incorporates insights gleaned over the course of previous collaborations and exchanges, notably with Massimo Bertolini, Pierre Charollois, Samit Dasgupta, Matthew Greenberg, and Victor Rotger. The spark for the present work was ignited when the authors became aware of the beautiful article of Bill Duke, Ozlem Imamoglu, and Arpad Toth [DIT] expressing the topological linking numbers of real quadratic modular geodesics on SL_{2}(ℤ)\SL_{2}(ℝ) in terms of RM values of multiplicative rational modular cocycles. It is a pleasure to thank all these mathematicians for the inspiration they have given us. Finally, the algorithms devised and implemented by James Rickards for computing the q-weighted topological intersection numbers of real quadratic geodesics on Shimura curves, which are a part of his ongoing PhD thesis, were invaluable in testing the Gross-Zagier style factorisations of Section 4.4.

1. Meromorphic cocycles of weight two

This chapter introduces additive counterparts of the rigid meromorphic cocycles and their associated rigid meromorphic period functions that were described in the introduction. These are referred to as rigid meromorphic cocycles and period functions of weight k ≥ 0. The main results of this chapter are Theorems 1.22 and 1.23, which together give the full classification of rigid meromorphic period functions of weight two up to rigid analytic period functions of the same weight. The techniques are greatly inspired by the classification of rational period functions carried out by Knopp [Kn], Ash [Ash] and Choie-Zagier [CZ].

1.1. The p-adic upper half-plane. We begin by recalling some facts about the p-adic upper half plane as a rigid analytic space.

Let T denote the Bruhat–Tits tree of PGL_{2}(ℚ_{p}), 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 representative lattices contained in each other with index p. Write T_{0}, T_{1}, and T_{1}^{∗} for the set of vertices, unordered edges, and ordered edges respectively of T . If e ∈ T_{1}^{∗} is an ordered edge, we denote by s(e) and t(e) ∈ T_{0} its source and target vertices respectively. The group Γ acts on T through its natural left action on ℚ_{p}^{2}, viewing the latter as column vectors. The “standard vertex” v_{0} ∈ T_{0} associated to the lattice ℤ_{p}^{2} has SL_{2}(ℤ) as its stabiliser for this action. A vertex is said to be even if its distance to v_{0} is even, and is said to be odd otherwise. The set of even and odd vertices are denoted T_{0}^{+} and T_{0}^{−} respectively. Likewise, an ordered edge in T_{1}^{∗} is said to be even if its source is even and odd if its source is odd. The subsets of even and odd oriented edges are denoted T_{1}^{+} and T_{1}^{−} respectively, so that we have the decompositions

T_{0} = T_{0}^{+} ⊔ T_{0}^{−}, T_{1}^{∗} = T_{1}^{+} ⊔ T_{1}^{−}.

The p-adic upper half plane H_{p} may be thought of as a tubular neighbourhood of the Bruhat–Tits tree T via the natural “reduction map”

red : H_{p} ⟶ T ,

which maps H_{p}(ℚ̂_{p}^{nr}) to T_{0}, where ℚ̂_{p}^{nr} is the completion of the maximal unramified extension of ℚ_{p}. The inverse image of a vertex v ∈ T_{0}, denoted A_{v}, is called a vertex affinoid in H_{p}, and the inverse image of an edge e ∈ T_{1} is an annulus denoted by W_{e}. The vertex affinoid corresponding to the standard vertex v_{0} is called the standard affinoid: it is the complement in ℙ_{1}(ℂ_{p}) of the p + 1 mod p residue discs around the points in ℙ_{1}(𝔽_{p}). The edge e_{0} ∈ T_{1} with stabiliser Γ_{0}(p) is the image under the reduction map of the standard annulus

W_{e_{0}} = {z ∈ ℂ_{p} with 1 < |z| < p}.

If v is a vertex and e_{1}, … e_{p+1} are the distinct edges in T_{1} having v as an endpoint, then the union

W_{v} := A_{v} ∪ ⋃_{j} W_{e_{j}} := red^{−1} (v ∪ ⋃_{j} e_{j})

is called the standard wide open subset attached to v.

Let T^{≤n} denote the subgraph of T spanned by the vertices of distance ≤ n from the standard vertex, and write H_{p}^{≤n} for the affinoid subdomain of H_{p} consisting of those points reducing to T^{≤n}. Likewise let T^{<n} denote the subgraph of T containing all the vertices of distance ≤ n − 1 as well as all the edges containing at least one of these vertices, and write H_{p}^{<n} for the wide open subspace of H_{p} consisting of those points reducing to T^{<n}. The subsets H_{p}^{≤n} define an admissible cover

H_{p} = ⋃_{n≥0} H_{p}^{≤n}

of the p-adic upper half-plane by affinoid subsets.

Of course, the actions of Γ on T and on H_{p} by Möbius transformations are compatible under the reduction map. In particular, for all γ ∈ Γ,

A_{γv} = γA_{v}, W_{γe} = γW_{e}.

A ℂ_{p}-valued function on H_{p} is said to be rigid analytic if its restriction to any affinoid subset A of H_{p} is a uniform limit, relative to the supremum norm, of rational functions on ℙ_{1}(ℂ_{p}) having poles outisde of A. The space O of rigid analytic functions on H_{p} is endowed with a natural topology arising from its expression as the inverse limit of the affinoid algebras O(H_{p}^{≤n}), which are Banach spaces for their supremum norms. Let M denote the fraction field of O. Its elements are called rigid meromorphic functions on H_{p}.

If τ ∈ H_{p} is an RM point, then there is a primitive integral binary quadratic form F_{τ} (x, y) satisfying F (τ, 1) = 0, which is unique up to sign. The discriminant of τ is the discriminant of this binary quadratic form. The discriminant is an invariant for the action of SL_{2}(ℤ), but not of Γ, which only preserves the prime-to-p part of disc(τ).

Proposition 1.1. If τ is an RM point of discriminant D_{0}p^{n}, where D_{0} is prime to p, then τ reduces to a point of T at distance n/2 from v_{0}. In particular:

(1) If n = 2m is even, then τ reduces to a vertex of T , and belongs to one of the affinoids in H_{p}^{≤m} − H_{p}^{<m}.

(2) If n = 2m + 1 is odd, then τ reduces to the midpoint of an edge of T , and belongs to one of the annuli in H_{p}^{<m+1} − H_{p}^{≤m}.

Proof. Let Ax^{2} + Bxy + Cy^{2} be the primitive integral binary quadratic form of discriminant D = D_{0}p^{n} having τ as a root. Let ϖ be an element of O_{ℂ_{p}} of normalised p-adic valuation n/2. The natural image of τ = ^{\frac{−B+\sqrt{D}}{2A}} in ℙ_{1}(O_{ℂ_{p}}/ϖ) agrees with the image of [−B : 2A] ∈ ℙ_{1}(ℚ_{p}) under the natural composition

ℙ_{1}(ℚ_{p}) ↪ ℙ_{1}(ℂ_{p}) = ℙ_{1}(O_{ℂ_{p}}) ⟶ ℙ_{1}(O_{ℂ_{p}}/ϖ),

while its image in ℙ_{1}(O_{ℂ_{p}}/ϖp^{ϵ}) for any ϵ > 0 does not lie in the image of ℙ_{1}(ℚ_{p}). The proposition follows. □

1.2. Modular symbols. The parabolic cohomology of Γ with values in a GL_{2}(ℚ_{p})-module Ω admits a concrete description in terms of Γ-invariant modular symbols, which will also provide a natural bridge between rigid meromorphic cocycles and the rigid meromorphic period functions evoked in the introduction.

The action of Γ on Ω shall be written (both on the right and on the left according to convenience) as

(m, γ) ↦ m|γ, (γ, m) ↦ γm := m|γ^{−1}, m ∈ Ω, γ ∈ Γ.

Definition 1.2. An Ω-valued modular symbol is a function

m : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ Ω,

satisfying

m{r, s} = −m{s, r},  m{r, s} + m{s, t} = m{r, t}  for all r, s, t ∈ ℙ_{1}(ℚ).

The space of Ω-valued modular symbols is denoted MS(Ω). It is endowed with a natural action of PGL_{2}(ℚ) by the rule

(m|γ){r, s} := (m{γr, γs}) |γ.

The space of Γ-invariant modular symbols, denoted

MS^{Γ}(Ω) := H^{0}(Γ, MS(Ω)),

is the set of modular symbols satisfying the Γ-invariance property

m{γr, γs} = m{r, s}|γ^{−1} = γm{r, s}, for all γ ∈ Γ.

It is equipped with the usual action of the Hecke operators T_{n} (for (n, p) = 1) defined in terms of the double coset

Γ (\begin{matrix}n & 0 \\ 0 & 1 \end{matrix}) Γ = ⊔_{j}Γγ_{j}

by setting

(m|T_{n}) = ∑_{j}(m|γ_{j}).

The normaliser of Γ in PGL_{2}(ℚ) is the group PGL_{2}(ℤ[1/p]), and the determinant induces an isomorphism

det : PGL_{2}(ℤ[1/p])/Γ ⟶ ℤ[1/p]^{×}/(ℤ[1/p]^{×})^{2} = {1, −1, p, −p} ≃ ℤ/2ℤ × ℤ/2ℤ.

The elements w_{∞} and w_{p} associated to classes of matrices of determinant −1 and p respectively generate this quotient, and give rise to involutions on MS^{Γ}(Ω) by the rules

(8) (m|w_{∞}){r, s} := (m{w_{∞}r, w_{∞}s}) |w_{∞},  (m|w_{p}){r, s} := (m{w_{p}r, w_{p}s}) |w_{p}.

A Γ-invariant modular symbol which is in the +1 (resp. −1) eigenspace for the involution w_{∞} is said to be even (resp. odd). Likewise, it is said to be p-even (resp. p-odd) if it is in the +1 (resp. −1) eigenspace for the involution w_{p}.

The following lemma relates Γ-invariant modular symbols to the corresponding parabolic cohomology groups.

Lemma 1.3. There is a natural exact sequence

(9)  0 ⟶ Ω^{Γ} ⟶ Ω^{Γ_{∞}} ⟶ MS^{Γ}(Ω) ⟶ H^{1}(Γ, Ω) ⟶ H^{1}(Γ_{∞}, Ω).

In particular, there is a canonical Hecke-equivariant surjection δ : MS^{Γ}(Ω) ⟶ H_{par}^{1}(Γ, Ω) which is an isomorphism when Ω^{Γ} = Ω^{Γ_{∞}}.

Proof. Let F(ℙ_{1}(ℚ), Ω) be the Γ-module of Ω-valued functions on ℙ_{1}(ℚ), equipped with the natural Γ-action arising from the action of Γ on ℙ_{1}(ℚ) by Möbius transformations. It fits into the exact sequence of ℤ[Γ]-modules

(10)  0 ⟶ Ω ⟶ F(ℙ_{1}(ℚ), Ω) ⟶^{d} MS(Ω) ⟶ 0,

where df{r, s} := f(s) − f(r). The lemma follows from taking the long exact Γ-cohomology sequence associated to this short exact sequence and invoking Shapiro’s Lemma to identify H^{i}(Γ, F(ℙ_{1}(ℚ), Ω)) with H^{i}(Γ_{∞}, Ω). □

The cohomology class Φ := δ(Φ_{0}) associated to Φ_{0} ∈ MS^{Γ}(Ω) is defined by choosing a base point r ∈ ℙ_{1}(ℚ) and setting

Φ(γ) = Φ_{0}{r, γr}.

The parabolic representative of Φ which vanishes on Γ_{∞} is obtained by choosing r = ∞ in this assignment.

Let

(11)  S = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ,  U = (\begin{matrix}0 & 1 \\ −1 & 1 \end{matrix}) ,  D = (\begin{matrix}p & 0 \\ 0 & 1/p \end{matrix})

denote the standard matrices in Γ satisfying S^{2} = U^{3} = −1. Given m ∈ MS^{Γ}(Ω), the element ω := m{0, ∞} ∈ Ω satisfies the so-called two and three-term relations

(12)  ω + Sω = 0,  ω + Uω + U^{2}ω = 0,

which follow from the modular symbol relations

m{0, ∞} + m{∞, 0} = 0, m{0, ∞} + m{∞, 1} + m{1, 0} = 0

after noting that S interchanges 0 and ∞ while U induces the cyclic permutation (0 1 ∞) on these three elements of ℙ_{1}(ℚ). Let Ω^{†} ⊂ Ω be the set of elements satisfying (12).

The proposition below is a well known assertion about the cohomology of SL_{2}(ℤ).

Proposition 1.4. The assigment m ↦ m{0, ∞} identifies MS^{SL_{2}(ℤ)}(Ω) with Ω^{†}.

Proof. A pair of elements (a/b, c/d) of ℙ_{1}(ℚ) (expressed in lowest terms, with the convention that ∞ = 1/0) is said to be unimodular if ad − bc = ±1. The fact that any two elements of ℙ_{1}(ℚ) can be inserted into a unimodular sequence, in which all consecutive terms form unimodular pairs, implies that a modular symbol is completely determined by its values on such pairs. The injectivity of the assignment m ↦ m{0, ∞} then follows from the fact that SL_{2}(ℤ) acts transitively on the set of unimodular pairs. To prove surjectivity, observe that any ω ∈ Ω^{†} determines a well-defined function m on the set of unimodular pairs by setting

m \{\frac{a}{b}, \frac{c}{d}\} := (\begin{matrix}c & a \\ d & b \end{matrix}) ω,

where (a/b, c/d) have been adjusted so that the matrix appearing on the right belongs to SL_{2}(ℤ). If r and s are arbitrary elements of ℙ_{1}(ℚ), the unimodular sequences joining r and s are far from unique, but the theory of Farey sequences implies that any two unimodular sequences admit a common refinement, where a refinement is obtained by making a finite number of replacements of the form

\frac{a}{b}, \frac{c}{d}, \frac{−a}{−b} ⇝ \frac{a}{b}, \frac{a}{b}, \frac{c}{d} ⇝ \frac{a}{b}, \frac{a + c}{b + d}, \frac{c}{d}.

The two and three-term relations satisfied by ω imply that

m \{\frac{a}{b}, \frac{c}{d}\} + m \{\frac{c}{d}, \frac{a}{b}\} = 0, m \{\frac{a}{b}, \frac{c}{d}\} = m \{\frac{a}{b}, \frac{a + c}{b + d}\} + m \{\frac{a + c}{b + d}, \frac{c}{d}\} ,

for all unimodular pairs (a/b, c/d). It follows that m extends uniquely to an SL_{2}(ℤ)-invariant Ω-valued modular symbol, and hence that the map MS^{SL_{2}(ℤ)}(Ω) → Ω^{†} is surjective. Proposition 1.4 follows. □

Let Ω^{‡} ⊂ Ω^{†} denote the image of the group MS^{Γ}(Ω) under the assignment m ↦ m{0, ∞}.

Lemma 1.5. If ω belongs to Ω^{†}, then ω satisfies the two and three term relations in (12) along with the further relation

(13)  Dω = ω.

Proof. This follows directly from the fact that both 0 and ∞ are fixed by the diagonal matrices. □

Remark 1.6. The equation (13) does not characterise Ω^{‡}: in general, its elements may need to satisfy further relations, which are less simple to write down explicitly and whose complexity presumably grows as a function of p.

1.3. Basic definitions. For all k ≥ 0, the continuous weight k action (cf. [ST, Section 1]) of the group Γ := SL_{2}(ℤ[1/p]) on O and on M is given by

(14) (f|_{k}γ)(τ) := (γ^{−1} ·_{k} f)(τ) := (cτ + d)^{−k} f(\frac{aτ + b}{cτ + d}) ,  where γ := ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

The underlying addditive groups of O and M endowed with this weight k action are denoted O_{k} and M_{k} respectively, with the convention that O and M will be used to denote O_{0} and M_{0} respectively.

The following is the additive counterpart of Definition 1 of the Introduction:

Definition 1.7. A rigid meromorphic (resp. analytic) cocycle of weight k ≥ 0 is a class in H_{par}^{1}(Γ, M_{k}) (resp. in H_{par}^{1}(Γ, O_{k})).

Remark 1.8. The multiplicative group H^{1}(Γ, M^{×}) should not be confused with the vector space H^{1}(Γ, M). Although elements of the latter can be evaluated at RM points just as well as their multiplicative counterparts, it will be shown later (cf. Prop. 2.14) that

H_{par}^{1}(Γ, M) = 0,

and hence no interesting class invariants for real quadratic fields are to be extracted from the additive theory.

Of greatest importance for our study are the rigid meromorphic cocycles of weight two, which are related to rigid meromorphic cocycles via the map

dlog : H_{f}^{1}(Γ, M^{×}) ⟶ H_{par}^{1}(Γ, M_{2})

arising from the logarithmic derivative

dlog : M^{×} ⟶ M_{2}, dlog(g) = g′/g,

which is compatible with the Γ-actions on source and target and therefore induces a map on the associated parabolic cohomology groups. The first step in classifying rigid meromorphic cocycles will be to do the same for their additive, weight two counterparts, with the advantage that the latter are endowed with a natural ℂ_{p}-linear structure.

Let us first specialise the discussion of the previous section on modular symbols to the setting where Ω = M^{×} or M_{k} for k ≥ 0.

Lemma 1.9. The Γ_{∞}-invariants of M^{×} and M_{k} are given by

H^{0}(Γ_{∞}, M^{×}) = ℂ_{p}^{×}, H^{0}(Γ_{∞}, M_{k}) = \{ \begin{matrix}ℂ_{p} & if k = 0; \\ 0 & if k > 0. \end{matrix}

Proof. This follows from the Weierstrass preparation theorem: any translation invariant rigid analytic function must be constant on the standard affinoid and hence everywhere by analytic continuation. □

Corollary 1.10. For all k ≥ 0, the map δ of Lemma 1.3 induces isomorphisms

(15)  MS^{Γ}(M^{×}) ⟶^{∼} H_{par}^{1}(Γ, M^{×}),  MS^{Γ}(M_{k}) ⟶^{∼} H_{par}^{1}(Γ, M_{k}).

Proof. Lemma 1.9 implies that

H^{0}(Γ, M^{×}) = H^{0}(Γ_{∞}, M^{×}), H^{0}(Γ, M_{k}) = H^{0}(Γ_{∞}, M_{k}),

and the corollary follows from Lemma 1.3. □

Corollary 1.10 allows us to work with elements of MS^{Γ}(M_{k}) in studying rigid meromorphic cocycles of weight k, with the advantage that many arguments tend to become more transparent when couched in the language of modular symbols.

Recall the multiplicative group R^{×} ⊂ M^{×} of rigid meromorphic period functions given after Definition 2 of the introduction, which is identified with MS^{Γ}(M^{×}) via the assignment J ↦ J{0, ∞}. The following is the additive counterpart of Definition 2:

Definition 1.11. A rigid meromorphic period function of weight k is the value at S of the parabolic representative of a rigid meromorphic cocycle of weight k.

Let R_{k} denote the ℂ_{p}-vector space of rigid meromorphic period functions of weight k. The assignment Φ ↦ φ := Φ{0, ∞} identifies MS^{Γ}(M_{k}) with R_{k}. Just as in the multiplicative setting, one has:

Lemma 1.12. A function φ ∈ R_{k} satisfies the two and three term relations

φ (−\frac{1}{z}) = −z^{k}φ(z), φ(z) + z^{−k}φ (\frac{z − 1}{z}) + (z − 1)^{−k}φ (\frac{−1}{z − 1}) = 0,

as well as the further linear relation

φ(p^{2}z) = p^{−k}φ(z).

In conclusion, we have obtained canonical maps

H_{par}^{1}(Γ, M^{×}) = MS^{Γ}(M^{×}) ⊂ R^{×}, H_{par}^{1}(Γ, M_{k}) = MS^{Γ}(M_{k}) = R_{k}.

1.4. Prelude: rational cocycles and period functions. The classification of rigid meromorphic cocycles and rigid meromorphic period functions of weight two which will be described in the next section parallels closely – and its proof is strongly inspired by – the classification of so called rational modular cocycles and their associated rational period functions that were introduced by Marvin Knopp and arise, notably, in the work of Knopp [Kn], Ash [Ash] , Choie–Zagier [CZ], and Duke–Imamoglu–Toth [DIT].

Let C be an algebraically closed field of characteristic 0, and let M_{k}^{rat} denote the C-vector space of rational functions on ℙ_{1}(C), endowed with the weight k action of SL_{2}(ℤ) as defined in (14) above.

Definition 1.13. A rational modular cocycle of weight k is a class in H_{par}^{1}(SL_{2}(ℤ), M_{k}^{rat}). A rational period function of weight k is an element of (M_{k}^{rat})^{†}, i.e., a rational function ϕ ∈ M_{k}^{rat} satisfying the two and three term relations

ϕ (−\frac{1}{z}) = −z^{k}ϕ(z), ϕ(z) + z^{−k}ϕ (\frac{z − 1}{z}) + (z − 1)^{−k}ϕ (\frac{−1}{z − 1}) = 0, for all z ∈ C.

Proposition 1.4 shows that the assignment Φ ↦ ϕ := Φ{0, ∞} identifies MS^{SL_{2}(ℤ)}(M_{k}^{rat}) with the space of rational period functions of weight k.

We now focus on the case k = 2, where the assignment r(z) ↦ r(z)dz identifies M_{2}^{rat} with the space Ω_{rat}^{1} of rational differentials on ℙ_{1/C}. We will henceforth view rational period functions of weight two interchangeably as elements of Ω_{rat}^{1} or as functions on H_{p} endowed with the weight two Γ-action.

We begin by giving some examples of rational period functions of weight two.

Lemma 1.14. The function ϕ_{∞}^{∘}(z) := \frac{1}{z} is a rational period function of weight two.

Proof. Consider the function

Φ_{∞}^{∘} : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ Ω_{rat}^{1}

given by

Φ_{∞}^{∘}{r, s} := ω{r, s},

where ω{r, s} is the unique rational differential with poles concentrated at r and s and residues 1 and −1 respectively at these points. The fact that Φ_{∞}^{∘} is an SL_{2}(ℤ)-invariant modular symbol follows directly from this description. For instance, its SL_{2}(ℤ) invariance can be seen from the calculation

Φ_{∞}^{∘}{γr, γs} = ω{γr, γs} = γω{r, s},

the last equality following from the fact that both differentials have the same poles and residues. The lemma follows after noting that

Φ_{∞}^{∘}\{0, ∞\} = \frac{dz}{z} = ϕ_{∞}^{∘}(z)dz.

□

To construct more interesting examples, let τ be any real quadratic irrationality in ℝ, and let F_{τ} (x, y) be the primitive integral binary quadratic form for which F (τ, 1) = 0. The SL_{2}(ℤ)-orbit of τ is dense in ℝ, but the subset

(16)  Σ_{τ}^{∘} := {w ∈ SL_{2}(ℤ) · τ such that ww′ < 0}

is finite and non-empty. This is because its elements are the roots of F (z, 1) were F (x, y) = Ax^{2}+Bxy+Cy^{2} is a primitive integral quadratic form in the same class as F_{τ} (x, y), and hence of a fixed positive discriminant, satisfying AC < 0, and the coefficients of such a quadratic form are bounded in absolute value.

The set Σ_{τ}^{∘} is endowed with a natural sign function δ_{∞} : Σ_{τ}^{∘} ⟶ ±1, which partitions Σ_{τ}^{∘} into its subsets of positive and negative elements respectively. These sets are of equal cardinality since they are interchanged by the involution z ↦ −1/z.

More generally, if r and s are elements of ℙ_{1}(ℚ), let γ(r, s) denote the geodesic on H̄ := H ∪ ℝ ∪ {∞} joining r to s and oriented in the direction from r to s. The complement of this geodesic in H̄ is partitioned into two disjoint connected subsets

H̄ − γ(r, s) := H^{+}(r, s) ∪ H^{−}(r, s),

labelled with the convention that, as one is travelling along γ(r, s) in the direction from r to s, the region H^{+}(r, s) is to one’s right and the region H^{−}(r, s) is to one’s left. If w ∈ SL_{2}(ℤ)·τ is any real quadratic irrationality in the SL_{2}(ℤ)-orbit of τ, we say that it is linked to γ(r, s) if it and its algebraic conjugate w′ belong to distinct connected components of H̄ − γ(r, s), and write Σ_{τ}^{∘}(r, s) ⊂ SL_{2}(ℤ) · τ for the set of w which are linked to γ(r, s) in this way. The set Σ_{τ}^{∘}(r, s) is endowed with the sign function δ_{r,s} defined by

(17)  δ_{r,s}(w) = \{ \begin{matrix}1 & if w ∈ H^{+}(r, s), \\ −1 & if w ∈ H^{−}(r, s), \end{matrix}

which partitions Σ_{τ}^{∘}(r, s) into its subsets of positive and negative elements respectively, and

Σ_{τ}^{∘}(0, ∞) = Σ_{τ}^{∘}, δ_{0,∞} = δ_{∞}.

Let Div(ℙ_{1}(C)) denote the free abelian group consisting of finite formal ℤ-linear combinations of points of ℙ_{1}(C), let Div^{0}(ℙ_{1}(C)) denote its subgroup of degree zero divisors, and define

(18)  Δ_{τ}^{∘}{r, s} := ∑_{w∈Σ_{τ}^{∘}(r,s)} δ_{r,s}(w)[w] ∈ Div(ℙ_{1}(C)).

Lemma 1.15. The function

Δ_{τ}^{∘} : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ Div(ℙ_{1}(C)),

is an element of MS^{SL_{2}(ℤ)}(Div^{0}(ℙ_{1}(C))).

Proof. To check the modular symbol property of Δ_{τ}^{∘}, observe that for all r, s, t ∈ ℙ_{1}(ℚ), the element w ∈ SL_{2}(ℤ)τ is linked to (r, t) if and only if it is linked to either (r, s) or to (s, t), but not to both, and that, in the latter case,

δ_{r,s}(w) + δ_{s,t}(w) = 0.

The SL_{2}(ℤ) equivariance

Δ_{τ}^{∘}{γr, γs} = γΔ_{τ}^{∘}{r, s}

follows directly from the definitions. Finally, since an SL_{2}(ℤ)-invariant modular symbol is completely determined by its value on the unimodular pair (0, ∞), all the divisors Δ_{τ}^{∘}{r, s} inherit from Δ_{τ}^{∘}{0, ∞} the property of being of degree zero. □

Lemma 1.16. The function

ϕ_{τ}^{∘}(z) := ∑_{w∈Σ_{τ}^{∘}} δ_{∞}(w)\frac{1}{z − w}

is a rational period function of weight two.

Proof. Consider the function

Φ_{τ}^{∘} : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ Ω_{rat}^{1}

given by

(19)  Φ_{τ}^{∘}\{r, s\} := ∑_{w∈Σ_{τ}^{∘}(r,s)} δ_{r,s}(w)\frac{dz}{z − w} =: ω_{τ}^{∘}\{r, s\}.

The differential ω_{τ}^{∘}{r, s} is the unique rational differential on ℙ_{1}(C) whose residual divisor is equal to Δ_{τ}^{∘}{r, s}. It follows from Lemma 1.15 that Φ_{τ}^{∘} defines an element of MS^{SL_{2}(ℤ)}(Ω_{rat}^{1}). Lemma 1.16 now follows from the fact that

ϕ_{τ}^{∘}(z)dz = Φ_{τ}^{∘}{0, ∞}.
□

Lemmas 1.14 and 1.16 have exhibited an explicit collection of distinct rational period functions ϕ_{τ}^{∘} of weight two, as τ ranges over the infinite index set

I = {∞} ∪ (SL_{2}(ℤ)\C^{RM}),

where C^{RM} denotes the collection of real quadratic irrationalities in C. The following classification of rational period functions of weight two, whose statement can be read off by setting k = 1 in Theorem 1 of [CZ], asserts that the ϕ_{τ}^{∘} form a basis for the C-vector space of rational period functions of weight two.

Theorem 1.17 (Knopp, Ash, Choie-Zagier). Any rational period function of weight two is a finite linear combination of the functions ϕ_{∞}^{∘} of Lemma 1.14 and ϕ_{τ}^{∘} of Lemma 1.16.

Since some of the steps of the proof will be used in our later classification of rigid meromorphic period functions, it is worth briefly recalling them here.

Let ϕ be any rational period function and let Σ_{ϕ} ⊂ C denote its set of poles. The two and three term relations satisfied by ϕ imply that

(20)  w ∈ Σ_{ϕ}  ⇒  S(w) ∈ Σ_{ϕ} and U(w) ∈ Σ_{ϕ} or U^{2}(w) ∈ Σ_{ϕ}.

Recall the sets Σ_{τ}^{∘} described in (16) for τ ∈ C^{RM}, and set Σ_{∞}^{∘} = {0, ∞}.

Lemma 1.18. If Σ_{ϕ} is any finite subset of C satisfying (20), then the set Σ_{ϕ} is a finite union of the sets of the form Σ_{τ}^{∘} with τ ranging over a finite subset I_{ϕ} ⊂ I.

Proof. This is just a restatement of Lemma 2 of [CZ], whose proof relies solely on the fact that Σ satisfies (20). Although it is formulated as a statement about rational period functions over ℂ, the argument carries over to the more abstract setting where ℂ is replaced by any algebraically closed field C of characteristic zero, by fixing an embedding C ⟶ ℂ. □

Concerning the behaviour of ϕ at its poles, one has the following:

Lemma 1.19. The differential ϕ(z)dz has only simple poles. Given any τ ∈ I for which Σ_{τ}^{∘} ⊂ Σ_{ϕ}, there is a λ_{τ} ∈ C satisfying

res_{w}ϕ(z)dz = \{ \begin{matrix}−λ_{τ} & if w < 0, \\ λ_{τ} & if w > 0, \end{matrix} for all w ∈ Σ_{τ}^{∘}.

Proof. The proof, which is described in Lemmas 4 and 5 of [CZ], exploits the invariance of the principal part of ϕ at w under any non-trivial matrix of SL_{2}(ℤ) which fixes w. More precisely, consider the Laurent expansion ϕ_{w}(z) around z = w, and write

ϕ_{w}(z) = PP_{w}(z) + O(1) = (z − w)^{−m} + O ((z − w)^{−m+1}) ,

where ℙ_{w}(z) denotes the principal part of ϕ at w, a polynomial of some degree m ≥ 1 in (z − w)^{−1} with no constant term. Let γ be a generator of the stabiliser of w in SL_{2}(ℤ). It is shown in Lemmas 4 and 5 of [CZ] that PP_{w}|_{2}γ = PP_{w}, while

(ϕ|_{2}γ)(z) = (rw + s)^{2−2m}(z − w)^{−m} + O ((z − w)^{−m+1}) , where γ = (\begin{matrix}p & q \\ r & s \end{matrix}) .

The quantity (rw + s) is a fundamental unit in an appropriate quadratic order of ℚ(w), and is hence non-torsion in C^{×}. It follows that 2 − 2m = 0, i.e., that m = 1 and therefore that ϕ has at most simple poles. The two and three term relations satisfied by ϕ immediately imply (in light of Lemma 1.20 below) that all of its residues are equal up to sign on any given Σ_{τ}^{∘}, and the two term relation shows that the sign of the residue depends only on the sign of w ∈ Σ_{τ}^{∘}. □

Proof of Theorem 1.17. Let ϕ be a rational period function. Write Σ_{ϕ} = ∪_{τ∈I_{ϕ}}Σ_{τ}^{∘}, where I_{ϕ} is the finite subset of I given in Lemma 1.18. Let (λ_{τ})_{τ∈I_{ϕ}} be the vector of scalars indexed by τ ∈ I_{ϕ} determined by Lemma 1.19. The difference ϕ − ∑_{τ∈I_{ϕ}} λ_{τ} ϕ_{τ}^{∘} is a rational function without singularities, and hence is constant. It follows that

ϕ = ∑_{τ∈I_{ϕ}} λ_{τ} ϕ_{τ}^{∘},

since there are no constant rational period functions of weight two. The theorem follows. We record the following closure property of the sets Σ_{τ}^{∘} refining (20):

Lemma 1.20. For all positive w ∈ Σ_{τ}^{∘}, (1) the negative element S(w) = −1/w also belongs to Σ_{τ}^{∘};

(2) the set {U(w), U^{2}(w)} contains exactly one element w^{♭} ∈ Σ_{τ}^{∘}, which is negative, and given by

w^{♭} = \{ \begin{matrix}U^{2}(w) = \frac{w−1}{w} & if 0 < w < 1; \\ U(w) = \frac{1}{1−w} & if w > 1. \end{matrix}

Proof. The first statement is clear. For the second, observe that U cyclically permutes the elements 0, 1, and ∞ ∈ ℙ_{1}(ℝ), and hence does the same to the open intervals (0, 1), (1, ∞), and (−∞, 0). It follows that, if (w′, w) belongs to (−∞, 0)×(0, 1), the translate U(w) has positive norm while U^{2}(w) is a negative element of Σ_{τ}^{∘}. Likewise, if (w′, w) belongs to (−∞, 0)×(1, ∞), the translate U^{2}(w) has positive norm while U(w) is a negative element of Σ_{τ}^{∘}. □

1.5. Classification of rigid meromorphic cocycles of weight two. We will now adapt the ideas of the previous section to classify elements of MS^{Γ}(M_{2}). Recall that the rigid meromorphic period function φ := Φ{0, ∞} attached to a rigid meromorphic cocycle Φ satisfies the properties

(21)  φ|(1 + S) = 0,  φ|(1 + U + U^{2}) = 0,  φ|D = φ,

where the matrices S, U and D ∈ Γ are defined in (11). The matrix P ∈ GL_{2}(ℤ[1/p]) defined by

P := (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})

satisfies P^{2} = D in PGL_{2}(ℤ[1/p]) and induces an involution on the space of rigid meromorphic period functions of weight two, defined by

ϖ_{p}(φ)(z) = −φ|P (z) = −pφ(pz).

(Note the presence of the minus sign in this definition.) A rigid meromorphic period function is said to be p-even (resp. p-odd) if is satisfies

(22)  ϖ_{p}(φ) = φ,  (resp. ϖ_{p}(φ) = −φ).

As in the previous section, a rigid meromorphic period function of weight two shall be viewed as an element of the space Ω^{1} of rigid meromorphic differentials on H_{p}, and a rigid meromorphic cocycle, as an element of MS^{Γ}(Ω^{1}).

We begin by constructing some basic examples of rigid meromorphic period functions of weight two, modelled on the construction of the rational period functions ϕ_{τ}^{∘} of Lemma 1.16. Let τ be any RM point in H_{p}, and fix an embedding of the real quadratic field ℚ(τ) into ℝ. Recall that the image of τ in T under the reduction map belongs either to T_{0} or is the midpoint of an edge in T_{1}. The Γ-orbit of τ is dense in H_{p} for the rigid analytic topology. As in (16), consider the subset

(23)  Σ_{τ} := {w ∈ Γ · τ such that ww′ < 0}.

It is endowed with the sign function δ_{∞} defined as in Section 1.4. Other notations and terminologies similar to those in Section 1.4 are also adopted. Notably, for each r, s ∈ ℙ_{1}(ℚ), let Σ_{τ} (r, s) ⊂ Γ · τ denote the set of w ∈ Γτ which are linked to γ(r, s) in the sense of that section, and let δ_{r,s} denote the same sign function as in (17). Finally, imitating (18), we let

(24)  Δ_{τ} {r, s} := ∑_{w∈Σ_{τ} (r,s)}δ_{r,s}(w)[w] ∈ Div(H_{p}).

The sum in this equation is to be viewed as an infinite formal sum of points in H_{p}, and Div(H_{p}) simply denotes the Γ-module of such formal sums. For the same reason as in Lemma 1.15, the function Δ_{τ} defines a Γ-invariant, Div(H_{p})-valued modular symbol.

A subset of H_{p} is said to be discrete if its intersection with each affinoid subset of H_{p} is finite. The module of divisors on H_{p} with discrete support is denoted Div^{†}(H_{p}).

Lemma 1.21. For all r, s ∈ ℙ_{1}(ℚ), the sets Σ_{τ} (r, s) are discrete. Furthermore, the finite intersection Σ_{τ} (r, s)∩A_{v} contains equal numbers of positive and negative elements, and likewise for Σ_{τ} (r, s) ∩ W_{v}.

Proof. Proposition 1.1 implies that the intersection Σ_{τ} ∩Hp^{≤n} is finite for all n ≥ 0, since it consists of RM points that are roots of reduced binary quadratic forms of bounded discriminant. Letting

Δ_{τ}^{v}{r, s} := ∑_{w∈Σ_{τ} {r,s}∩A_{v}} δ_{r,s}(w)[w],  for v ∈ T_{0}, r, s ∈ ℙ_{1}(ℚ),

it follows that Δ_{τ}^{v}{0, ∞} has finite support, for all v ∈ T_{0}. The Γ-equivariance property

γΔ_{τ}^{v}{0, ∞} = Δ_{τ}^{γv}{γ0, γ∞}

then implies that Δ_{τ}^{v}{r, s} has finite support for all v ∈ T_{0} and all unimodular pairs (r, s) of elements of ℙ_{1}(ℚ), since the group Γ acts transitively on the latter. But then the same conclusion must hold for all pairs (r, s), by the additivity properties of modular symbols. The discreteness of Σ_{τ} (r, s) follows. To verify the second assertion in the lemma, consider the functions

deg_{A_{v}}, deg_{W_{v}} : Div^{†}(H_{p}) ⟶ ℤ, deg_{A_{v}}(Δ) := deg(Δ|_{A_{v}}), deg_{W_{v}}(Δ) := deg(Δ|_{W_{v}}).

These functions are Γ_{v} := Stab_{Γ}(v)-equivariant and hence induce maps

deg_{A_{v}}, deg_{W_{v}} : MS^{Γ}(Div^{†}(H_{p})) ⟶ MS^{Γ_{v}}(ℤ).

Since Γ_{v} ≃ SL_{2}(ℤ), there are no non-trivial Γ_{v}-invariant modular symbols, and hence, for all v ∈ T_{0},

deg_{A_{v}}(Δ_{τ} {r, s}) = deg(Δ_{τ}^{v}{r, s}) = 0.

The lemma follows since the degree of Δ_{τ}^{v}{r, s} is precisely the difference between the number of positive and negative elements in Σ_{τ} (r, s)∩A_{v}, and likewise when A_{v} is replaced by W_{v}. □

The following lemma is the natural extension of Lemma 1.16 to the setting of rigid meromorphic period functions:

Theorem 1.22. For any τ ∈ Γ\H_{p}^{RM}, the infinite sum

φ_{τ} (z) := ∑_{w∈Σ_{τ}} δ_{∞}(w)\frac{1}{z − w}

converges to a rigid meromorphic period function of weight two.

Proof. The infinite sum in the statement of the theorem is the limit as h ⟶ ∞ of the rational functions

(25)  ∑_{w∈Σ_{τ}^{≤h}} δ_{∞}(w)\frac{1}{z − w} ,  where Στ^{≤h} := Σ_{τ} ∩ Hp^{≤h} .

Assume first for simplicity that τ, and hence all w ∈ Γτ, reduce to vertices of T . By Lemma 1.21, the functions in (25) are sums of terms of the form

\frac{1}{z − w_{1}} − \frac{1}{z − w_{2}} ,

where w_{1} and w_{2} belong to the same vertex affinoid A_{v}, and v ∈ T_{0} is of distance N ≤ h from v_{0}. This term is regular on H_{p}^{≤n} if N > n, and for z ∈ H_{p}^{≤n} it satisfies the inequalities

|\frac{1}{z − w_{1}} | ≤ p^{n},  |\frac{1}{z − w_{2}} | ≤ p^{n},  |\frac{1}{z − w_{1}} − \frac{1}{z − w_{2}} | ≤ p^{2n−N}.

For all n ≥ 0, the restrictions of the rational functions in (25) to H_{p}^{≤n} therefore form a Cauchy sequence relative to the sup norm on this affinoid, and hence converge to a rigid meromorphic function on H_{p}. By the same reasoning, the infinite sums

(26)  Φ_{τ} \{r, s\} := ∑_{w∈Σ_{τ} (r,s)} δ_{r,s}(w)\frac{dz}{z − w}

converge to rigid meromorphic differentials on H_{p}, with residual divisor equal to Δ_{τ} {r, s}. In particular, the function

Φ_{τ} : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ Ω^{1}

is an Ω^{1}-valued modular symbols, just as in the proof of Lemma 1.16. Theorem 1.22 now follows from the fact that

φ_{τ} (z) = Φ_{τ} {0, ∞}.

To handle the case when τ (and hence all w in its Γ-orbit) reduce to midpoints of edges of T rather than to vertices, which happens precisely when τ is defined over a real quadratic field in which p is ramified, it suffices to replace the system of affinoids {A_{v}}_{v∈T_{0}} by the system of wide open subsets {W_{v}}_{v∈T_{0}^{+}} in the above argument. □

Theorem 1.22 provides an explicit collection of rigid meromorphic period functions φ_{τ} of weight two, as τ ranges over the infinite index set

I^{(p)} = Γ\H_{p}^{RM}.

These functions are linearly independent, since their residual divisors have disjoint support. The following result extends Theorem 1.17 to the rigid meromorphic setting:

Theorem 1.23. Any rigid meromorphic period function of weight two is a finite linear combination of the functions φ_{τ} of Theorem 1.22 and of a rigid analytic period function of weight two.

Proof. Let φ be a rigid meromorphic period function of weight two. Any such φ can be written as the average of φ^{+} := φ + ϖ_{p}(φ) and φ^{−} := φ − ϖ_{p}(φ), which are p-even and p-odd respectively. Hence we may assume without loss of generality that φ satisfies (22), and even, for the sake of definiteness, that it is p-even, since the case where it is p-odd will be disposed of by the same argument. Let Σ_{φ} ⊂ H_{p} be the set of poles of φ. While the invariance of φ under the matrix D shows that Σ_{φ} is either empty or infinite, the intersection

Σφ^{<1} := Σ_{φ} ∩ Hp^{<1}

is finite, since a rigid differential on H_{p} has finitely many poles when restricted to any affinoid. Since Hp^{<1} is preserved by the action of SL_{2}(ℤ), the two and three term relations satisfied by φ imply that the set Σφ^{<1} satisfies the closure properties of (20). It follows from Lemma 1.18 that

Σφ^{<1} = ∪_{τ∈I_{φ}}Σ_{τ}^{∘},

where

I_{φ} ⊂ SL_{2}(ℤ)\(H_{p}^{RM} ∩ Hp^{<1} )

is a finite set, and Σ_{τ}^{∘} is defined as in (16), but is now being viewed as a subset of H_{p}. Lemma 1.19, whose proof applies just as well, mutatis mutandis, to the setting where φ is a rigid meromorphic period function, shows that φ has only simple poles on Hp^{<1} , and that for each τ ∈ I_{φ}, there is a λ_{τ} ∈ ℂ_{p} satisfying

res_{w}φ(z)dz = \{ \begin{matrix}λ_{τ} & if w > 0 \\ −λ_{τ} & if w < 0, \end{matrix} for all w ∈ Σ_{τ} .

The difference

φ − ∑_{τ∈I_{φ}} λ_{τ} φ_{τ}^{+}

is a p-even rigid meromorphic period function having no singularities in Hp^{<1} . Theorem 1.23 now follows from Proposition 1.24 below. □

Proposition 1.24. Let φ be any rigid meromorphic period function of weight two. Assume that it satisfies (22), i.e., that it is either p-odd or p-even. If φ is regular on Hp^{<1} , then it is regular everywhere.

Proof. Suppose that φ has a pole at τ ∈ H_{p}, and hence at all w ∈ Σ_{τ}^{∘}. Since τ does not belong to Hp^{<1} , Proposition 1.1 implies that it is an RM point of discriminant D_{0}p^{n} with n ≥ 2 and p ∤ D_{0}. Let

D = \{ \begin{matrix}D_{0} & if n is even, \\ D_{0}p & if n is odd . \end{matrix} m = [n/2] ≥ 1.

The set Σ_{τ}^{∘} then contains an element of the form p^{m}w_{0}, where w_{0} is an RM point in H_{p} of discriminant D. The invariance property (22) of φ shows that φ is singular at w_{0} as well.

But w_{0} belongs to Hp^{<1} by Proposition 1.1, contradicting the regularity assumption that was made on φ. □

Theorem 1.23 classifies the rigid meromorphic period functions only up to rigid analytic period functions of weight two. The latter will be classified in turn in Chapter 2.

The following describes the action of the Hecke operators on the cocycles Φ_{τ}:

Lemma 1.25. If ℓ is a prime not dividing p or the discriminant D of τ ∈ H_{p}^{RM}, then T_{ℓ}(Φ_{τ}) is a linear combination of cocycles of the form Φ_{τ′} where τ′ is of discriminant D or Dℓ^{2}, and involves at least one Φ_{τ′} in which τ′ is of discriminant Dℓ^{2}.

The proof of this lemma is by a direct calculation, and proceeds along the same lines as in [Ge], where the argument is explained in the setting of rational period functions. As in loc. cit., Lemma 1.25 has the following important corollary when combined with Theorem 1.23:

Corollary 1.26. Any finite-dimensional subspace of MS^{Γ}(M_{2}) which is stable under the Hecke operators is contained in MS^{Γ}(O_{2}). In particular, if θ is a non-zero Hecke operator and Φ ∈ MS^{Γ}(M_{2}) belongs to the kernel of θ, then Φ belongs to MS^{Γ}(O_{2}).

For future reference, it is also worth recording the following corollary of the fact that rigid meromorphic period functions of weight two have at worst simple poles:

Corollary 1.27. Any rigid meromorphic modular cocycle of weight zero is analytic, i.e., the natural inclusion MS^{Γ}(O) ⟶ MS^{Γ}(M) is an isomorphism.

Proof. The image of the derivative d : M ⟶ M_{2} consists of rigid meromorphic functions with vanishing residues, and the image of the induced map

d : MS^{Γ}(M) ⟶ MS^{Γ}(M_{2})

on Γ-invariant modular symbols is therefore contained in MS^{Γ}(O_{2}). It follows that any Mvalued Γ-invariant modular symbol is necessarily O-valued, as claimed. □

2. Analytic cocycles of weight two

This chapter completes Theorem 1.23 of Chapter 1 by describing the space of rigid analytic cocycles of weight two. The main result is Theorem 2.12 of Section 2.5, which shows that H_{par}^{1}(Γ, O_{2}) is finite dimensional and closely related to the space of classical modular forms of weight two for the Hecke congruence group Γ_{0}(p). The techniques used to prove this theorem differ markedly from those of the previous chapter, relying heavily on ideas of Stevens and Schneider-Teitelbaum. The reader with a single minded interest in the applications of rigid meromorphic cocycles to explicit class field theory may elect to skip this chapter on a first reading, since none of the material it contains is required to understand the subsequent chapters.

2.1. Rigid analytic functions and boundary distributions. The classification of elements of MS^{Γ}(O_{k}) = H_{par}^{1}(Γ, O_{k}) rests on the fact that O_{k} is isomorphic to a space of locally analytic distributions on the boundary ℙ_{1}(ℚ_{p}) of H_{p}. Assume henceforth that k = 2 for simplicity, although the results described below can certainly be extended to more general positive even weights.

The dual of the space of locally analytic functions on ℙ_{1}(ℚ_{p}), equipped with the strong topology of uniform convergence on compact open subsets, is called the space of locally analytic distributions on ℙ_{1}(ℚ_{p}), and is denoted D(ℙ_{1}(ℚ_{p})). Given μ ∈ D(ℙ_{1}(ℚ_{p})), the notation

μ(h) =: ∫_{ℙ_{1}(ℚ_{p})} h(t)dμ(t)

shall be adopted. More generally, if U is a compact open subset of ℙ_{1}(ℚ_{p}) and 1_{U} is its characteristic function, we define

∫_{U} h(t)dμ(t) := ∫_{ℙ_{1}(ℚ_{p})} 1_{U}(t)h(t)dμ(t).

A distribution μ ∈ D(ℙ_{1}(ℚ_{p})) satisfying μ(1) = 0, where 1 denotes the constant function 1 on ℙ_{1}(ℚ_{p}), is said to be of total volume zero, and the space of such locally analytic distributions is denoted D_{0}(ℙ_{1}(ℚ_{p})).

The group PGL_{2}(ℚ_{p}) acts naturally on D(ℙ_{1}(ℚ_{p})) and on D_{0}(ℙ_{1}(ℚ_{p})) via the weight zero action on locally analytic functions on ℙ_{1}(ℚ_{p}), defined as in (14). More precisely,

∫_{ℙ_{1}(ℚ_{p})} h(t)d(μ|γ)(t) = ∫_{ℙ_{1}(ℚ_{p})} h(γt)dμ(t), where γt := \frac{at + b}{ct + d}.

To any rigid analytic function f ∈ O_{2}, we attach a locally analytic distribution μ_{f} on ℙ^{1}(ℚ_{p}) by setting, for all analytic functions h(t) on a compact open U_{e} ⊂ ℙ_{1}(ℚ_{p}),

∫_{U_{e}} h(t)dμ_{f}(t) := res_{e}(f(z)h(z)dz).

Here, res_{e} denotes the p-adic annular residue along the oriented annulus W_{e}. The distribution μ_{f} is called the boundary distribution attached to f. It is a direct consequence of the residue theorem that μ_{f} belongs to D_{0}(ℙ_{1}(ℚ_{p})).

Proposition 2.1. The map f ↦ μ_{f} induces a topological isomorphism

BD : O_{2} ⟶^{∼} D_{0}(ℙ_{1}(ℚ_{p})),

which is compatible with the PGL_{2}(ℚ_{p})-actions on both sides.

Proof. Setting k = 2 in the statement of Theorem 2.2.1 of [DT], the dual of the map denoted I_{2} in loc.cit. induces an isomorphism

D_{0}(ℙ_{1}(ℚ_{p})) ⟶ O_{2},

in light of the fact that O_{2} is a reflexive Frechet space and hence is identified with its double dual. The “boundary distribution map” BD is just the inverse of this isomorphism. □

The map BD induces an isomorphism on the parabolic cohomology groups, denoted by the same symbol by a slight abuse of notation:

(27)  BD : MS^{Γ}(O_{2}) ⟶^{∼} MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))).

This reduces the problem of understanding MS^{Γ}(O_{2}) to that of classifying the Γ-invariant modular symbols with values in D_{0}(ℙ_{1}(ℚ_{p})). An element μ of the latter is simply a collection of distributions μ{r, s} on ℙ_{1}(ℚ_{p}), indexed by elements r, s ∈ ℙ_{1}(ℚ), which satisfy the usual modular symbol relations

μ{r, s} = −μ{s, r}, μ{r, s} + μ{s, t} = μ{r, t},

together with the equivariance property

(28)  ∫_{γB} h(t)dμ{r, s}(t) = ∫_{B} h(γt)dμ{γ^{−1}r, γ^{−1}s}(t),  for all γ ∈ Γ.

If μ is an eigensymbol for the involutions w_{∞} and w_{p} defined in (8), then the invariance property of (28) even holds for all γ ∈ PGL_{2}(ℤ[1/p]), up to a sign which depends on the determinant of γ and on the parity and the p-parity of μ:

(29) ∫_{γB} h(t)dμ{r, s}(t) = ± ∫_{B} h(γt)dμ{γ^{−1}r, γ^{−1}s}(t),  for all γ ∈ PGL_{2}(ℤ[1/p]).

2.2. Restriction to ℤ_{p}. The compact open subset ℤ_{p} ⊂ ℙ_{1}(ℚ_{p}) is a ball whose stabiliser in Γ is the usual congruence group Γ_{0}(p). The restriction map D_{0}(ℙ_{1}(ℚ_{p})) ⟶ D(ℤ_{p}) to the space of distributions on ℤ_{p} therefore induces a map on modular symbols:

res_{ℤ_{p}} : MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))) ⟶ MS^{Γ_{0}(p)}(D(ℤ_{p})).

The target of this map is called the space of overconvergent modular symbols of weight two and level p.

Lemma 2.2. The map res_{ℤ_{p}} is injective.

Proof. The matrix ι_{p} := (\begin{matrix}0 & −1 \\ p & 0 \end{matrix}) interchanges ℤ_{p} and its complement ℤ′_{p} := ℙ_{1}(ℚ_{p}) − ℤ_{p},

and normalises Γ_{0}(p). It therefore induces mutually inverse isomorphisms

MS^{Γ_{0}(p)}(D(ℤ_{p})) ⟷^{ι_{p}} MS^{Γ_{0}(p)}(D(ℤ′_{p}))

for which the diagram

\xymatrix{MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))) \ar[r]^{w_{p}} \ar[d]_{_{res_{ℤp}}} & MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))) \ar[d]_{_{res_{ℤ′p}}} \\ MS^{Γ_{0}(p)}(D(ℤ_{p})) \ar[r]^{ι_{p}} & MS^{Γ_{0}(p)}(D(ℤ′_{p}))}

commutes. In particular, the involution w_{p} interchanges the kernels of res_{ℤ_{p}} and of res_{ℤ′_{p}}, and it suffices to show that res_{ℤ′_{p}} is injective. If μ is in the kernel of res_{ℤ′_{p}}, then

(30)  μ{r, s}|_{ℤ′_{p}} = 0, for all r, s ∈ ℙ_{1}(ℚ).

The domain ℙ_{1}(ℚ_{p}) admits a decomposition as a disjoint union of p + 1 open balls,

(31)  ℙ_{1}(ℚ_{p}) = B_{0} ⊔ B_{1} ⊔ ⋯ ⊔ B_{p−1} ⊔ ℤ′_{p},

where B_{j} ⊂ ℤ_{p} is the mod p residue disc of −j. The group SL_{2}(ℤ) acts transitively on the collection {B_{0}, B_{1}, …, ℤ′_{p}}. Let γ_{j} ∈ SL_{2}(ℤ) be a matrix satisfying ℤ′_{p} = γ_{j}B_{j}. Then for all j = 0, …, p − 1, and for all r, s ∈ ℙ_{1}(ℚ),

μ{r, s}|_{B_{j}} = μ{r, s}|_{γ_{j}^{−1}ℤ′_{p}} = (μ{γ_{j}r, γ_{j}s}|_{ℤ′_{p}})|γ_{j} = 0,

where the last equality follows from (30). It now follows from (31) that μ{r, s} = 0 as a distribution on ℙ_{1}(ℚ_{p}), for all r, s ∈ ℙ_{1}(ℚ). The lemma follows. □

The space of overconvergent modular symbols is equipped with a Hecke operator U_{p}, defined explicitly by

(32) ∫_{ℤ_{p}} h(t)d(U_{p}μ)\{r, s\}(t) := ∑_{j=0}^{p−1} ∫_{ℤ_{p}} h(α_{j}^{−1}t)dμ\{α_{j}r, α_{j}s\}(t),  where α_{j} = (\begin{matrix}1 & j \\ 0 & p \end{matrix}) .

The space MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p})) decomposes as a direct sum

MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p})) = MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))^{+} ⊕ MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))^{−},

where MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))^{ϵ} denotes, for ϵ ∈ {+, −}, the ϵ-eigenspace for the action of the involution w_{p}. Let MS^{Γ_{0}(p)}(D(ℤ_{p}))^{U_{p}=ϵ} denote the space of overconvergent modular symbols on which U_{p} acts as multiplication by ϵ.

Proposition 2.3. The map res_{ℤ_{p}} induces Hecke-equivariant inclusions

res_{ℤ_{p}} : MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))^{ϵ} ↪ MS^{Γ_{0}(p)}(D(ℤ_{p}))^{U_{p}=ϵ}.

Proof. For j = 0, 1, …, p − 1, let B_{j} ⊂ ℤ_{p} denote, as in the proof of Lemma 2.2, the residue class of −j modulo p, so that

ℤ_{p} = B_{0} ⊔ B_{1} ⊔ ⋯ ⊔ B_{p−1}, α_{j}B_{j} = ℤ_{p},

with α_{j} as in (32). By the additivity of the distribution μ{r, s} ∈ D(ℤ_{p}), we have, for any locally analytic function h on ℤ_{p}:

∫_{ℤ_{p}} h(t)dμ{r, s}(t) = ∑_{j=0}^{p−1} ∫_{B_{j}} h(t)dμ{r, s}(t) = ∑_{j=0}^{p−1} ∫_{α_{j}^{−1}ℤ_{p}} h(t)dμ{r, s}(t)

= ϵ ∑_{j=0}^{p−1} ∫_{ℤ_{p}} h(α_{j}^{−1}t)dμ{α_{j}r, α_{j}s}(t),

where the last equality follow from (29) in light of the fact that the matrices α_{j} ∈ PGL_{2}(ℤ[1/p]) have determinant p. The proposition now follows from the definition of the U_{p} operator given in (32). □

Proposition 2.3 shows that U_{p} preserves the image of res_{ℤ_{p}} and that the minimal polynomial of its restriction to this space divides x^{2} − 1. Composing the map BD of (27) with the restriction map res_{ℤ_{p}} thus gives an injection

(33)  MS^{Γ}(O_{2}) ↪ MS^{Γ_{0}(p)}(D(ℤ_{p}))^{U_{p}^{2}=1}.

2.3. Stevens’ control theorem. The “total measure” map D(ℤ_{p}) ⟶ ℂ_{p} which sends μ to μ(1) induces a “weight two specialisation map”

(34)  ρ : MS^{Γ_{0}(p)}(D(ℤ_{p})) ⟶ MS^{Γ_{0}(p)}(ℂ_{p}),

which is compatible with the actions of the Hecke operators on both sides.

Theorem 2.4 (Stevens). The weight two specialisation map ρ induces an isomorphism

(35)  ρ : MS^{Γ_{0}(p)}(D(ℤ_{p}))^{U_{p}^{2}=1} ⟶ MS^{Γ_{0}(p)}(ℂ_{p}).

Proof. The ordinary subspace of a Hecke module M is the direct summand of it on which the U_{p} operator acts with slope zero, and is denoted M^{ord}. The control theorem for overconvergent modular symbols (cf. the case k = 0 of Theorem 1.1 of [PS]) asserts that ρ induces an isomorphism

ρ^{ord} : MS^{Γ_{0}(p)}(D(ℤ_{p}))^{ord} ⟶ MS^{Γ_{0}(p)}(ℂ_{p})^{ord}.

But MS^{Γ_{0}(p)}(ℂ_{p}) is isomorphic as a Hecke module to the direct sum of an “Eisenstein line” with two copies of the space of modular forms of weight two on Γ_{0}(p). Since all such modular forms are new at p, it follows that U_{p}^{2} acts as the identity on this space, and that

MS^{Γ_{0}(p)}(ℂ_{p})^{ord} = MS^{Γ_{0}(p)}(ℂ_{p})^{U_{p}^{2}=1} = MS^{Γ_{0}(p)}(ℂ_{p}).

The theorem follows. □

Corollary 2.5. The map

η := ρ ∘ res_{ℤ_{p}} ∘ BD : MS^{Γ}(O_{2}) ↪ MS^{Γ_{0}(p)}(ℂ_{p})

is injective. Proof. This follows from Propositions 2.1 and 2.3 combined with Theorem 2.4. □

Our goal in the remainder of this chapter is to show that the map η is surjective as well.

2.4. The residue map. Let Ω be a Γ-module. Definition 2.6. A function c : T_{1}^{∗} ⟶ Ω is said to be harmonic if it satisfies

c(ē) = −c(e), for all e ∈ T_{1}^{∗}, and ∑_{s(e)=v} c(e) = 0, for all v ∈ T_{0}.

The space of harmonic functions on T_{1}^{∗} with values in Ω is denoted C_{har}(Ω). The action of Γ on T induces a natural right action of Γ on the space C_{har}(Ω). In what follows we will be primarily interested in the case where Ω = ℤ or a bounded subgroup of ℂ_{p}, equipped with the trivial action of Γ. Remark 2.7. Elsewhere in the literature (e.g., in [Te]) it is cusomary to refer to harmonic functions as harmonic cocycles. Since the noun “cocycle” already being used in its more standard form in this article, a more transparent terminology was chosen for Definition 2.6.

One can associate to any f ∈ O_{2} a harmonic function c_{f} ∈ C_{har}(ℂ_{p}) by the rule

c_{f}(e) = ∂_{e}f(z)dz, for e ∈ T_{1}^{∗},

where

∂_{e} : O(W_{e}) ⟶ ℂ_{p}

is the p-adic annular residue on the space of rigid differentials on the oriented annulus W_{e}. The PGL_{2}(ℚ_{p})-equivariant map

∂ : O_{2} ⟶ C_{har}(ℂ_{p})

sending f to c_{f} is called the residue map. The same notation and terminology is used to describe the induced map

(36)  ∂ : MS^{Γ}(O_{2}) ⟶ MS^{Γ}(C_{har}(ℂ_{p}))

on modular symbols. Let e_{0} denote the standard edge of T^{∗}, whose stabiliser is Γ_{0}(p) and whose associated open ball in ℙ_{1}(ℚ_{p}) is ℤ_{p}. The evaluation at e_{0}

ev_{e_{0}} : C_{har}(ℂ_{p}) ⟶ ℂ_{p}

is Γ_{0}(p)-equivariant, and hence induces a map

(37)  ev_{e_{0}} : MS^{Γ}(C_{har}(ℂ_{p})) ⟶ MS^{Γ_{0}(p)}(ℂ_{p}).

Our strategy to show the surjectivity of η will be to prove the surjectivity of the maps ev_{e_{0}} and ∂ that fit into the following commutative diagram:

(38)  [figure: MS^{Γ}(O_{2}); ∂; BD; η; MS^{Γ}(C_{har}(ℂ_{p})); _{ev_{e_{0}}}; MS^{Γ}(D_{0}(ℙ_{1}(ℚ_{p}))); _{res_{ℤp}}; MS^{Γ_{0}(p)}(D(ℤ_{p})); ρ; MS^{Γ_{0}(p)}(ℂ_{p}).]