Gross–Stark units, Stark–Heegner points, and class fields of real

quadratic fields

by Samit Dasgupta

A.B. (Harvard University) 1999

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in Mathematics

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge: Professor Kenneth Ribet, Chair

Professor Bjorn Poonen Professor David Wagner

Spring 2004 The dissertation of Samit Dasgupta is approved:

Chair Date \overline{Date} \overline{Date}

University of California, Berkeley Spring 2004 Gross–Stark units, Stark–Heegner points, and class fields of real

quadratic fields

Copyright 2004 by Samit Dasgupta

Abstract

Gross–Stark units, Stark–Heegner points, and class fields of real quadratic fields

by

Samit Dasgupta Doctor of Philosophy in Mathematics

University of California, Berkeley

Professor Kenneth Ribet, Chair

We present two generalizations of Darmon’s construction of Stark–Heegner points on elliptic curves defined over Q. First, we provide a lifting of Stark–Heegner points from elliptic curves to certain modular Jacobians which parameterize them. This construction involves a generalization of a theorem of Greenberg and Stevens that proves the Mazur–Tate–Teitelbaum conjecture.

Next, we replace the modular symbols attached to an elliptic curve with those

attached to a modular unit α of level N > 1. For a real quadratic field K in which the rational prime p is inert, this allows the definition of certain numbers u(α, τ) ∈ K_{p}^{×} attached to α and τ ∈ K − Q. The elements u(α, τ) are analogous to classical elliptic units arising from α. In this vein, we conjecture that the elements u(α, τ) belong to specific abelian extensions of K. Although this conjecture is still open, we prove a formula relating the p-adic valuation and p-adic logarithm of u(α, τ) to the leading terms at s = 0 of certain partial zeta functions (classical and p-adic, respectively). The existence of units satisfying these properties is equivalent to the p-adic Gross–Stark conjecture; thus our construction gives an analytic construction of Gross’s units, minus a proof of their algebraicity.

Professor Kenneth Ribet

Dissertation Committee Chair

i

To Baba

Contents

List of Tables iv

I Stark–Heegner Points on Modular Jacobians 1

1 Introduction 3

2 Previous uniformization results 6 2.1 Archimedean uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Generalities on p-adic measures . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Mumford curves and Schottky groups . . . . . . . . . . . . . . . . . . . . . 13

3 Properties of the multiplicative double integral 17 3.1 Behavior of the integral under conjugation . . . . . . . . . . . . . . . . . . . 17 3.2 An exact sequence in homology . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 The valuation of the integration map . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Choosing a system of representatives . . . . . . . . . . . . . . . . . . . . . . 22

4 An arithmetic uniformization and Stark–Heegner points 23 4.1 The universal modular symbol . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Statement of the uniformization . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Stark–Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Hecke actions 32 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Hecke equivariance of the integration map . . . . . . . . . . . . . . . . . . . 36 5.3 The homology of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4 The lattice L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Proof of the uniformization 47 6.1 Connection with the Mazur–Tate–Teitelbaum conjecture . . . . . . . . . . . 50 6.2 Hida families and the definition of ℒ_{p} . . . . . . . . . . . . . . . . . . . . . 52 6.3 Proof that ℒ_{p} is the ℒ -invariant of T/L . . . . . . . . . . . . . . . . . . . . 54 6.4 Proof that ℒ_{p} is the ℒ -invariant of J . . . . . . . . . . . . . . . . . . . . . 63

II Gross–Stark units 67

7 Introduction 69

8 Definition of the units 71 8.1 Eisenstein “minus” modular symbols . . . . . . . . . . . . . . . . . . . . . . 72 8.2 The unit u(α, τ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3 The action of complex conjugation and of U_{p} . . . . . . . . . . . . . . . . . 75

9 Special values of zeta functions 77 9.1 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.2 Dedekind sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.3 The p-adic valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 9.4 Values at negative integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 9.5 Connection with the Gross–Stark conjecture . . . . . . . . . . . . . . . . . . 83

10 A Kronecker limit formula 85 10.1 Measures associated to Eisenstein series . . . . . . . . . . . . . . . . . . . . 85 10.2 Construction of the p-adic L-function . . . . . . . . . . . . . . . . . . . . . . 87 10.3 An explicit splitting of the one-cocycle . . . . . . . . . . . . . . . . . . . . . 88 10.4 Generalized Dedekind sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 10.5 Measures on Z_{p} × Z_{p} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 10.6 A partial modular symbol of measures on Z_{p} × Z_{p} . . . . . . . . . . . . . . 94 10.7 From Z_{p} × Z_{p} to X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.8 The measures μ̃ and Γ-invariance . . . . . . . . . . . . . . . . . . . . . . . . 99 10.9 Integrality of the measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

11 Computations 104 11.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 11.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Bibliography 113

List of Tables

11.1 Characteristic Polynomial of u(α, τ) for p = 3 . . . . . . . . . . . . . . . . . 109 11.2 Characteristic Polynomial of u(α, τ) for p = 5 . . . . . . . . . . . . . . . . . 110 11.3 Characteristic Polynomial of u(α, τ) for p = 7 . . . . . . . . . . . . . . . . . 111 11.4 Characteristic Polynomial of u(α, τ) for p = 11 . . . . . . . . . . . . . . . . 112

Acknowledgments

It is an honor to thank Prof. Henri Darmon of McGill University for the guidance he provided me over the course of my graduate career. I first met Darmon at a conference during the summer of 2001 and became fascinated with his work; our ensuing e-mail correspondence grew into fruitful collaboration. Darmon went beyond my greatest expectations in guiding my mathematical development. I am continually amazed by the energy and passion that he brings to all his endeavors, and I have benefited greatly from my interaction with him. He is an unbelievable mentor, colleague, and friend. I am also truly indebted to my adviser Prof. Kenneth Ribet of University of California-Berkeley, who provided an extraordinary amount of support for me during my graduate school experience. Ken’s infallible ability to lead me in the right direction during moments of confusion—mathematical and otherwise— is genuinely remarkable.

Learning during graduate school is necessarily and fortunately a collaborative

project, undertaken by those with a common esoteric passion. I am grateful for the companionship and friendship of John Voight and Ronald van Luijk during our mutual struggles to forge a clearing through the forested wilderness of number theory.

The last four years have also witnessed dramatic social changes in my life and my

community, and I am blessed to have had the company and support of an amazing circle of friends and family. My mother has always served as a model of strength, love, and tactful jocundity. My brother and Priya have demonstrated by example the virtues of resilience, passion, and humor. My father’s values, open-heartedness, and optimism continue to serve as an inspiration to all of us. In Berkeley and beyond, I have been sustained during the last four years by the warmth and friendship of Daniel Biss, Kevin Amer, Nirit Sandman, Danielle Gordon, Saurabh Bhargava, and Grace Lyo.

I would like to acknowledge William Stein for granting me access to the MECCAH

cluster at Harvard University, on which the MAGMA programs of Chapter 11 were executed. Finally, I am indebted to Profs. Bjorn Poonen, Kenneth Ribet, and Henri Darmon for many detailed suggestions which greatly improved the quality of this thesis. vi

Part I

Stark–Heegner Points on Modular Jacobians

Chapter 1

Introduction

The theory of complex multiplication allows the construction of a collection of

points on arithmetic curves over Q, defined over abelian extensions of quadratic imaginary fields. Foremost among these are Heegner points on modular curves, as described for example in [15]. By embedding a modular curve in its Jacobian (typically by sending a rational cusp to the origin), one may transfer Heegner points on the curve to each factor of its Jacobian. A study of the arithmetic properties of the points constructed in this fashion has yielded many striking results, most notably, the theorems of Gross–Zagier [18], Kolyvagin [25], and Kolyvagin–Logachëv [26].

The goal of [5] was to define certain points on elliptic curves analogous to Heegner

points, except that they would be defined over abelian extensions of real quadratic fields instead of imaginary quadratic fields. In the setting considered, the existence of such points is predicted by the conjecture of Birch and Swinnerton-Dyer. Darmon constructs these “Stark–Heegner points” analytically by replacing complex integration with a certain p-adic integral. The conjecture that Stark–Heegner points are defined over global number fields remains open.

The goal of part I of this thesis is to lift the construction of Stark–Heegner points

from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an elliptic curve E of conductor Np (a key tool in [5]) with the universal modular symbol for Γ_{0}(Np). We then construct a certain torus T over Q_{p} and sub-lattice L of T, and prove that the quotient T/L is isogenous to the maximal toric quotient J_{0}(Np)^{p-new} of the Jacobian of X_{0}(Np). This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum [35] on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens [12], [13]. Indeed, our proof borrows greatly from theirs.

Our isogeny theorem allows us to define Stark–Heegner points on the abelian

variety J_{0}(Np)^{p-new}. The points we define map to the Stark–Heegner points on E under the projection J_{0}(Np)^{p-new} → E. We conjecture that they satisfy the same algebraicity properties. One interesting difference from the case of classical Heegner points is that our points, while lying on modular Jacobians, do not appear to arise from points on the modular curves themselves.

Although the construction of Stark–Heegner points is the most significant arith-

metic application of our isogeny theorem, the result is interesting in its own right because it allows the practical computation of the p-adic periods of J_{0}(Np)^{p-new}.

In Chapter 2 we summarize known uniformization results, beginning with the

complex analytic construction of J_{0}(N) and classical Heegner points. We then discuss p-adic uniformization of Mumford curves via Schottky groups, and present the Manin–Drinfield theorem on the uniformization of the Jacobian of a Mumford curve in the language of p-adic integration. In Chapter 4 we construct our analytic space T/L and state the isogeny theorem. We then use the isogeny theorem to define Stark–Heegner points on J_{0}(Np)^{p-new}. The remainder of part I is devoted to proving the isogeny theorem. Section 6.1 describes precisely how our result generalizes the Mazur–Tate–Teitelbaum conjecture.

There are some differences to note between our presentation and that of [12]. First,

by dealing with the entire Jacobian rather than a component associated to a particular newform, we avoid some technicalities arising in Hida theory. Furthermore, the role of −2a′_{p}(2) in [12] is played by

ℒ_{p} := the “derivative” of 1 − U_{p}^{2},

as defined in Section 6.2; accordingly we treat the cases of split and non-split reduction simultaneously. The proof that the ℒ -invariant of T/L is equal to ℒ_{p} is somewhat different from (though certainly bears commonalities with) what appears in [12]. Indeed, the space T/L is constructed from the group

Γ := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ PSL_{2}(Z[1/p]) such that N|c\}

and a study of its cohomology. The construction of Stark–Heegner points is contingent on the splitting of a certain 2-cocycle for Γ, which is proven by lifting measures on P^{1}(Q_{p}) to the Z_{p}^{×}-bundle X = (Z_{p} ×Z_{p})′ of primitive vectors over this space. The connection between integrals on X and p-adic L-functions is described in [2], for which the calculations of part II of this thesis (in particular, Section 10.2) served as the motivation.

Chapter 2

Previous uniformization results

The classical theory of Abel–Jacobi gives a complex analytic uniformization of the

Jacobian of a nonsingular proper curve over C. We begin this chapter by recalling this construction for X_{0}(N) and giving the definition of Heegner points on J_{0}(N) using this uniformization. Manin and Drinfield have also given a p-adic uniformization for the Jacobians of Mumford curves. We give a restatement of their result in the language of p-adic integration, which may thus be viewed as a p-adic Abel–Jacobi theory. Unfortunately, the p-adic uniformization of J_{0}(p) that arises in this fashion does not allow the natural construction of Heegner-type points in an obvious manner. By applying the general integration theory in a more arithmetic setting, we find an alternate p-adic uniformization of J_{0}(Np)^{p-new} and use it to construct Stark–Heegner points on this modular Jacobian.

2.1 Archimedean uniformization

The Abel–Jacobi theorem states that the Jacobian of a nonsingular proper curve

X over C is analytically isomorphic to the quotient of the dual of its space of 1-forms by the image of the natural integration map from H_{1}(X(C), Z). To execute this uniformization in practice, one often wants to understand the space of 1-forms and the first homology group of X explicitly. A general approach to this problem is given by Schottky uniformization (see [38] for the original work and [22] for a modern summary and generalization). The “retrosection” theorem of [24] states that there exists a Schottky group Γ ⊂ PGL_{2}(C) and an open set H_{Γ} ⊂ P^{1}(C) such that X(C) is analytically isomorphic to H_{Γ}/Γ. Among its other properties, the group Γ is free of rank g, the genus of the curve X. Under certain convergence conditions, one may describe the Jacobian of X as the quotient of a split torus (C^{×})^{g} by the image of an explicit homomorphism from Γ.

While Schottky uniformization is useful as a general theory, it does not necessarily

provide a method of constructing rational points on X or its Jacobian in cases of arithmetic interest. Furthermore, if the parameterizing group Γ cannot be found explicitly, one may not even be able to calculate the periods of X in practice.

In our case of study, namely the modular curves X_{0}(N), it is essential to exploit the

“arithmeticity” given by modularity. By its moduli description, the set of complex points of X_{0}(N) can be identified with the quotient of the extended upper half plane H^{∗} = H∪P^{1}(Q) by the discrete group Γ_{0}(N) acting on the left via linear fractional transformations:

X_{0}(N)(C) ≅ H^{∗}/Γ_{0}(N).  (2.1)

Let X_{0}(N) have genus g, and let S_{2}(N) denote the space of cusp forms of level N. For any τ_{1}, τ_{2} ∈ H^{∗}, we can define a homomorphism denoted ∫_{τ_{1}}^{τ_{2}} from S_{2}(N) to C via a complex line integral:

∫_{τ_{1}}^{τ_{2}} : f ↦ 2πi ∫_{τ_{1}}^{τ_{2}} f(z) dz.

Since f is a modular form of level N, this value is unchanged if τ_{1} and τ_{2} are replaced by γτ_{1} and γτ_{2}, respectively, for γ ∈ Γ_{0}(N). Thus if Div_{0} H^{∗} denotes the group of degree-zero divisors on the points of the extended upper half plane, we obtain a homomorphism

(Div_{0} H^{∗})_{Γ_{0}(N)} → Hom(S_{2}(N), C)  (2.2)
[τ_{1}] − [τ_{2}] ↦ (f ↦ 2πi ∫_{τ_{1}}^{τ_{2}} f(z) dz) .

The short exact sequence

0 → Div_{0} H^{∗} → Div H^{∗} → Z → 0

gives rise to a boundary map in homology:

δ : H_{1}(Γ_{0}(N), Z) → (Div_{0} H^{∗})_{Γ_{0}(N)}.  (2.3)

Denote the composition of the maps in (2.2) and (2.3) by

Φ_{1} : H_{1}(Γ_{0}(N), Z) → Hom(S_{2}(N), C),

and let L denote the image of Φ_{1}. The group L is free abelian of rank 2g and is Heckestable. For x ∈ H^{∗}, let x̃ represent the image of x in X_{0}(N)(C) = H^{∗}/Γ_{0}(N). Under these notations, the Abel–Jacobi theorem may be stated as follows:

Theorem 2.1.1. The map [x̃] − [ỹ] ↦ ∫_{y}^{x} induces a complex analytic uniformization of the Jacobian of X_{0}(N):

J_{0}(N)(C) ≅ Hom(S_{2}(N), C)/L.

Let τ ∈ H^{∗} lie in an imaginary quadratic subfield K of C. Then

P_{τ} := ∫_{∞}^{τ} ∈ Hom(S_{2}(N), C)/L = J_{0}(N)(C)

is a Heegner point on J_{0}(N). The theory of complex multiplication shows that this analytically defined point is actually defined over an abelian extension of K, and it furthermore prescribes the action of the Galois group of K on this point.

The goal of the remainder of this chapter is to present the theory of p-adic uni-

formization of Jacobians of degenerating curves via Schottky groups, as studied by Tate, Mumford, Manin, and Drinfeld, in the language of p-adic integration. The standard presentation of this subject (see [11], for example) involves certain theta functions that often have no direct analogue in the complex analytic situation because of convergence issues. Thus our new notation, inspired by [1] (the ideas we have drawn from in [1] appear in the construction of certain p-adic L-functions), allows one to draw a more direct parallel between the complex analytic and p-adic settings.

2.2 Generalities on p-adic measures

To describe a p-adic analogue of the theory described in section 2.1, we first define

the p-adic integrals that will play the role of complex line integrals. Let K be a local field (a locally compact field, complete with respect to a discrete valuation). Let Γ be a subgroup of PGL_{2}(K), which acts on P = P_{K}^{1} by linear fractional transformations. Denote by C the completion of an algebraic closure of K.

Definition 2.2.1. The functor of limit points L_{Γ} associates to each complete field extension F of K contained in C the set of p ∈ P(F) such that there exists q ∈ P(F) and distinct γ_{n} ∈ Γ with lim γ_{n}q = p. The group Γ is said to be discontinuous if L_{Γ} ≠ P.

Since K is a local field, Γ is discontinuous if and only if it is discrete as a subgroup

of PGL_{2}(K) and in this case L_{Γ}(C) ⊂ P(K) ([11, 1.6.4]). We will not in general be assuming that Γ is discontinuous. Let L_{Γ}(F) have the induced topology from F. The space L_{Γ}(F) is a closed subset of P(F), and hence compact if F is a local field.

Definition 2.2.2. Let H be a free abelian group of finite rank (endowed with the trivial Γ-action). The group Meas(L_{Γ}(F), H) of additive measures on L_{Γ}(F) with values in H is the group of maps μ that assign to each compact open subset U of L_{Γ}(F) an element μ(U) of H, such that

• μ(U) + μ(V) = μ(U ∪ V) for disjoint open compacts U and V, and

• μ(L_{Γ}(F)) = 0.

The group Meas(L_{Γ}(F), H) has a natural Γ action, given by (γμ)(U) := μ(γ^{−1}U)

(here and throughout, we will be viewing Γ as an ordinary group, rather than a topological group, when considering Γ-modules and cohomology). Let M be a Γ-module, and suppose we have a Γ-invariant homomorphism from M to Meas(L_{Γ}(K), H):

μ ∈ Hom(M, Meas(L_{Γ}(K), H))^{Γ}.

Denote the image of m under μ by μ_{m}.

Definition 2.2.3. Let d ∈ Div_{0} H_{Γ}(C) be a degree-zero divisor, and let m ∈ M. Choose a rational function f_{d} on P^{1}(C) with divisor d, and define the multiplicative double integral:

× ∫_{d} ∫_{m}ω_{μ} := × ∫_{L_{Γ}(K)} f_{d}(t) dμ_{m}(t)
:= lim_{||U||→0} ∏_{U∈U} f_{d}(t_{U}) ⊗ μ_{m}(U) ∈ C^{×} ⊗_{Z} H.  (2.4)

Here the limit is taken over uniformly finer disjoint covers U of L_{Γ}(K) by nonempty open compact subsets U, and t_{U} is an arbitrarily chosen point of U.

Remark 2.2.4. The products in (2.4) are finite since L_{Γ}(K) is compact. The limit converges since μ_{m} is a measure.

Also, since μ_{m}(L_{Γ}(K)) = 0, the multiplicative integral on L_{Γ}(K) of a constant

(with respect to μ_{m}) vanishes, so Definition 2.2.3 is independent of the choice of f_{d}.

For a complete field extension F of K lying in C, denote by H_{Γ}(F) the space

P^{1}(F) − L_{Γ}(K). It is clear that if d ∈ Div_{0} H_{Γ}(F), then

× ∫_{d} ∫_{m}ω_{μ} ∈ F^{×} ⊗ H.

Proposition 2.2.5. The multiplicative double integral is Γ-invariant:

× ∫_{d} ∫_{m}ω_{μ} = × ∫_{γd} ∫_{γm}ω_{μ} ∈ C^{×} ⊗ H

for d ∈ H_{Γ}(C), m ∈ M, and γ ∈ Γ.

Proof. From the Γ-invariance of μ, we have

× ∫_{γd} ∫_{γm}ω_{μ} = × ∫_{L_{Γ}(K)} f_{γd}(t) dμ_{γm}(t) = × ∫_{L_{Γ}(K)} f_{γd}(γu) dμ_{m}(u),  (2.5)

where u = γ^{−1}t. Since we may choose f_{γd}(γu) = f_{d}(u), the result follows.

Thus the multiplicative double integral defines a map

× ∫∫ ω_{μ} : ((Div_{0} H_{Γ}) ⊗ M)_{Γ} → G_{m} ⊗ H  (2.6)

Here we view H_{Γ} and G_{m} as functors on the category of complete field extensions of K contained in C.

Remark 2.2.6. If τ_{1}, τ_{2} ∈ H_{Γ}(C) and τ_{i} ≠ ∞, we write

× ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{μ} = × ∫_{[τ_{2}]−[τ_{1}]} ∫_{m}ω_{μ} = × ∫_{L_{Γ}(K)} (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{m}(t)

as in [5].

As we now describe, for each Γ-module M there is a universal group H admitting

a Γ-invariant homomorphism μ : M → Meas(L_{Γ}(K), H), in the sense that for any other group H′ admitting a Γ-invariant homomorphism μ′ : M → Meas(L_{Γ}(K), H′), there exists a unique homomorphism f_{μ′} : H → H′ such that μ′_{m}(U) = f_{μ′}(μ_{m}(U)).

We first recall the Bruhat–Tits tree T of PGL_{2}(K). Denote by O the ring of

integers of K, by π a uniformizer of O, and by k = O/πO the residue field. The vertices of T are equivalence classes of free rank-two O-submodules of K ⊕K, where two such modules are considered equivalent if they are homothetic by an element of K^{×}. Two vertices are connected by an edge if they can be represented by modules M and N with N ⊂ M and M/N ≅ k; this is clearly a symmetric relation. The unoriented graph T that results from these definitions is a regular tree of degree #P^{1}(k). The group PGL_{2}(K) acts naturally on the tree.

Let v^{∗} denote the vertex corresponding to O ⊕ O and let w^{∗} denote the vertex

corresponding to O ⊕ πO. The stabilizer of v^{∗} in PGL_{2}(K) is PGL_{2}(O). The matrix P = (\begin{matrix}π & 0 \\ 0 & 1 \end{matrix}) sends w^{∗} to v^{∗}, and hence the stabilizer of w^{∗} is P^{−1}PGL_{2}(O)P. Let e^{∗}

denote the oriented edge from w^{∗} to v^{∗}. The stabilizer of e^{∗} in PGL_{2}(K) is the intersection of the stabilizers of v^{∗} and w^{∗}, namely, the set of matrices of PGL_{2}(O) that are upper triangular modulo π. This group equals the stabilizer of O in PGL_{2}(K) under linear fractional transformations. Thus if we associate to the oriented edge e^{∗} the compact open set U_{e^{∗}} := O ⊂ P^{1}(K), this extends to an assignment of a compact open subset of P^{1}(K) to each oriented edge of the tree via PGL_{2}(K)-invariance:

U_{γe^{∗}} := γO for all γ ∈ PGL_{2}(K).

We note some essential properties of this assignment:

• For an oriented edge e, the oppositely oriented edge ē satisfies Uē = P^{1}(K) − U_{e}.

• For each vertex v, the sets U_{e} as e ranges over the edges emanating from v form a disjoint cover of P^{1}(K).

• The sets U_{e} form a basis of compact open subsets of P^{1}(K).

Definition 2.2.7. The Bruhat–Tits tree of Γ is the subtree T_{Γ} ⊂ T spanned by all edges such that both open sets corresponding to the two possible orientations of the edge contain an element of L_{Γ}(K).

The group Γ acts on the tree T_{Γ}. To each oriented edge e of T_{Γ} we associate the

compact open set U_{e}(Γ) = U_{e} ∩ L(K). The sets U_{e}(Γ) satisfy the properties above with P^{1}(K) replaced by L_{Γ}(K).

An end of a tree is a path without backtracking that is infinite in exactly one

direction, modulo the relation that two such paths are equivalent if they are eventually equal^{1}. The ends of T_{Γ} are naturally in bijection with L_{Γ}(K), by sending an end to the unique point in the intersection of all U_{e}(Γ) for the oriented edges e of the end.

The space H_{Γ} may be viewed as a thickening of the tree T_{Γ} by means of the

reduction map

red : H_{Γ} → T_{Γ}.

1Rigorously, an end is an infinite sequence v_{0}, v_{1}, … of distinct vertices of the tree such that (v_{i}, v_{i+1})

is an oriented edge, modulo the relation that {v_{i}} ∼ {w_{i}} if there exist n, m such that v_{n+i} = w_{m+i} for all i ≥ 0.

We will define the reduction map only on the points of H_{Γ} defined over finite unramified extensions F of K. In this case, the tree T of PGL_{2}(K) is naturally a subtree of the Bruhat–Tits tree T_{F} of PGL_{2}(F), and hence the tree T_{Γ} may be viewed as a subtree of T_{F} as well. A point u ∈ H_{Γ}(F) corresponds to an end of T_{F}; this end may be represented by a unique path originating from a vertex v_{u} in T_{Γ} and intersecting T_{Γ} only at v_{u}. The vertex v_{u} is defined to be the reduction of u.

The Bruhat–Tits tree of Γ allows one to understand measures on L_{Γ}(K) combina-

torially. Denote by E_{Γ} (resp. V_{Γ}) the set of all oriented edges (resp. vertices) of the tree T_{Γ}. Denote by C_{E} the group Div E_{Γ}/(e + \overline{e}), the abelian group generated freely by the oriented edges of T_{Γ} modulo the relation that oppositely oriented edges add to zero. Denote by C_{V} the group Div V_{Γ}. Define a trace map Tr : C_{V} → C_{E} by sending each vertex v to the sum of the edges of T_{Γ} with source vertex v. The trace map is injective, and we define C_{Q} by the short exact sequence

0 ⟶ C_{V} ⟶^{Tr} C_{E} ⟶ C_{Q} ⟶ 0.  (2.7)

The correspondence e ↦ U_{e}(Γ) shows that for each H, the group Meas(L_{Γ}(K), H)

is the kernel of the dual of the trace map:

0 ⟶ Meas(L_{Γ}(K), H) ⟶ Hom(C_{E}, H) ⟶^{Tr^{∗}} Hom(C_{V} , H) ⟶ 0. (2.8)

Thus Meas(L_{Γ}(K), H) ≅ Hom(C_{Q}, H), which is called the group of H-valued harmonic cocycles on the tree T_{Γ}. For each Γ-module M,

Hom(M, Meas(L_{Γ}(K), H)) ≅ Hom(C_{Q} ⊗ M, H).

In all the cases we will consider, the maximal torsion-free quotient of (C_{Q} ⊗ M)_{Γ} will be free of finite rank. Hence this group will be the universal free abelian group H admitting a Γ-invariant homomorphism from M to Meas(L_{Γ}(K), H)).

Definition 2.2.8. Denote by H̃ the maximal torsion-free quotient of (C_{Q} ⊗ M)_{Γ}, and by T̃ the torus G_{m} ⊗ H̃. Denote by μ^{univ} the universal Γ-invariant homomorphism from M to Meas(L_{Γ}(K), H̃)), with corresponding multiplicative double integral

× ∫∫ ω_{univ} : ((Div_{0} H_{Γ}) ⊗ M)_{Γ} → T̃.  (2.9)

Consider the short exact sequence of Γ-modules defining Div_{0} H_{Γ}:

0 → Div_{0} H_{Γ} → Div H_{Γ} → Z → 0.

After tensoring with M, the long exact sequence in homology yields a boundary map

δ : H_{1}(Γ, M) → ((Div_{0} H_{Γ}) ⊗ M)_{Γ}.  (2.10)

Definition 2.2.9. Let Φ_{1} denote the composition of the homomorphisms in (2.9) and (2.10):

Φ_{1} : H_{1}(Γ, M) → K^{×} ⊗ H̃.  (2.11)

Each element in the image of Φ_{1} may be expressed in terms of double integrals

involving divisors d ∈ H_{Γ}(F) for any nontrivial extension F of K. By the independence of the integral from the choice of F, it follows that the image of Φ_{1} indeed lies in K^{×} ⊗ H̃.

2.3 Mumford curves and Schottky groups

A discrete subgroup Γ of PGL_{2}(K) is called a Schottky group if it is finitely

generated and has no nontrivial elements of finite order. Such a group Γ acts freely on its Bruhat–Tits tree T_{Γ} and is hence free [11, §I.3]. A curve X over K is called a Mumford curve if the stable reduction of X contains only rational curves over k. Mumford proved [36] that for such a curve there exists a Schottky group Γ ⊂ PGL_{2}(K) and a Gal(C/K)-equivariant rigid analytic isomorphism

X(C) ≅ H_{Γ}(C)/Γ.  (2.12)

Furthermore, the Schottky group Γ satisfying (2.12) is unique up to conjugation by an element of PGL_{2}(K). It is free of rank g, the genus of X.

We will apply the definitions of the previous section with the trivial Γ-module

M = Z. In this case we see that H̃ = (C_{E})_{Γ}/ Tr((C_{V})_{Γ}). But (C_{E})_{Γ} ≅ C_{E′} and (C_{V})_{Γ} ≅ C_{V′} where C_{E′} and C_{V′} are the corresponding groups for the finite quotient graph T_{Γ}/Γ. Define an inner product on C_{E′} with values in Z by declaring that two oriented edges e, e′ are orthogonal unless they are equal as unoriented edges, in which case their inner product is 1 (resp. −1) if e′ = e (resp. e′ = \overline{e}, the oppositely oriented edge). One easily checks that under this inner product, the quotient C_{E′}/ Tr(C_{V′}) is dual to the kernel of the boundary map

∂ : C_{E′} → C_{V′} defined by ∂([e]) = [t(e)] − [s(e)],

sending an edge to the difference of its terminal and source vertices. Moreover, basic combinatorics identifies this kernel with the topological first homology group (with Z coefficients) of the quotient T_{Γ}/Γ. Since Γ acts freely on the tree T_{Γ}, this homology group is canonically identified with the abelianization of Γ. (Alternatively, this follows directly from the long exact sequence (3.3) that we describe later.) Thus we have

H̃ ≅ Hom(H_{1}(Γ, Z), Z) = Hom(Γ, Z).

Thus our multiplicative integral (2.6) becomes a map

(Div_{0} H_{Γ})_{Γ} → Hom(Γ, K^{×}).

Recall from [11, §II.2] the definition of the theta function Θ(a, b; z) for a, b, z ∈ H_{Γ}(C) and z ∉ Γa, Γb:

Θ(a, b; z) = ∏_{γ∈Γ} \frac{z − γa}{z − γb} ∈ C^{×}.

The function Θ satisfies various automorphy properties, which we do not list here. The following result relates our multiplicative integral to values of theta functions. (The ideas of the proof are present already in [1].)

Proposition 2.3.1. Let a, b be elements of H_{Γ}(F). Viewing the universal multiplicative integral as a homomorphism from Γ to F^{×}, we have

(× ∫_{b}^{a} ω_{univ}) (δ) = \frac{Θ(a, b; δz)}{Θ(a, b; z)}  (2.13)

for any z ∈ H_{Γ}(C) − (Γa ∪ Γb) and δ ∈ Γ. (The aforementioned automorphy properties of Θ imply that the right-hand side is independent of z.)

Proof. For a vertex v of the Bruhat–Tits tree of PGL_{2}(K), define the level ℓ(v) to be the length of the unique path from v to the central vertex v^{∗}. Define ℓ(e) for an edge e to be the maximum level of its two boundary vertices. Write ⟨·, ·⟩ for the inner product on (C_{E})_{Γ} defined above. Choose a vertex v_{0} of T_{Γ} and let P be a path from v_{0} to δv_{0}. Let f(t) = \frac{t−a}{t−b}. Then writing P̄ for the image of P in (C_{E})_{Γ}, we have

(× ∫_{b}^{a} ω_{univ}) (δ) = lim_{n→∞} ∏_{{e∈T_{Γ} \atop ℓ(e)=n}} f(t_{e})^{⟨e,P̄⟩},  (2.14)

where e is oriented outwards from v^{∗} and t_{e} is an arbitrarily chosen point of U_{e}(Γ).

To calculate the right side of (2.13), let z_{0} be an element of H_{Γ}(C) reducing to the

vertex v_{0}. We have

\frac{Θ(a, b; δz_{0})}{Θ(a, b; z_{0})} = ∏_{γ∈Γ} \frac{δz_{0} − γa}{δz_{0} − γb} · \frac{z_{0} − γb}{z_{0} − γa}
= ∏_{γ∈Γ} \frac{γ^{−1}δz_{0} − a}{γ^{−1}δz_{0} − b} · \frac{γ^{−1}z_{0} − b}{γ^{−1}z_{0} − a}  (2.15)
= ∏_{γ∈Γ} \frac{γδz_{0} − a}{γδz_{0} − b} · \frac{γz_{0} − b}{γz_{0} − a},  (2.16)

where (2.15) follows from the invariance of cross-ratios under linear fractional transformations. Let the vertices in the path P be v_{0}, v_{1}, …, v_{s} = δv_{0} and let the corresponding edges be written e_{1}, …, e_{s}. For each vertex v_{i}, choose an element z_{i} ∈ H_{Γ} reducing to v_{i}, such that z_{s} = δz_{0}. The product (2.16) may be broken up as

∏_{γ∈Γ} \frac{f(γδz_{0})}{f(γz_{0})} = ∏_{γ∈Γ} ∏_{i=1}^{s} \frac{f(γz_{i})}{f(γz_{i−1})}.  (2.17)

Each of the products

∏_{γ∈Γ} \frac{f(γz_{i})}{f(γz_{i−1})}

converges since Γ acts freely on the tree, so the values ℓ(γe_{i}) increase as γ ranges over Γ and the quotients of the values of f on the endpoints of these edges tend to 1. Thus we may rewrite (2.17) as

∏_{i=1}^{s} ∏_{γ∈Γ} \frac{f(γz_{i})}{f(γz_{i−1})}.

In fact for fixed n, if Γ(i) denotes the set of γ ∈ Γ such that ℓ(γe_{i}) ≤ n, then we have

∏_{γ∈Γ} \frac{f(γz_{i})}{f(γz_{i−1})} ≡ ∏_{γ∈Γ(i)} \frac{f(γz_{i})}{f(γz_{i−1})} (mod^{∗} π^{n}),  (2.18)

where a ≡ b (mod^{∗} π^{n}) if \frac{a}{b} − 1 is divisible by π^{n}. The right side of (2.18) may be further decomposed as

∏_{ℓ(γv_{i−1})<n} f(γz_{i−1})^{−1} ∏_{ℓ(γv_{i})<n} f(γz_{i}) ∏_{{ℓ(γv_{i})=n−1 \atop ℓ(γv_{i−1})=n}} f(γz_{i−1})^{−1} ∏_{{ℓ(γv_{i})=n \atop ℓ(γv_{i−1})=n−1}} f(γz_{i}).  (2.19)

In the product over i = 1, …, s, the first two terms of (2.19) cancel out. The last two yield

∏_{i=1}^{s} ∏_{{γ∈Γ \atop ℓ(γe_{i})=n}} f(γz_{∗})^{⟨γe_{i},P̄⟩},  (2.20)

where z_{∗} ∈ {z_{i−1}, z_{i}} and γz_{∗} reduces to the endpoint of γe_{i} of level n. Expression (2.20) is equivalent to (2.14) mod^{∗} π^{n}, and the result follows.

For x ∈ H_{Γ}(C), let x̃ represent the image of x in X(C) = H_{Γ}(C)/Γ.

Theorem 2.3.2 (Manin–Drinfeld). The map

[x̃] − [ỹ] ↦ × ∫_{y}^{x} ω_{univ}

induces a Gal(C/K)-equivariant rigid analytic isomorphism between the rigid analytic space associated to the C-valued points of the Jacobian of the curve X and (C^{×} ⊗ H̃)/ Im(Φ_{1}).

Proof. The original statement of the Manin–Drinfeld theorem is given in terms of theta functions [11], [30]. Proposition 2.3.1 relates the classical version to our formulation.

Theorem 2.3.2 is the p-adic analogue of the Abel–Jacobi theorem. However, it

does not allow the obvious construction of Heegner-type points in the case X = X_{0}(p) over Q_{p}. In fact, since Mumford’s group Γ is not given in an explicit way and is probably not an arithmetic group, the question of computing the periods of J_{0}(p) (i.e., calculating the lattice Im(Φ_{1})) using this uniformization seems difficult. (See [9] for the calculation of this lattice modulo p using this theory, however.) Accordingly, one needs to find an alternative p-adic uniformization for modular Jacobians that uses arithmetic groups in a crucial way; this is taken up in Chapter 4.

Chapter 3

Properties of the multiplicative

double integral

In this chapter we establish some fundamental properties of the multiplicative

double integral.

3.1 Behavior of the integral under conjugation

Let δ ∈ PGL_{2}(K), and let Γ′ = δΓδ^{−1}. The Γ-module M gives rise to a Γ′-module

M′; the module M′ is isomorphic to M as an abelian group, and Γ′ acts on M′ by

δγδ^{−1} : m ↦ γm,

where the action on the right side is given by the action of Γ on M. In this setting, we have

L_{Γ′}(K) = δL_{Γ}(K) and T_{Γ′} = δT_{Γ}.

We also have an isomorphism

c_{δ} : H̃ = (C_{Q} ⊗ M)_{Γ} ≅ H̃′ = (C_{Q′} ⊗ M′)_{Γ′}

given by e ⊗ m ↦ δe ⊗ m. We omit the proof of the following proposition, which shows how the multiplicative double integral behaves under conjugation.

Proposition 3.1.1. For τ_{1}, τ_{2} ∈ H_{Γ} and m ∈ M, we have

(Id ⊗c_{δ}) (× ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{univ}) = × ∫_{δτ_{1}}^{δτ_{2}}∫_{m}ω_{univ},

where the integral on the left is with the universal measure for (Γ, M), and the integral on the right is with the universal measure for (Γ′, M′).

3.2 An exact sequence in homology

Continuing with the notation of Section 2.2, define homomorphisms

∂ : C_{E} → C_{V} , ∂([y]) = [t(y)] − [s(y)],  (3.1)
ϵ : C_{V} → Z, ϵ([x]) = 1.

Since T_{Γ} is a tree, the sequence

0 ⟶ C_{E} ⟶^{∂} C_{V} ⟶^{ϵ} Z ⟶ 0,  (3.2)

is exact. Since Z is free, we may tensor with M without losing exactness, and then taking homology gives the the long exact sequence

⋯ → H_{i}(Γ, C_{E} ⊗ M) → H_{i}(Γ, C_{V} ⊗ M) → H_{i}(Γ, M) → H_{i−1}(Γ, C_{E} ⊗ M) → · · · . (3.3)

We would like to give an explicit description of the map

ϕ : H_{1}(Γ, M) → ker(∂ : H_{0}(Γ, C_{E} ⊗ M) → H_{0}(Γ, C_{V} ⊗ M))  (3.4)

given in (3.3). We first recall the presentation of H_{1}(Γ, M) given by “dimension shifting.” The short exact sequence defining Div_{0} Γ, upon tensoring with M, becomes:

0 → (Div_{0} Γ) ⊗ M → (Div Γ) ⊗ M → M → 0,  (3.5)

giving the long exact sequence in homology:

⋯ → H_{1}(Γ, (Div Γ) ⊗ M) → H_{1}(Γ, M) → ((Div_{0} Γ) ⊗ M)_{Γ} → ((Div Γ) ⊗ M)_{Γ} → · · · .

(3.6)

Now (Div Γ) ⊗ M being an induced module, the first term of (3.6) vanishes. Furthermore, the map

B : ((Div Γ) ⊗ M)_{Γ} → M, γ ⊗ m ↦ γ^{−1}m

is an isomorphism of abelian groups. Writing B again for the induced map

((Div_{0} Γ) ⊗ M)_{Γ} → M,

we hence obtain

H_{1}(Γ, M) ≅ ker(B : ((Div_{0} Γ) ⊗ M)_{Γ} → M).  (3.7)

Remark 3.2.1. An element k ∈ (Div_{0} Γ) ⊗ M may be written in the form

k = ∑_{γ∈Γ}([γ^{−1}] − [Id]) ⊗ m_{γ}.  (3.8)

In the literature, elements of H_{1}(Γ, M) are often represented by formal linear combinations ∑_{γ∈Γ} m_{γ} ·[γ], where all but finitely many of the m_{γ} ∈ M are 0 and ∑(γm_{γ} −m_{γ}) = 0; such an expression is called a 1-cycle, and corresponds to the element k ∈ H_{1}(Γ, M) in (3.8).

Remark 3.2.2. If

∑([α_{i}] − [β_{i}]) ⊗ m_{i} ∈ H_{1}(Γ, M),

the expression

∑([α_{i}γ] − [β_{i}γ]) ⊗ m_{i}

for γ ∈ Γ represents the same element of H_{1}(Γ, M), since their difference equals

∑([α_{i}] − [α_{i}γ]) ⊗ m_{i} − ∑([β_{i}] − [β_{i}γ]) ⊗ m_{i}
= ∑([Id] − [γ]) ⊗ α_{i}^{−1}m_{i} − ∑([Id] − [γ]) ⊗ β_{i}^{−1}m_{i}
= ([Id] − [γ]) ⊗ ∑(α_{i}^{−1}m_{i} − β_{i}^{−1}m_{i})
= ([Id] − [γ]) ⊗ 0 = 0.

We now describe the map ϕ of (3.4) explicitly. Fix v ∈ V_{Γ} and consider the

following commutative diagram between the short exact sequences of Γ-modules in (3.2) and (3.5):

\xymatrix{0 \ar[r] & Div_{0} Γ \ar[r] \ar[d]^{ι_{v}} & Div Γ \ar[r] \ar[d]^{κ_{v}} & Z \ar[r] \ar[d]^{Id} & 0 \\ 0 \ar[r] & C_{E} \ar[r]^{∂} & C_{V} \ar[r]^{ϵ} & Z \ar[r] & 0.}  (3.9)

Here the map κ_{v} : Div Γ → C_{V} is defined by [γ] ↦ [γ]v := [γv]. The map ι_{v} is defined by the commutativity of the diagram; for each divisor G ∈ Div_{0} Γ, there exists a unique G′ ∈ C_{E} such that ∂(G′) = Gv; we define ι_{v}(G) = G′. After tensoring all the terms in the diagram (3.9) with M, there is a commutative diagram between the corresponding long exact sequences (3.3) and (3.6). Part of this diagram reads:

\xymatrix{0 \ar[r] \ar[d] & H_{1}(Γ, M) \ar[r] \ar[d]^{Id} & ((Div_{0} Γ) ⊗ M)_{Γ} \ar[r] \ar[d]^{ι_{v}} & M \ar[d] \\ H_{1}(Γ, C_{V} ⊗ M) \ar[r] & H_{1}(Γ, M) \ar[r]^{ϕ} & H_{0}(Γ, C_{E} ⊗ M) \ar[r]^{∂} & H_{0}(Γ, C_{V} ⊗ M).} (3.10)

Consider an element

k = ∑_{i} G_{i} ⊗ m_{i} ∈ ((Div_{0} Γ) ⊗ M)_{Γ}.

For each element G_{i}, there is a unique C_{i} ∈ C_{E} such that G_{i}v = ∂(C_{i}). The image of k in under the map ι_{v} of (3.10) is:

ι_{v}(k) = ∑_{i} C_{i} ⊗ m_{i} ∈ (C_{E} ⊗ M)_{Γ}.  (3.11)

One checks that when

k ∈ ker(B : (Div_{0} Γ) ⊗ M)_{Γ} → M) ≅ H_{1}(Γ, M),

this value of ι_{v}(k) is independent of the choice of vertex v. Equation (3.11) provides an explicit formula for the map ϕ.

3.3 The valuation of the integration map

Let the valuation v_{K} of K be normalized so that the valuation of the uniformizer

π is 1. In this section we will analyze the composite map

v_{K} ∘ Φ_{1} : H_{1}(Γ, M) ⟶^{Φ_{1}} K^{×} ⊗ H̃ ⟶^{v_{K}⊗Id} Z ⊗ H̃ = H̃.

Since H̃ as defined in Definition 2.2.8 is a quotient of

(C_{Q} ⊗ M)_{Γ} = (C_{E} ⊗ M)_{Γ}/ Tr((C_{V} ⊗ M)_{Γ}),

we have natural maps

H_{1}(Γ, M) ⟶^{ϕ} (C_{E} ⊗ M)_{Γ} ⟶ H̃,  (3.12)

with ϕ as in (3.4). Denote by ι the composite map H_{1}(Γ, M) → H̃ in (3.12).

Proposition 3.3.1. The map v_{K} ∘ Φ_{1} is equal to ι.

We first prove a lemma.

Lemma 3.3.2. Let τ_{1}, τ_{2} ∈ H_{Γ} reduce to vertices connected by an oriented edge e of T_{Γ}:

∂(e) = [red(τ_{2})] − [red(τ_{1})].

Extend the valuation v_{K} to the maximal unramified extension of K. Then we have

v_{K} (× ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{univ}) = μ_{m}(U_{e}).

Proof. The group PGL_{2}(K) acts transitively on the edges of its Bruhat–Tits tree, and the reduction map is PGL_{2}(K)-equivariant. Thus Proposition 3.1.1 implies that it suffices to consider the case where τ_{1} reduces to the standard vertex corresponding to O_{K} ⊕O_{K} and τ_{2} reduces to the vertex corresponding to O_{K} ⊕πO_{K}. In this case, we have U_{e} = L_{Γ}(K)−O_{K}. Let τ_{1} ∈ P^{1}(F) for an unramified extension F of K. The fact that τ_{1} reduces to v_{∗} implies that the image of τ_{1} in P^{1}(k_{F}) does not equal the image of any point t ∈ L_{Γ}(K) in P^{1}(k_{F}), where k_{F} is the residue field of F. In particular, the fact that the edge e connecting τ_{2} and τ_{1} lies in T_{Γ} implies that τ_{1} ∈ O_{F}. Thus for t ∈ L_{Γ}(K), we have

v_{K}(τ_{1} − t) = \begin{cases}0 & if t ∈ O_{K} \\ v_{K}(t) & otherwise. \end{cases}

Similarly,

v_{K}(τ_{2} − t) = \begin{cases}−1 & if t ∈ O_{K} \\ v_{K}(t) & otherwise. \end{cases}

Without loss of generality, in the definition of the multiplicative integral, we need consider only open coverings U that refine the open covering {Uē, U_{e}}. For each U in such a covering, the previous calculation shows that v_{K}((t_{U} − τ_{2})/(t_{U} − τ_{1})) equals −1 or 0 according to whether U ⊂ Uē or not. Thus the valuation of each product inside the limit defining the multiplicative integral equals −μ_{m}(Uē) = μ_{m}(U_{e}).

By combining Lemma 3.3.2 with the explicit description of ϕ given in Section 3.2,

we can now prove Proposition 3.3.1. Let v be a vertex of T_{Γ}, and let τ ∈ H_{Γ} reduce to v. Let

k = ∑_{i} G_{i} ⊗ m_{i} ∈ ker(B : ((Div_{0} Γ) ⊗ M)_{Γ} → M) ≅ H_{1}(Γ, M).

Tracing through definitions, one finds that

Φ_{1}(k) = × ∫_{G_{i}τ} ∫_{m_{i}}ω_{univ} ∈ K^{×} ⊗ H̃.

Let C_{i} ∈ C_{E} be such that G_{i}v = ∂(C_{i}). For an oriented edge e of T_{Γ}, μ_{m}(U_{e}) is defined to be the image in H̃ of the element e ⊗ m ∈ (C_{E} ⊗ M)_{Γ}. Lemma 3.3.2 thus implies that

∑_{i} v_{K} (× ∫_{G_{i}τ} ∫_{m_{i}}ω_{univ})

is equal to the image of ∑_{i} C_{i} ⊗ m_{i} in H̃. Equation (3.11) shows that this is precisely the value ι(k), thereby proving Proposition 3.3.1.

3.4 Choosing a system of representatives

In (2.9) we defined a universal multiplicative double integral that is canonically

associated to a group Γ ⊂ PGL_{2}(K) and a Γ-module M. In practice, it will be convenient to make certain choices that allow one to be more explicit with the homology groups that arise. Thus we choose a system of representatives (E, V) for Γ acting on T_{Γ}. This is a subset E of the edges and V of the vertices of T_{Γ} such that every edge (resp. vertex) of T_{Γ} is equivalent modulo Γ to exactly one element of E (resp. V). It need not be the case that the endpoints of an edge in E lie in V. Denote by 𝔬 an orientation on each edge of E. Assume that Γ acts without inversion on T_{Γ}, meaning that no oriented edge is in the same Γ-orbit as its opposite. Then extending 𝔬 by Γ-invariance gives an orientation on each edge of the tree T_{Γ}.

Denote by Γ_{e} the stabilizer of the edge e. The Γ-module C_{E} is a sum of induced

modules:

⨁_{e∈E} Div Γ/Γ_{e} ≅ C_{E}, (∑_{i} a_{i}γ_{e,i})_{e∈E} ↦ ∑_{i,e} a_{i}[γ_{e,i}e],

where on the right side the edge e has the orientation given by 𝔬. Similarly, we have an isomorphism ⨁_{v∈V} Div Γ/Γ_{v} ≅ C_{V}. Accordingly, for each Γ-module M, Shapiro’s Lemma gives isomorphisms

H_{i}(Γ, C_{E} ⊗ M) ≅ ⨁_{e∈E} H_{i}(Γ_{e}, M) and H_{i}(Γ, C_{V} ⊗ M) ≅ ⨁_{v∈V} H_{i}(Γ_{v}, M).  (3.13)

Thus the long exact sequence (3.3) becomes

⋯ → ⨁_{e∈E} H_{i}(Γ_{e}, M) → ⨁_{v∈V} H_{i}(Γ_{v}, M) → H_{i}(Γ, M) → ⨁_{e∈E} H_{i−1}(Γ_{e}, M) → · · · . (3.14)

In particular, H̃ is the maximal torsion-free quotient of

⨁_{e∈E} H_{0}(Γ_{e}, M)/ Tr (⨁_{v∈V} H_{0}(Γ_{v}, M)) .  (3.15)

We will discuss the maps Tr and ∂ in this presentation of our homology groups with an explicit example in the next chapter.

Chapter 4

An arithmetic uniformization and

Stark–Heegner points

Let p be a prime number and N ≥ 1 an integer not divisible by p. Write M = Np.

In this chapter we will present a p-adic uniformization of the maximal quotient of J_{0}(M) with toric reduction at p. A key idea, suggested by the definitions of [5], is that the p-adic arithmetic of J_{0}(M) is intimately linked with the group

Γ = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ PSL_{2}(Z[1/p]) such that N|c\}  (4.1)

and its homology. The group Γ is not discrete as a subgroup of PGL_{2}(K) and hence acts with dense orbits on P^{1}. In this setting, the limit point set equals L := L_{Γ}(Q_{p}) = P^{1}(Q_{p}). We also have

H_{Γ}(C_{p}) = H_{p} := P^{1}(C_{p}) − P^{1}(Q_{p}),

where C_{p} is the completion of an algebraic closure of Q_{p}.

A measure on P^{1}(Q_{p}) is given by a harmonic cocycle on the entire Bruhat–Tits

tree T of PGL_{2}(Q_{p}). From (3.15) with M = Z, one finds that there are no non-trivial Γ-invariant measures on P^{1}(Q_{p}) (see (4.3) below with M = Z). This problem can be remedied by introducing a Γ-invariant measure-valued modular symbol as follows.

Let M := Div_{0} P^{1}(Q) be the group of degree-zero divisors on P^{1}(Q), viewed as

cusps of the complex upper half plane. The group M is defined by the exact sequence

0 → M → Div P^{1}(Q) → Z → 0.  (4.2)

The group Γ acts on M via its action on P^{1}(Q) by linear fractional transformations. For a free abelian group H, a Meas(P^{1}(Q_{p}), H)-valued modular symbol is a homomorphism

M → Meas(P^{1}(Q_{p}), H), m ↦ μ_{m}.

The group of modular symbols μ has a Γ-action given by

(γ^{−1}μ)_{m}(U) = μ_{γm}(γU).

In Chapter 2 we constructed a universal Γ-invariant Meas(P^{1}(Q_{p}), H̃)-valued modular symbol. In the next section we will explore the group H̃.

4.1 The universal modular symbol

The action of Γ on the tree T is particularly easy to describe ([40, §II]). Each

oriented edge of T is equivalent to either e^{∗} or \overline{e^{∗}}; each vertex is equivalent to either v^{∗} or w^{∗}. The stabilizers of v^{∗} and w^{∗} in Γ are Γ_{0}(N) and P^{−1}Γ_{0}(N)P, respectively, where P is

the matrix (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}). The stabilizer of e^{∗} is the intersection of these, namely Γ_{0}(M). Hence

from (3.15) we have that H̃ is the maximal torsion-free quotient of

M_{Γ_{0}(M)}/ Tr(M_{Γ_{0}(N)} ⊕ M_{P^{−1}Γ_{0}(N)P}).  (4.3)

Furthermore, the map m ↦ P m defines an isomorphism

H_{∗}(P^{−1}Γ_{0}(N)P, M) → H_{∗}(Γ_{0}(N), M).

One can interpret the group of co-invariants

M_{Γ_{0}(M)} = H_{0}(Γ_{0}(M), M)

geometrically as follows. Given a divisor [x] − [y] ∈ M, consider a path from x to y in H^{∗}. If we make the identification H^{∗}/Γ_{0}(M) = X_{0}(M)(C), the image of this path gives a well defined element of H_{1}(X_{0}(M), cusps, Z), the singular homology of the Riemann surface X_{0}(M)(C) relative to the cusps. Manin [29] proves that this map induces an isomorphism between H_{1}(X_{0}(M), cusps, Z) and the maximal torsion-free quotient of M_{Γ_{0}(M)}. This maximal torsion-free quotient will be denoted H_{0}(Γ_{0}(M), M)_{T}. The torsion of H_{0}(Γ_{0}(M), M) is finite and supported at 2 and 3. The projection

M → M_{Γ_{0}(M)} → H_{1}(X_{0}(M), cusps, Z)

is called the universal modular symbol for Γ_{0}(M).

The points of X_{0}(M) over a scheme S correspond to isomorphism classes of pairs

(E, C_{M}) of generalized elliptic curves E/S equipped with a cyclic subgroup C_{M} ⊂ E of order M. To such a pair we can associate two points of X_{0}(N), namely the points corresponding to the pairs (E, C_{N}) and (E/C_{p}, C_{M}/C_{p}), where C_{p} and C_{N} are the subgroups of C_{M} of order p and M, respectively. This defines two morphisms of curves

f_{1} : X_{0}(M) → X_{0}(N) and f_{2} : X_{0}(M) → X_{0}(N),  (4.4)

each of which is defined over Q. The map f_{2} is the composition of f_{1} with the Atkin–Lehner involution W_{p} on X_{0}(M). Write f_{∗} = f_{1}_{∗} ⊕ f_{2}_{∗} and f^{∗} = f_{1}^{∗} ⊕ f_{2}^{∗} (resp. f_{∗}^{c} and f_{c}^{∗}) for the induced maps on (relative) singular homology:

f_{∗} : H_{1}(X_{0}(M), Z) → H_{1}(X_{0}(N), Z)^{2},
f^{∗} : H_{1}(X_{0}(N), Z)^{2} → H_{1}(X_{0}(M), Z),
f_{∗}^{c} : H_{1}(X_{0}(M), cusps, Z) → H_{1}(X_{0}(N), cusps, Z)^{2},
f_{c}^{∗} : H_{1}(X_{0}(N), cusps, Z)^{2} → H_{1}(X_{0}(M), cusps, Z).

Via the universal modular symbol, the last two maps are identified with maps^{1}

f_{∗}^{c} : H_{0}(Γ_{0}(M), M) → H_{0}(Γ_{0}(N), M)^{2}
f_{c}^{∗} : H_{0}(Γ_{0}(N), M)^{2} → H_{0}(Γ_{0}(M), M).

The map f_{c}^{∗} is precisely the map Tr of (4.3). Hence the universal Γ-invariant

modular symbol of measures on P^{1}(Q_{p}) takes values in the cokernel of f_{c}^{∗}:

H̃ ≅ (H_{1}(X_{0}(M), cusps, Z)/f_{c}^{∗}(H_{1}(X_{0}(N), cusps, Z)^{2}))_{T} .

Define also

H := (H_{1}(X_{0}(M), Z)/f^{∗}(H_{1}(X_{0}(N), Z)^{2}))_{T} .  (4.5)

The abelian variety J_{0}(M)^{p-new} is defined to be the quotient of J_{0}(M) by the sum

of the images of the Picard maps on Jacobians associated to f_{1} and f_{2}. This is the abelian variety with purely toric reduction at p for which we will provide a uniformization (up to isogeny).

1In purely homological terms, f_{1}_{∗} is corestriction, and f_{2}_{∗} is the composition f_{1}_{∗} ∘ W_{p}; similarly for f^{∗} and restriction.

Proposition 4.1.1. If we write g for the dimension of J_{0}(N)^{p-new}, the free abelian groups H and H̃ have ranks 2g and 2g+1 respectively, and the natural map H → H̃ is an injection.

Proof. It is well known that f^{∗} is injective and that H has rank 2g. Consider the following commutative diagram of relative homology sequences:

\xymatrix{0 \ar[r] & H_{1}(X_{0}(N), Z)^{2} \ar[r] \ar[d]^{f^{∗}} & H_{1}(X_{0}(N), cusps, Z)^{2} \ar[r] \ar[d]^{f_{c}^{∗}} & C(N)^{2} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & H_{1}(X_{0}(M), Z) \ar[r] & H_{1}(X_{0}(M), cusps, Z) \ar[r] & C(M) \ar[r] & 0.}  (4.6)

Here C(N) and C(M) denote the groups of degree-zero divisors on the set of cusps of X_{0}(N) and X_{0}(M), respectively. If c denotes the number of cusps of X_{0}(N), these are free abelian groups of rank c−1 and 2c−1, respectively. Above each cusp of X_{0}(N) (under the map f_{1}) lies two cusps of X_{0}(M), one of which has ramification index p and the other one of which is unramified. The map W_{p} on X_{0}(M) interchanges these two cusps. This implies that the map C(N)^{2} → C(M) of (4.6) is injective and that the torsion subgroup of its cokernel has exponent dividing p^{2} − 1. Since H and H̃ are the cokernels of f^{∗} and f_{c}^{∗}, the snake lemma yields the proposition. (Note that we have also shown that f_{c}^{∗} is injective.)

4.2 Statement of the uniformization

As noted in Proposition 4.1.1, the module H̃ contains H with corank 1. To define

a modular symbol that takes values in H rather than H̃, we choose a map

ψ : H̃ ⟶ H.

We will require two properties of the map ψ, whose uses will later become evident:

• The groups H̃ and H have natural Hecke actions, described in detail in Chapter 5. We assume that the map ψ is Hecke-equivariant.

• We assume that the composition of ψ with the inclusion H ⊂ H̃ is an endomorphism of H with finite cokernel.

Associated to ψ we have the multiplicative double integral

× ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{ψ} = (Id ⊗ψ) (× ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{univ}) ∈ T := G_{m}^{×} ⊗ H.

The long exact sequence in homology associated to the sequence (4.2) defining M gives a boundary map

δ : H_{2}(Γ, Z) → H_{1}(Γ, M).  (4.7)

Denote the composition of Φ_{1} : H_{1}(Γ, M) → T (Q_{p}) with (4.7) by

Φ_{2} : H_{2}(Γ, Z) → T (Q_{p}).

Let L be the image of Φ_{2}.

We now state the main result of part I of the thesis. The torus T inherits a Hecke

action from H.

Theorem 4.2.1. Let K_{p} denote the quadratic unramified extension of Q_{p}. The group L is a discrete, Hecke-stable subgroup of T (Q_{p}) of rank 2g. The quotient T/L admits a Heckeequivariant isogeny over K_{p} to the rigid analytic space associated to the product of two copies of J_{0}(M)^{p-new}.

During the course of proving Theorem 4.2.1, we will give some control over the

set of primes appearing in the degree of this isogeny. Also, we will see that if one lets the nontrivial element of Gal(K_{p}/Q_{p}) act on T/L by the Hecke operator U_{p} (defined in Chapter 5), this isogeny is defined over Q_{p}.

Remark 4.2.2. If we had not used the auxiliary projection ψ : H̃ → H and continued our construction with integrals valued in G_{m} ⊗ H̃, the corresponding quotient T̃/L̃ would be isogenous to two copies of J_{0}(M)^{p-new}, with one copy of G_{m}, arising from the Eisenstein quotient H̃/H. However, as the projections to this G_{m} of the Stark–Heegner points we will define later bear little arithmetic interest (see Chapter 8), we lose little in employing the projection ψ in exchange for the technical simplicity gained. The Eisenstein quotient H̃/H has eigenvalue +1 for complex conjugation. In part II of the thesis, partial modular symbols are used to construct Eisenstein quotients where complex conjugation acts as −1, and the resulting projections of the Stark–Heegner points to G_{m} are analyzed.

Remark 4.2.3. The module H can be expressed up to finite index as a sum H^{+} ⊕ H^{−}, where the modules H^{+} and H^{−} are the subgroups on which complex conjugation (denoted W_{∞}) acts as 1 or −1, respectively; these each have rank g over Z. This decomposition of H explains the two components of T/L described in Theorem 4.2.1.

Remark 4.2.4. In Chapter 6.1, we will show how Theorem 4.2.1 is a generalization of the Mazur–Tate–Teitelbaum conjecture [35, Conjecture II.13.1] proven by Greenberg and Stevens [12], [13].

Granting theorem 4.2.1, we next describe the construction of Stark–Heegner points

on J_{0}(M)^{p-new}.

4.3 Stark–Heegner points

Fix τ ∈ H_{p} and x ∈ P^{1}(Q). Consider the 2-cocycle in Z^{2}(Γ, T (C_{p})) given by
d_{τ,x}(γ_{1}, γ_{2}) := × ∫_{τ}^{γ_{1}τ}∫_{γ_{1}x}^{γ_{1}γ_{2}x}ω_{ψ} := × ∫_{τ}^{γ_{1}τ}∫_{[γ_{1}x]−[γ_{1}γ_{2}x]}ω_{ψ},  (4.8)

where here as always Γ acts trivially on T. It is an easy verification that the image d of d_{τ,x} in H^{2}(Γ, T (C_{p})) is independent of the choice of τ and x. Since T (C_{p}) is divisible and H_{1}(Γ, Z) is finite (see Proposition 4.3.1 below), the universal coefficient theorem identifies d with a homomorphism

H_{2}(Γ, Z) → T (C_{p});

this homomorphism is precisely Φ_{2}. Thus L, which was defined to be the image of Φ_{2}, is the minimal subgroup of T (C_{p}) such that the image of d in H^{2}(Γ, T (C_{p})/L) is trivial.

Thus there exists a map β_{τ,x} : Γ → T/L such that

β_{τ,x}(γ_{1}γ_{2}) − β_{τ,x}(γ_{1}) − β_{τ,x}(γ_{2}) = × ∫_{τ}^{γ_{1}τ}∫_{γ_{1}x}^{γ_{1}γ_{2}x}ω_{ψ} (mod L).  (4.9)

The 1-cochain β_{τ,x} is defined uniquely up to an element of Hom(Γ, T/L). The following proposition allows us to deal with this ambiguity.

Proposition 4.3.1. The abelianization of Γ is finite, and every prime dividing its size divides 6φ(N)(p^{2} − 1).

Proof. This is a result of Ihara [23]; we provide a quick sketch. In Chapter 3 (see (3.14)) we described an exact sequence that identifies H_{1}(Γ, Z) with the cokernel of the natural map

H_{1}(Γ_{0}(M), Z) → H_{1}(Γ_{0}(N), Z)^{2}.

Since Γ_{0}(N) acts on the complex upper half plane H with isotropy groups supported at the primes 2 and 3, the group H_{1}(Γ_{0}(N), Z) may be identified with the corresponding singular homology of Y_{0}(N)(C) = H/Γ_{0}(N) outside of a finite torsion group supported at 2 and 3. Hence we must show that

f_{∗}^{Y} : H_{1}(Y_{0}(M), Z) → H_{1}(Y_{0}(N), Z)^{2}  (4.10)

has finite cokernel.

Poincaré duality identifies H_{1}(Y_{0}(N), Z) with the Z-dual of the relative homology

group H_{1}(X_{0}(N), cusps, Z). We are thus led to reconsider the diagram (4.6) of Proposition 4.1.1. The injectivity of f_{c}^{∗} implies that the cokernel of (4.10) is finite; furthermore, this cokernel is isomorphic to a subgroup of the cokernel of f_{c}^{∗}. A result of of Ribet [37] implies that the torsion subgroup of the cokernel of f^{∗} is supported on the set of primes dividing φ(N). We saw in the proof of Proposition 4.1.1 that the torsion subgroup of the cokernel of Z^{2c−2} → Z^{2c−1} has exponent p^{2} − 1. The snake lemma completes the proof.

We may now define Stark–Heegner points on J_{0}(M)^{p-new}. Define the ring

R := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(Z[1/p]) such that N divides c\} .  (4.11)

Let K be a real quadratic field such that p is inert in K; choose an embedding σ of K into R, and also an embedding of K into C_{p}. For each τ ∈ H_{p} ∩ K, consider the collection O_{τ} of matrices g ∈ R satisfying

g\binom{τ}{1} = λ_{g}\binom{τ}{1} for some λ_{g} ∈ K.  (4.12)

The ring O_{τ} is isomorphic to a Z[1/p]-order in K, via the map g ↦ λ_{g} = cτ + d. The group of units in O_{τ}^{×} of norm 1 is a free abelian group of rank 1. Let γ_{τ} be the generator such that σ(λ_{γ_{τ}}) > 1 if σ(τ) > σ(τ′), and such that σ(λ_{γ_{τ}}) < 1 if σ(τ) < σ(τ′). Here τ′ denotes the conjugate of τ over Q; the definition of γ_{τ} is independent of choice of σ. Finally, choose an x ∈ P^{1}(Q), and let t denote the exponent of the abelianization of Γ.

Definition 4.3.2. The Stark–Heegner point associated to τ is given by

Φ(τ) := t · β_{τ,x}(γ_{τ}) ∈ T (K_{p})/L.

Multiplication by t ensures that this definition is independent of choice of β_{τ,x}

satisfying (4.9), and one also checks that Φ(τ) is independent of x. Furthermore, the point Φ(τ) depends only on the Γ-orbit of τ, so we obtain a map

Φ : (H_{p} ∩ K)/Γ → T (K_{p})/L.

Let us now denote by ν_{±} the two maps T/L → J_{0}(M)^{p-new} of Theorem 4.2.1, where the ± sign denotes the corresponding eigenvalue of complex conjugation on H Composing Φ with the maps ν_{±}, we obtain

Φ_{±} : (H_{p} ∩ K)_{Γ} → J_{0}(M)^{p-new}(K_{p}).

The images of Φ_{±} are the Stark–Heegner points on J_{0}(M)^{p-new}.

As in [5] and [6], we conjecture that the images of Φ_{±} satisfy explicit algebraicity

properties. Fix a Z[1/p]-order O in K; let us assume that the discriminant of O is prime to M. Let K_{+}^{×} denote the multiplicative group of elements of K of positive norm. Define the narrow Picard group Pic^{+}(O) to be the group of projective rank one O-submodules of K modulo homothety by K_{+}^{×}. Class field theory canonically identifies Pic^{+}(O) with the Galois group of an extension H^{+} of K called the narrow ring class field of K attached to O:

rec : Pic^{+}(O)⟶Gal(H^{+}/K).

Denote by H_{p}^{O} the set of τ ∈ H_{p} ∩ K such that O_{τ} ≅ O. The basic fundamental conjecture regarding Stark–Heegner points is:

if τ ∈ H_{p}^{O}, then Φ_{±}(τ) lies in J_{0}(M)^{p-new}(H^{+}).

We now proceed to refine this statement into a “Shimura reciprocity law” for

Stark–Heegner points. A Z[1/p]-lattice in K is a Z[1/p]-submodule of K that is free of rank 2. Define

Ω_{N}(K) = \{(Λ_{1}, Λ_{2}), with \begin{matrix}Λ_{j} a Z[1/p]-lattice in K, \\ Λ_{1}/Λ_{2} ≃ Z/NZ. \end{matrix} \} /K_{+}^{×}.

There is a natural bijection \underline{τ} from Ω_{N}(K) to (H_{p} ∩ K)/Γ, which to x = (Λ_{1}, Λ_{2}) assigns

\underline{τ}(x) = ω_{1}/ω_{2},

where ⟨ω_{1}, ω_{2}⟩ is a Z[1/p]-basis of Λ_{1} satisfying

ω_{1}ω′_{2} − ω′_{1}ω_{2} > 0, ord_{p}(ω_{1}ω′_{2} − ω′_{1}ω_{2}) ≡ 0 (mod 2),

and Λ_{2} = ⟨Nω_{1}, ω_{2}⟩. Here we have written ω ↦ ω′ for the action of the nontrivial automorphism of Gal(K/Q). Denote by Ω_{N}(O) the set of pairs (Λ_{1}, Λ_{2}) ∈ Ω_{N}(K) such that O is the largest Z[1/p]-order of K preserving both Λ_{1} and Λ_{2}. Note that \underline{τ}(Ω_{N}(O)) = H_{p}^{O}/Γ. The group Pic^{+}(O) acts naturally on Ω_{N}(O) by translation:

𝔞 : (Λ_{1}, Λ_{2}) ↦ (𝔞Λ_{1}, 𝔞Λ_{2}),

and hence it also acts on \underline{τ}(Ω_{N}(O)) = H_{p}^{O}/Γ. Denote this latter action by

(𝔞, τ) ↦ 𝔞 ⋆ τ, for 𝔞 ∈ Pic^{+}(O), τ ∈ H_{p}^{O}/Γ.

Our conjectural reciprocity law then states:

Conjecture 4.3.3. If τ ∈ H_{p}^{O}, then Φ_{±}(τ) ∈ J_{0}(M)^{p-new}(H^{+}), and

Φ_{±}(𝔞 ⋆ τ) = rec(𝔞)^{−1}Φ_{±}(τ)

for all 𝔞 ∈ Pic^{+}(O).

Remark 4.3.4. Since H^{+} is a ring class field, the complex conjugation associated to either real place of K is the same in Gal(H^{+}/K). Let 𝔞_{∞} denote an element of Pic^{+}(O) corresponding to this complex conjugation. Then for either choice of sign ϵ = ±, we have

Φ_{ϵ}(𝔞_{∞} ⋆ τ) = ϵΦ_{ϵ}(τ).

The proof of this fact is identical to Proposition 5.13 of [5], since the map ν_{ϵ} factors through a torus on which the Hecke operator W_{∞} (see Definition 5.1.5) acts as ϵ.

The general conjecture that Stark–Heegner points are defined over global fields,

and certainly the full Conjecture 4.3.3 are very much open. However, theoretical evidence is provided in [2] and by the analogous constructions of part II. Computational evidence is provided in [7]. Theorem 4.2.1 is proven over the course of the next two chapters. We start by analyzing the Hecke structure of the torus T.

Chapter 5

Hecke actions

The homology groups we have already considered are endowed with natural Hecke

actions, which we now describe.

5.1 Definitions

Let Δ_{Q} = PGL_{2}(Q), and let Δ denote one of the groups Γ, Γ_{0}(N), or Γ_{0}(M),

considered as a subgroup of Δ_{Q}. For α ∈ PGL_{2}(Q), define

ΔαΔ = ⨆_{i} α_{i}Δ, α_{i} ∈ Δ_{Q}  (5.1)

be a decomposition of the indicated double coset into left cosets.

Definition 5.1.1. Let A be a Δ_{Q}-module, and let α ∈ Δ_{Q}. Define the Hecke operator T (α) on the group of Δ-co-invariants of A as follows. Let m ∈ A represent the element m̃ ∈ H_{0}(Δ, A), and let

T (α)m̃ := ∑_{i} { \atop α_{i}^{−1}m} ∈ H_{0}(Δ, A).

This definition is clearly independent of the choice of α_{i}. Also, for each γ ∈ Δ and

each α_{i}, there exist unique j and γ_{i} ∈ Δ such that

γ^{−1}α_{i} = α_{j}γ_{i}^{−1}.  (5.2)

For δ fixed, the correspondence i ↦ j is a permutation. This implies that the definition of T (α) is independent of choice of representative m for m̃.

We define the Hecke operators on H_{1}(Δ, A) for i > 0 by dimension-shifting. Ten-

soring the exact sequence

0 → Div_{0} Δ_{Q} → Div Δ_{Q} → Z → 0

with A and taking Δ-homology yields

H_{1}(Δ, A) ≅ ker(H_{0}(Δ, (Div_{0} Δ_{Q}) ⊗ A) → H_{0}(Δ, (Div Δ_{Q}) ⊗ A)),  (5.3)
H_{i}(Δ, A) ≅ H_{i−1}(Δ, (Div_{0} Δ_{Q}) ⊗ A) for i > 1  (5.4)

since the induced modules (Div Δ_{Q}) ⊗ A have trivial homology H_{i}(Δ, (Div Δ_{Q}) ⊗ A) for i ≥ 1. The kernel in (5.3) is stable under T (α) for α ∈ Δ_{Q}. Thus the isomorphisms (5.3) and (5.4) allow one to inductively define the Hecke operators T (α) on H_{i}(Δ, A) for any Δ_{Q}-module A and i ≥ 0, with Definition 5.1.1 as the base case.

Remark 5.1.2. In (3.7) we described H_{1}(Δ, A) (with Δ = Γ and A = M) by dimensionshifting with Div Δ rather than Div Δ_{Q}. We obtained

H_{1}(Δ, A) ≅ ker(H_{0}(Δ, (Div_{0} Δ) ⊗ A) → A).  (5.5)

The purpose of using Div Δ_{Q} rather than Div Δ in the present context is that the former group has the structure of a Δ_{Q}-module, whereas the latter does not. However, the Hecke operator T (α) on H_{1}(Δ, A) may be described in terms of the isomorphism (5.5) as follows. Let

k = ∑([γ] − [δ]) ⊗ m

represent an element of H_{1}(Δ, A) as in (5.5). Then

T (α)k = ∑ ∑_{i}([γ_{i}] − [δ_{i}]) ⊗ α_{i}^{−1}m,

where the α_{i} and γ_{i} are as in (5.1) and (5.2), and the δ_{i} are obtained from δ as in (5.2).

Lemma 5.1.3. Given a short exact sequence of Δ_{Q}-modules

0 → A → B → C → 0,

the associated long exact sequence in Δ-homology is equivariant for the Hecke operators.

Proof. This is immediate except for possibly the boundary map δ : H_{i}(Δ, C) → H_{i−1}(Δ, A). By dimension-shifting, it suffices to consider the case i = 1. In this case, the map δ has the following explicit description. Let

d = ∑([γ] − [δ]) ⊗ c ∈ H_{1}(Δ, C)

as in (5.5). Lift each c to an element b of B. The element ∑ γ^{−1}b − δ^{−1}b ∈ B maps to 0 in C and hence comes from an element of A; the image of this element in H_{0}(Δ, A) is independent of all choices and is the value δ(d). We find that

T (α)(δ(d)) = ∑ ∑_{i} α_{i}^{−1}γ^{−1}b − α_{i}^{−1}δ^{−1}b  (5.6)

while

δ(T (α)(d)) = ∑ ∑_{i} γ_{i}^{−1}α_{i}^{−1}b − δ_{i}^{−1}α_{i}^{−1}b.  (5.7)

In view of (5.2) and the comments following, these values are equal in H_{0}(Δ, A).

When Δ = Γ_{0}(M) or Γ and ℓ is prime, we write T_{ℓ} or U_{ℓ} for T (\begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix}), according

to whether ℓ divides M or not. Recall from (4.3) we have

H̃ = (H_{0}(Γ_{0}(M), M)/ Tr(H_{0}(Γ_{0}(N), M) ⊕ H_{0}(P^{−1}Γ_{0}(N)P, M)))_{T} .

Define also

H̃′ = ker ∂ : H_{0}(Γ_{0}(M), M) → H_{0}(Γ_{0}(N), M) ⊕ H_{0}(P^{−1}Γ_{0}(N)P, M)

with the boundary map ∂ as in Section 3.2. We wish to define Hecke operators on H̃ and H̃′.

Proposition 5.1.4. The maps ∂ and Tr are equivariant with respect to the Hecke operators T_{ℓ} for ℓ ∤ M and U_{ℓ} for ℓ|N.

Proof. The map m ↦ P m defines an isomorphism

H_{0}(P^{−1}Γ_{0}(N)P, M) ≅ H_{0}(Γ_{0}(N), M).  (5.8)

One easily checks that this isomorphism is equivariant for all the operators T_{ℓ}. Letting j run over the elements of P^{1}(F_{p}), we may write

Γ_{0}(N) = ⨆_{j} γ_{j}Γ_{0}(M), P^{−1}Γ_{0}(N)P = ⨆_{j} P^{−1}β_{j}P Γ_{0}(M)

where γ_{j} = (\begin{matrix}a_{j} & b_{j} \\ Nc_{j} & d_{j} \end{matrix}) (resp. β_{j}) are elements of Γ_{0}(N) such that j = [a_{j} : c_{j}N] ∈ P^{1}(F_{p})

(resp. j = [b_{j} : d_{j}]). With the identification of (5.8), the maps ∂ and Tr become:

∂ : H_{0}(Γ_{0}(M), M) → H_{0}(Γ_{0}(N), M) ⊕ H_{0}(Γ_{0}(N), M)  (5.9)
m̃ ↦ (m̃, −P̃ m)  (5.10)

and

Tr : H_{0}(Γ_{0}(N), M) ⊕ H_{0}(Γ_{0}(N), M) → H_{0}(Γ_{0}(M), M)  (5.11)
(m̃, ñ) ↦ ∑ { \atop γ_{j}^{−1}m} − ∑ { \atop P^{−1}β_{j}^{−1}n}.  (5.12)

Write ∂ = ∂_{1} ⊕ ∂_{2} and Tr = Tr_{1} ⊕ Tr_{2} . It is clear that ∂_{1} is T_{ℓ}-equivariant. As for ∂_{2}, one checks that for each α_{i}, there exists a unique α_{j} such that α_{i}P^{−1} ∈ P^{−1}α_{j}Γ_{0}(M), and the correspondence i ↦ j is a permutation. This implies that ∂_{2} is T_{ℓ}-equivariant.

For Tr_{1} one similarly checks that for each α_{i} and γ_{j}, there exists a unique γ_{k} and

α_{h} such that α_{i}γ_{j} ∈ γ_{k}α_{h}Γ_{0}(M), and the map (i, j) ↦ (k, h) is a permutation. The same is true for the γ_{j} replaced by β_{j}P.

The proposition shows that the Hecke operators T_{ℓ} for ℓ ∤ M and U_{ℓ} for ℓ|N on

H_{0}(Γ_{0}(M), M) induce actions on H̃ and on H̃′. The situation for Hecke operators at p is more subtle. Let N denote the normalizer of Γ_{0}(M) in

R^{×}/U = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ PGL_{2}(Z[1/p]) such that N divides c\} ,  (5.13)

where U = Z[1/p]^{×} is embedded in R^{×} via scalar matrices. The determinant on R^{×}/U maps to U/U^{2}, which is a Klein 4-group. When restricted to N, the determinant map induces an isomorphism

N /Γ_{0}(M) ≅ U/U^{2} ≅ Z/2Z × Z/2Z.

Let α_{p} denote a matrix in N that maps to the image of p under the determinant map, and let α_{∞} be a matrix that maps to the image of −1. To be explicit, we may take

α_{p} = (\begin{matrix}p & y \\ M & px \end{matrix}) , α_{∞} = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) ,  (5.14)

where px − Ny = 1.

Definition 5.1.5. Let A be a PGL_{2}(Q)-module. The Atkin–Lehner involution at p acting on the homology groups of Γ_{0}(M) is given by W_{p} := T (α_{p}). The Atkin–Lehner involution at infinity is defined by W_{∞} := T (α_{∞}).

Lemma 5.1.6. The operator U_{p} + W_{p} on H_{0}(Γ_{0}(M), M) has image contained in the subgroup Tr_{2}(H_{0}(Γ_{0}(N), M)).

Proof. The image of m̃ under U_{p} + W_{p} is

(U_{p} + W_{p})m̃ = ∑_{u=0}^{p−1} { \atop α_{u}^{−1}m} + { \atop α_{p}^{−1}m},

where α_{u} = (\begin{matrix}p & u \\ 0 & 1 \end{matrix}) and α_{p} is as in (5.14). Now one checks that for each β_{i}, i ∈ P^{1}(F_{p}),

there exists a unique j = 0, …, p such that β_{i}P ∈ α_{j}Γ_{0}(M), and the map i ↦ j is a bijection. This implies that (U_{p} + W_{p})m̃ = Tr_{2}(m̃).

Lemma 5.1.7. The operator W_{p} interchanges the images of Tr_{1} and Tr_{2} in H_{0}(Γ_{0}(M), M).

Proof. For each i ∈ P^{1}(F_{p}), there is a unique j such that β_{i}P ∈ γ_{j}α_{p}Γ_{0}(M), and the map i ↦ j is a permutation.

Lemmas (5.1.6) and (5.1.7) imply that the image of Tr is preserved under both U_{p}

and W_{p}. Hence these operators have well defined induced actions on H̃, and their sum is the zero operator on H̃. A similar argument gives the same result for H̃′:

Lemma 5.1.8. The operator U_{p} +W_{p} is zero on the subspace ker ∂_{1} of H_{0}(Γ_{0}(M), M). The involution W_{p} interchanges ker ∂_{1} and ker ∂_{2}. Hence U_{p} and W_{p} have well defined induced actions on H̃′ = ker ∂, and U_{p} + W_{p} = 0 on H̃′.

Definition 5.1.9. Write W for the involutions U_{p} = −W_{p} on H̃ and H̃′. For Δ = Γ, (5.1) implies that the operator U_{p} = W_{p} is an involution; we write W for this involution as well.

5.2 Hecke equivariance of the integration map

In Section 4.2 we defined an integration map:

× ∫∫ ω_{univ} : ((Div_{0} H_{p}) ⊗ M)_{Γ} → T̃ = G_{m} ⊗ H̃.  (5.15)

In the previous section we defined a Hecke action on ((Div_{0} H_{p}) ⊗ M)_{Γ} and also on T̃. For this section, we will write μ for μ^{univ}.

Proposition 5.2.1. The integration map (5.15) is equivariant for T_{ℓ}, U_{ℓ} for ℓ ≠ p, and W_{∞}.

Proof. The first observation is that since ℓ ≠ p, we may take the same set of α_{i} in defining the Hecke operators for Γ and H̃. Also, we have α_{i}e^{∗} = e^{∗} for the distinguished edge e^{∗}. We now show that this implies

∑_{i=0}^{ℓ} μ_{α_{i}^{−1}m}(α_{i}^{−1}U) = T_{ℓ}(μ_{m}(U)).  (5.16)

For γ ∈ Γ, write γ^{−1}α_{i} = α_{γ(i)}γ_{i}^{−1} for some index γ(i) and γ_{i} ∈ Γ. Then

∑_{i=0}^{ℓ} μ_{α_{i}^{−1}m}(α_{i}^{−1}U_{γe^{∗}}) = ∑_{i=0}^{ℓ} μ_{α_{i}^{−1}m}(γ_{i}U_{e}^{∗})
= ∑_{i=0}^{ℓ} γ_{i}^{−1}α_{i}^{−1}m
= ∑_{j=0}^{ℓ} α_{j}^{−1}γ^{−1}m

This proves equation (5.16).

We now calculate, for k = ([τ_{2}] − [τ_{1}]) ⊗ m ∈ ((Div_{0} H_{p}) ⊗ M)_{Γ},

(× ∫∫ ω_{μ}) (T_{ℓ}k) = ∏_{i} × ∫_{α_{i}^{−1}τ_{1}}^{α_{i}^{−1}τ_{2}}∫_{m}ω_{μ}  (5.17)
= ∏_{i} lim_{||U||→0} ∏_{U∈U} (\frac{t_{U} − α_{i}^{−1}τ_{2}}{t_{U} − α_{i}^{−1}τ_{1}} ) ⊗ μ_{α_{i}^{−1}m}(U)
= ∏_{i} lim_{||U||→0} ∏_{U∈U} (\frac{α_{i}^{−1}t_{U} − α_{i}^{−1}τ_{2}}{α_{i}^{−1}t_{U} − α_{i}^{−1}τ_{1}} ) ⊗ μ_{α_{i}^{−1}m}(α_{i}^{−1}U)
= ∏_{i} lim_{||U||→0} ∏_{U∈U} (\frac{t_{U} − τ_{2}}{t_{U} − τ_{1}} ) ⊗ μ_{α_{i}^{−1}m}(α_{i}^{−1}U)  (5.18)
= lim_{||U||→0} ∏_{U∈U} (\frac{t_{U} − τ_{2}}{t_{U} − τ_{1}} ) ⊗ T_{ℓ}(μ_{m}(U)).  (5.19)

Equation (5.18) uses the fact that μ_{m}(P^{1}(Q_{p})) = 0, and (5.19) uses (5.16). The right-hand side of (5.19) is

T_{ℓ} ((× ∫∫ ω_{μ}) (k)) ,

as desired.

Proposition 5.2.2. The integration map (5.15) is W -equivariant.

Proof. The key to this proposition is that the matrix α_{p} defining W_{p} on H interchanges the vertices v^{∗} and w^{∗}, and hence sends the edge e^{∗} to its opposite. The action of W on the Γ-coinvariants of a PGL_{2}(Q)-module is given by the action of a matrix in R of determinant p. Thus for each γ ∈ Γ, if we let β = α_{p}^{−1}γα_{p} ∈ Γ, we find that

μ_{α_{p}^{−1}m}(α_{p}^{−1}U_{γe^{∗}}) = μ_{α_{p}^{−1}m}(α_{p}^{−1}γZ_{p}) = μ_{α_{p}^{−1}m}(βα_{p}^{−1}Z_{p})
= −μ_{α_{p}^{−1}m}(βZ_{p}) = −β^{−1}α_{p}^{−1}m
= −α_{p}^{−1}γ^{−1}m = W (μ_{m}(U_{γe^{∗}})).

The proof of the proposition now follows along the proof of Proposition 5.2.1.

5.3 The homology of M

In this section we study the Γ-module M. Two fractions x and y in P^{1}(Q) are

said to be adjacent if they can be written as x = a/b and y = c/d in P^{1}(Q) with a, b, c, d integral and bc − ad = 1. (This is a symmetric relation since we can write y = c/d and x = (−a)/(−b).)

Lemma 5.3.1. For adjacent x and y in P^{1}(Q), let D_{x,y} = [x]−[y] ∈ M. The D_{x,y} generate M, and the following relations generate all those satisfied by the D_{x,y}:

D_{x,y} + D_{y,x} = 0  (5.20)
D_{\frac{a}{b} , \frac{a+c}{b+d}} + D_{\frac{a+c}{b+d} , \frac{c}{d}} + D_{\frac{c}{d} , \frac{a}{b}} = 0.  (5.21)

Proof. The fact that the D_{x,y} generate M is standard: between any two fractions x_{0} < y_{0} we may take a Farey sequence with large enough denominator as a sequence of adjacent fractions from x_{0} to y_{0}. Thus the divisors of the form [x_{0}] − [y_{0}], and hence all degree-zero divisors, are generated by the D_{x,y}.

Consider a relation among the D_{x,y}. Using relation (5.20), we may assume that

all the coefficients in the relation are positive. It is clear that such a relation can be written as a sum of relations of the form

r : ∑_{i=0}^{n−1} D_{x_{i},x_{i+1}} = 0

where the indices x_{i} are taken modulo n. If n = 2 then r is evidently of the form (5.20); assume that n > 2. We claim that if x_{0} = a/b and x_{n−1} = c/d with bc − ad = 1, then either (a + c)/(b + d) or (a − c)/(b − d) appears among the x_{i}. By applying the transformation x ↦ (bx−a)/(−dx+c), which sends adjacent fractions to adjacent fractions, we may assume that x_{0} = 0 and x_{n−1} = ∞. Now 0 is adjacent to fractions of the form 1/d, which lie in the interval [−1, 1], and ∞ is adjacent to the integers. Thus it suffices to show that adjacent fractions in Q cannot contain an integer between them; for then 1 or −1 appears among the x_{i}, which is equivalent to (a + c)/(b + d) or (a − c)/(b − d) appearing among the original x_{i}. Hence suppose we have a/b < m < c/d with bc − ad = 1 and m an integer. Multiplying by bd we obtain

ad < bdm < bc or ad > bdm > bc.

Both possibilities contradict bc − ad = 1.

Thus we have shown that there is some x_{j} equal to (a+c)/(b+d) or (a−c)/(b−d).

We can then write our original relation as:

r : ∑_{i=0}^{j−1} (D_{x_{i},x_{i+1}} + D_{x_{j},x_{0}}) + (D_{x_{0},x_{j}} + D_{x_{j},x_{n−1}} + D_{x_{n−1},x_{0}})
+ ∑_{i=j}^{n−2} (D_{x_{i},x_{i+1}} + D_{x_{n−1},x_{j}}) − (D_{x_{j},x_{0}} + D_{x_{0},x_{j}}) − (D_{x_{j},x_{n−1}} + D_{x_{n−1},x_{j}}) . (5.22)

The second term in (5.22) is of the form (5.20), while the last two are of the form (5.21). The other two terms are relations that are shorter in length than r; continuing to decompose the relations in this fashion gives the desired result.

In fact, the relations of Lemma 5.3.1 are independent. Write Θ := PSL_{2}(Z), and

let S = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) and T = (\begin{matrix}−1 & 1 \\ −1 & 0 \end{matrix}) be the standard generators of Θ with order 2 and 3,

respectively. Let S\Θ and T \Θ denote the sets of right cosets of the subgroups generated by S and T. Define maps

Z[S\Θ] ⊕ Z[T \Θ] ⟶^{s} Z[Θ] ⟶^{ϵ} D,

where s is summation:

s({θ, Sθ}) := [θ] + [Sθ] and s({θ, T θ, T^{2}θ}) := [θ] + [T θ] + [T^{2}θ],

and ϵ is defined by ϵ(θ) := θ^{−1}(0) − θ^{−1}(∞).

Proposition 5.3.2. The relations of Lemma 5.3.1 are independent. That is, the sequence of right Θ-modules

0 ⟶ Z[S\Θ] ⊕ Z[T \Θ] ⟶^{s} Z[Θ] ⟶^{ϵ} M ⟶ 0

is exact.

Proof. Lemma 5.3.1 implies exactness except for injectivity on the left. To prove the proposition, construct the undirected Cayley graph for Θ with respect to the subset consisting of {S, T, T^{2}}. This is the graph whose vertices are the elements of Θ and whose edges connect elements in the same right coset S\Θ or T \Θ. Now suppose we have a dependence among the relations. Since each simple divisor [θ] appears in exactly two relations, namely one with the other member of its S-coset and one with the other two members of its T -coset, the coefficients of these two relations in our dependence must be negatives of each other. Label each vertex θ with the coefficient a_{θ} of its S-relation. By definition, the vertex corresponding to Sθ has the same label: a_{Sθ} = a_{θ}. Since the T -relation of θ has coefficient −a_{θ}, the vertex corresponding to T θ must also be labelled a_{θ}. Continuing in this fashion, we see that the entire connected component of θ must have the same label. But S and T generate Θ, so the graph is connected; as Θ is infinite, we must have a_{θ} = 0 for all θ. This proves the proposition.

Proposition 5.3.3. The group H_{1}(Θ, M) is isomorphic to Z. A generator for H_{1}(Θ, M) is given by q_{Θ} = ([ST] − [T^{2}] + [ST S] − [T^{2}S]) ⊗ ([0] − [∞]).

Proof. Taking the Θ-homology of the short exact sequence of Proposition 5.3.2 yields

0 ⟶ H_{1}(Θ, M) ⟶ Z ⊕ Z ⟶^{s} Z ⟶ 0.

This shows immediately that H_{1}(Θ, M) ≅ Z. Furthermore, the map s sends (1, 0) to 2 and (0, 1) to 3, so the kernel is generated by (3, −2); in other words, H_{1}(Θ, M) is generated by each element of the form

(∑_{i=1}^{3}([θ_{i}] + [Sθ_{i}]) − ∑_{i=1}^{2}([θ′_{i}] + [T θ′_{i}] + [T^{2}θ′_{i}])) ⊗ ([0] − [∞]).

for any θ_{i} and θ′_{i} in Θ. The expression for q_{Θ} given in the proposition involves choosing these matrices to obtain as much cancellation as possible.

Lemma 5.3.4. The group H_{1}(Γ_{0}(N), M) is isomorphic to Z, and its image in H_{1}(Θ, M) corresponding to the inclusion Γ_{0}(N) ⊂ Θ is c_{N} · δ_{N} · H_{1}(Θ, M), where

c_{N} = #P^{1}(Z/NZ) = N ∏_{ℓ|N} (1 + \frac{1}{ℓ})

and

δ_{N} = (\begin{cases}1 & if − 1 is a square in Z/NZ \\ 1/2 & otherwise \end{cases}) · (\begin{cases}1 & if − 3 is a square in Z/NZ \\ 1/3 & otherwise \end{cases}) .

Proof. The coset space Θ/Γ_{0}(N) is identified with P^{1}(Z/NZ) by

(\begin{matrix}a & b \\ c & d \end{matrix}) ⟷ [a : c].

Taking the homology sequence for Γ_{0}(N) corresponding to the short exact sequence in Proposition 5.3.2 we obtain

[figure: 0; H_{1}(Γ_{0}(N), M); Z[S\P^{1}(Z/NZ)] ⊕ Z[T \P^{1}(Z/NZ)]; s; Z[P^{1}(Z/NZ)]; H_{0}(Γ_{0}(N), M); 0.]  (5.23)

Here S\P^{1}(Z/NZ) represents the quotient of P^{1}(Z/NZ) by the left action of S; the elements of S\P^{1}(Z/NZ) should be thought of as multisets (in particular, if i ∈ P^{1}(Z/NZ) such that Si = i then the image of [i] under s is 2[i] ∈ Z[P^{1}(Z/NZ)]).

Consider an element r in the kernel of s. The graph theory argument given in

the proof of Proposition 5.3.2 applies here again to show that the coefficients of all the T -orbits in r must be equal to a constant a (except for T -orbits that involve a fixed point of T, whose coefficients are a/3) and the coefficients of the S-orbits in r must equal −a (except for S-fixed points, which have coefficient −a/2). Unlike the case of Proposition 5.3.2, this does not imply that a = 0 because here our graph is finite. Indeed, this shows that H_{1}(Γ_{0}(N), M) is isomorphic to Z and gives its image in H_{1}(Θ, M). For example, if neither −1 nor −3 are squares in Z/NZ, then there are no fixed points for the action of S and T on P^{1}(Z/NZ). Then a generator for H_{1}(Γ_{0}(N), M) is the sum of all c_{N}/3 of the T -orbits minus the sum of all c_{N}/2 of the S-orbits; hence the image of H_{1}(Γ_{0}(N), M) in H_{1}(Θ, M) is c_{N} · H_{1}(Θ, M). The other cases are similar.

Lemma 5.3.5. The operator W_{p} acts as the identity on H_{1}(Γ_{0}(p), M). The operator W_{∞} acts as −1 on H_{1}(Γ_{0}(p), M) and H_{1}(Θ, M).

Proof. Taking the long exact sequence in homology associated to the sequence defining M, we obtain

H_{2}(Γ_{0}(p), Z) ⟶ H_{1}(Γ_{0}(p), M) ⟶^{ν} H_{1}(Γ_{0}(p), Div P^{1}(Q)) ⟶^{ι} H_{1}(Γ_{0}(p), Z).

(5.24)

Up to torsion, we can identify the ends of this sequence with the corresponding

Betti homology groups of the open curve Y_{0}(p):

H_{∗}(Γ_{0}(p), Z)_{T} ≅ H_{∗}(Y_{0}(p)(C), Z).

The manifold Y_{0}(p)(C) is a genus(X_{0}(p))-holed torus with two points removed.

There are two Γ_{0}(p)-equivalence classes of cusps, namely those of 0 and ∞. These

two cusps have infinite cyclic stabilizers U_{0} and U_{∞}, generated by

u_{0} = ( \begin{matrix}1 & 0 \\ p & 1 \end{matrix} ) and u_{∞} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

The Γ_{0}(p)-module Div P^{1}(Q) is the direct sum of two induced trivial modules, one from U_{0} and one from U_{∞}. Shapiro’s lemma then yields

H_{1}(Γ_{0}(p), Div P^{1}(Q)) = H_{1}(U_{0}, Z) ⊕ H_{1}(U_{∞}, Z) = U_{0} ⊕ U_{∞} ≅ Z ⊕ Z.

In terms of the topological interpretation, the map ι in (5.24) is the inclusion of simple loops around the two cusps (i.e., the removed points in the torus) in H_{1}(Y_{0}(p)(C), Z). As the difference of these two loops is clearly a boundary, they are equivalent in H_{1}. Thus the kernel of ι is generated by (u_{0}, u_{∞}^{−1}). ^{1}

The action of W_{p} is now easy to calculate. Unwinding the identifications in

Shapiro’s Lemma, we find that the element (u_{0}, u_{∞}^{−1}) corresponds to the element

([u_{0}] − [Id]) ⊗ [0] − ([u_{∞}] − [Id]) ⊗ [∞] ∈ H_{1}(Γ_{0}(p), Div P^{1}(Q)).

1One might object that the statement about the action of W_{p} on H_{1}(Γ_{0}(p), Z) is clearly correct up to a sign because it is an involution, and we have not been careful in checking orientations and signs; but to show that our signs are correct in describing the kernel of ι, it suffices to check that there is a group homomorphism Γ_{0}(p) → Z that takes the same non-zero value on u_{0} and u_{∞}. Indeed such a homomorphism exists; Mazur [32, §II.2] calls it the Dedekind–Rademacher homomorphism.

Conjugation by α_{p} takes u_{0} to u_{∞}^{−1} and u_{∞} to u_{0}^{−1}. Hence on U_{0} ⊕ U_{∞}, the involution W_{p} acts as (a, b) ↦ (−b, −a). Thus the action of W_{p} fixes elements of the form (x, −x). This proves that W_{p} acts as the identity on H_{1}(Γ_{0}(p), M). The proof that W_{∞} acts as −1 on H_{1}(Γ_{0}(p), M) and H_{1}(Θ, M) is much simpler and will be omitted.

Consider now the exact sequence (3.14) in the case where N = 1. We obtain

H_{1}(Γ_{0}(p), M) → H_{1}(Θ, M)^{2} → H_{1}(Γ, M).

Lemmas 5.3.4 and 5.3.5 imply that the first map sends a generator of H_{1}(Γ_{0}(p), M) to

(−(p + 1) · δ_{p} · q_{Θ}, (p + 1) · δ_{p} · q_{Θ}).

In particular, the map H_{1}(Θ, M) → H_{1}(Γ, M) induced by the inclusion Θ ⊂ Γ is an injection.

Lemma 5.3.6. The Hecke module H_{1}(Γ_{0}(N), M) is Eisenstein; that is, T_{ℓ} acts as multiplication by ℓ + 1 for ℓ ∤ N.

Proof. Since H_{1}(Γ_{0}(N), M) injects into H_{1}(Θ, M) in a T_{ℓ}-equivariant manner for ℓ ∤ N, it suffices to consider the case N = 1. Let α_{i} be as in (5.1) giving the action of T_{ℓ}. Write q_{Θ} = B ⊗ ([0] − [∞]) with

B = [ST] − [T^{2}] + [ST S] − [T^{2}S],

and let

B_{i} = [(ST)_{i}] + [(ST S)_{i}] − [(T^{2})_{i}] − [(T^{2}S)_{i}],

where the matrices with index i are as in (5.2). Since H_{1}(Θ, M) ≅ Z, we may write T_{ℓ}q_{Θ} = a_{ℓ}q_{Θ} for some integer a_{ℓ}. Let Γ_{ℓ} = SL_{2}(Z[1/ℓ]), and denote by q_{ℓ} the image of q_{Θ} in H_{1}(Γ_{ℓ}, M). We have

a_{ℓ}q_{ℓ} = the image of T_{ℓ}q_{Θ} in H_{1}(Γ_{ℓ}, M)
= ∑_{i=0}^{ℓ} B_{i} ⊗ ([α_{i}^{−1}0] − [α_{i}^{−1}∞])
= ∑_{i=0}^{ℓ} α_{ℓ}^{−1}α_{i}B_{i} ⊗ ([α_{ℓ}^{−1}0] − [α_{ℓ}^{−1}∞])  (5.25)

where α_{ℓ} ∈ Γ_{ℓ} is a fixed matrix of determinant ℓ, since then α_{ℓ}^{−1}α_{i} ∈ Γ_{ℓ}. Since α_{ℓ} gives the action of W_{ℓ} on H_{1}(Γ_{ℓ}, M), (5.25) equals

W_{ℓ} ∑_{i=0}^{ℓ} α_{i}B_{i}α_{ℓ}^{−1} ⊗ ([0] − [∞]) = W_{ℓ} ∑_{i=0}^{ℓ} Bα_{i}α_{ℓ}^{−1} ⊗ ([0] − [∞])
= W_{ℓ} ∑_{i=0}^{ℓ} B ⊗ ([0] − [∞])  (5.26)
= (ℓ + 1)W_{ℓ}(q_{ℓ}) = (ℓ + 1)q_{ℓ}.  (5.27)

Here (5.27) uses Lemmas 5.3.4 and 5.3.5, along with the W_{ℓ}-equivariance of

H_{1}(Γ_{0}(ℓ), M) → H_{1}(Γ_{ℓ}, M),

which is immediate from the definitions. Equation (5.26) results from Remark 3.2.2. Since H_{1}(Θ, M) injects into H_{1}(Γ_{ℓ}, M), we see that a_{ℓ} = ℓ + 1 as desired.

5.4 The lattice L

From (3.14) we have the exact sequence

H_{1}(Γ_{0}(N), M)^{2} → H_{1}(Γ, M) → H_{0}(Γ_{0}(M), M) → H_{0}(Γ_{0}(N), M)^{2}.  (5.28)

The map H_{1}(Γ, M) → H_{0}(Γ_{0}(M), M) is the map ϕ in (3.12). Hence (5.28) combined with Proposition 3.3.1 implies that the image of H_{1}(Γ_{0}(N), M)^{2} in H_{1}(Γ, M) under the integration map Φ_{1} has trivial p-adic valuation. Thus, for the image of the integration map to be discrete in T, it must be the case that the image of H_{1}(Γ_{0}(N), M)^{2} in T is finite. We will prove this by exploiting Hecke actions.

Define a modified Eisenstein ideal I of the (abstract) Hecke algebra by letting I

be generated by T_{ℓ} − (ℓ + 1) for ℓ ∤ M, (p + 1)W − (p + 1), and W_{∞} + 1.

Lemma 5.4.1. The ideal I annihilates the image of H_{1}(Γ_{0}(N), M)^{2} under the integration map Φ_{1}.

Proof. The map H_{1}(Γ_{0}(N), M)^{2} → H_{1}(Γ, M) is Hecke-equivariant for T_{ℓ}, ℓ ∤ M, and W_{∞} by basic computation; from Lemma 5.1.3 and Section 5.2 it follows that Φ_{1} is Heckeequivariant. Also, the action of T_{p} on H_{1}(Γ_{0}(N), M) is given by p + 1 matrices α_{i} of determinant p, hence the action of T_{p} on H_{1}(Γ_{0}(N), M) becomes that of (p + 1)W on its image in H_{1}(Γ, M). The result now follows from Lemmas 5.3.5 and 5.3.6.

Let H^{∗} = Hom(H, Z) denote the dual of H, so T (Q_{p}) = Hom(H^{∗}, Q_{p}^{×}). It is a

standard fact that H^{∗}/IH^{∗} is finite: after tensoring with C, one obtains the space of holomorphic Eisenstein series that are cusp forms (of which there are none).^{2} Thus Lemma 5.4.1 implies that the image of H_{1}(Γ_{0}(N), M)^{2} under the integration map Φ_{1} is finite.

Proposition 5.4.2. The group of periods L ⊂ T (Q_{p}) is a Hecke-stable lattice of rank 2g.

Proof. Lemma 5.1.3 implies the Hecke-equivariance of the boundary maps

H_{2}(Γ, Z) → H_{1}(Γ, M) and H_{1}(Γ, M) → H_{0}(Γ, (Div_{0} H_{p}) ⊗ M).

Hence the propositions of Section 5.2 imply that Φ_{2} is Hecke-equivariant, and thus that L is Hecke-stable.

From (5.28), the kernel of the map

H_{1}(Γ, M) → H̃′ = ker ∂ ≅ ker f_{∗}^{c}

is the image of H_{1}(Γ_{0}(N), M)^{2}, which has finite image under the integration map. Thus it remains to show that the image of H_{2}(Γ, Z) in H̃′ injects into H and has rank 2g. The group H_{2}(Γ, Z) may be understood using the sequence (3.3) again:

H_{2}(Γ_{0}(N), Z)^{2} → H_{2}(Γ, Z) → H_{1}(Γ_{0}(M), Z) → H_{1}(Γ_{0}(N), Z)^{2}.  (5.29)

As in the proof of Proposition 4.3.1, the homology of Γ_{0}(N) may be identified with that of Y_{0}(N) outside of 2 and 3-torsion. Since Y_{0}(N) is a non-compact Riemann surface, the group H_{2}(Y_{0}(N), Z) vanishes. And the right-hand arrow of (5.29) may be identified with

f_{∗}^{Y} : H_{1}(Y_{0}(M), Z) → H_{1}(Y_{0}(N), Z)^{2}.

Thus the image of H_{2}(Γ, Z) in H̃′ = ker f_{∗}^{c} is precisely H′ := ker f_{∗}. We need to show that H′ injects into H = coker f^{∗} and has finite cokernel. Yet the endomorphism f_{∗} ∘ f^{∗} of H_{1}(X_{0}(N), Z)^{2} can be given explicitly by the matrix

f_{∗} ∘ f^{∗} = (\begin{matrix}p + 1 & T_{p} \\ T_{p} & p + 1 \end{matrix}) .

Since the eigenvalues of T_{p} are bounded by 2\sqrt{p}, this endomorphism is injective and has finite cokernel; the result follows.

2In the case where N = 1, Mazur [31] has conducted a detailed analysis of the group H^{∗}/IH^{∗}. When I includes the element W − 1 rather than just (p + 1)(W − 1) as in our setting, Mazur finds that H^{∗}/IH^{∗} is cyclic of size n, where n is the numerator of the fraction (p − 1)/12.

Remark 5.4.3. The finite group H/ ker f^{∗} reflects congruences between modular forms of level N and M. In [3], the images of Stark–Heegner points in this finite Hecke module under the p-adic valuation map is explored.

Chapter 6

Proof of the uniformization

Let 𝒯 denote the Hecke algebra of H (that is, the subring of the ring of endo-

morphisms of the group H generated over Z by T_{ℓ} for ℓ ∤ M, U_{ℓ} for ℓ|N, and W). In Proposition 3.3.1 we gave a combinatorial description of the map ord_{p} : L → H ⊗ Z_{p}:

L ⊂ T (Q_{p}) ⟶^{ord_{p}} H ⊗ Z ⟶ H ⊗ Z_{p}.

Consider now the logarithm log_{p} : L → H ⊗ Z_{p} given by

L ⊂ T (Q_{p}) ⟶^{log_{p}} H ⊗ Z_{p},

where now and throughout this thesis we choose the branch of the logarithm for which log_{p}(p) = 0. We will define an element ℒ_{p} ∈ 𝒯 ⊗ Z_{p}, called the ℒ -invariant of T/L, such that

ℒ_{p} ord_{p} λ = log_{p} λ for all λ ∈ L.  (6.1)

Remark 6.0.4. Let H_{±} = H/(W_{∞} ∓ 1) be the maximal quotients of H on which complex conjugation acts as a scalar ±1. The fact that there is an element ℒ_{p} in 𝒯 ⊗ Q_{p} satisfying (6.1) for each of the factors H_{−} ⊗ Q_{p} and H_{+} ⊗ Q_{p} follows from the fact that each of these modules is free of rank one over 𝒯 ⊗ Q_{p}. The fact that the same ℒ_{p} works on each factor, and that this element is integral, follows from our specific construction and proof.

Our goal is to connect T/L with the abelian variety J = J_{0}(M)^{p-new}. This abelian

variety has purely toric reduction at p, and its p-adic uniformization can be described as follows. Let S denote the set of supersingular points in characteristic p on X_{0}(N), and let X := Div_{0} S denote the group of degree-zero divisors on S. The group X has a natural Hecke action: by T_{ℓ} for ℓ ∤ M (and U_{ℓ} for ℓ|M, including ℓ = p) by sending a supersingular point on X_{0}(N) to the formal sum of the ℓ + 1 (resp. ℓ) ℓ-isogenous supersingular points, counted with multiplicity; the operator W = U_{p} has order two and is also given by the action of Gal(F_{p^{2}}/F_{p}) on the supersingular points. It is well known that the Hecke algebra of X equals that of H; in other words, there is a ring homomorphism 𝒯 → End(X) sending T_{ℓ} ↦ T_{ℓ}, etc.^{1} Thus we may consider X as a module for 𝒯 . Let G_{p} = Gal(Q̄_{p}/Q_{p}) act on X (via the Gal(F_{p^{2}}/F_{p})-action on S) and on Hom(X, Q̄_{p}^{×}) by σ(h)(x) = σ(h(σ^{−1}(x))). There is a rigid analytic 𝒯 [G_{p}]-equivariant isomorphism

J(Q̄_{p}) ≅ Hom(X, Q̄_{p}^{×})/X,

where the inclusion

Q : X → Hom(X, Q_{p}^{×}) = X^{∗} ⊗ Q_{p}^{×}

is given by a symmetric pairing

X × X → Q_{p}^{×}.  (6.2)

(Here we have written X^{∗} = Hom(X, Z).) The Hecke operators are self-adjoint for this pairing. Composing (6.2) with the p-adic valuation gives a Z-valued pairing (the “monodromy pairing”) on X which is non-degenerate, and hence yields an injection

ord_{p} Q : X → X^{∗}.

Similarly, composing Q with log_{p} yields

log_{p} Q : X → X^{∗} ⊗ Z_{p}.

We will then show, with ℒ_{p} as in (6.1), the relation

ℒ_{p} ord_{p} Q(x) = log_{p} Q(x) for all x ∈ X.  (6.3)

Equations (6.1) and (6.3) imply theorem 4.2.1 as follows. For a set of primes P,

we say that two analytic spaces are P-isogenous if there is an isogeny between them whose degree is supported on the elements of P. Let π_{±} : H → H_{±} denote the natural projections.

1This is because after tensoring with C, the spectrum of T_{ℓ} on X consists of the ℓ-th Fourier coefficients of a basis of p-new forms of level M; the spectrum of T_{ℓ} on H consists of each these eigenvalues repeated twice.

Since Φ_{2} is equivariant for W_{∞}, and H → H_{−} ⊕ H_{+} has cokernel supported at 2, it follows that T/L is {2}-isogenous to

(T/L)_{−} ⊕ (T/L)_{+} := (G_{m} ⊗ H_{−})/π_{−}L ⊕ (G_{m} ⊗ H_{+})/π_{+}L.

We will show that each of (T/L)_{±} is isogenous to J.

Recall the Hecke-equivariant map ψ : H̃ → H from Section 4.2 used to define our

modular symbol valued in H, and denote by ψ_{−} : H̃_{−} → H_{−} the induced map obtained by modding out by W_{∞} + 1. Recall that H′ = ker f_{∗} and let H′_{−} be its corresponding quotient.

Since all of the groups below are free of rank 1 over 𝒯 ⊗ Q after tensoring with Q,

it is possible to find Hecke-equivariant maps ξ_{−} and ξ′_{−} fitting into a commutative diagram

\xymatrix{H′_{−} \ar[r] \ar[d]_{ξ′_{−}} & H̃_{−} \ar[r]^{ψ_{−}} & H_{−} \ar[d]^{ξ_{−}} \\ X \ar[rr]^{ord_{p} Q} &  & X^{∗},}  (6.4)

where the horizontal arrow H′_{−} → H̃_{−} is the natural inclusion. Recall that

1. The map Φ_{2} : H_{2}(Γ, Z) → T factors through Φ_{1} : H_{1}(Γ, M) → T.

2. The composite

H_{1}(Γ_{0}(N), M)^{2} ⟶ H_{1}(Γ, M) ⟶^{Φ_{1}} T

has finite image.

3. The image of H_{2}(Γ, Z) in H_{1}(Γ, M)/H_{1}(Γ_{0}(N), M)^{2} is canonically identified with H′ (see the proof of Proposition 5.4.2).

The identification of (3) implies that ξ′_{−} induces a map H_{2}(Γ, Z)_{−} → X, also denoted ξ′_{−}. Consider the diagram

\xymatrix{H_{2}(Γ, Z)_{−} \ar[r]^{π_{−}Φ_{2}} \ar[d]_{ξ′_{−}} & Q_{p}^{×} ⊗ H_{−} \ar[d]^{ξ_{−}} \\ X \ar[r]^{Q} & Q_{p}^{×} ⊗ X^{∗}.}  (6.5)

Proposition 3.3.1 implies that the composition of the top row of (6.5) with ord_{p} is equal to the composition of H_{2}(Γ, Z)_{−} → H′_{−} with the top row of (6.4). Thus the commutativity of (6.4) implies the commutativity of the p-adic valuation of (6.5). Since all the maps in (6.5) are Hecke-equivariant, equations (6.1) and (6.3) show that the commutativity of the ord_{p} of (6.5) automatically implies the commutativity of the log_{p} of (6.5). Thus the diagram (6.5) itself commutes, up to elements in the kernel of both log_{p} and ord_{p}; these elements are torsion of order dividing p − 1 (or 2 if p = 2).

Hence the map

Id ⊗ξ_{−} : G_{m} ⊗ H_{−} → G_{m} ⊗ X^{∗}

induces an isogeny (T/Λ)_{−} → J. Furthermore, the kernel of this isogeny is identified with the cokernel of

ξ′_{−} : H′_{−} → X.

A similar argument for H_{+} then proves Theorem 4.2.1 and furthermore bounds

the primes dividing the degree of the isogeny to lie in

P = {ℓ : ℓ divides 2(p − 1) or the size of either coker ξ′_{±} : H′_{±} → X}.

6.1 Connection with the Mazur–Tate–Teitelbaum conjecture

Let E be an elliptic curve with conductor Np and split multiplicative reduction

at p. In this chapter, we show that Theorem 4.2.1 implies the Mazur–Tate–Teitelbaum conjecture for E.

Let I_{E}^{+} denote the ideal of the Hecke algebra of H̃ corresponding to E and the

“plus” modular symbol (that is, the ideal generated by T_{ℓ} − a_{ℓ} for ℓ ∤ Np, W_{p} − 1, and W_{∞} −1, where a_{ℓ} = ℓ+1−#E(F_{ℓ})). The quotient H̃/I_{E}^{+} has rank 1 over Z; the projection

Ψ_{E}^{+} : H̃ → H_{E}^{+} := (H̃/I_{E}^{+})_{T} ≅ Z

is the plus modular symbol attached to E. We retain the notation of [35] and write

λ_{E}(a, M) := Ψ_{E}^{+} ([−\frac{a}{M} ] − [∞]) .

After tensoring with G_{m}, the projection Ψ_{E}^{+} yields a map φ : T̃ → G_{m}. According to Theorem 4.2.1, the quotient G_{m}/φ(L) is an analytic space isogenous to the elliptic curve E. This implies that every element of φ(L) is commensurable with the Tate period q_{E} of E. By evaluating a particular element of φ(L), we will deduce the MTT conjecture. In what follows, we denote

φ (× ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω_{univ})

by

× ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω_{E}^{+} ∈ C_{p}^{×}.

Consider the class c ∈ H_{1}(Γ, M) represented by the element

([(\begin{matrix}p & 0 \\ 0 & 1/p \end{matrix})] − [Id]) ⊗ ([0] − [∞]).

The element c is in the image of the boundary map from H_{2}(Γ, Z), since a simple calculation shows that the image of c in the next term of the exact sequence

H_{2}(Γ, Z) → H_{1}(Γ, M) → H_{1}(Γ, Div P^{1}(Q)) = Γ_{∞}^{ab}  (6.6)

vanishes (this is true for any class represented by ([γ] − [Id]) ⊗ ([x] − [y]), for γ stabilizing x and y). Here Γ_{∞} denotes the stabilizer of ∞ in Γ, and the equality of (6.6) follows from Shapiro’s Lemma. Thus the double integral

q′ := × ∫_{τ}^{p^{2}τ}∫_{0}^{∞}ω_{E}^{+} ∈ Q_{p}^{×}

lies in φ(L), and in particular does not depend on the choice of τ. In fact, since W_{p} = W_{∞} = 1 on H_{E}^{+}, it follows (see [5, Proposition 5.13]) that the multiplicative double integral is invariant under the full group R^{×}/U (defined in (5.13)). In particular, one finds that

q := × ∫ τ_{\frac{τ}{p}}∫_{0}^{∞}ω_{E}^{+} ∈ Q_{p}^{×}

is independent of τ and that q′ = q^{2}. Thus q is commensurable with q_{E} as well, so

\frac{log_{p}(q)}{ord_{p}(q)} = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})}.  (6.7)

To evaluate q, choose τ to reduce to the standard vertex v^{∗} of the tree T . Since

the matrix P^{−1} sends v^{∗} to w^{∗}, Lemma 3.3.2 implies that

ord_{p}(q) = Ψ_{E}^{+}([0] − [∞]) = λ_{E}(0, 1).  (6.8)

Let n ≥ 1; for a = 0, …, p^{n} − 1 define U_{a} := a + p^{n}Z_{p}. To evaluate log_{p} q ∈ Z_{p}

modulo p^{n}, it suffices to take a cover of P^{1}(Q) by the sets

U_{∞} := {t ∈ P^{1}(Q_{p}) : ord_{p}(t) < −n},  (6.9)
\frac{1}{p^{n}} U_{a}, \frac{1}{p^{n−1}} U_{a}, …, \frac{1}{p}U_{a} for a ≢ 0 (mod p),  (6.10)

and U_{a} for all a = 0, …, p^{n} − 1. The contributions to the integral defining log_{p} q from each of these terms are as follows:

U_{∞} : 0,
\frac{1}{p^{k}} U_{a} : log_{p} (\frac{a/p^{k} − τ}{a/p^{k} − τ/p}) λ_{E}(a, p^{n}) = log_{p} (\frac{a − p^{k}τ}{a − p^{k−1}τ} ) λ_{E}(a, p^{n}), (6.11)
U_{a} : log_{p} (\frac{a − τ}{a − τ/p}) λ_{E}(a, p^{n}) = log_{p} (\frac{a − τ}{pa − τ} ) λ_{E}(a, p^{n}).  (6.12)

Summing these values (the terms of (6.11) for k = 1, …, n telescope, and the distribution relation

λ_{E}(a′, p^{n−1}) = ∑_{{a mod p^{n} \atop a≡a′ (mod p^{n−1})}} λ_{E}(a, p^{n})

allows one to cancel terms in the denominator of (6.12) for all a with the terms in the numerator for p|a) one obtains

log_{p}(q) ≡ ∑_{{a=1 \atop (a,p)=1}}^{p^{n}−1} log_{p}(a)λ_{E}(a, p^{n}) (mod p^{n}).  (6.13)

Equations (6.7), (6.8), and (6.13) yield Conjecture II.13.1 of [35] for the trivial

character. The more general statement for a character of conductor c > 1 may be obtained by repeating our analysis above for each (v, c) = 1 with the element

([γ] − [Id]) ⊗ ([v/c] − [∞]) ∈ H_{1}(Γ, M),

with γ ∈ Γ stabilizing v/c and ∞. We omit the details (see [5, §2.3]).

6.2 Hida families and the definition of ℒ_{p}

In this chapter we define the element ℒ_{p} ∈ 𝒯 ⊗ Z_{p} that we later show satisfies

(6.1) and (6.3). Let X_{0} = X_{0}(Np), and define a tower of curves X_{r} above X_{0} corresponding to the congruence subgroups Γ_{r} := Γ_{0}(N) ∩ Γ_{1}(p^{r}), for r ≥ 1:

X_{r}(C) = H^{∗}/Γ_{r}.

The points of X_{r} classify triples (E, C, P), where E is an elliptic curve, C a cyclic subgroup of size N, and P a point of order p^{r}. The natural maps X_{r+1} → X_{r} send (E, C, P) ↦ (E, C, pP) for r ≥ 1, and the map X_{1} → X_{0} sends (E, C, P) ↦ (E, ⟨C, P ⟩). Composing X_{r} → X_{0} with the two maps X_{0} → X_{0}(N) from (4.4), we obtain two degeneracy maps X_{r} → X_{0}(N) for each r ≥ 0 (there are actually r + 1 natural degeneracy maps for each r ≥ 1, but we will be interested in only these two). Let f_{r}^{∗} denote the pullback on homology from the two copies of X_{0}(N) to X_{r}, and define

H_{r} = [H_{1}(X_{r}, Z_{p})/f_{r}^{∗}(H_{1}(X_{0}(N), Z_{p})^{2})]_{T} ,

so that H_{0} = H ⊗ Z_{p}. Let 𝒯_{r} denote the Hecke algebra of H_{r}, generated over Z_{p} by T_{ℓ} for ℓ ∤ M, U_{ℓ} for ℓ|M, and the diamond operators ⟨d⟩ for d ∈ (Z/p^{r}Z)^{×}. Here 𝒯_{0} = 𝒯 ⊗ Z_{p}. Define the Hida Hecke algebra

T := lim_{⟵} 𝒯_{r},

which has the structure of Λ := Z_{p}[[Z_{p}^{×}]]-module, where the group elements of Z_{p}^{×} act via the diamond operators. Let I denote the augmentation ideal of Λ. If T^{o} denotes the image of T under Hida’s ordinary projector, Hida has shown:

Theorem 6.2.1 (Hida, [20], Corollary 3.2). The Λ-module T^{o} is free of finite rank and

T^{o}/IT^{o} ≅ 𝒯_{0}.

Remark 6.2.2. The map T^{o} → 𝒯_{0} of Theorem 6.2.1 is the natural projection. It is clear that IT^{o} lies in the kernel; Hida’s “control theorem” 6.2.1 is that IT^{o} is the entire kernel.

The standard identification ⟨d⟩ − 1 ↦ d yields an isomorphism

I/I^{2} ≅ lim_{←} Z_{p}^{×}/(Z_{p}^{×})^{p^{n}},

since the group Z_{p}^{×} is abelian. Composing with log_{p} : Z_{p}^{×} → Z_{p}, we obtain a map also denoted log_{p} : I/I^{2} → Z_{p}.

Let t be an element of T whose image in 𝒯_{0} vanishes. By Hida’s Theorem 6.2.1,

t^{o} lies in IT^{o} (where t^{o} is the image of t under the ordinary projector). Consider the image of t^{o} in

IT^{o}/I^{2}T^{o} = I/I^{2} ⊗_{Λ} T^{o}.

Using the map

log_{p} ⊗ Id : I/I^{2} ⊗_{Λ} T^{o} → Z_{p} ⊗_{Λ} T^{o}

we further map our element t to

Z_{p} ⊗_{Λ} T^{o} = Λ/I ⊗_{Λ} T^{o} = T^{o}/IT^{o} = 𝒯_{0}.

The image of t under this series of maps is denoted t′ ∈ 𝒯_{0}, to reflect the intuition that it represents the derivative of t in the direction of the level (i.e., the fact that t^{o} ∈ IT^{o} means that the “value” of the “function” t is 0 at the base of the tower, so its image in IT^{o}/I^{2}T^{o} is its “derivative”).

Since U_{p} = −W_{p} on H, and W_{p} is an involution, the element 1 − U_{p}^{2} vanishes in

𝒯 . We define:

ℒ_{p} := (1 − U_{p}^{2})′ ∈ 𝒯_{0}.

6.3 Proof that ℒ_{p} is the ℒ -invariant of T/L

Let τ ∈ H_{p} lie in the quadratic unramified extension K_{p} of Q_{p}, and assume further

that τ reduces to the central vertex v^{∗} of the tree T . Consider the map

β_{ℒ_{p}} : K_{p}^{×} ⊗ H → K_{p} ⊗ H, k ⊗ h ↦ log_{p}(k) ⊗ h − ℒ_{p}(ord_{p}(k) ⊗ h).

Composing the 2-cocycle d_{τ,x} ∈ Z^{2}(Γ, T (K_{p})) from Section 4.3 with β_{ℒ_{p}} yields

d_{τ,x}^{ℒ_{p}} ∈ Z^{2}(Γ, K_{p} ⊗ H), d_{τ,x}^{ℒ_{p}}(γ_{1}, γ_{2}) := β_{ℒ_{p}} (× ∫_{τ}^{γ_{1}τ}∫_{γ_{1}x}^{γ_{1}γ_{2}x}ω_{ψ}) .

As in Section 4.3, the lattice β_{ℒ_{p}} (L) is the smallest subgroup of K_{p} ⊗ H such that the cocycle d_{τ,x}^{ℒ_{p}} splits in the quotient; thus to prove equation (6.1), it suffices to prove that d_{τ,x}^{ℒ_{p}} splits.

We will in fact show a stronger result. Define a 1-cocycle

c_{τ} ∈ Z^{1}(Γ, Hom(M, T (K_{p})))

by the rule

c_{τ} (γ)(m) := × ∫_{τ}^{γτ}∫_{m}ω_{ψ}.

Composing c_{τ} with β_{ℒ_{p}} , we obtain a 1-cocycle

c_{τ}^{ℒ_{p}} ∈ Z^{1}(Γ, Hom(M, K_{p} ⊗ H)).

It is a basic calculation^{2} that the splitting of c_{τ}^{ℒ_{p}} implies the splitting of d_{τ,x}^{ℒ_{p}}; the splitting of c_{τ}^{ℒ_{p}} is in fact what we will show.

2If the 0-chain ρ_{τ} splits c_{τ}^{L_{p}}, then the 1-chain η_{τ,x} defined by η_{τ,x}(γ) = ρ_{τ} ([x] − [γx]) splits d_{τ,x}^{L_{p}}.

The main idea for splitting the cocycle c_{τ,x}^{ℒ_{p}} is to lift the modular symbol μ of

measures on P^{1}(Q_{p}) to a modular symbol of measures μ̃ on a Z_{p}^{×}-bundle X over P^{1}(Q_{p}). The space X := (Z_{p} × Z_{p})′ is defined to be the set of pairs (a, b) ∈ Z_{p} × Z_{p} such that a and b are not both divisible by p; this set of “primitive vectors” makes an appearance in the earlier work of Greenberg and Stevens [13]. The space X admits a map

π : X → P^{1}(Q_{p}), (a, b) ↦ a/b.

The fibers of π are principal homogeneous spaces for Z_{p}^{×}. If we consider the elements of X as column vectors and let GL_{2}(Z_{p}) act on the left, the map π is GL_{2}(Z_{p})-equivariant. In this chapter, we will consider the groups Γ_{0}(N), Γ_{r}, Γ, etc. as subgroups of GL_{2} (rather than PGL_{2} as in previous chapters). Remark 6.3.1. If the function f(t) = t−τ were integrable on P^{1}(Q_{p}), a formal calculation would show that

ρ_{τ} (m) = ∫_{P^{1}(Q_{p})}log_{p}(t − τ) dμ_{m}(t)

is an explicit splitting of the cocycle c_{τ}, i.e. that dρ_{τ} = c_{τ} . However, this is not the case since f(t) has a pole at t = ∞. This explains the role of the space X: the function f(a, b) = a−bτ is integrable on X, has a zero along the fiber over τ, and no poles. Proposition 6.3.2. There exists a Γ_{0}(N)-invariant Meas(X, H ⊗Z_{p})-valued modular symbol μ̃ such that

μ̃_{m}(Z_{p}^{×} × pZ_{p}) = μ_{m}(P^{1}(Q_{p}) − Z_{p}) = −ψ(m)  (6.14)

for all m ∈ M.

(Recall that the map ψ was defined in Section 4.2 to create a modular symbol

valued in H rather than H̃.) Remark 6.3.3. Since the Γ_{0}(N)-translates of P^{1}(Q_{p}) − Z_{p} form a disjoint open cover of P^{1}(Q_{p}), the Γ_{0}(N)-invariance of μ̃ combined with (6.14) implies that μ̃ has the same total measure as μ, namely 0:

μ̃_{m}(X) = 0 for all m ∈ M.

Proof. Our methods follow those of Greenberg and Stevens [13, pg. 203]. Let A denote a free Z_{p}-module of finite rank, viewed as a trivial Γ_{0}(N)-module. A Γ_{0}(N)-invariant Meas(X, A)-valued modular symbol is an element of

M(A) := H^{0}(Γ_{0}(N), Hom(M, Meas(X, A))).  (6.15)

For each r ≥ 1, let Γ_{r} = Γ_{0}(N) ∩ Γ_{1}(p^{r}) as in Section 6.2. The Γ_{0}(N)-module Meas(X, A) is isomorphic to an inverse limit of induced modules from the groups Γ_{r} as follows. Let X_{r} := (Z/p^{r}Z × Z/p^{r}Z)′, the set of primitive vectors in (Z/p^{r}Z)^{2}. Then we have

Meas(X, A) = lim_{⟵} Meas(X_{r}, A),

where the maps Meas(X_{r+1}, A) → Meas(X_{r}, A) are given by μ_{r+1} ↦ μ_{r}, where

μ_{r}(x) = ∑_{y≡x (mod p^{r})} μ_{r+1}(y).

The map Γ_{0}(N)/Γ_{r} → X_{r} given by

(\begin{matrix}a & b \\ c & d \end{matrix}) ↦ \binom{a}{c}

is a bijection and hence induces an isomorphism

Meas(X_{r}, A) ≅ Ind_{Γ_{r}}^{Γ_{0}(N)}(A).

Thus by Shapiro’s Lemma and the universal coefficient theorem, (6.15) can be identified with

lim_{⟵} Hom(H_{0}(Γ_{r}, M), A).  (6.16)

Concretely, an element of (6.16) is a sequence of maps φ_{r} : H_{0}(Γ_{r}, M) → A, compatible in the sense that

φ_{r}(m) = ∑_{i} φ_{r+1}(γ_{i}^{−1}m) ∈ A,

for all m ∈ M, where the γ_{i} range over a set of coset representatives for Γ_{r}/Γ_{r+1}. The sequence {φ_{r}} defines a Γ_{0}(N)-invariant Meas(X, A)-valued modular symbol by the rule:

μ̃_{m} (\{x ∈ X : x ≡ \binom{a}{c} (mod p^{r})\}) = φ_{r}(γ^{−1}m)

where γ is a matrix in Γ_{0}(N) that is equivalent to (\begin{matrix}a & ∗ \\ c & ∗ \end{matrix}) modulo p^{r}.

For an element μ̃ of (6.16) representing an element of M(A), the measure of the

compact open set Z_{p}^{×} × pZ_{p} (the inverse image under π of P^{1}(Q_{p}) − Z_{p}) is given by the image of μ̃ in Hom(H_{0}(Γ_{0}(Np), M), A).

Extending our maps via Z_{p}-linearity, and identifying H_{0}(Γ_{r}, M)_{T} geometrically

with H_{1}(X_{r}, cusps, Z), we may write

M(A) = lim_{⟵} Hom_{Z_{p}}(H_{1}(X_{r}, cusps, Z_{p}), A).

In relating the module M(A) to the work of Hida, it will be convenient to dualize the description above. Denote by Ǎ the Z_{p}-dual Hom_{Z_{p}}(A, Z_{p}); then for two finite free Z_{p}-modules A and B, it is clear that Hom_{Z_{p}}(A, B) = Hom_{Z_{p}}(B̌, Ǎ); we write the map corresponding to f ∈ Hom_{Z_{p}}(A, B) as f̌ ∈ Hom_{Z_{p}}(B̌, Ǎ), so f̌(g)(a) := g ∘ f(a). Identifying the dual of H_{1}(X_{r}, cusps, Z_{p}) with H_{1}(Y_{r}, Z_{p}) (where Y_{r} = X_{r} − cusps) via Poincaré duality, we then have

M(A) = lim_{⟵} Hom_{Z_{p}}(Ǎ, H_{1}(Y_{r}, Z_{p})).

The statement of the proposition is that there exists an element of M(H ⊗ Z_{p}) such that its image in Hom_{Z_{p}}(Ȟ, H_{1}(Y_{0}, Z_{p})) is precisely ψ̌. Such an element exists since the maps H_{1}(Y_{r}, Z_{p}) → H_{1}(Y_{r−1}, Z_{p}) are surjective.

In the course of the above proof, we showed M(A) = Hom_{Z_{p}}(M, A), where

M = lim_{⟶} H_{1}(X_{r}, cusps, Z_{p}).

As usual M has a Hecke algebra generated over Z_{p} by the diamond operators and the operators T_{ℓ}, etc.

Remark 6.3.4. A point of caution is in order: since the tower defining M is (essentially) dual to the tower of Section 6.2, one must correspondingly take the dual Hecke operators.

In other words, T_{ℓ} is given by T (\begin{matrix}1 & 0 \\ 0 & ℓ \end{matrix}) in the notation of Chapter 5, since these are the

operators that are compatible with the maps in the direct limit defining M. In particular,

the action of U_{p} is given by T (\begin{matrix}1 & 0 \\ 0 & p \end{matrix}), for which the α_{i} in (5.1) can be chosen to be

\{(\begin{matrix}1 & 0 \\ Np^{r}i & p \end{matrix})\}_{i=0}^{p−1}

for the group Δ = Γ_{r}.

Define the ordinary part M(A)^{o} ⊂ M(A) to be the set of homomorphisms that

factor through the ordinary projector M → M^{o} ⊂ M. In order to have U_{p} invertible, we will always assume that the modular symbol μ̃ arises from M^{o}.

Let the modular symbol μ̃ correspond to a Z_{p}-module homomorphism f : M^{o} → A.

For each element t ∈ End(M^{o}) of the Hecke algebra of M^{o}, the map f ∘ t : M^{o} → A yields another measure-valued modular symbol, which we denote by μ̃^{t}.

We also extend all measures on X to the larger space Y := Q_{p}^{2} − 0 by imposing

invariance under multiplication by p:

μ̃_{m}(pU) = μ̃_{m}(U)

for all compact opens U ⊂ Y; this extension is well defined because X forms a fundamental domain for the action of multiplication by p on Y. The purpose of this extension is that Y (considered as column vectors) has a natural action of Γ by left multiplication, whereas X

does not. Recall that P denotes the matrix (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) .

Proposition 6.3.5. Let U be an compact open subset of X and μ̃ as above. We have

μ̃_{P m}(P U) = \begin{cases}μ̃_{m}^{U_{p}}(U) & if U ⊂ Z_{p}^{×} × pZ_{p} \\ μ̃_{m}^{U_{p}^{−1}}(U) & if U ⊂ Z_{p} × Z_{p}^{×} \end{cases} .

Proof. Let U(a, c; r, s) denote the basic compact open set

U(a, c; r, s) = \{\binom{x}{y} ∈ X : x ≡ a (mod p^{r}), y ≡ c (mod p^{s})\} .

We will demonstrate the first case of the proposition by considering U = U(a, c; r, r) with Np|c and (a, p) = 1. Then

μ̃_{P m}(P U) = μ̃_{P m} (U (a, \frac{c}{p}; r, r − 1))
= ∑_{i=0}^{p−1} μ̃_{P m} (U (a, \frac{c}{p} + Np^{r−1}i; r, r))
= φ_{r} (∑_{i=0}^{p−1} (\begin{matrix}a & ∗ \\ \frac{c}{p} + Np^{r−1}i & ∗ \end{matrix})^{−1} P m)
= φ_{r} (∑_{i=0}^{p−1} (\begin{matrix}1 & 0 \\ Np^{r}i & p \end{matrix})^{−1} (\begin{matrix}a & ∗ \\ c & ∗ \end{matrix})^{−1} m)
= μ̃_{m}^{U_{p}}(U),

where the ∗’s are chosen so that the resulting matrices lie in Γ_{0}(N). This proves the first case of the proposition. Similarly, one shows that

μ̃_{P^{−1}m}(P^{−1}V) = μ̃_{m}^{U_{p}}(V)

for V ⊂ pZ_{p} × Z_{p}^{×}; letting U = P^{−1}V proves the second case of the proposition.

Corollary 6.3.6. The push forward under π of the modular symbol μ̃ is precisely μ, i.e.,

μ̃_{m}(π^{−1}(U)) = μ_{m}(U)

for all m ∈ M and all compact open U ⊂ P^{1}(Q_{p}).

Proof. The fact that μ̃ is Γ_{0}(N)-invariant means that

μ̃_{γm}(γU) = μ̃_{m}(U)  (6.17)

for all γ ∈ Γ_{0}(N). Furthermore, we showed in Proposition 6.3.5 that we can choose a Γ-stable basis of compact opens U of X satisfying

μ̃_{P m}(P U) = μ̃_{m}^{U_{p}^{±1}}(U).  (6.18)

Combining (6.17) and (6.18) we find

μ̃_{P^{−1}γP m}(P^{−1}γP U) = μ̃_{m}^{U_{p}^{e}}(U)  (6.19)

for γ ∈ Γ_{0}(N), where e denotes some even power depending on γ. Since Γ is generated by its subgroups Γ_{0}(N) and P^{−1}Γ_{0}(N)P (our description of the action of Γ on the tree T in Section 4.1 shows that Γ is the amalgam of these two subgroups with respect to their intersection Γ_{0}(M), cf. [40, §II.1.4]), equations (6.17) and (6.19) imply that

μ̃_{γm}(γU) = μ̃_{m}^{U_{p}^{e}}(U)  (6.20)

for all γ ∈ Γ. Let us apply this rule with U = Z_{p}^{×} × pZ_{p}:

μ̃_{m}(γU) = μ̃_{γ^{−1}m}^{U_{p}^{e}}(U)
= −U_{p}^{e}(ψ(γ^{−1}m))  (6.21)
= μ_{m}(γ(P^{1}(Q_{p}) − Z_{p})).  (6.22)

Here (6.21) follows from property (6.14) defining μ̃; (6.22) uses the fact that U_{p}^{2} = 1 on H, and the definition of μ. Since the Γ translates of P^{1}(Q_{p}) − Z_{p} and its complement form a basis of compact opens for P^{1}(Q_{p}), the result follows.

The modular symbol μ̃ can be used to split the cocycle c_{τ}^{ℒ_{p}} explicitly. Define a

0-chain ρ_{τ} ∈ C^{0}(Γ, Hom(M, K_{p} ⊗ H)) by the rule

ρ_{τ} (m) := ∫_{X} log_{p}(a − bτ) dμ̃_{m}(a, b)
:= lim_{||U||→0} ∑_{U∈U} log_{p}(a_{U} − b_{U}τ) ⊗ μ̃_{m}(U),

where the limit is over uniformly finer covers U of X by disjoint compact opens U, and (a_{U}, b_{U}) is an arbitrary point of U. We will show in stages that dρ_{τ} = c_{τ}^{ℒ_{p}} . Proposition 6.3.7. If γ ∈ Γ_{0}(N), then

ρ_{τ} (γ^{−1}m) − ρ_{τ} (m) = log_{p} (× ∫_{τ}^{γτ}∫_{m}ω_{ψ})
= c_{τ}^{ℒ_{p}} (γ)(m).

Proof. Recall our assumption that τ reduces to the distinguished vertex v^{∗} of the Bruhat– Tits tree of PGL_{2}(Q_{p}). Since Γ_{0}(N) preserves this vertex, Lemma 3.3.2 shows that

ord_{p} (× ∫_{τ}^{γτ}∫_{m}ω_{ψ}) = 0,

and hence the first equality of the proposition implies the second. Write γ = (\begin{matrix}a & b \\ c & d \end{matrix}).

Using the Γ_{0}(N) invariance of μ̃ we calculate ρ_{τ} (γ^{−1}m) − ρ_{τ} (m):

∫_{X} log_{p}(x − yτ) dμ̃_{m}(γ(x, y)) − ∫_{X} log_{p}(x − yτ) dμ̃_{m}(x, y)

= ∫_{X} [log_{p}((dx − by) − (−cx + ay)τ) − log_{p}(x − yτ)] dμ̃_{m}(x, y)
= ∫_{X} log_{p} (\frac{(dx/y − b) − (−cx/y + a)τ}{x/y − τ} ) dμ̃_{m}(x, y).

Since the integrand depends only on x/y and the push forward of μ̃ is μ, the above expression equals

∫_{P^{1}(Q_{p})} log_{p} (\frac{(dt − b) − (−t + a)τ}{t − τ} ) dμ_{m}(t)

= ∫_{P^{1}(Q_{p})} log_{p} (\frac{t(d + cτ) − (aτ + b)}{t − τ}) dμ_{m}(t)
= ∫_{P^{1}(Q_{p})} log_{p} (\frac{t − γτ}{t − τ} ) dμ_{m}(t) + ∫_{P^{1}(Q_{p})} log_{p}(d + cτ) dμ_{m}(t)
= log_{p} (× ∫_{τ}^{γτ}∫_{m}ω_{ψ}) ,

where the last equality follows since μ has total measure zero.

For the matrix P ∉ Γ_{0}(N), the situation is somewhat different.

Proposition 6.3.8.

∫_{X} log_{p}(x − yτ) dμ̃_{P m}^{U_{p}}(x, y) − ∫_{X} log_{p}(x − yτ) dμ̃_{m}(x, y)  (6.23)
= (log_{p} −ℒ_{p} ord_{p}) (× ∫_{τ}^{P^{−1}τ}∫_{m}ω_{ψ})

Proof. We use the change of variables (x, y) ↦ (px, y) and the decomposition

P^{−1}X = (Z_{p} × Z_{p}^{×}) ⨆ (\begin{matrix}p & 0 \\ 0 & p \end{matrix})^{−1} (Z_{p}^{×} × pZ_{p})

to break up the first integral (note also log_{p}(px − yτ) = log_{p}(x − \frac{yτ}{p})):

∫_{X} log_{p}(x − yτ) dμ̃_{P m}^{U_{p}}(x, y) = ∫_{Z_{p}×Z_{p}^{×}} log_{p} (x − \frac{yτ}{p} ) dμ̃_{P m}^{U_{p}}(px, y)

+ ∫_{Z_{p}^{×}×pZ_{p}} log_{p} (x − \frac{yτ}{p} ) dμ̃_{P m}^{U_{p}}(x, y/p)

= ∫_{Z_{p}×Z_{p}^{×}} log_{p} (x − \frac{yτ}{p} ) dμ̃_{m}(x, y)

+ ∫_{Z_{p}^{×}×pZ_{p}} log_{p} (x − \frac{yτ}{p} ) dμ̃_{m}^{U_{p}^{2}}(x, y),

by Proposition 6.3.5. Thus the left-hand side of (6.23) becomes

∫_{X} log_{p} (\frac{x − \frac{yτ}{p}}{x − yτ} ) dμ̃_{m}(x, y) − ∫_{Z_{p}^{×}×pZ_{p}} log_{p} (x − \frac{yτ}{p} ) dμ̃_{m}^{1−U_{p}^{2}}(x, y).  (6.24)

The first integral of (6.24) can be pushed forward to P^{1}(Q_{p}) as in the proof of Proposition 6.3.7 and equals

log_{p} (× ∫_{τ}^{P^{−1}τ}∫_{m}ω_{ψ}) .

The second integral of (6.24) may be further decomposed:

∫_{Z_{p}^{×}×pZ_{p}} log_{p} (x − \frac{yτ}{p} ) dμ̃_{m}^{1−U_{p}^{2}}(x, y) = ∫_{Z_{p}^{×}×pZ_{p}} log_{p}(x) dμ̃_{m}^{1−U_{p}^{2}}(x, y)
+ ∫_{Z_{p}^{×}×pZ_{p}} log_{p} (1 − \frac{yτ}{px}) dμ̃_{m}^{1−U_{p}^{2}}(x, y). (6.25)

The second integral of (6.25) is again a push forward to P^{1}(Q_{p})−Z_{p}; since the push forward of μ̃_{m}^{1−U_{p}^{2}} is evidently the zero measure, this integral vanishes. Thus the proposition results from the lemma below (and Lemma 3.3.2).

Recall the notation of Section 6.2.

Lemma 6.3.9. Let t be an element of T whose image in 𝒯_{0} vanishes. We have

∫_{Z_{p}^{×}×pZ_{p}} log_{p}(x) dμ̃_{m}^{t}(x, y) = −t′ · ψ(m).

Proof. By Hida’s Theorem 6.2.1, we need only consider elements of the form t = (⟨d⟩ − 1)h, with h ∈ T. Modulo p^{r},

∫_{Z_{p}^{×}×pZ_{p}} log_{p}(x) dμ̃_{m}^{t}(x, y) ≡ ∑_{γ∈Γ_{0}(Np)/Γ_{r}} log_{p}(a)φ_{r}(h(⟨d⟩ − 1)γ^{−1}m),  (6.26)

where a represents the upper left entry of the matrix γ. The action of ⟨d⟩ is given by a matrix γ_{d} ∈ Γ_{0}(N) such that

γ_{d} ≡ (\begin{matrix}d^{−1} & ∗ \\ 0 & d \end{matrix}) (mod p^{r}),

so (6.26) becomes

∑_{γ∈Γ_{0}(Np)/Γ_{r}} log(a)φ_{r}(hγ_{d}^{−1}γ^{−1}m) − ∑_{γ∈Γ_{0}(Np)/Γ_{r}} log(a)φ_{r}(hγ^{−1}m).  (6.27)

As γ ranges through coset representatives for Γ_{0}(Np)/Γ_{r}, the matrices γγ_{d} do as well; the change of variables γ ↦ γγ_{d} in the first sum of (6.27) simplifies the entire expression to

∑_{γ∈Γ_{0}(Np)/Γ_{r}} (log_{p}(ad) − log_{p}(a)) φ_{r}(hγ^{−1}m) = log_{p}(d) ∑_{γ∈Γ_{0}(Np)/Γ_{r}} φ_{r}(hγ^{−1}m)
= log_{p}(d)μ̃_{m}^{h}(Z_{p}^{×} × pZ_{p})
= − log_{p}(d) · h · ψ(m).

This proves the desired result.

Propositions 6.3.7 and 6.3.8 together imply that for for γ ∈ Γ_{0}(N), we have

ρ_{τ} (P^{−1}γP m) − ρ_{τ} (m) = c_{τ}^{ℒ_{p}} (P^{−1}γP)(m).  (6.28)

Since the groups Γ_{0}(N) and P^{−1}Γ_{0}(N)P generate Γ, Propositions 6.3.7 and equa-

tion (6.28) yield the following proposition, which implies equation (6.1):

Proposition 6.3.10. The chain ρ_{τ} splits the 1-cocycle c_{τ}^{ℒ_{p}} for the group Γ, i.e., dρ_{τ} = c_{τ}^{ℒ_{p}} .

6.4 Proof that ℒ_{p} is the ℒ -invariant of J

The methods of this chapter follow very closely those of [12], but we include the

argument for completeness. The map Q yields an exact sequence of 𝒯 [G_{p}]-modules:

0 → X → Hom(X, Q̄_{p}^{×}) → J(Q̄_{p}) → 0.  (6.29)

The image of the first nontrivial map above lies in Hom(X, Q_{p}^{×}). Definition 6.4.1. The ℒ -invariant of J is the element of End(X ⊗ Q_{p}) such that

log_{p} Q(x) = ord_{p} Q(ℒ x) for all x ∈ X,

where log_{p} Q and ord_{p} Q have been extended via Q_{p}-linearity.

Following Greenberg and Stevens, we will interpret the ℒ -invariant of J as arising

from the deformation theory of the Galois action on its Tate module Ta_{p} J. Let rec : Q_{p}^{×} → G_{p}^{ab} be the Artin reciprocity map. Write Frob_{p} for rec(p)^{−1}; this is a lifting to G_{p}^{ab} of the Frobenius map on the maximal unramified extension of Q_{p}.

From (6.29) one finds (by connecting (6.29) with itself via the multiplication by

p^{r} map, employing the snake lemma, and taking the inverse limit over all r):

0 → Hom(X, Ta_{p} Q̄_{p}^{×}) → Ta_{p} J → X ⊗ Z_{p} → 0.

Twist the above sequence by the unramified character φ : G_{p} → 𝒯 ^{×} that sends Frob_{p} to U_{p} (so the module X(φ) has trivial G_{p}-action), and tensor with Q_{p}. We then have

0 → Hom(X(φ), V_{p} Q̄_{p}^{×}) → V_{p} J(φ) → X(φ) ⊗ Q_{p} → 0,  (6.30)

where V_{p} denotes Q_{p} ⊗ Ta_{p}. We will denote the three terms in this sequence by A, B, and C, respectively. As noted by Greenberg and Stevens, the ℒ -invariant of J can be deduced from knowledge about deformations of the sequence (6.30). A deformation of the module A is a T^{o}[G_{p}]-module A such that A/IA ≅ A as (T^{o}/IT^{o})[G_{p}] = 𝒯_{0}[G_{p}]-modules, where I is the augmentation ideal of Λ as in Section 6.2. Suppose we have a deformation of the sequence (6.30), that is, a commutative diagram

\begin{aligned}0 & ⟶ & IA & ⟶ & IB & ⟶ & IC & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & A & ⟶ & B & ⟶ & C & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & A & ⟶ & B & ⟶ & C & ⟶ & 0, \end{aligned}  (6.31)

where we have omitted the 0 terms on both ends of the vertical short exact sequences. Suppose further that A is a trivial deformation, in the sense that the action of G_{p} on A is given by the cyclotomic character (as it is on A). Let Ψ : G_{p} → End(C) define the Galois action on C. Since G_{p} acts trivially on C, for each c ∈ C we have σ(c) − c ∈ IC. Consider the image of σ(c) − c in

IC/I^{2}C = I/I^{2} ⊗_{Λ} C.  (6.32)

As in Section 6.2, we map this via log_{p} ⊗ Id to

Z_{p} ⊗_{Λ} C = Λ/I ⊗_{Λ} C = C/IC = C.  (6.33)

Thus to each σ ∈ G_{p} and c ∈ C, we have associated an element denoted Ψ′(σ)(c). Furthermore Ψ′(σ)(IC) = 0, so Ψ′(σ) factors through the quotient C/IC = C, and may thus be viewed as an element of End(C). It is trivial to check that Ψ′(σ) depends only on the image of σ ∈ G_{p}^{ab}. We now relate the ℒ -invariant of J to Ψ′.

Proposition 6.4.2. Let u ∈ 1 + pZ_{p} be a nontrivial unit. Then we have

Ψ′(Frob_{p}) = ℒ ∘ \frac{1}{log_{p} u}Ψ′(rec(u))  (6.34)

as elements of End(C) = End(X ⊗ Q_{p}).

Proof. Denote by Δ_{C} : IC → C the composition of the maps in (6.32) and (6.33). Define Δ_{A} similarly, and use the same notation for the induced maps on cohomology. A basic calculation verifies the commutativity of the following diagram of G_{p}-cohomology groups:

\begin{matrix}H^{0}(C) & ⟶^{δ_{C}} & H^{1}(IC) & ⟶^{Δ_{C}} & H^{1}(C) \\ ^{−δ_{1}}↓ & & ^{δ_{2}}↓ & & ^{δ_{3}}↓ \\ H^{1}(A) & ⟶^{δ_{A}} & H^{2}(IA) & ⟶^{Δ_{A}} & H^{2}(A) \end{matrix}  (6.35)

All of the maps labelled δ arise from coboundary maps in (6.31). Now G_{p} acts trivially on C, so H^{0}(C) = C and H^{1}(C) = Hom_{cont}(G_{p}^{ab}, C). Furthermore, for each nontrivial unit u ∈ 1 + pZ_{p}, the map H^{1}(C) → C ⊕ C given by

π : ξ ↦ (ξ(Frob_{p}), \frac{1}{log_{p} u}ξ(rec(u)))  (6.36)

is an isomorphism independent of u.

By definition, Ψ′(σ)(c) = Δ_{C} ∘ δ_{C}(c)(σ). Also, since A is a trivial deformation,

δ_{A} = 0. Thus from the commutativity of (6.35), we have

δ_{3} (Ψ′(Frob_{p})(c), \frac{1}{log_{p} u}Ψ′(rec(u))(c)) = 0  (6.37)

for all c ∈ C. We will determine the kernel of δ_{3} via the perfect pairings of Tate duality:

H^{i}(C) × H^{2−i}(A) → H^{2}(Q_{p}(1)) = Q_{p}, i = 0, 1, 2.

This pairing may be described explicitly for i = 1 as follows. For each element ĉ ∈ Hom(X, Q_{p}^{×}), we define an element γĉ ∈ H^{1}(A); choose a p^{n}th root ĉ^{1/p^{n}} of ĉ in Hom(X, Q̄_{p}^{×}) for each n, compatible in the sense that (ĉ^{1/p^{n+1}})^{p} = ĉ^{1/p^{n}}. The assignment

σ ↦ (σ − 1)(ĉ^{1/p^{n}}) ∈ Hom(X, Ta_{p} Q̄_{p}^{×})

is a cocycle representing a class denoted γĉ ∈ H^{1}(A). The definition of γĉ is independent of the choice of p^{n}th roots of ĉ. If

ĉ = ∑ f ⊗ q ∈ X^{∗} ⊗ Q_{p}^{×} = Hom(X, Q_{p}^{×})

and ξ ∈ H^{1}(C), the Tate duality pairing is given by

⟨ξ, γĉ⟩ = ∑ f(ξ(rec(q))) ∈ Q_{p}.

From this description, it one verifies that the dual π^{∗} of the isomorphism π from (6.36) satisfies

(π^{∗})^{−1} : H^{1}(A) → C^{∗} ⊕ C^{∗}
γĉ ↦ (− ord_{p}(ĉ), log_{p}(ĉ)).  (6.38)

By the self-duality of (6.30), the kernel of δ_{3} is dual to the image of δ_{1}. Yet δ_{1} is

the map c ↦ γ_{Q(c)}. Thus (6.38) implies that the kernel of δ_{3} consists of elements of the form (ℒ a, a); hence (6.37) yields the result.

The following theorem of Mazur and Wiles is the main arithmetic ingredient to-

wards proving ℒ = ℒ_{p}. Theorem 6.4.3 (Mazur-Wiles, [34]). There exists a deformation sequence as in (6.31) with the Galois action on C given by Ψ : G_{p} → T^{o} → End(C) satisfying

Ψ(Frob_{p}) = U_{p}^{2} ∈ T^{o} and Ψ(rec(u)) = ⟨u⟩

for u ∈ 1 + pZ_{p}.

Proof. This is Proposition 2 of Chapter 8 in [34]; see the comments at the end of that chapter for a description of the Galois action. Although [34] deals only with N = 1, the constructions of [33] from which the result is derived are carried out for higher level N as well.

We may now verify equation (6.3):

Proposition 6.4.4. The ℒ -invariant of J is equal to ℒ_{p}.

Proof. With Ψ as in Theorem 6.4.3, it is clear that Ψ′(Frob_{p}) = ℒ_{p} and Ψ′(rec(u)) = log_{p} u. Thus the proposition follows from Proposition 6.4.2.

Part II

Gross–Stark units

Chapter 7

Introduction

When the modular Jacobian J_{0}(N) is replaced by the group of units O^{×} in the

ring of algebraic numbers, the role of Heegner points on J_{0}(N) is played by elliptic units in O^{×}. Elliptic units are obtained as special values of modular units. A modular unit is a meromorphic function on X_{0}(N) whose zeroes and poles lie only at the cusps. An example of such a function is

α(z) = ∏_{d|N} Δ(dz)^{n_{d}}, where ∑_{d|N} n_{d} = 0.

For fixed N, the collection of modular units α of this form is a subgroup of finite index in the group of all modular units. Elliptic units are obtained by evaluating modular units at complex numbers τ ∈ H^{∗}/Γ_{0}(N) that lie in quadratic imaginary extensions K of Q. The theory of complex multiplication shows that α(τ) for such τ lies in a specific class field H of K, and describes the action of Gal(H/K) on the collection of α(τ). The goal of part II of this thesis is to present a conjectural formulation of an analogous construction when K is a real quadratic field. See [6] for a description of the classical theory of elliptic units in the form that motivates our current presentation, and [27] for a detailed study of the group of modular units.

Let K be a real quadratic field and p a prime that is inert in K. The completion

K_{p} is a quadratic unramified extension of Q_{p}. We begin by applying the constructions of part I to associate to a modular unit α and τ ∈ K − Q an element u(α, τ) ∈ K_{p}^{×}. The construction of u(α, τ) is obtained by using the same group

Γ = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ PSL_{2}(Z[1/p]) such that N|c\}  (7.1)

as in Chapter 4, but choosing a different module M whose universal torus T̃ admits Eisenstein quotients in which complex conjugation acts as −1. The image of the lattice L = Im Φ_{2} in such an Eisenstein quotient of T̃ is trivial. The elements u(α, τ) are simply the images of Stark–Heegner points in the corresponding G_{m} quotient of T̃/L. Motivated by Conjecture 4.3.3, Conjecture 8.2.5 states that u(α, τ) is a p-unit in a specific narrow ring class field of K depending on τ and denoted H_{τ}.

While this conjecture is still open, Chapter 9 relates the p-adic valuation of u(α, τ)

to the value at 0 of a certain zeta function ζ(α, τ, s). Gross has shown (as a consequence of the analytic class number formula) that there are unique p-units u_{GS} in H_{τ} satisfying this exact relation, and has given a precise conjecture (analogous to those of Stark in the archimedean setting) relating the p-adic logarithm of N_{K_{p}/Q_{p}} u_{GS} to the first derivative of a partial p-adic zeta function ζ_{p}(α, τ, s) [16]. This p-adic zeta function is obtained by interpolating ζ(α, τ, s) at negative integers s. The main result of part II is Theorem 10.0.1, which shows that the u(α, τ) satisfy Gross’s relationship. This result may be viewed as an analogue for real quadratic fields of the Kronecker limit formula. Conjecture 8.2.5, which implies that u_{GS} = u(α, τ), therefore implies Gross’s conjecture for H_{τ} /K at the prime p.

It should be stressed that Conjecture 8.2.5 leads to a genuine strengthening of the

Gross–Stark conjectures of [16] in the setting of ring class fields of real quadratic fields, and also of the refinement of these conjectures proposed in [17]. Indeed, the latter exploits the special values at s = 0 of abelian L-series attached to K, as well as derivatives of the corresponding p-adic zeta functions, to recover the images of Gross–Stark units in K_{p}^{×}/Ō_{K}^{×}, where Ō_{K}^{×} is the topological closure in K_{p}^{×} of the unit group of K. Conjecture 8.2.5 proposes an explicit formula for the Gross–Stark units themselves. It would be interesting to see whether other instances of the Stark conjectures (both classical, and p-adic) are susceptible to similar refinements^{1}.

1In a purely archimedean context, recent work of Ren and Sczech on the Stark conjectures for a complex cubic field suggests a positive answer to this question.

Chapter 8

Definition of the units

With Γ as in (7.1) and M = Div_{0} P^{1}(Q), Proposition 4.1.1 implies that the

maximal torsion-free quotient of H̃/H has rank 1 over Z. As a Hecke-module, this quotient is Eisenstein. Let ψ : H̃ ↦ Z denote the projection onto this quotient. Concretely, we have

ψ([x] − [y]) = \begin{cases}1 & if x ∼ 0 and y ∼ ∞ \\ 0 & if x ∼ y \\ −1 & if x ∼ ∞ and y ∼ 0, \end{cases}

where the equivalence relation ∼ has two classes, Z_{p} ∩ Q and its complement. A simple computation shows that in this case the multiplicative double integral becomes the crossratio:

× ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω_{ψ} = \frac{τ_{2} − y}{τ_{1} − y} · \frac{τ_{1} − x}{τ_{2} − x} ∈ Q_{p}(τ_{1}, τ_{2})^{×}.

The 2-cocycle d defined in (4.8) splits modulo ψ(L) = ⟨−1, p⟩, and one may define a Stark– Heegner point attached to τ as in (4.3.2). However, one checks that the point defined in this way is simply a fundamental unit of norm 1 in the ring O_{τ}. In other words, no non-trivial extensions of K arise from this construction. The main problem with this simple setting is that the modular symbol ψ has eigenvalue +1 for complex conjugation. Since G_{m} has a single period 2πi which is purely imaginary, one expects to construct interesting Stark– Heegner points only by using modular symbols with eigenvalue −1 for complex conjugation. To define such a modular symbol, we retain the group Γ but shrink the support of the module M to only contain one Γ-orbit of cusps. This creates Eisenstein quotients of H̃ in which complex conjugation acts as −1.

8.1 Eisenstein “minus” modular symbols

Consider the Γ-module

M = Div_{0} Γ∞ = Div_{0} \{\frac{a}{c} such that N|c\} .

Since M only consists of divisors supported on one Γ-orbit of cusps, homomorphisms from M are called partial modular symbols. Chapter 2 produces a universal multiplicative double integral with values in the torus G_{m} ⊗ H̃, where H̃ is the maximal torsion-free quotient of

H_{0}(Γ_{0}(Np), M)/f^{∗}(H_{0}(Γ_{0}(N), M)^{2}).

We will be interested in projections of H̃ onto certain Eisenstein factors, as follows. Let

α(z) = ∏_{d|N} Δ(dz)^{n_{d}}  (8.1)

be a modular unit of level N with neither pole nor zero at the cusp ∞ of Γ_{0}(N). Then we have

∑_{d|N} n_{d} = 0, ∑_{d|N} n_{d} · d = 0.  (8.2)

From α we produce a modular unit

α^{∗}(z) := α(z)/α(pz)

of level Np, which again has no zeroes or poles on Γ∞.

The logarithmic derivative of α defines a meromorphic differential on X_{0}(N) that

is regular on the Γ-orbit of ∞. The differentials dlog α and dlog α^{∗} can be written as

dlog α(z) = 2πiF_{2}(z) dz,  dlog α^{∗}(z) = 2πiF_{2}^{∗}(z) dz,  (8.3)

where F_{2}(z) and F_{2}^{∗}(z) are the weight two Eisenstein series on Γ_{0}(N) and Γ_{0}(Np), respectively, given by the formulae

F_{2}(z) = −24 ∑_{d|N} dn_{d}E_{2}(dz),  F_{2}^{∗}(z) = F_{2}(z) − pF_{2}(pz),  (8.4)

and E_{2}(z) is the standard Eisenstein series of weight 2:

E_{2}(z) = \frac{2}{(2πi)^{2}} ∑_{m,n=−∞}^{∞}′ \frac{1}{(mz + n)^{2}} = −\frac{1}{24} + ∑_{n=1}^{∞} σ_{1}(n)q^{n}, q = e^{2πiτ} .  (8.5)

(We remark that the above series is not absolutely convergent and the resulting expression is not invariant under SL_{2}(Z).)

Define a homomorphism ψ : M → Z by the complex line integral

ψ([x] − [y]) := \frac{1}{2πi} ∫_{x}^{y} dlog α ∈ Z.  (8.6)

The homomorphism ψ is clearly Γ_{0}(N)-invariant. It can be viewed as giving the winding number around the other cusps of the image in X_{0}(N) of a path in H^{∗} from x to y, with each cusp weighted by the corresponding order of zero or pole of α. Similarly, integrating dlog α^{∗} yields a Γ_{0}(Np)-invariant homomorphism ψ^{∗} : M → Z satisfying

ψ^{∗}(m) = ψ(m) − ψ(P · m),  (8.7)

where as usual P is the matrix representing multiplication by p.

The homomorphism ψ^{∗} factors through H̃. Indeed, to check this for the image

under f^{∗} of the first copy of H_{0}(Γ_{0}(N), M), let {γ_{i}}_{i=1}^{p+1} be a set of coset representatives for Γ_{0}(N)/Γ_{0}(Np). Then

ψ^{∗}(f^{∗}(m)) = ∑_{i} ψ(γ_{i}^{−1}m) − ψ(P γ_{i}^{−1}m)
= (p + 1)ψ(m) − (T_{p}ψ)(m) = 0,

as the q-expansion of the Eisenstein series F_{2} shows that ψ has eigenvalue p + 1 for T_{p}. A similar calculation holds for the image of the other copy of H_{0}(Γ_{0}(N), M) under f^{∗}. Thus the homomorphism ψ^{∗} : H̃ → Z allows us to define a Γ-invariant Meas(P^{1}(Q_{p}), Z)-valued partial modular symbol μ = ψ^{∗} ∘ μ^{univ} and yields a multiplicative double integral, written formally as the integral of dlog α:

× ∫_{τ_{1}}^{τ_{2}}∫_{m}dlog α := × ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{μ} = (Id ⊗ψ^{∗}) (× ∫_{τ_{1}}^{τ_{2}}∫_{m}ω_{univ}) ∈ G_{m}.

8.2 The unit u(α, τ)

Let τ ∈ H_{p} lie in the quadratic unramified extension K_{p} of Q_{p}. As in Section 6.3,

define a 1-cocycle

c_{τ} ∈ Z^{1}(Γ, Hom(M, K_{p}^{×}))

by the rule

c_{τ} (γ)(m) := × ∫_{τ}^{γτ}∫_{m}dlog α.

Proposition 8.2.1. The cocycle c_{τ} splits, i.e., there is a 0-chain ρ_{τ} : M → K_{p}^{×} such that

dρ_{τ} (γ)(m) = ρ_{τ} (γ^{−1}m) − ρ_{τ} (m) = c_{τ} (γ)(m)

for all γ ∈ Γ.

Proposition 8.2.1 will be proved over the course of the next three chapters.

Remark 8.2.2. A chain splitting the cocycle c_{τ} is unique only up to an element of the group of co-invariants M_{Γ}. This group is finite, but not necessarily trivial. However, we will be constructing a particular splitting ρ_{τ}, and this choice of ρ_{τ} will remain fixed.

Following [5], we denote the chain ρ_{τ} with an “indefinite integral”: × ∫ ^{τ}∫_{m}dlog α := ρ_{τ} (m).

The indefinite integral is Γ-invariant. This notation indicates the purpose of splitting the cocycle c_{τ} —it allows us to define an integral that depends only on one variable τ and not two endpoints of integration.

Recall the notation of Section 4.3, and Definition 4.3.2 in particular.

Definition 8.2.3. The unit u(α, τ) is defined by

u(α, τ) := × ∫ ^{τ}∫_{x}^{γ_{τ} x}dlog α ∈ K_{p}^{×},

where x ∈ Γ∞.

This definition is independent of choice of x.

Remark 8.2.4. In Chapter 4 we defined Stark–Heegner points via the splitting of a 2-cocycle d_{τ,x}. However, as remarked in Section 6.3, the splitting of c_{τ} implies the splitting of d_{τ,x}, and the corresponding points coincide.

With the notation as in Conjecture 4.3.3, we propose

Conjecture 8.2.5. If τ belongs to H_{p}^{O}/Γ, then u(α, τ) belongs to O_{H}[1/p]^{×}, and in fact,

u(α, 𝔞 ⋆ τ) = rec(𝔞)^{−1}u(α, τ),  (8.8)

for all 𝔞 ∈ Pic^{+}(O).

8.3 The action of complex conjugation and of U_{p}

The partial modular symbol ψ used to define u(α, τ) is odd in the sense that it has

eigenvalue −1 for complex conjugation W_{∞}:

ψ([−x] − [−y]) = −ψ([x] − [y]) for all x, y ∈ Γ∞

(cf. [32], Chapter II, §3). The parity of m_{α} implies the following behavior of the elements u(α, τ) under the action of complex conjugation, denoted as in Remark 4.3.4 by 𝔞_{∞}.

Proposition 8.3.1. Assume Conjecture 8.2.5. For all τ ∈ H_{p}^{O},

𝔞_{∞}u(α, τ) = u(α, τ)^{−1}.

Proof. The fact that the partial modular symbol ψ is odd implies that the sign denoted w_{∞} in proposition 5.13 of [5] is equal to −1. The proof of Proposition 8.3.1 is then identical to the proof of Proposition 5.13 of [5].

Remark 8.3.2. In the context of Stark–Heegner points from part I, we obtained two sets of points on J, having eigenvalue +1 or −1 for W_{∞}, corresponding to the choice of real or imaginary periods attached to J, respectively. In the situation treated here, where J is replaced by the multiplicative group, only the odd modular symbol ψ remains available, in harmony with the fact that the multiplicative group has a single period, 2πi, which is purely imaginary.

Remark 8.3.3. Suppose that O has a fundamental unit of negative norm. Then equivalence of ideals in the strict and usual sense coincide, so the narrow ring class field H associated to O is equal to the ring class field taken in the non-strict sense, which is totally real. Conjecture 8.2.5 predicts that 𝔞_{∞} should act trivially on u(α, τ) in this case, and that the p-units u(α, τ) should be trivial. In fact it can be shown, independently of any conjectures, that

u(α, τ) = 1, for all τ ∈ H_{p}^{O}.

This suggests that interesting elements of H^{×} are obtained only when H is a totally complex extension of K. This explains why it is essential to work with equivalence of ideals in the narrow sense and with narrow ring class fields to obtain useful invariants.

The q-expansion of the Eisenstein series dlog α^{∗} shows that it has eigenvalue 1

for the U_{p} operator. Similarly to the proof of Proposition 8.3.1, this implies that the sign denoted w in Proposition 5.13 of [5] equals 1. Thus the indefinite integral is invariant under all of

Γ̃ = \{γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ PGL_{2}(Z[1/p]) such that det(γ) > 0 and N|c\} ⊃ Γ.

In particular, the element u(α, τ) depends only on the Γ̃ orbit of τ. Since Γ̃ acts transitively on the vertices of the Bruhat–Tits tree T of PGL_{2}(Q_{p}), we may assume from now on that

τ reduces to the central vertex v^{∗} of T .  (8.9)

Chapter 9

Special values of zeta functions

Consider the decomposition

K_{p}^{×} ≅ Z × O_{K_{p}}^{×}
k ↦ (ord_{p}(k), u_{p}(k) := \frac{k}{p^{ord_{p}(k)}} ) .

We will define the splitting ρ_{τ} by defining its components ord_{p} ρ_{τ} and u_{p}ρ_{τ}. In this chapter we define ord_{p} ρ_{τ} and study its connection with zeta functions.

9.1 The zeta function

Let τ ∈ H_{p}^{O} satisfy assumption (8.9). There is a unique primitive integral binary

quadratic form Q_{τ} such that Q_{τ} (τ, 1) = 0 and

Q_{τ} (x, y) = Ax^{2} + Bxy + Cy^{2}, with A > 0.  (9.1)

The quadratic form Q_{τ} is indefinite. We will assume that the discriminant D of the Z[1/p]- order O is relatively prime to N (by convention, the integer D is taken to be prime to p); this implies N|A. Furthermore, by (8.9) it follows that B^{2} − 4AC = D (rather than D times some power of p). The assumption (8.9) also implies that the generator γ_{τ} of Γ_{τ} /⟨±1⟩ belongs to Γ_{0}(N). The matrix γ_{τ} fixes the quadratic form Q_{τ} under the usual action of SL_{2}(Z) on the set of binary quadratic forms. Furthermore, the simplifying assumption that gcd(D, N) = 1 implies that γ_{τ} = γ̃_{τ}, where the latter matrix is taken to be the stabilizer of the form Q_{τ} in SL_{2}(Z).

For an indefinite binary quadratic form Q, we let γ_{Q} be a generator of its stabilizer

in SL_{2}(Z). Note that Q takes on both positive and negative integer values, and that each value in the range of Q is taken on infinitely often, since Q is constant on γ_{Q}-orbits in Z^{2}. Thus we set

W := (Z^{2} − {0})/⟨γ_{Q}⟩,  (9.2)

and let

ζ_{Q}(s) = ∑_{(m,n)∈W} sign(Q(m, n))|Q(m, n)|^{−s}, Re(s) > 1,  (9.3)

where sign(x) = ±1 denotes the sign of a non-zero real number x. The function ζ_{Q}(s) extends by analytic continuation to a meromorphic function on C that is holomorphic except for a simple pole at s = 1.

With α as in (8.1) and (8.2), define

ζ_{τ} (s) := ζ_{Q_{τ}} (s),  ζ(α, τ, s) := ∑_{d|N} n_{d}d^{s}ζ_{dτ} (s).  (9.4)

The zeta functions ζ_{dτ} (s) appearing in the definition of ζ(α, τ, s) are attached to the integral quadratic forms Q_{dτ}, each of which has the same discriminant D. Note that ζ(α, τ, s) depends only on the Γ_{0}(N)-orbit of the element τ.

The main formula of this chapter is:

Theorem 9.1.1. Suppose that τ belongs to H_{p}^{O}, and is normalized by the action of Γ̃ to satisfy (8.9). Then

ζ(α, τ, 0) = \frac{1}{12} · ord_{p}(u(α, τ)).

Remark 9.1.2. Let A_{K} denote the ring of adeles of K. A finite order idèle class character

χ = ∏_{v} χ_{v} : A_{K}^{×}/K^{×}⟶C^{×}

is called a ring class character if it is trivial on A_{Q}^{×}. If χ is such a character, then its two archimedean components χ_{∞_{1}} and χ_{∞_{2}} attached to the real places of K are either both trivial or both equal to the sign character. In the former case χ is called even and in the latter, it is said to be odd. A ring class character can be interpreted as a character on the narrow Picard group G_{O} := Pic^{+}(O) of narrow ideal classes attached to a fixed order O of K whose conductor is equal to the conductor of χ.

The zeta functions ζ_{τ} (s) with τ ∈ H_{p}^{O} can be interpreted as partial zeta functions

encoding the zeta function of K twisted by odd ring class characters of G_{O}. Letting τ_{0} be an element of H_{p}^{O} that is equivalent to \sqrt{D} under the action of SL_{2}(Z), we have:

∑_{σ∈G_{O}} χ(σ)ζ_{σ⋆τ_{0}}(s) = \{ \begin{matrix}0 & if χ is even, \\ L(K, χ, s) & if χ is odd. \end{matrix}  (9.5)

9.2 Dedekind sums

The line integral in (8.6) defining the partial modular symbol ψ can be expressed

in terms of classical Dedekind sums

D (\frac{a}{m}) := ∑_{x=1}^{m−1} B_{1} (\frac{x}{m}) B_{1} (\frac{ax}{m}) , for gcd(a, m) = 1, m > 0,

where

B_{1}(x) = {x} − 1/2 = x − [x] − 1/2

is the first Bernouilli polynomial made periodic. Rademacher proved:

ψ ([∞] − [\frac{a}{Nc}]) = \frac{1}{2πi} ∫_{∞}^{a/Nc} dlog α = −12 ∑_{d|N} n_{d} · D (\frac{a}{Nc/d}) .  (9.6)

The equation ∑ n_{d}d = 0 simplifies the behavior of the Dedekind–Rademacher homomorphism, eliminating the extra terms appearing in Equation (2.1) of [32].

Meyer has proven a formula expressing the special values of partial zeta functions

attached to real quadratic fields at s = 0 in terms of Dedekind sums (Cf. [42], equation (4.1) for a statement of Meyer’s formula in the case where D is fundamental; the general case can be derived from equation (18) in §5 of [4], for example.) This formula can be used to derive the identity

ζ(α, τ, 0) = − ∑_{d|N} n_{d} · D (\frac{a}{Nc/d}) ,  (9.7)

where γ_{τ} = (\begin{matrix}a & ∗ \\ Nc & ∗ \end{matrix}). Equations (9.6) and (9.7) yield

ζ(α, τ, 0) = \frac{1}{12} · ψ ([∞] − [γ_{τ} ∞]) .  (9.8)

9.3 The p-adic valuation

Define a homomorphism ψ_{v} : M → Z for each vertex v of the tree T by assigning

ψ_{v^{∗}} := ψ for the central vertex and extending by Γ̃-invariance:

ψ_{γv}(m) = ψ_{v}(γ^{−1}m).

This is well defined since ψ is invariant under the stabilizer of v in Γ̃, namely Γ_{0}(N). From (8.7), the measure of the open set U_{e^{∗}} = Z_{p} under μ_{m} is ψ_{v^{∗}}(m) − ψ_{w^{∗}}(m), where w^{∗} = P^{−1}v^{∗}, and e^{∗} is the oriented edge from w^{∗} to v^{∗}. Thus by Γ̃-invariance, we have that

μ_{m}(U_{e}) = ψ_{t(e)}(m) − ψ_{s(e)}(m)

for all oriented edges e of T . Hence by Lemma 3.3.2, we see that

ord_{p} (× ∫_{τ}^{γτ}∫_{m}dlog α) = ψ_{γv^{∗}}(m) − ψ_{v^{∗}}(m)
= ψ(γ^{−1}m) − ψ(m).

We have thus shown

Proposition 9.3.1. The chain ord_{p} ρ_{τ} ∈ C^{0}(Γ, Hom(M, Z)) defined by ord_{p} ρ_{τ} (m) := ψ(m) splits the cocycle ord_{p} c_{τ}.

Since u(α, τ) = ρ_{τ} ([∞] − [γ_{τ} ∞]), Proposition 9.3.1 and (9.8) complete the proof

of Theorem 9.1.1.

9.4 Values at negative integers

In this section, we generalize formula (9.8) by expressing ζ(α, τ, 1 − r) in terms of

periods of certain Eisenstein series of weight 2r, for odd r ≥ 1.

The Eisenstein series of (8.4) and (8.5) are part of a natural family of Eisenstein

series of varying weights. For even k ≥ 2, consider the standard Eisenstein series of weight k:

E_{k}(z) = \frac{2(k − 1)!}{(2πi)^{k}} ∑_{m,n=−∞}^{∞}′ \frac{1}{(mz + n)^{k}} = −\frac{B_{k}}{2k} + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}.  (9.9)

Define likewise, as a function of the coefficients n_{d} used to define the modular unit α, the higher weight Eisenstein series

F_{k}(z) = −24 ∑_{d|N} n_{d} · d · E_{k}(dz)
= −\frac{48(k − 1)!}{(2πi)^{k}} ∑_{m,n=−∞}^{∞}′ (\frac{1}{(mz + n)^{k}} ∑_{d|(N,m)} n_{d}d)
= −24 ∑_{n=1}^{∞} σ_{k−1}(n) ∑_{d|N} n_{d}dq^{nd}.  (9.10)

The F_{k} are modular forms of weight k on Γ_{0}(N), holomorphic on the upper half plane. These Eisenstein series have no constant term and hence are holomorphic at the cusp ∞. We also define, for the purposes of p-adic interpolation, the function

F_{k}^{∗}(z) = F_{k}(z) − p^{k−1}F_{k}(pz).

We extend the definition of E_{k}(z) and F_{k}(z) to all k ≥ 2 by letting E_{k} = F_{k} = 0 for k odd.

Recall the standard right action of GL_{2}^{+}(R) on the space of modular forms of

weight k, given by

F |_{γ}(z) = \frac{det(γ)}{(cz + d)^{k}} F (γz) when γ = (\begin{matrix}a & b \\ c & d \end{matrix}) .

The definition of F_{k}^{∗} can be written

F_{k}^{∗} = F_{k} − p^{k−2}F_{k}|_{P} (z), where P = (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) .

The following proposition expresses ζ(α, τ, 1 − r) in terms of periods of F_{2r}.

Proposition 9.4.1. For all odd integers r > 0,

12 · ζ(α, τ, 1 − r) = ∫_{∞}^{γ_{τ} ∞} Q_{τ} (z, 1)^{r−1}F_{2r}(z) dz.

Proof. Let k ≥ 2 be a positive integer and let Ẽ_{k} denote the weight k Eisenstein series

Ẽ_{k}(z) = ∑_{m,n}′ \frac{1}{(mz + n)^{k}} .

By Hilfsatz 1 of [39], letting z_{0} ∈ H be an arbitrary base point, the following identity holds for all integers r > 1:

∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}Ẽ_{2r}(z) dz = (−1)^{r−1} \frac{(r − 1)!^{2}}{(2r − 1)!}D^{r− \frac{1}{2}} ∑_{W} Q_{τ} (m, n)^{−r},

with W as in (9.2). Suppose that r > 1 is an odd integer. Then

∑_{W} Q_{τ} (m, n)^{−r} = ∑_{W} sign(Q_{τ} (m, n))|Q_{τ} (m, n)|^{−r} = ζ_{τ} (r),

so

∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}Ẽ_{2r}(z) dz = \frac{(r − 1)!^{2}}{(2r − 1)!}D^{r− \frac{1}{2}} ζ_{τ} (r).  (9.11)

On the other hand, it follows from the relation (9.5) and from the functional equation for L(K, χ, s) for odd characters (cf. [28], Corollary 1 after Theorem 14 of §8, Chapter XIV) that ζ_{τ} (s) satisfies the functional equation

ζ_{τ} (1 − s) = \frac{D^{s− \frac{1}{2}}}{π^{2s−1}} Γ (\frac{s + 1}{2})^{2} Γ (\frac{2 − s}{2})^{−2} ζ_{τ} (s).

Hence if r ≥ 2 is an even positive integer, ζ_{τ} (1 − r) = 0, while if r ≥ 1 is odd,

ζ_{τ} (1 − r) = \frac{4D^{r− \frac{1}{2}}}{(2π)^{2r}} (r − 1)!^{2}ζ_{τ} (r).

Combining this functional equation with (9.11), we obtain

∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}Ẽ_{2r}(z) dz = \frac{(2π)^{2r}}{4(2r − 1)!}ζ_{τ} (1 − r).

Since

E_{k}(z) = \frac{2(k − 1)!}{(2πi)^{k}} Ẽ_{k}(z),

it follows that

∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}E_{2r}(z) dz = −\frac{1}{2}ζ_{τ} (1 − r).  (9.12)

From the definition of F_{k}, we have

∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}F_{2r}(z) dz = −24 ∑_{d|N} n_{d} · d · ∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}E_{2r}(dz) dz.

Making the change of variables w = d · z, we obtain

d · ∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}E_{2r}(dz) dz = ∫_{d·z_{0}}^{d·γ_{τ} z_{0}} Q_{τ} (\frac{w}{d} , 1)^{r−1} E_{2r}(w) dw.

Recall that γ̃_{dτ} denotes the generator of the stabilizer of dτ in SL_{2}(Z), chosen in such a way that γ_{dτ} is a positive power of γ̃_{dτ}. We have

dγ_{τ} z_{0} = γ̃_{dτ} (dz_{0}), Q_{τ} (\frac{w}{d} , 1) = \frac{1}{d}Q_{dτ} (w, 1).

Hence

d · ∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}E_{2r}(dz) dz = ∫_{dz_{0}}^{γ̃_{dτ} (dz_{0})} \frac{1}{d^{r−1}} Q_{dτ} (w)^{r−1}E_{2r}(w) dw.

The expression on the right is equal to −\frac{d^{1−r}}{2} ζ_{dτ} (1 − r), by (9.12). It follows that, for all odd r > 1,

∫_{z_{0}}^{γ_{τ} z_{0}} Q_{τ}^{r−1}F_{2r}(z) dz = 12 ∑_{d|N} n_{d}d^{1−r}ζ_{dτ} (1 − r) = 12 · ζ(α, τ, 1 − r).

The integrand in the left-hand expression involves an Eisenstein series that is holomorphic at ∞, hence we may replace the base point z_{0} ∈ H by the cusp ∞.

In the next chapter we will generalize Meyer’s formula (corresponding to the case

r = 1) by expressing the periods of F_{2k} in terms of generalized Dedekind sums.

9.5 Connection with the Gross–Stark conjecture

A general result of Deligne and Ribet (cf. the discussion in [16], §2) implies the

existence of a p-adic meromorphic function ζ_{p}(α, τ, s) of the variable s ∈ Z_{p} characterized by its values on a dense set of negative integers:

ζ_{p}(α, τ, n) = (1 − p^{−2n})ζ(α, τ, n),  for all n ≤ 0, n ≡ 0 (mod 2(p − 1)). (9.13)

Let U_{H,p} denote the group of p-units of H defined as in [16, Proposition 3.8]:

U_{H,p} := {ϵ ∈ H^{×} : ||ϵ||_{𝔇} = 1 for all places 𝔇 that do not divide p}.

Since the places 𝔇 involved in the definition of U_{H,p} include all the archimedean ones, it follows that U_{H,p} is infinite only when H has no real embeddings, and that images of the elements of U_{H,p} under all the complex embeddings of H lie on the unit circle.

Proposition 8.3.1 implies that the p-unit u(α, τ) belongs to U_{H,p} (assuming, of

course, Conjecture 8.2.5). Theorem 9.1.1 states that

ord_{p}(u(α, τ)) = 12 · ζ(α, τ, 0).

Thus the Gross-Stark conjecture, stated in Conjecture 2.12 of [16] (cf. the formulation given in Proposition 3.8 of [16]), suggests that one should have

log_{p} N_{K_{p}/Q_{p}}(u(α, τ)) = −12 · ζ′_{p}(α, τ, 0).  (9.14)

In fact, the relation (9.14) is essentially equivalent (by varying α appropriately) to the Gross–Stark conjecture for H/K, assuming Conjecture 8.2.5. The next chapter is devoted to the explicit construction of ζ_{p}(α, τ, s) and to a proof of (9.14).

Chapter 10

A Kronecker limit formula

The first three sections of this chapter give an explicit construction of the p-adic

zeta function ζ_{p}(α, τ, s) satisfying the interpolation property (9.13). The remainder of the chapter proves the main theorem of part II: Theorem 10.0.1. Suppose that τ belongs to H_{p}^{O}, and is normalized by the action of Γ̃ to satisfy (8.9). Then

ζ′_{p}(α, τ, 0) = −\frac{1}{12} · log_{p} N_{K_{p}/Q_{p}}(u(α, τ)).

Note the clear analogy between this formula and the classical Kronecker limit

formula stated in [6]. Theorem 10.0.1 allows us to deduce the Gross–Stark conjecture for H/K from Conjecture 8.2.5. It should be pointed out that Conjecture 8.2.5 is stronger and more precise than Gross’s conjecture in this setting, since it gives a formula for the Gross–Stark unit u(α, τ) itself, and not just its “absolute value.”

10.1 Measures associated to Eisenstein series

We recall the definitions of Section 6.3. Once again the key to splitting the cocycle

c_{τ} is lifting μ to a bundle over P^{1}(Q_{p}). Let

X := (Z_{p} × Z_{p})′ ⊂ (Q_{p} × Q_{p} − {0})

(considered as column vectors) be the set of vectors (x, y) ∈ Z_{p}^{2} such that x and y are not both divisible by p in Z_{p}. The space Q_{p}^{2} − {0} is endowed with a natural action of Γ by left multiplication. There is a Z_{p}^{×}-bundle map

π : X → P^{1}(Q_{p}), (x, y) ↦ x/y.

The crucial technical ingredient in the construction of ζ_{p}(α, τ, s) and in the proof

of Theorem 10.0.1 is the following result, which can be viewed as an extension of the construction of μ from F_{2} to the family of Eisenstein series introduced in the previous section.

Theorem 10.1.1. There is a unique Γ̃-invariant Meas(Q_{p}^{2} −{0}, Z)-valued partial modular symbol μ̃ satisfying the following properties:

(1) For every homogeneous polynomial h(x, y) ∈ Z_{p}[x, y] of degree k − 2 and every

pair of cusps r, s ∈ Γ∞,

∫_{X} h(x, y) dμ̃_{[r]−[s]}(x, y) = Re ((1 − p^{k−2}) ∫_{r}^{s} h(z, 1)F_{k}(z) dz) .  (10.1)

(2) The measures μ̃_{m} are invariant under multiplication by p:

μ̃_{m}(pU) = μ̃_{m}(U).

Furthermore this measure satisfies:

(3) For every homogeneous polynomial h(x, y) ∈ Z_{p}[x, y] of degree k − 2,

∫_{Z_{p}×Z_{p}^{×}} h(x, y) dμ̃_{[r]−[s]}(x, y) = Re (∫_{r}^{s} h(z, 1)F_{k}^{∗}(z) dz) .

The proof of Theorem 10.1.1 will be given at the end of this chapter (beginning

with Section 10.4).

The following lemma shows how the measures μ̃_{m} are related to the measures μ_{m}.

Lemma 10.1.2. For all compact opens U ⊂ P^{1}(Q_{p}),

μ̃_{m}(π^{−1}U) = μ_{m}(U)  (10.2)

Proof. Define a partial modular symbol of measures π_{∗}μ̃ on P^{1}(Q_{p}) by the rule

π_{∗}μ̃_{m}(U) := μ̃_{m}(π^{−1}U).

To see that π_{∗}μ̃ is Γ̃-invariant, we calculate

π_{∗}μ̃_{γm}(γU) = μ̃_{γm}(π^{−1}(γU)) = μ̃_{γm}(γπ^{−1}(U)),

where the last equality follows from the fact that both π^{−1}(γU) and γπ^{−1}(U) are fundamental regions for the action of ⟨p⟩ on the inverse image of γU in Q_{p}^{2} − {0}. Hence

π_{∗}μ̃_{γm}(γU) = μ̃_{m}(π^{−1}(U)) = π_{∗}μ̃_{m}(U).

Now, property (3) of Theorem 10.1.1 with h(x, y) = 1 implies that π_{∗}μ̃ agrees with

μ on the compact open set Z_{p}. Since the Γ̃-translates of Z_{p} form a basis of compact opens for P^{1}(Q_{p}), we have π_{∗}μ̃ = μ.

10.2 Construction of the p-adic L-function

The special values of ζ(α, τ, s) at certain even negative integers can be expressed

in terms of the measure μ̃ described in Section 10.1. For notational convenience we will write μ̃{r → s} for μ̃_{[r]−[s]}.

Lemma 10.2.1. For all odd integers r > 0

12(1 − p^{2r−2}) · ζ(α, τ, 1 − r) = ∫_{X} Q_{τ} (x, y)^{r−1} dμ̃{∞ → γ_{τ} ∞}(x, y).

Proof. This follows directly from Proposition 9.4.1 and property (1) of Theorem 10.1.1.

Suppose that the odd integer r is congruent to 1 modulo p − 1. By Lemma 10.2.1,

12(1 − p^{2r−2}) · ζ(α, τ, 1 − r) = ∫_{X}⟨Q_{τ} (x, y)⟩^{r−1} dμ̃{∞ → γ_{τ} ∞}(x, y),  (10.3)

where for x ∈ Z_{p}^{×}, the expression ⟨x⟩ denotes the unique element in 1 + pZ_{p} that differs multiplicatively from x by a (p − 1)st root of unity. The advantage of the expression (10.3) is that it interpolates p-adically, expressing ζ(α, τ, 1 − r) with its Euler factor at p removed as a function of the p-adic variable r. This leads us to define

ζ_{p}(α, τ, s) = \frac{1}{12} ∫_{X}⟨Q_{τ} (x, y)⟩^{−s} dμ̃\{∞ → γ_{τ} ∞\}(x, y)

for all s ∈ Z_{p}. One recovers the p-adic L-function introduced in Section 9.5 that is uniquely characterized by the interpolation property (9.13).

In terms of this explicit definition of ζ_{p}(α, τ, s), we have

Lemma 10.2.2. The derivative ζ′_{p}(α, τ, s) at s = 0 is given by

ζ′_{p}(α, τ, 0) = −\frac{1}{12} ∫_{X} log_{p} (Q_{τ} (x, y)) dμ̃\{∞ → γ_{τ} ∞\}(x, y).

Proof. This is a direct consequence of the definition.

10.3 An explicit splitting of the one-cocycle

We now turn to the definition of the indefinite integral.

Proposition 10.3.1. Let μ̃ be as in Theorem 10.1.1. Then

u_{p}ρ_{τ} (m) := × ∫_{X} u_{p}(x − yτ) dμ̃_{m}(x, y)

provides an explicit splitting of u_{p}c_{τ}. Proof. The proof follows the arguments of Section 6.3, with great simplification since the partial modular symbol μ̃ is now Γ̃-invariant. Let γ ∈ Γ̃. Since γX and X are both fundamental domains for multiplication by p on Q_{p}^{2} − {0}, and u_{p} is constant on cosets of p^{Z} in K_{p}^{×}, it follows that

× ∫_{X} u_{p}(a − bτ) dμ̃_{γ^{−1}m}(a, b) = × ∫_{X} u_{p}(a − bτ) dμ̃_{m}(x, y)

where \binom{a}{b} = γ^{−1}\binom{x}{y}. Hence

\frac{ρ_{τ} (γ^{−1}m)}{ρ_{τ} (m)} = × ∫_{X} u_{p} (\frac{a − bτ}{x − yτ} ) dμ̃_{m}(x, y).  (10.4)

Since (\frac{a−bτ}{x−yτ} ) is a constant multiple of (\frac{x−yγτ}{x−yτ} ) and μ̃_{m} has total measure 0, equation (10.4) becomes

× ∫_{X} u_{p} (\frac{x − yγτ}{x − yτ} ) dμ̃_{m}(x, y) = × ∫_{P^{1}(Q_{p})} u_{p} (\frac{t − γτ}{t − τ} ) dμ_{m}(t)
= u_{p}c_{τ} (γ)(m),

since the push forward of μ̃ to P^{1}(Q_{p}) is μ.

We can now prove theorem 10.0.1:

Proof of Theorem 10.0.1. Since our chosen branch of log_{p} factors through u_{p}, Proposition 10.3.1 imples that

log_{p} N_{K_{p}/Q_{p}}(u(α, τ)) = log_{p} (× ∫_{X} u_{p}((x − yτ)(x − yτ′)) dμ̃{∞ → γ_{τ} ∞}(x, y))
= ∫_{X} log_{p} Q_{τ} (x, y) dμ̃{∞ → γ_{τ} ∞}(x, y),  (10.5)

where (10.5) follows since Q_{τ} (x, y) is proportional to (x − τy)(x − τ′y) and μ̃(X) = 0. The result now follows from Lemma 10.2.2.

The remainder of this chapter is devoted to the proof of Theorem 10.1.1.

10.4 Generalized Dedekind sums

In this section we evaluate the integrals appearing in the right of (10.1) in Theorem

10.1.1, which characterize the partial modular symbol μ̃. The computations of this section are not new, but we include them for completeness and notational consistency. Let f denote a modular form and let a_{f}(0) denote the constant term of its q-expansion at ∞. For relatively prime integers a and c with c ≥ 1, the function

A_{f}(s; a, c) = e^{πis/2}c^{s−1} ∫_{0}^{∞} (f(it + a/c) − a_{f}(0)) t^{s−1} dt

is well defined for Re(s) large enough, and has a meromorphic continuation to all of C (see [10]). For the Eisenstein series E_{2k} with k > 1, this continuation is given by

A_{E_{2k}}(s; a, c) = e^{πis/2}c^{2k−2} \frac{Γ(s)}{(2π)^{s}} ∑_{h=1}^{c} [ζ(s + 1 − 2k, h/c) ∑_{m=1}^{∞} \frac{1}{m^{s}} e^{2πimha/c}] ,  (10.6)

where ζ(s, b) denotes the Hurwitz zeta function. This is a relatively standard computation, carried out for example in Proposition 3.1 of [10]. Let us calculate the real part of this expression for s an integer, 1 ≤ s ≤ 2k − 1. When s = 1, the term for h = c in (10.6) is taken to be

lim_{s→1} ζ(s + 1 − 2k)ζ(s) ∈ R.

The Hurwitz zeta function has the well known value ζ(1 − n, b) = −B_{n}(b)/n. Furthermore, for a real number x,

Re (\frac{i^{s}(s − 1)!}{(2π)^{s}} ∑_{m=1}^{∞} \frac{1}{m^{s}} e^{2πimx}) = \frac{i^{s}(s − 1)!}{2(2π)^{s}} ∑_{m≠0^{m=−∞}}^{∞} \frac{1}{m^{s}} e^{2πimx}
= \frac{(−1)^{s+1}}{2} · \frac{B̃_{s}(x)}{s},

where

B̃_{s}(x) := \begin{cases}0 & if s = 1 and x ∈ Z \\ B_{s}(\{x\}) = B_{s}(x − [x]) & otherwise, \end{cases}

and B_{s}(x) is the standard Bernoulli polynomial of degree s (see [19, Section II]). We obtain

Re (A_{E_{2k}}(s; a, c)) = c^{2k−2} \frac{(−1)^{s}}{2} ∑_{h=1}^{c} \frac{B_{2k−s}(h/c)}{2k − s} · \frac{B̃_{s}(ha/c)}{s}.  (10.7)

We would like to replace the term B_{2k−s}(h/c) by B̃_{2k−s}(h/c) in the sum above. Only the term for h = c, which we now consider, may cause difficulty. If s is even, then B_{2k−s}(1) = B_{2k−s}(0) since in general one has

B_{n}(1 − x) = (−1)^{n}B_{n}(x).

If s is odd then the other term in the product is B̃_{s}(a) = 0. Thus in either case we may replace the term B_{2k−s}(h/c) by B̃_{2k−s}(h/c). This motivates the following definition. Definition 10.4.1. Let s, t ≥ 0. For a and c relatively prime and c ≥ 1, the generalized Dedekind sum D̃_{s,t}(a/c) is defined by

D̃_{s,t}(a/c) := c^{s−1} ∑_{h=1}^{c} B̃_{s}(h/c)B̃_{t}(ha/c).

The sum defining D̃ may be taken over any complete set of representatives h mod c. For s, t ≥ 1, define

D_{s,t}(a/c) := \frac{D̃_{s,t}(a/c)}{st}.

Remark 10.4.2. When s = t = 1, we have

D_{1,1}(a/c) = D̃_{1,1}(a/c) = D(a/c),

where D(a/c) is as in Section 9.2.

Equation (10.7) may be written in terms of generalized Dedekind sums as

Re (A_{E_{k}}(s; a, c)) = c^{s−1} \frac{(−1)^{s}}{2} D_{k−s,s}(a/c).  (10.8)

This formula continues to hold when k is odd, since then the Dedekind sum D_{k−s,s}(a/c) vanishes (using the relation B̃_{s}(−x) = (−1)^{s}B̃_{s}(x).)

From the definition of F_{k}, we find

A_{F_{k}}(s; a, Nc) = −24 ∑_{d|N} n_{d}A_{E_{k}}(s; a, Nc/d).  (10.9)

We are now ready to evaluate the integrals appearing in (10.1). Let 0 ≤ n ≤ k − 2. Using the change of variables z = it + a/(Nc), we have

∫_{a/Nc}^{i∞} z^{n}F_{k}(z) dz = ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (Nc)^{−ℓ}A_{F_{k}}(ℓ + 1; a, Nc)

= −24 ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (Nc)^{−ℓ} ∑_{d|N} n_{d}A_{E_{k}} (ℓ + 1; a, \frac{Nc}{d}) .  (10.10)

In view of (10.8), the real part of (10.10) is equal to

12 ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (−1)^{ℓ} ∑_{d|N} n_{d}d^{−ℓ}D_{k−ℓ−1,ℓ+1} (\frac{a}{Nc/d}) .  (10.11)

As we now check, formula (10.11) remains valid for k = 2. In this case n = 0 and the desired formula simplifies to

Re (∫_{a/Nc}^{i∞} F_{2}(z) dz) = 12 ∑_{d|N} n_{d}D_{1,1} (\frac{a}{Nc/d}) ,

which is nothing but equation (9.6) in light of Remark 10.4.2 and (8.3).

10.5 Measures on Z_{p} × Z_{p}

Let ξ = \frac{a}{Nc} ∈ Γ∞, and assume that p does not divide c. In this section we prove

the following crucial lemma.

Lemma 10.5.1. Let ξ ∈ Γ∞ have denominator not divisible by p. There exists a unique Z_{p}-valued measure ν_{ξ} on Z_{p} × Z_{p} such that for all k ≥ 2,

∫_{Z_{p}×Z_{p}} h(x, y) dν_{ξ}(x, y) = Re ((1 − p^{k−2}) ∫_{ξ}^{i∞} h(z, 1)F_{k}(z) dz)  (10.12)

for every homogeneous polynomial h(x, y) ∈ Z[x, y] of degree k − 2.

Equation (10.12) is equivalent to the statement that

∫_{Z_{p}×Z_{p}} x^{n}y^{m} dν_{ξ}(x, y) = Re ((1 − p^{n+m}) ∫_{a/Nc}^{i∞} z^{n}F_{n+m+2}(z) dz)
= 12(1 − p^{n+m}) ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (−1)^{ℓ} ∑_{d|N} n_{d}d^{−ℓ}D_{n+m−ℓ+1,ℓ+1} (\frac{a}{Nc/d})  (10.13)

for all integers n, m ≥ 0. Denote the last expression appearing in equation (10.13) by I_{n,m} ∈ Q. The key tool in showing the existence and uniqueness of ν_{ξ} is the following result, which is the two-variable version of a classical theorem of Mahler (see Theorem 3.3.1 of [21]).

Lemma 10.5.2. Let b_{n,m} ∈ Z_{p} be constants indexed by integers n, m ≥ 0. There exists a unique Z_{p}-valued measure ν on Z_{p} × Z_{p} such that

∫_{Z_{p}×Z_{p}} \binom{x}{n}\binom{y}{m} dν(x, y) = b_{n,m}.

Thus to prove Lemma 10.5.1, we must show that the rational numbers

J_{n,m} := ∑_{i=0}^{n} ∑_{j=0}^{m} c_{n,i}c_{m,j}I_{i,j}

lie in Z_{p}, where the integers c_{n,i} are defined by the equation

\binom{x}{n} = ∑_{i=0}^{n} c_{n,i}x^{i}.

Our proof of this fact will follow the proof of the existence of p-adic Dirichlet L-functions, as in Section 3.4 of [21].

Consider the rightmost term appearing in the definition (10.13) of I_{n,m} (here k =

n + m + 2):

d^{−ℓ}D_{k−ℓ−1,ℓ+1} (\frac{a}{Nc/d}) = d^{−ℓ} (\frac{Nc}{d})^{k−ℓ−2} ∑_{h=1}^{Nc/d} \frac{B̃_{k−ℓ−1}(\frac{h}{Nc/d})}{k − ℓ − 1} \frac{B̃_{ℓ+1}(\frac{ha}{Nc/d})}{ℓ + 1}
= (\frac{Nc}{d})^{k−ℓ−2} ∑_{h=1}^{Nc} \frac{B̃_{k−ℓ−1}(\frac{h}{Nc/d})}{k − ℓ − 1} \frac{B̃_{ℓ+1}(\frac{ha}{Nc})}{ℓ + 1},  (10.14)

where (10.14) follows from the distribution relation for Bernoulli polynomials. For each h = 1, …, Nc, write θ = {ha/Nc}. Let x be a formal variable and write u = e^{x}. Then the Bernoulli polynomials are given by the power series

\frac{u^{θ}}{u − 1} − \frac{1}{x} + F_{h} = ∑_{s=0}^{∞} \frac{B̃_{s+1}(\frac{ha}{Nc})}{(s + 1)!}x^{s},  (10.15)

where F_{h} = 1/2 when h = Nc and F_{h} = 0 otherwise (the error term F_{h} deals with the discrepancy between B̃_{1}(0) and B_{1}(0)). Similarly, write β_{d} = {hd/Nc}, let y be a formal variable and write v = e^{y}; we then have

∑_{d|N} n_{d} \frac{v^{β_{d}/d}}{v^{1/d} − 1} + G_{h} = ∑_{t=0}^{∞} ∑_{d|N} n_{d} \frac{B̃_{t+1}(\frac{hd}{Nc})}{(t + 1)!} (\frac{y}{d})^{t} ,  (10.16)

where G_{h} is a constant in \frac{1}{2}Z. Multiplying (10.15) and (10.16), and summing over all h, we obtain

H(u, v) := ∑_{h=1}^{Nc} (∑_{d|N} n_{d} \frac{v^{β_{d}/d}}{v^{1/d} − 1} + G_{h}) (\frac{u^{θ}}{u − 1} + F_{h})  (10.17)
= ∑_{s,t=0}^{∞} ∑_{h=1}^{Nc} ∑_{d|N} n_{d} \frac{B̃_{s+1}(\frac{ha}{Nc})}{(s + 1)!} \frac{B̃_{t+1}(\frac{hd}{Nc})}{(t + 1)!}x^{s} (\frac{y}{d})^{t} .  (10.18)

The −1/x terms from (10.15) have dropped out in (10.17) since summing (10.16) over all h gives the value 0. By the same reasoning, we may replace \frac{u^{θ}}{u−1} in equation (10.17) defining H(u, v) by \frac{u^{θ}−1}{u−1} (this will be useful in later computations). Recalling that u = e^{x} and v = e^{y}, we define the commuting differential operators

D_{u} = u\frac{∂}{∂u} = \frac{∂}{∂x} and D_{v} = v \frac{∂}{∂v} = \frac{∂}{∂y} .

Using (10.14) and (10.18), we then have

(1 − p^{n+m}) ∑_{d|N} n_{d}d^{−ℓ} \frac{D_{n+m−ℓ+1,ℓ+1} (\frac{a}{Nc/d})}{(n + m − ℓ + 1)(ℓ + 1)} =

(Nc)^{n+m−ℓ}(D_{u}^{ℓ}D_{v}^{n+m−ℓ}H^{∗}(u, v))|_{(u,v)=(1,1)},

where

H^{∗}(u, v) := H(u, v) − H(u^{p}, v^{p}).

We thus find that

I_{n,m} = 12 ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (−1)^{ℓ}(Nc)^{n+m−ℓ}(D_{u}^{ℓ}D_{v}^{n+m−ℓ}H^{∗}(u, v))|_{(1,1)}
= (NcD_{v})^{m}(aD_{v} − D_{u})^{n}(12H^{∗}(u, v))|_{(1,1)}.  (10.19)

If we define a change of variables (u, v) = (z^{−1}, w^{Nc}z^{a}), then D_{w} = NcD_{u} and D_{z} = aD_{u} − D_{v}. Hence we obtain

J_{n,m} = \binom{D_{w}}{m}\binom{D_{z}}{n}(12H^{∗}(u, v))|_{(1,1)}.

The following lemma will allow us to prove that these rational numbers lie in Z_{p}.

Lemma 10.5.3. Consider the subset R of Z_{p}(u^{1/Nc}, v^{1/Nc}) defined by

R := \{\frac{P}{Q} where P, Q ∈ Z_{p}[u^{1/Nc}, v^{1/Nc}] and Q(1, 1) ∈ Z_{p}^{×}\} .

Then R is a ring stable under the operators \binom{D_{w}}{m} and \binom{D_{z}}{n}.

Proof. The proof of this proposition follows exactly as in Lemma 3.4.2 of [21], except for the subtlety that we must check that Z_{p}[u^{1/Nc}, v^{1/Nc}] is stable under the given differential operators; for this it suffices to check that for example

\binom{D_{z}}{n}(v^{1/Nc}) = \frac{z^{n}}{n!} \frac{∂^{n}}{∂z^{n}} (wz^{a/Nc}) = \binom{a/Nc}{n}wz^{a/Nc},

which lies in Z_{p}[u^{1/Nc}, v^{1/Nc}] because p does not divide Nc; similarly for the other cases.

Thus to prove that J_{n,m} ∈ Z_{p}, it suffices to prove that H^{∗}(u, v) is an element of

R, and for this it suffices to prove that H(u, v) ∈ R. Writing

Ψ_{d}(v) := 1 + v^{1/d} + ⋯ + v^{(d−1)/d},

we have

∑_{d|N} n_{d} \frac{v^{β_{d}/d}}{v^{1/d} − 1} = \frac{1}{v − 1} ∑_{d|N} n_{d}v^{β_{d}/d}Ψ_{d}(v)
= \frac{1}{Ψ_{Nc}(v)} · \frac{∑_{d|N} n_{d}v^{β_{d}/d}Ψ_{d}(v)}{v^{1/Nc} − 1}.  (10.20)

Since the numerator of the rightmost term in (10.20) is a polynomial in v^{1/Nc} that vanishes when v^{1/Nc} = 1, the rightmost term itself is a polynomial in v^{1/Nc}. Since we are assuming that p does not divide Nc, equation (10.20) then implies that

∑_{d|N} n_{d} \frac{v^{β_{d}/d}}{v^{1/d} − 1} ∈ R.

Similarly one shows that \frac{u^{θ}−1}{u−1} ∈ R, and it follows that H(u, v) ∈ R. This concludes the proof of Lemma 10.5.1.

10.6 A partial modular symbol of measures on Z_{p} × Z_{p}

In this section, we use the measures ν_{ξ} to construct a partial modular symbol of

measures on Z_{p} × Z_{p} encoding the periods of F_{k}. Note that Z_{p} × Z_{p} is stable under the action of Γ_{0}(N).

Lemma 10.6.1. There exists a unique Γ_{0}(N)-invariant partial modular symbol ν of Z_{p}- valued measures on Z_{p} × Z_{p} such that

∫_{Z_{p}×Z_{p}} h(x, y) dν{r → s}(x, y) = Re ((1 − p^{k−2}) ∫_{r}^{s} h(z, 1)F_{k}(z) dz)  (10.21)

for r, s ∈ Γ∞, and every homogeneous polynomial h(x, y) ∈ Z[x, y] of degree k − 2.

Proof. Uniqueness follows from Lemma 10.5.2; we must show existence. Let M denote the Γ-module of degree-zero divisors on the set Γ∞. Let M′ ⊂ M be the set of divisors m for which there exists a Z_{p}-valued measure ν_{m} on Z_{p} × Z_{p} such that

∫_{Z_{p}×Z_{p}} h(x, y) dν_{m}(x, y) = Re ((1 − p^{k−2}) ∫_{m} h(z, 1)F_{k}(z) dz) .

We must show that M′ = M.

It is clear that M′ is a subgroup of M. We will show that M′ is a Γ_{0}(N)-stable

submodule. Let m ∈ M′ and γ = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Γ_{0}(N) ; for compact open U ⊂ Z_{p} × Z_{p}

define

ν_{γm}(U) := ν_{m}(γ^{−1}U).

Define a right action of Γ_{0}(N) on the space of polynomials in two variables by

h|_{γ}(x, y) = h(Ax + By, Cx + Dy).

We calculate

∫_{Z_{p}×Z_{p}} h(u, v) dν_{γm}(u, v) = ∫_{Z_{p}×Z_{p}} h(u, v) dν_{m}(γ^{−1}(u, v))
= ∫_{Z_{p}×Z_{p}} h|_{γ}(x, y) dν_{m}(x, y)
= Re ((1 − p^{k−2}) ∫_{m} h|_{γ}(z, 1)F_{k}(z) dz)
= Re ((1 − p^{k−2}) ∫_{γm} h(u, 1)F_{k}(u) du),  (10.22)

where equation (10.22) uses the change of variables u = γz and the fact that F_{k}|_{γ^{−1}} = F_{k}. Therefore, M′ is a Γ_{0}(N)-stable submodule of M. Lemma 10.5.1 shows that [a/Nc] − [∞] lies in M′ when p does not divide c. Since the Γ_{0}(N)-module generated by these elements is all of M, we indeed have M′ = M. Furthermore, the Γ_{0}(N)-invariance of ν follows from uniqueness and the calculation of (10.22) above.

10.7 From Z_{p} × Z_{p} to X

In this section we show that the measures ν{x → y} of Lemma 10.6.1 are supported

on the set X ⊂ Z_{p} × Z_{p} of primitive vectors.

Lemma 10.7.1. Let r, s ∈ Γ∞ and let k ≥ 2. We have

∫_{Z_{p}×Z_{p}^{×}} h(x, y) dν{r → s}(x, y) = Re (∫_{r}^{s} h(z, 1)F_{k}^{∗}(z) dz)

for every homogeneous polynomial h(x, y) ∈ Z[x, y] of degree k − 2.

Proof. The characteristic function of the open set Z_{p}×Z_{p}^{×} is lim_{j→∞} y^{(p−1)p^{j}}. For notational simplicity, let g = (p − 1)p^{j} throughout the remainder of this section. Then for n, m ≥ 0 and k = n + m + 2, we have

∫_{Z_{p}×Z_{p}^{×}} x^{n}y^{m} dν \{\frac{a}{Nc} → ∞\} (x, y)

= lim_{j→∞} ∫_{Z_{p}×Z_{p}} x^{n}y^{m+g} dν \{\frac{a}{Nc} → ∞\} (x, y)
= lim_{j→∞} 12(1 − p^{k+g−2}) ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (−1)^{ℓ} ×

∑_{d|N} \frac{n_{d}}{d^{ℓ}} D_{k+g−ℓ−1,ℓ+1} (\frac{a}{Nc/d})

= 12 ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (−1)^{ℓ} ∑_{d|N} \frac{n_{d}}{d^{ℓ}} lim_{j→∞} D_{k+g−ℓ−1,ℓ+1} (\frac{a}{Nc/d}). (10.23)

Meanwhile we calculate

Re (∫ i∞_{\frac{a}{Nc}} z^{n}F_{k}^{∗}(z) dz)
= Re (∫ i∞_{\frac{a}{Nc}} z^{n}F_{k}(z) dz − p^{k−n−2} ∫ i∞_{\frac{pa}{Nc}} z^{n}F_{k}(z) dz)
= 12 ∑_{ℓ=0}^{n} \binom{n}{ℓ} (\frac{a}{Nc})^{n−ℓ} (−1)^{ℓ} ×
∑_{d|N} \frac{n_{d}}{d^{ℓ}} [D_{k−ℓ−1,ℓ+1} (\frac{a}{Nc/d}) − p^{k−ℓ−2}D_{k−ℓ−1,ℓ+1} (\frac{pa}{Nc/d})]. (10.24)

The following lemma implies that (10.23) and (10.24) are equal, and finishes the proof.

Lemma 10.7.2. Let s, t ≥ 0. For a rational number x, we have in Q_{p}:

lim_{j→∞} D̃_{s+g,t}(x) = D̃_{s,t}(x) − p^{s−1}D̃_{s,t}(px).  (10.25)

Proof. This essentially follows from the generalized Kummer congruences for Bernoulli polynomials. Let x = a/c and assume first that p does not divide c. Let b denote an integer such that abp ≡ 1 (mod c). Then

D̃_{s,t}(a/c) = c^{s−1} ∑_{ℓ=1}^{c} B̃_{s}(ℓbp/c)B̃_{t}(ℓ/c).  (10.26)

Similarly

D̃_{s+g,t}(a/c) = c^{s+g−1} ∑_{ℓ=1}^{c} B̃_{s+g}(ℓbp/c)B̃_{t}(ℓ/c)

and

D̃_{s,t}(pa/c) = c^{s−1} ∑_{ℓ=1}^{c} B̃_{s}(ℓb/c)B̃_{t}(ℓ/c).

Write y = {ℓbp/c} and y′ = {ℓb/c}. Since c^{g} → 1, it suffices to prove that

lim_{j→∞} B_{s+g}(y) = B_{s}(y) − p^{s−1}B_{s}(y′).

For s > 0, this follows from the proof of Theorem 3.2 of [41], which applies for our purposes even in the case s ≡ 0 (mod p − 1). For s = 0, the desired equality follows from the fact that the p-adic L-function L_{p}(s, χ) for a Dirichlet character χ is analytic at s = 1 unless χ = 1, in which case L_{p} has a simple pole with residue 1 − 1/p. This completes the proof for the case x ∈ Z_{p}.

We now handle the case x ∉ Z_{p}. From equation (10.26), one sees that

D̃_{s,t}(a/c) = c^{s−t}D̃_{t,s}(bp/c).

Thus the result proved above is that

lim_{j→∞} D̃_{t,s+g}(bp/c) = D̃_{t,s}(bp/c) − p^{s−1}D̃_{t,s}(b/c)  (10.27)

whenever p ∤ c. By switching indices in a similar fashion, equation (10.25) for x = a/bp becomes

lim_{j→∞}(bp)^{s+g−t}D̃_{t,s+g}(c/bp) = (bp)^{s−t}D̃_{t,s}(c/bp) − p^{s−1}b^{s−t}D̃_{t,s}(c/b)  (10.28)

where ac ≡ 1 (mod bp). We will reduce equation (10.28) to equation (10.27) by means of the reciprocity law for these generalized Dedekind sums, given in Theorem 2 of [19]. Let b > 0; the reciprocity law then states

b^{s−t}D̃_{t,s}(c/b) = sign(c) ∑_{ℓ=0}^{t} \frac{s}{s + ℓ}\binom{t}{ℓ}(−1)^{s+ℓ}b^{−ℓ}c^{s−t+ℓ}D̃_{t−ℓ,s+ℓ}(b/c)  (10.29)
+ ∑_{σ=0}^{s+t} \begin{matrix}\binom{s+t−σ−1}{t−1} & \binom{s+t}{σ} \\ ( & {s+t \atop t} & ) \end{matrix}(−1)^{σ}b^{σ−t}c^{s−σ}D̃_{t+s−σ,σ}(0)  (10.30)
+ \begin{cases}− sign(c)/4 & if s = t = 1 \\ 0 & otherwise. \end{cases}

The sum in (10.29) is taken to be 0 if s = 0. We will call the terms in the sum on line (10.29) “type I” terms and those on line (10.30) “type II” terms. Using the Dedekind reciprocity law on each of the terms in (10.28), one easily checks that the desired limit holds for the type I terms by (10.27). The same is true for each of the type II terms with σ = 0, …, s + t. To conclude the proof, one checks that each of the type II terms for σ = s + t + 1, …, s + t + g arising from the reciprocity law for (bp)^{s+g−t}D̃_{t,s+g}(c/bp) has ord_{p} greater than ord_{p}(g) minus some constant depending only on s and t. Thus in the limit, the sum of these terms vanishes.

We can now prove:

Lemma 10.7.3. The measures ν{r → s} are supported on X.

Proof. Let γ ∈ Γ_{0}(N). As in (10.22) above, we calculate for a homogeneous polynomial h(x, y) of degree k − 2,

∫_{γ(Z_{p}×Z_{p}^{×})} h(x, y) dν{r → s}(x, y) = Re (∫_{r}^{s} h(z, 1)F_{k}^{∗}|_{γ^{−1}}(z) dz) .  (10.31)

Let {γ_{i}}_{i=1}^{p+1} be a set of left coset representatives for Γ_{0}(N)/Γ_{0}(Np). Then

⋃_{i=1}^{p+1} γ_{i}(Z_{p} × Z_{p}^{×})

is a degree p cover of X. Hence from (10.31) we find that

p ∫_{X} h(x, y) dν{r → s}(x, y) = ∑_{i=1}^{p+1} Re (∫_{r}^{s} h(z, 1)F_{k}^{∗}|_{γ_{i}^{−1}}(z) dz) .  (10.32)

Now

∑_{i=1}^{p+1} F_{k}^{∗}|_{γ_{i}^{−1}} = ∑_{i=1}^{p+1} (F_{k}|_{γ_{i}^{−1}} − p^{k−2}F_{k}|_{P γ_{i}^{−1}})
= (p + 1)F_{k} − T_{p}F_{k} = (p − p^{k−1})F_{k},

since F_{k} is evidently an eigenform for T_{p} with eigenvalue 1 + p^{k−1}. Thus (10.32) becomes

∫_{X} h(x, y) dν{r → s}(x, y) = Re ((1 − p^{k−2}) ∫_{r}^{s} h(z, 1)F_{k}(z) dz) .

Therefore, the integral on X of every polynomial h(x, y) equals the integral on Z_{p} × Z_{p} of h(x, y); this implies that the measure ν{r → s} is supported on X.

10.8 The measures μ̃ and Γ-invariance

The compact open set X is a fundamental domain for the action of multiplication

by p on Q_{p}^{2} − {0}. Hence if we define for compact open U ⊂ X:

μ̃{r → s}(U) := ν{r → s}(U),

then μ̃ extends uniquely to a Γ_{0}(N)-invariant partial modular symbol of Z_{p}-valued measures on Q_{p}^{2} − {0} that is invariant under the action of multiplication by p:

μ̃{r → s}(pU) = μ̃{r → s}(U)

for all compact open U ⊂ Q_{p}^{2} − {0}. Lemmas 10.6.1, 10.7.1, and 10.7.3 show that μ̃ satisfies properties (1) and (3) of Theorem 10.1.1. Furthermore, property (2) is satisfied by construction. Thus to complete the proof of Theorem 10.1.1, it remains to show that the partial modular symbol of measures μ̃ is Γ̃-invariant and Z-valued.

Lemma 10.8.1. The partial modular symbol μ̃ is invariant under Γ̃.

Proof. Since Γ̃ is generated by Γ_{0}(N) and P = (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}), it suffices to show that μ̃ is

invariant for the action of P. For a homogeneous polynomial h(x, y) of degree k − 2, we have

∫_{X} h(x, y) dμ̃{P^{−1}r → P^{−1}s}(P^{−1}(x, y))
= ∫_{P^{−1}X} h(pu, v) dμ̃ \{\frac{r}{p} → \frac{s}{p}\} (u, v).  (10.33)

Writing P^{−1}X as a disjoint union

P^{−1}X = (Z_{p} × Z_{p}^{×}) ⨆ (\begin{matrix}p & 0 \\ 0 & p \end{matrix})^{−1} (Z_{p}^{×} × pZ_{p})

and using the invariance of μ̃ under multiplication by p, (10.33) becomes

∫_{Z_{p}×Z_{p}^{×}} h(pu, v) dμ̃ \{\frac{r}{p} → \frac{s}{p}\} (u, v) + ∫_{Z_{p}^{×}×pZ_{p}} h(u, v/p) dμ̃ \{\frac{r}{p} → \frac{s}{p}\} (u, v).

By the homogeneity of h, one simplifies the above expression:

p^{2−k} ∫_{X} h(pu, v) dμ̃ \{\frac{r}{p} → \frac{s}{p}\} (u, v)

+(1 − p^{2−k}) ∫_{Z_{p}×Z_{p}^{×}} h(pu, v) dμ̃ \{\frac{r}{p} → \frac{s}{p}\} (u, v)

= Re (p^{2−k}(1 − p^{k−2}) ∫ \frac{s}{p} ; \frac{r}{p} h(pz, 1)F_{k}(z) dz+

(1 − p^{2−k}) ∫ \frac{s}{p} ; \frac{r}{p} h(pz, 1)F_{k}^{∗}(z) dz)

= Re ((p^{2−k} − 1) ∫ \frac{s}{p} ; \frac{r}{p} h(pz, 1)p^{k−1}F_{k}(pz) dz)  (10.34)
= Re ((1 − p^{k−2}) ∫_{r}^{s} h(u, 1)F_{k}(u) du) ,  (10.35)

where (10.34) uses the definition of F_{k}^{∗} and (10.35) uses the change of variables u = pz. Since this equals the integral over X of h(x, y) against the measure μ̃{r → s}, we find that μ̃ is indeed invariant for the action of P.

10.9 Integrality of the measures

In this section we prove that the measures μ̃_{m}, which a priori are only Z_{p}-valued,

actually take on integer values. We begin by reviewing the single-variable measures arising from Bernoulli polynomials. Let e ≥ 1 be a positive integer divisible by N but not by p, and let

Z = lim_{←} Z/ep^{n}Z ≅ Z/eZ × Z_{p}.

For each integer k ≥ 1, we define a Z_{p}-valued measure F_{k} on Z corresponding to the Eisenstein series F_{2k} by the rule

F_{k}(a + ep^{n} · Z) := ∑_{d|N} n_{d} (\frac{ep^{n}}{d} )^{k−1} · \frac{1}{k} · B̃_{k} (\frac{a}{ep^{n}/d})

for each integer a. The distribution relation for Bernoulli polynomials demonstrates that this is indeed a distribution for each k ≥ 1. Furthermore, the proof of the following proposition shows that these measures are actually Z_{p}-valued. For x ∈ Z, let x_{p} denote the projection of x onto Z_{p}.

Proposition 10.9.1. For every compact open set U ⊂ Z we have

F_{k}(U) = ∫_{U} x_{p}^{k−1} dF_{1}(x).

Proof. It suffices to consider U of the form U = a + ep^{n}Z for integers a. We will prove that

F_{k}(U) ≡ a^{k−1}F_{1}(U) (mod p^{n−ϵ}Z_{p})  (10.36)

where ϵ depends only on k. The key fact is that the Bernoulli polynomial B_{k}(x) begins x^{k} − \frac{1}{2}kx^{k−1} + · · ·. Therefore

F_{k}(U) ≡ ∑_{d|N} n_{d} (\frac{ep^{n}}{d} )^{k−1} \frac{1}{k} ((\frac{da}{ep^{n}} − [\frac{da}{ep^{n}} ])^{k} − \frac{k}{2} (\frac{da}{ep^{n}} − [\frac{da}{ep^{n}} ])^{k−1}) (10.37)

modulo p^{n−ϵ}Z_{p}, where ϵ is the largest power of p appearing in the denominators of the coefficients of B_{k}(x)/k. The congruence (10.37) yields:

F_{k}(U) ≡ ∑_{d|N} \frac{n_{d}}{k} ((\frac{d}{ep^{n}} ) a^{k} − ka^{k−1} [\frac{da}{ep^{n}} ] − \frac{k}{2} a^{k−1}) (mod p^{n−ϵ}Z_{p})
≡ − ∑_{d|N} n_{d}a^{k−1} [\frac{da}{ep^{n}} ] ,  (10.38)

where (10.38) uses ∑ n_{d} = ∑ n_{d}d = 0. Meanwhile we find

a^{k−1}F_{1}(U) = a^{k−1} ∑_{d|n} n_{d} (\frac{d}{ep^{n}} − [\frac{da}{ep^{n}} ]) = − ∑_{d|N} n_{d}a^{k−1} [\frac{da}{ep^{n}} ] .  (10.39)

Equations (10.38) and (10.39) yield (10.36), proving the proposition.

The measures F_{k} may be used to calculate the modular symbol of measures μ̃.

Let the fraction a/Nc be fixed; we will write ν for μ̃{a/Nc → ∞}.

Let V be a compact open subset of Z_{p}^{×}, and let f_{i} = ∑_{n=0}^{d_{i}} c_{n}(i)y^{n} be a sequence

of polynomials such that lim_{i→∞} f_{i}(x) is the characteristic function of V. Then equation (10.13) for the moments of ν yields

ν(Z_{p} × V) = lim_{i→∞} 12(1 − p^{d_{i}}) ∑_{d|N} n_{d} ∑_{n=0}^{d_{i}} c_{n}(i) · D_{n+1,1} (\frac{a}{Nc/d}) .  (10.40)

Now as in (10.14), we have

D_{n+1,1} (\frac{a}{Nc/d}) = (\frac{Nc}{d})^{n} ∑_{h=1}^{Nc} \frac{B̃_{n+1} (\frac{h}{Nc/d})}{n + 1} · B̃_{1} (\frac{ha}{Nc}) .

Hence (10.40) becomes

ν(Z_{p} × V) = lim_{i→∞} 12 ∑_{h=1}^{Nc} B̃_{1} (\frac{ha}{Nc}) ∑_{n=0}^{d_{i}} ∑_{d|N} n_{d} (\frac{Nc}{d})^{n} \frac{B̃_{n+1} (\frac{h}{Nc/d})}{n + 1}c_{n}(i).  (10.41)

Write Nc = ep^{r} with p not dividing e. Then N divides e, and in terms of the measure F_{1} above we have

∑_{n=0}^{d_{i}} ∑_{d|N} n_{d} (\frac{Nc}{d})^{n} \frac{B̃_{n+1} (\frac{h}{Nc/d})}{n + 1}c_{n}(i) = ∫_{h+ep^{r}Z} f_{i}(x_{p}) dF_{1}(x)

by Proposition 10.9.1. Therefore (10.41) becomes

ν(Z_{p} × V) = 12 ∑_{h=1}^{Nc} B̃_{1} (\frac{ha}{Nc}) F_{1}(\{x ∈ h + ep^{r}Z : x_{p} ∈ V \}).

Let us now specify V of the form V = b + p^{s}Z_{p}, with s ≥ r and b ∈ Z_{p}^{×}. Then

ν(Z_{p} × V) = 12 ∑_{{h=1 \atop h∈b+p^{r}Zp}}^{Nc} B̃_{1} (\frac{ha}{ep^{r}} ) ∑_{d|N} n_{d}B̃_{1} (\frac{y}{ep^{s}/d})  (10.42)

where y is an integer such that y ≡ h (mod e) and y ≡ b (mod p^{s}). Fixing one such y for each h we obtain

ν(Z_{p} × V) = −12 ∑_{{h=1 \atop h∈b+p^{r}Zp}}^{Nc} B̃_{1} (\frac{ha}{ep^{r}} ) ∑_{d|N} n_{d} [\frac{y}{ep^{s}/d}]  (10.43)
≡ −12\frac{a}{Nc} ∑_{{h=1 \atop h∈b+p^{r}Zp}}^{Nc}h ∑_{d|N} n_{d} [\frac{y}{ep^{s}/d}] (mod Z),

where (10.43) uses ∑ n_{d} = ∑ n_{d}d = 0. Hence to prove integrality, it suffices to consider the case a = 1. For this purpose, we return to (10.42) with a = 1 and rewrite the expression in terms of a generalized Dedekind sum:

ν(Z_{p} × V) = 12 ∑_{d|N} n_{d} ∑_{{h=1 \atop h∈b+p^{r}Zp}}^{ep^{r}/d} B̃_{1} (\frac{h}{ep^{r}/d}) B̃_{1} (\frac{y}{ep^{s}/d}) .  (10.44)

The inner sum is the generalized Dedekind Sum denoted C(1, 1, p^{s−r}, e/d, \frac{ke/d}{p^{s}}, 0) in [19], where k is an integer chosen so that ke/d ≡ b (mod p^{s}). The reciprocity law governing these Dedekind sums ([19, Theorem 2]) shows that this value equals

∑_{{h=1 \atop h∈b+p^{r}Zp}}^{ep^{r}/d} B̃_{1} (\frac{h}{ep^{r}/d}) B̃_{1} (\frac{y}{ep^{s}/d}) = \frac{(e/d)^{2}}{2p^{s−r}} − ∑_{i=1}^{p^{s−r}} B̃_{1} (\frac{i}{p^{s−r}} ) B̃_{1} (\frac{(ip^{r} − k)e/d}{p^{s−r}})
+\frac{p^{s−r}}{2e/d}B̃_{2} (−\frac{b}{p^{r}} ) + \frac{p^{s−r}}{2e/d}B̃_{2} (\frac{b}{p^{s}} )  (10.45)
−B̃_{1} (\frac{b}{p^{s}} ) .  (10.46)

Using this expression in equation (10.44), the terms from lines (10.45) and (10.46) vanish since ∑ n_{d}d = 0. The remaining line yields only terms in Z[1/p]. Since we know that ν is Z_{p}-valued, we thus conclude that ν(Z_{p} × V) ∈ Z. Since the Γ̃ translates of the sets Z_{p} × V form a basis of compact opens for Q_{p}^{2} −{0}/p^{Z} ≅ X, the Γ̃-invariance of μ̃ therefore implies:

Proposition 10.9.2. The modular symbol of measures μ̃ is Z-valued.

This completes the proof of Theorem 10.1.1.

Chapter 11

Computations

The formulas of Chapter 10 may be used to calculate the units u(α, τ) to a high

p-adic accuracy. In this chapter we describe some computations that supply empirical evidence for conjecture 8.2.5.

11.1 Method

As in Chapter 10, let K denote a real quadratic field in which p is inert, so that the

completion K_{p} is the quadratic unramified extension of Q_{p}. Let β be a primitive (p^{2} − 1)st root of unity in K_{p}^{×}, and let log_{β} denote the discrete logarithm with base β:

log_{β} : K_{p}^{×} → Z/(p^{2} − 1)Z, where \frac{x}{p^{ord_{p}(x)}β^{log_{β}(x)}} ∈ 1 + pO_{K_{p}}.

We then have the decomposition

K_{p}^{×} ≅ Z × Z/(p^{2} − 1)Z × O_{K_{p}} given by x ↦ (ord_{p}(x), log_{β}(x), log_{p}(x)).

For x = u(α, τ) and γ_{τ} = (\begin{matrix}a & ∗ \\ Nc & ∗ \end{matrix}) , these three components are given by the formulas

ord_{p}(u(α, τ)) = −12 ∑_{d|N} n_{d} · D (\frac{a}{Nc/d}) ,  (11.1)
log_{β}(u(α, τ)) = ∫_{X} log_{β}(x − yτ) dμ̃{∞ → a/Nc}(x, y),  (11.2)
log_{p}(u(α, τ)) = ∫_{X} log_{p}(x − yτ) dμ̃{∞ → a/Nc}(x, y).  (11.3)

The computations of (11.1) and (11.2) are easy to execute in practice (note that for (11.2) it suffices to take a cover of X in which x and y are determined modulo p), so we only elaborate upon the computation of (11.3).

Suppose we are content to calculate (11.3) to an accuracy of M p-adic digits. Let

m = [∞] − [a/Nc] ∈ M. Then log_{p}(u(α, τ)) equals

∫_{Z_{p}×Z_{p}^{×}} log_{p}(y) dμ̃_{m}(x, y) + ∫_{Z_{p}×Z_{p}^{×}} log_{p} (\frac{x}{y} − τ) dμ̃_{m}(x, y)  (11.4)
+ ∫_{Z_{p}^{×}×pZ_{p}} log_{p}(x) dμ̃_{m}(x, y) + ∫_{Z_{p}^{×}×pZ_{p}} log_{p} (1 − \frac{yτ}{x} ) dμ̃_{m}(x, y).  (11.5)

The first term of (11.4) is independent of τ. To evaluate this term to an accuracy of p^{M}, one finds a polynomial f(y) that is congruent to log_{p}(y) modulo p^{M} for all y ∈ Z_{p}^{×}. This can be done as follows. For each i = 1, …, p − 1, let

g_{i}(y) = ∏_{{j=1 \atop j≠i}}^{p−1}(y − j)^{M},

and let h_{i}(y) denote the power series of log_{p}(y)/g_{i}(y) on the residue disc i + pZ_{p}, truncated after M + log M terms (the extra log M terms account for the denominators divisible by powers of p in the power series of log_{p}). Then letting f_{i}(y) = g_{i}(y)h_{i}(y) and f(y) = ∑_{i=1}^{p−1} f_{i}(y) produces the desired polynomial; it has degree p(M + log M). The first term of (11.4) may then be evaluated be replacing log_{p}(y) by f(y) and using (10.24) to evaluate the integral of y^{n} on Z_{p} × Z_{p}^{×} against the measure μ̃_{m}.

The second term of (11.4) may be recognized as a push forward from X to P^{1}(Q_{p}),

and equals

∫_{Z_{p}} log_{p}(t − τ) dμ_{m}(t) = ∑_{i=0}^{p−1} ∫_{i+pZ_{p}} log_{p}(t − i + (i − τ)) dμ_{m}(t)
= ∑_{i=0}^{p−1} [log_{p}(τ − i)μ_{m}(i + pZ_{p}) + ∫_{i+pZ_{p}}log_{p} (1 − \frac{t − i}{τ − i}) dμ_{m}(t)] .

The last integrand may be expanded as a power series in the residue disc i+pZ_{p}, and hence to calculate the integral to an accuracy of p^{M} it suffices to calculate the moments

∫_{i+pZ_{p}}(t − i)^{n} dμ_{m}(t) = p^{n} ∫_{Z_{p}} u^{n} dμ_{P_{i}^{−1}m}(u)  (11.6)

for n = 0, …, M −1, where P_{i} = (\begin{matrix}p & i \\ 0 & 1 \end{matrix}), and (11.6) uses the invariance of μ under P_{i} ∈ Γ̃.

Writing P_{i}^{−1}m = w̃ = [∞] − [w], we calculate (11.6) by pulling back to X:

∫_{Z_{p}} u^{n} dμw̃(u) = ∫_{Z_{p}×Z_{p}^{×}} x^{n}y^{−n} dμ̃w̃(x, y)
= lim_{{j→∞ \atop g=(p−1)p^{j}}} ∫_{Z_{p}×Z_{p}^{×}} x^{n}y^{g−n} dμ̃w̃(x, y)
= − lim_{{j→∞ \atop g=(p−1)p^{j}}}12 ∑_{ℓ=0}^{n} \binom{n}{l}w^{n−ℓ}(−1)^{ℓ} ∑_{d|N} n_{d}d^{−ℓ}D_{g−ℓ+1,ℓ+1}(dw), (11.7)

using (10.24). Writing w = b/ep^{r} (with p ∤ e and N|e) and employing the distribution relation for Bernoulli polynomials as in (10.14), the expression (11.7) maybe expressed in terms of the single-variable measures of Section 10.9:

lim_{{j→∞ \atop g=(p−1)p^{j}}} ∑_{d|N} n_{d}d^{−ℓ}D_{g−ℓ+1,ℓ+1} (\frac{bd}{ep^{r}} ) = ∑_{h=1}^{ep^{r}} \frac{B̃_{ℓ+1}(\frac{hb}{ep^{r}})}{ℓ + 1} lim_{{j→∞ \atop g=(p−1)p^{j}}}F_{g−ℓ+1}(h + ep^{r} · Z)
= ∑_{h=1}^{ep^{r}} \frac{B̃_{ℓ+1}(\frac{hb}{ep^{r}})}{ℓ + 1} ∫_{h+ep^{r}Z}x^{−ℓ} dF_{1}(x),  (11.8)

by Proposition 10.9.1. The integrals of (11.8) may be computed modulo p^{M} by expanding x^{−ℓ} as a power series, and using Proposition 10.9.1 to calculate the moments of F_{1}.

The terms of (11.5) may be calculated similarly using the methods described above

for (11.4). Our method has broken down the computation of (11.3) into two parts. The first is the calculation of the following integrals, independent of τ:

1. ∫_{Z_{p}×Z_{p}^{×}} log_{p}(y) dμ̃_{m}(x, y),

2. ∫_{i+pZ_{p}}(t − i)^{n} dμ_{m}(t), i = 0, …, p = 1, n = 0, …, M − 1,

3. ∫_{Z_{p}^{×}×pZ_{p}} log_{p}(x) dμ̃_{m}(x, y),

4. ∫_{P^{1}(Q_{p})−Z_{p}} t^{−n} dμ_{m}(t), n = 0, … M − 1.

(The last moment arises in the computation of (11.5).) Hence our algorithm is to execute one program, which calculates for a given α, p, and M, each of these integrals once and for all as m ranges over a Γ_{0}(N)-module basis for M, to an accuracy of p^{M}. (Using the Γ_{0}(N)- invariance of the indefinite integral, it suffices to calculate (1)–(4) for a Γ_{0}(N)-module basis of M in order to evaluate the indefinite integral for all m ∈ M.) This program executes O(pM^{2}) operations. The output is stored in a file.

A second program is then run, inputting the integrals (1)–(4) from the output file

of the first program, and calculating u(α, τ) to an accuracy of p^{M} as described above. This calculation executes O(pM) operations, and hence is rather quick even when M is large. Thus to compute the p-units u(α, τ) to a high accuracy for various real quadratic fields K, it suffices to execute the (much slower) first program only once.

11.2 Results

The methods of Section 11.1 were used with the modular unit

α(z) = Δ(z)^{2}Δ(2z)^{−3}Δ(4z)

of level N = 4 and various p. A p-adic accuracy of M = 50 digits was used. In our calculations, we restricted to fields K whose maximal orders embedded into R (see 4.11), so that the resulting units u(α, τ) would be (conjecturally) defined over the narrow Hilbert class field H^{+} of K. Since R admits a ring homomorphism to Z/NZ, the restriction in this case is that the discriminant D of K is congruent to 1 modulo 8. For each p we considered all D < 500 for which p is inert in K and O_{K} contains no unit of norm −1.

The programming language MAGMA was used for the computations. In each case,

representatives for each of the h classes of quadratic forms were used to produce a τ_{i} (see [5, §5.2]) and the corresponding u(α, τ_{i}) ∈ K_{p}^{×}. Conjecture 8.2.5 predicts that the conjugate of u(α, τ_{i}) ∈ H^{+} over H is u(α, τ_{i})^{−1}. Thus the characteristic polynomial of the u(α, τ_{i}) ∈ H^{+} over K will be

P (x) = ∏_{i=1}^{h}(x − u(α, τ_{i}))(x − u(α, τ_{i})^{−1}).

The polynomial P (x) is computed in K_{p}[x] to an accuracy of 50 p-adic digits, and a simple algorithm involving shortest lattice vectors (see [8, §1.6]) is used to recognize the resulting p-adic numbers as elements of K.

Remark 11.2.1. The modular symbol ψ attached to α actually takes values in 3Z, since α is the cube of the modular function

η(z)^{8}η(2z)^{−12}η(4z)^{4}

of level 4. In order to minimize the heights of the points u(α, τ), it is preferable to replace ψ with ψ/3.

Furthermore, after executing our algorithm, it was clear that in most cases our

p-units were still powers of smaller units. If the integers ord_{p}(u(α, τ_{i})) and

μ̃{∞ → a/Nc}((u + pZ_{p}) × (v + pZ_{p}))

for (u, v) ∈ X are divisible by a common integer r relatively prime to p, then formulas (11.1)–(11.3) yield a canonical rth root of u(α, τ) in K_{p}^{×}, by replacing ψ by ψ/r. In each case where ord_{p} u(α, τ) ≠ 0, we calculated the largest r for which this was the case.

The tables below present our results; we list for each discriminant the class number

h of O_{K} (so [H^{+} : K] = 2h), the maximal value of r as described in Remark 11.2.1, the values \frac{1}{r} ord_{p} u(α, τ), and the polynomial P (x) of the u(α, τ)^{1/r} scaled to clear powers of p from the denominator. In each case the polynomials produced are indeed characteristic polynomials of p-units in H^{+}. In many cases, the units listed are powers of smaller p-units in H^{+}; in these cases, the polynomial P (x) of the largest root lying in H^{+} is listed in the table on the following line (with the root taken implied by the value of r). This root is not necessarily uniquely defined, depending on the presence of roots of unity in H^{+}.

Remark 11.2.2. Since the units we produce conjecturally have trivial valuation at each place not lying above p, they are determined uniquely by their valuations at the places above p. In particular, when the class number of K is 1 and ζ(α, τ, 0) = 0, we expect u(α, τ) to be a root of unity. To produce non-trivial units in this case, one must work with a different modular unit α to avoid the “accidental zero” caused by the linear combination of zeta functions weighted by n_{d} used to define ζ(α, τ, s).

Similarly, if K has class number 2 and the zeta values ζ(α, τ, 0) for the two equiv-

alence classes of binary quadratic forms are equal, then we expect the corresponding units to be equal, and our polynomial P (x) to factor as a square. A different modular unit must be used to generate the full narrow Hilbert class field. These features of the construction are evident in the tables.

Table 11.1: Characteristic Polynomial of u(α, τ) for p = 3

D h r ord_{p} u(α, τ)^{1/r}  P (x)
161 1 6  ±0  x^{2} − 2x + 1
209 1 6  ±6  729x^{2} + 1358x + 729
36  ±1  3x^{2} + 5x + 3
305 2 6  ±2, ±4  6561x^{4} − ^{\frac{675\sqrt{D}+3987}{2}}x^{3} + ^{\frac{75\sqrt{D}+4607}{2}}x^{2}−
\frac{675\sqrt{D}+3987}{2}x + 6561
12  ±1, ±3  81x^{4} − ^{\frac{9\sqrt{D}+345}{2}}x^{3} + ^{\frac{15\sqrt{D}+419}{2}}x^{2} − ^{\frac{9\sqrt{D}+345}{2}}x + 81
329 1 6  ±0  x^{2} − 2x + 1
377 2 12  ±1, ±3  81x^{4} − ^{\frac{21\sqrt{D}+207}{2}}x^{3} + ^{\frac{21\sqrt{D}+499}{2}}x^{2} − ^{\frac{21\sqrt{D}+207}{2}}x + 81
473 3 6  ±2, ±2, ±6  3^{10}x^{6} + ^{\frac{15795\sqrt{D}+101493}{2}}x^{5} + ^{\frac{12285\sqrt{D}+620541}{2}}x^{4}+
\frac{34905\sqrt{D}+336763}{2}x^{3} + ^{\frac{12285\sqrt{D}+620541}{2}}x^{2}+
\frac{15795\sqrt{D}+101493}{2}x + 3^{10}
12 ±1, ±1, ±3 243x^{6} + ^{\frac{−9\sqrt{D}+945}{2}}x^{5} + ^{\frac{15\sqrt{D}+1167}{2}}x^{4} + ^{\frac{21\sqrt{D}+815}{2}}x^{3}+
\frac{15\sqrt{D}+1167}{2}x^{2} + ^{\frac{−9\sqrt{D}+945}{2}}x + 243
497 1 6  ±0  x^{2} − 2x + 1

Table 11.2: Characteristic Polynomial of u(α, τ) for p = 5

D h r ord_{p} u(α, τ)^{1/r}  P (x)
33 1 6  ±2  25x^{2} + ^{\frac{3\sqrt{D}−49}{2}}x + 25
12  ±1  5x^{2} + ^{\frac{3\sqrt{D}−1}{2}}x + 5
57 1 12  ±1  5x^{2} + ^{\frac{−\sqrt{D}+9}{2}}x + 5
177 1 6  ±6  5^{6}x^{2} + ^{\frac{4011\sqrt{D}+5231}{2}}x + 5^{6}
12  ±3  125x^{2} + ^{\frac{21\sqrt{D}−191}{2}}x + 125
217 1 6  ±0  x^{2} − 2x + 1
273 2 6  ±2, ±2  (25x^{2} + ^{\frac{3\sqrt{D}−41}{2}}x + 25)^{2}
12  ±1, ±1  (5x^{2} + ^{\frac{−\sqrt{D}+3}{2}}x + 5)^{2}
297 1 12  ±3  125x^{2} − 74x + 125
36  ±1  5x^{2} + x + 5
377 2 6  ±2, ±6  5^{8}x^{4} + ^{\frac{30375\sqrt{D}+533925}{2}}x^{3} + ^{\frac{76545\sqrt{D}+102167}{2}}x^{2}

+^{\frac{30375\sqrt{D}+533925}{2}}x + 5^{8}

12 ±1, ±3 625x^{4} + ^{\frac{75\sqrt{D}−655}{2}}x^{3} + ^{\frac{−15\sqrt{D}+1447}{2}}x^{2}
+^{\frac{75\sqrt{D}−655}{2}}x + 625
393 1 6  ±10  5^{10}x^{2} + 2275534x + 5^{10}
12  ±5  3125x^{2} + 4154x + 3125
417 1 6  ±6  5^{6}x^{2} − ^{\frac{2109\sqrt{D}+18929}{2}}x + 5^{6}
12  ±3  125x^{2} + ^{\frac{19\sqrt{D}+111}{2}}x + 125
473 3 6  ±2, ±2, ±6  5^{10}x^{6} + ^{\frac{−253125\sqrt{D}−4501875}{2}}x^{5}

+^{\frac{496125\sqrt{D}+5836125}{2}}x^{4} + ^{\frac{−59535\sqrt{D}−13546883}{2}}x^{3}+ \frac{496125\sqrt{D}+5836125}{2}x^{2} + ^{\frac{−253125\sqrt{D}−4501875}{2}}x + 5^{10}

12 ±1, ±1, ±3  3125x^{6} + ^{\frac{−1125\sqrt{D}−1475}{2}}x^{5}

+^{\frac{225\sqrt{D}+47345}{2}}x^{4} + ^{\frac{−2655\sqrt{D}−6797}{2}}x^{3}+ \frac{225\sqrt{D}+47345}{2}x^{2} + ^{\frac{−1125\sqrt{D}−1475}{2}}x + 3125

497 1 6  ±0  x^{2} − 2x + 1

Table 11.3: Characteristic Polynomial of u(α, τ) for p = 7

D h r ord_{p} u(α, τ)^{1/r}  P (x)
33 1 6  ±2  49x^{2} + 94x + 49
12  ±1  7x^{2} + 2x + 7
129 1 12  ±1  7x^{2} − 2x + 7
201 1 6  ±2  49x^{2} + 94x + 7
12  ±1  7x^{2} + 2x + 7
209 1 12  ±3  343x^{2} − 610x + 343
36  ±1  7x^{2} + 5x + 7
297 1 6  ±6  7^{6}x^{2} + 153502x + 7^{6}
36  ±1  7x^{2} + 2x + 7
321 3 6  ±2, ±2, ±6  7^{10}x^{6} − ^{\frac{1188495\sqrt{D}+567084987}{2}}x^{5}+

\frac{−557865\sqrt{D}+433702773}{2}x^{4} + ^{\frac{5083155\sqrt{D}−475485877}{2}}x^{3} +^{\frac{−557865\sqrt{D}+433702773}{2}}x^{2} − ^{\frac{1188495\sqrt{D}+567084987}{2}}x + 7^{10}

12 ±1, ±1, ±3  7^{5}x^{6} − ^{\frac{2205\sqrt{D}+53361}{2}}x^{5}+

\frac{3465\sqrt{D}+48699}{2}x^{4} − ^{\frac{4455\sqrt{D}+21791}{2}}x^{3} +^{\frac{3465\sqrt{D}+48699}{2}}x^{2} − ^{\frac{2205\sqrt{D}+53361}{2}}x + 7^{5}

377 2 6 ±2, ±6 7^{8}x^{4} + ^{\frac{−1210545\sqrt{D}+3900253}{2}}x^{3} + ^{\frac{−172935\sqrt{D}+31066815}{2}}x^{2}

+^{\frac{−1210545\sqrt{D}+3900253}{2}}x + 7^{8}

12 ±1, ±3 2401x^{4} + ^{\frac{315\sqrt{D}+10017}{2}}x^{3} + ^{\frac{405\sqrt{D}+15155}{2}}x^{2}

+^{\frac{315\sqrt{D}+10017}{2}}x + 2401

465 2 6  ±4 , ±4  (2401x^{2} − 4034x + 2401)^{2}
24  ±1 , ±1  (7x^{2} + 2x + 7)^{2}
489 1 6  ±2  49x^{2} + 94x + 49
12  ±1  7x^{2} + 2x + 7

Table 11.4: Characteristic Polynomial of u(α, τ) for p = 11

D h r ord_{p} u(α, τ)^{1/r}  P (x)
57 1 6  ±2  121x^{2} − ^{\frac{15\sqrt{D}+233}{2}}x + 121
12  ±1  11x^{2} − ^{\frac{5\sqrt{D}+3}{2}}x + 11
105 2 6  ±2, ±2  (121x^{2} − ^{\frac{39\sqrt{D}+73}{2}}x + 121)^{2}
12  ±1, ±1  (11x^{2} − ^{\frac{3\sqrt{D}+13}{2}}x + 11)^{2}
129 1 6  ±2  121x^{2} + ^{\frac{−21\sqrt{D}+199}{2}}x + 121
12  ±1  11x^{2} + ^{\frac{\sqrt{D}+21}{2}}x + 11
161 1 6  ±0  x^{2} − 2x + 1
217 1 6  ±0  x^{2} − 2x + 1
249 1 6  ±6  11^{6} − ^{\frac{167295\sqrt{D}+3198553}{2}}x + 11^{6}
12  ±3  11^{3} − ^{\frac{285\sqrt{D}+587}{2}}x + 11^{3}
305 2 6  ±2, ±6  11^{8}x^{4} + ^{\frac{10372725\sqrt{D}+344443077}{2}}x^{3} + ^{\frac{23917275\sqrt{D}−61466353}{2}}x^{2}

+^{\frac{10372725\sqrt{D}+344443077}{2}}x + 11^{8}

12 ±1, ±3 11^{4}x^{4} + ^{\frac{−2475\sqrt{D}+6853}{2}}x^{3} + ^{\frac{−225\sqrt{D}+44467}{2}}x^{2}

+^{\frac{−2475\sqrt{D}+6853}{2}}x + 11^{4}

321 3 6 ±2, ±2, ±6 11^{10}x^{6} − ^{\frac{1967882169\sqrt{D}+60603418095}{2}}x^{5}+

\frac{10953497049\sqrt{D}+178199983335}{2}x^{4} −^{\frac{13842699651\sqrt{D}+210615242059}{2}}x^{3} +^{\frac{10953497049\sqrt{D}+178199983335}{2}}x^{2}− \frac{1967882169\sqrt{D}+60603418095}{2}x + 11^{10}

12 ±1, ±1, ±3  11^{5}x^{6} + ^{\frac{−4719\sqrt{D}+427251}{2}}x^{5}+

\frac{−37257\sqrt{D}+801537}{2}x^{4} + ^{\frac{−55935\sqrt{D}+531929}{2}}x^{3} +^{\frac{−37257\sqrt{D}+801537}{2}}x^{2} + ^{\frac{−4719\sqrt{D}+427251}{2}}x + 11^{5}

329 1 6  ±0  x^{2} − 2x + 1
393 1 6  ±10  11^{10}x^{2} − 50395911602x + 11^{10}
12  ±5  11^{5}x^{2} − 319798x + 11^{5}
417 1 6  ±6  11^{6}x^{2} + ^{\frac{174795\sqrt{D}−2882153}{2}}x + 11^{6}
12  ±3  11^{3}x^{2} + ^{\frac{215\sqrt{D}−813}{2}}x + 11^{3}
497 1 6  ±0  x^{2} − 2x + 1

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