Non-archimedean integration and special values of L-functions for elliptic curves over function fields

Ignazio Longhi

Department of Mathematics and Statistics

McGill University, Montreal March 2000

A thesis submitted to the Faculty of Graduate Studies and Research in partial

fulfillment of the requirements of the

degree of Doctor of Philosophy \begin{matrix}c \\ ◯ \end{matrix}Ignazio Longhi 2000

Fratri minori Yupso, (etsi indefesse laboret ut vulgus ipsum putet

testadirapam)

Contents

Abstract i

Résumé iii

Acknowledgements v

Introduction vii

Notations xi

Chapter 1. Prolegomena 1 1. Ω and T 1 2. Explicit analytic uniformization of E 4 3. Integration 7 4. The Poisson formula 8

Chapter 2. De integralibus quibusdam 11 1. Embeddings 11 2. Galois interpretation 13 3. Main results 14

Chapter 3. Hypotheses finguntur 19 1. Iwasawa algebras 19 2. Towards a p-adic L-function 20

Chapter 4. Paralipomena 23 1. The inert case 23 2. The cyclotomic case 25

Conclusion 29

Bibliography 31

Abstract

Let F be a global function field of characteristic p, ∞ a fixed place of F. The analytic uniformization of elliptic curves in characteristic p by means of Drinfeld modular curves is used to associate to an elliptic curve E over F with split multiplicative reduction at ∞ a measure μ_{E} on ℙ^{1}(F_{∞}). By choosing an appropriate embedding of a quadratic extension K/F into the matrix algebra M_{2}(F), the measure μ_{E} is pushed forward to a measure on the p-adic group G, isomorphic to an anticyclotomic Galois group over the Hilbert class field of K. It is proven that integration of this measure yields the L-invariant of E at ∞, when ∞ is split in K, and a Heegner point in E if ∞ is inert. An interpretation of these results in terms of special values of L-functions is suggested.

Résumé

Soit F un corps de fonctions de caractéristique p, ∞ une place de F. L’uniformisation des courbes elliptiques sur F par des courbes modulaires de Drinfeld est utilisée pour associer à une telle courbe elliptique E/F une mesure sur ℙ^{1}(F_{∞}). Après avoir choisi un plongement convenable d’une extension quadratique K/F dans M_{2}(F), la mesure μ_{E} donne lieu à une mesure sur un groupe p-adique G, isomorphe au groupe de Galois d’une extension anticyclotomique du corps de classes de Hilbert de K. On démontre que certaines intégrales de cette mesure donnent l’invariant L de E si ∞ est déployé dans K, et donnent un point de Heegner si ∞ est inerte. Nous suggérons une interprétation de nos résultats en terme de valeurs spéciales de fonctions-L.

Acknowledgements

I would like to express my gratitude to my supervisor, professor Henri Darmon, for constant support and advice throughout these years, and for his patience in the rush of these last days.

I also thank professor Massimo Bertolini, who directed my first steps in the fascinating field of arithmetic geometry and gave me the opportunity to come to Montréal.

Thanks to Andreas Schweizer for many useful explanations; and in general to all the people who were in the number theory group in CICMA in these years, for having helped to create a stimulating atmosphere to learn arithmetic.

This is probably a good occasion to express my gratitude to all those who helped to develop my mathematical interests, and supported them, starting with my parents.

My thanks to Indam, which supported me for most of my stay in Montréal.

Finally, thanks to my friends, who were so much of help in that most difficult task of civilized persons, to “lose and neglect the creeping hours of time”, discussing mathematics (and, sometimes, a few other things).

Introduction

Let E be an elliptic curve over a global field F. The most prominent open problem in the arithmetic theory of elliptic curves is probably the conjecture of Birch and Swinnerton-Dyer, which, in its coarsest form, asserts the equality between the order of vanishing at s = 1 of the complex function L(E/F, s) associated to E/F and the rank of the finitely generated abelian group E(F). In a more refined version, the conjecture predicts that the leading coefficient of the Taylor expansion of L(E/F, s) at 1 encodes various arithmetically meaningful quantities associated to E.

The number field case. Since the work of Iwasawa it has been a fruitful line of investigation to replace L(E/F, s) by a function of a padic variable L_{p}(E/F, s). Starting from the seventies, studies have been conducted on such p-adic L-functions L_{p}(E, s) associated to modular elliptic curves E/ℚ (by work of Mazur, Manin and many others; see e.g. [35]). These functions have been constructed as Mellin transforms of modular forms (more precisely, as Mellin transforms of measures obtained from modular forms via modular symbols); they interpolate special values of the classical L-functions. On this basis, p-adic versions of the Birch and Swinnerton-Dyer conjecture were formulated by Mazur, Tate and Teitelbaum ([36]). The most noteworthy novelty they found is the fact that, when E has split multiplicative reduction at p, the p-adic L-function always vanishes at 1: in that case, as conjectured in [36] and proven in [27], the relation

(1)  L′_{p}(E, 1) = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})}L_{alg}(E, 1)

holds. (Here q_{E} is the Tate period of E at p, log_{p} a p-adic logarithm and L_{alg} a suitable normalization of L_{ℂ}.) The coefficient on the right-hand side of (1) is called the L-invariant L_{p}(E).

A different, purely p-adic, approach to the definition of a p-adic L-function was suggested by Schneider in [41]. He considers the p-adic uniformization of a modular elliptic curve E/ℚ by means of a Shimura curve: then the rigid analytic automorphic form corresponding to the canonical differential on E yields a harmonic cocycle on a certain tree T, which can be viewed as a measure on the set of ends of T; the latter is identified with ℙ^{1}(ℚ_{p}). Schneider’s L-function is then the Mellin transform of the restriction of this measure to ℤ_{p}^{∗}. Notice that this construction requires that E has split multiplicative reduction at p and that it is a quotient of the Jacobian of a Shimura curve.

Schneider’s approach has been used to attack the conjectures of [36] and to formulate analogous ones (see e.g. [32] - a work, however, flawed by some mistakes: cf. the remark in [6], §4.1).

Let E/ℚ be an elliptic curve and K/ℚ a quadratic imaginary field such that (cond(E), disc(K)) = 1. In [3] Bertolini and Darmon introduced a p-adic L-function L_{p}(E/K, s) associated to the anticyclotomic ℤ_{p}-tower above K and formulated conjectures similar to the ones of Mazur-Tate-Teitelbaum. In particular, for elliptic curves with split multiplicative reduction at p, they were later able to prove that when p is split in K an analogue of (1) holds ([5]), while when p is inert their L-function yields a Heegner point of E(K), by evaluating the Tate map Φ_{Tate} : ℂ_{p}^{∗} → E(ℂ_{p}) at L′_{p}(E/K, 1) ([4]). In this last case, it follows by the work of Kolyvagin [33] that E(K) has rank 1. The calculations of [28] and [8] suggest that L_{p}(E/K, s) interpolates special values of twists of the complex L-function L(E/K, s).

As first realized by Iovita (and independently by Spiess), the construction of Bertolini and Darmon is better understood as an anticyclotomic version of Schneider’s L-function: this approach has been explained in [2] and used in [6] to study L-functions associated to modular forms of higher weight. The idea is to consider the Cerednik-Drinfeld p-adic uniformization of E by a Shimura curve to get a distribution on ℙ^{1}(ℚ_{p}) and then to push forward this distribution to the anticyclotomic Galois group over K, by looking at the action of K^{∗} on ℙ^{1}(ℚ_{p}) induced by an embedding of K into the appropriate quaternion algebra. Explicit computations of the special value L_{p}(E/K, 1) are carried out thanks to Teitelbaum’s Poisson formula ([46]).

The present work is an application of the methods of [2] to the function field setting.

The function field case. The Birch and Swinnerton-Dyer conjecture has been generalized to elliptic curves defined over a function field F of transcendence degree 1 over a finite field 𝔽_{q} ([44]). In this case the phenomenon is better understood: it is known that L(E/F, s) is a rational function in q^{−s} (a polynomial if j(E) is non-constant) and Tate has proved that the rank of E(K) is at most equal to the vanishing order of the complex L-function at 1.

Besides, it has been known since the appearance of Drinfeld’s great work [11] that in this setting all elliptic curves are modular, i.e., that they appear as factors of the Jacobian of some Drinfeld modular curve \overline{M}_{0}(𝔫). More recently, Gekeler and Reversat ([22]) have provided an explicit analytic construction of the map \overline{M}_{0}(𝔫) → E.

The relation between measures and rigid analytic modular forms has been investigated by Teitelbaum, who, in [47], generalized to the function field case his own work [46].

A proposal for a p-adic, and characteristic p-valued, L-function in this context was made by Goss ([26]): he constructs Mellin transforms of measures attached to modular forms by the Schneider-Teitelbaum method (see also [48], p. 290-292, where a conjecture very similar to our main result, theorem 14 of the present work, is formulated).

The anticyclotomic approach of Bertolini and Darmon has inspired Pál’s thesis ([37]), which carries over to function fields the results of [4], and suggests a geometric interpretation of Heegner points in terms of level structure at ∞.

This thesis. Let F be a function field of transcendence degree 1 over 𝔽_{q}, with q a power of p, and fix a place ∞ of F. Fix a separable quadratic extension K of F, inert or split at ∞; denote by A (resp. B) the ring of functions in F (respectively K) regular away from ∞ (resp. the places of K above ∞). Consider an elliptic curve E/F, having conductor 𝔫∞ and with split multiplicative reduction at ∞. (This is not a heavy restriction, since, unless E is defined over the constant field, j(E) has a pole at some place of F and one can obtain the required properties by taking a finite extension F′/F where the conductor of E has the desired form.) In this situation, the results of [22] apply. For simplicity, E is understood to be the strong Weil curve in its isogeny class. We will also assume that 𝔫 is coprime to the discriminant of K/F.

Chapter 1 consists of preliminaries. In the first section a few basic facts about rigid analysis on the Drinfeld upper half-plane are recalled. Section 2 gives a short summary of some results from [22] needed in this work. Section 3 introduces the main tool for our investigations, the (multiplicative) integral over ℙ^{1}(F_{∞}). In section 4, a multiplicative version of Teitelbaum’s Poisson formula is proven: this is the only part in the chapter with some claim to novelty (of course the result comes from [47], and the multiplicative approach was suggested in [2]; the idea of using van der Put’s map r, which greatly simplifies the proof, seems to be original).

Chapter 2 contains the main result. In section 1, embeddings of K into M_{2}(F) are considered: they allow to push forward to the group G := (K ⊗_{F} F_{∞})^{∗}/F_{∞}^{∗}B^{∗} the measure μ_{E} on ℙ^{1}(F_{∞}) associated to E in the first chapter; the stress is put on the split case (i.e., when ∞ splits in K). In section 2, some class field theory is recalled and used to identify G with an “anticyclotomic” Galois group. Finally, in section 3, an explicit relation between the integral ∫_{G} gdμ_{E} and the Tate period of E at ∞ is proven to hold in the split case (theorem 14); this result should be regarded as an analogue of equation (1).

In chapter 3, a brief recall of the interpretation of non-archimedean measures in terms of the Iwasawa algebra allows to define ∫_{G} gdμ_{E} as special value of an L-function. A naive attempt, still very tentative, to construct such a function is explained in the second section.

Finally, chapter 4 sketches how in the inert case our techniques can be used to construct Heegner points; this was already done by Pál, in much more detail, but following the lines of [4], in which Schneider’s construction and the Poisson formula of Teitelbaum are implicit rather than explicit. In the last section of the chapter, worked out between submission and defense, a slightly different technique of p-adic integration is used to evaluate the Gekeler-Reversat theta functions at cusps.

The assumption is made throughout the thesis that F is the rational function field 𝔽_{q}(T) and ∞ the usual place at ∞; it should be possible to generalize this to any global function field F and any place ∞.

Notations

For the convenience of the reader, we list some notations which will be used throughout this work:

• F is a function field, with constant field 𝔽_{q}, q power of prime p;

• ∞ a fixed place of F;

• A ⊂ F the ring of regular functions away from ∞;

• F_{∞} the completion of F at ∞, O_{∞} its ring of integers;

• K a separable quadratic extension of F, inert or split at ∞;

• B the integral closure of A in K;

• C_{∞} the completion of an algebraic closure of F_{∞};

• G the group (K ⊗ F_{∞})^{∗}/F_{∞}^{∗}B^{∗};

• Ω := ℙ^{1}(C_{∞}) − ℙ^{1}(F_{∞}) Drinfeld’s upper half-plane;

• T the Bruhat-Tits tree of P GL_{2}(F_{∞}), E⃗(T ) its oriented edges;

• U_{e} set of ends of T exiting from e (also U(e), according to notational convenience);

• Γ := Γ_{0}(𝔫) ⊂ GL_{2}(A), 𝔫 a (non-zero) ideal of A;

• R the A-subalgebra of M_{2}(F) generated by Γ;

• M_{0}(𝔫) the affine Drinfeld modular curve;

• \overline{M}_{0}(𝔫) compactification of M_{0}(𝔫) by cusps;

• J_{0}(𝔫) the Jacobian of \overline{M}_{0}(𝔫);

• ∂X the boundary of a subset X of a topological subspace Y: in particular, ∂Ω = ℙ^{1}(F_{∞}).

In general, for v a place of F or K (the context should make clear which is the field involved), O_{v} will denote the respective ring of integers, with maximal ideal 𝔪_{v}, π_{v} a uniformizer at v in F or K and ν_{v} the corresponding valuation, with absolute value |.|_{v} (normalized so that |π_{v}|_{v} = ♯(O_{v}/𝔪_{v})^{−1}). In the case v = ∞, we will often just write |.|, ν, π. For v ≠ ∞, put 𝔭_{v} := A ∩ 𝔪_{v}.

For the time being, assume F = 𝔽_{q}(T), A = 𝔽_{q}[T].

CHAPTER 1 Prolegomena

1. Ω and T

For a more detailed exposition of most of the contents of this section, see [22], section 1.

Drinfeld’s upper half-plane is defined as Ω := ℙ^{1}(C_{∞}) − ℙ^{1}(F_{∞}); it has a canonical rigid analytic structure. In particular, the algebra of globally holomorphic functions on Ω is obtained by choosing a covering system by increasing affinoids. To fix ideas, define the imaginary absolute value on Ω as the distance from F_{∞}:

|z|_{i} := inf{|z − x| | x ∈ F_{∞}}

and let

U_{n} := {z ∈ Ω | q^{−n} ≤ |z|_{i}, |z| ≤ q^{n}}.

A function f : U_{n} → C_{∞} is holomorphic if it is the uniform limit of a sequence of rational functions without poles in U_{n}. Then O_{Ω}(U_{n}) is a Banach algebra over C_{∞}, with the norm

∥f∥_{U_{n}} := sup{|f(z)| | z ∈ U_{n}}

and O_{Ω}(Ω) := lim_{←} O_{Ω}(U_{n}) is a Fréchet space (for more detail, cf. [12], [7]).

As usual, GL_{2}(F_{∞}) acts on Ω by Möbius transformations.

Let V be a 2-dimensional vector space over F_{∞}: the Bruhat-Tits tree T is defined having as vertices homothety classes of O_{∞}-lattices in V; the distance between two such vertices is given by the minimal length of a Jordan-Hölder sequence between two representatives (see [42], II, 1.1; another useful reference is [34], ch. 3, sections 4-6). Let V(T ), E(T ), E⃗(T ) denote respectively the sets of vertices, edges and oriented edges of T ; given e ∈ E⃗(T ), \overline{e} will be its opposite, e^{∗} the underlying non-oriented edge, o(e) the origin and t(e) the terminus.

Let E_{∞}(T ) be the set of ends of T (i.e., infinite half-lines, identified if equal outside some finite set of edges). There is a canonical bijection E_{∞}(T ) ↔ ℙ(V) ≃ ℙ^{1}(F_{∞}): given an end 𝔈, choose O_{∞}-lattices Λ_{i} ⊂ V representing its vertices and contained one in each other with index q; then ∩Λ_{i} is an O_{∞}-module of rank 1, which generates a line in ℙ(V) (see e.g. [19], lemma 25; cf. also [42], p. 72). In particular, the ends exiting from an oriented edge e correspond to the points of a compact open U_{e} and in this way one obtains a basis {U_{e}}_{e∈E⃗(T )} for the topology of ℙ^{1}(F_{∞}).

The equality ∂Ω = ℙ^{1}(F_{∞}) suggests a relation between Ω and T . This is established by the building map λ : Ω → T : its construction will be briefly recalled, together with some properties to be used in the following; for more detail, see [22] 1.5 or [19] §9.

Let T (ℝ) be the realization of T , obtained identifying edges in E(T ) with copies of the segment [0, 1] ⊂ ℝ. By a theorem of Goldman and Iwahori ([24], explained also in [10], ch. 3), points in T (ℝ) classify norms on V: the basic idea is that a lattice in V can be seen as the unit ball of a certain norm.

Fix coordinates on V, so as to get an identification V = F_{∞}^{2}; let v_{0} ∈ V(T ) be the vertex corresponding to the lattice O_{∞} ⊕ O_{∞}. The building map λ : Ω → T (ℝ) is defined associating to z ∈ Ω the norm ν_{z}(u, v) = |uz + v|. In particular, one finds that λ^{−1}(v_{0}) = U_{0}.

Following [22], define an action of GL_{2}(F_{∞}) on T so that λ is GL_{2}(F_{∞})-equivariant^{1}:

λ(γz) = γλ(z) ∀ γ ∈ GL_{2}(F_{∞}).

Then, since GL_{2}(F_{∞}) acts transively on V(T ), for any vertex v the set λ^{−1}(v) = λ^{−1}(γv_{0}) = γλ^{−1}(v_{0}) is an affinoid isomorphic to U_{0}.

The action of GL_{2}(F_{∞}) is well-behaved with respect to the identification E_{∞}(T ) = ℙ^{1}(F_{∞}): if the end 𝔈 = {e_{i}} corresponds to a ∈ ℙ^{1}(F_{∞}), then γ𝔈 = {γe_{i}} is the end associated to γa.^{2}

Let T_{n} := λ(U_{n}): it is a finite subtree.

Associate to v ∈ V(T ) the spectral norm ∥.∥_{v} on O_{Ω}(λ^{−1}(v)):

∥f∥_{v} = sup{|f(z)| | z ∈ λ^{−1}(v)}.

The seminorms {∥.∥_{v}}_{v∈V(T )} generate the same topology on O_{Ω}(Ω) as the covering {U_{n}}: it is clear that max{∥f∥_{v} | v ∈ λ(U_{n})} ≤ ∥f∥_{U_{n}}; vice versa, ∥f∥_{U_{n}} = max{∥f∥_{v} | v ∈ ∂T_{n}} follows from the Mittag-Leffler decomposition on O_{Ω}(U_{n}) ([12], I.1.3). Notice also that spectral norms

1Notice that this is not the same action as, e.g., in [42], because to have a right action on the set of norms one needs a left action on V.

2Again, this requires taking a twist of the usual correspondence E_{∞}(T ) ↔ ℙ(V)

- namely, composing it with the map z ↦ −z^{−1} on ℙ^{1}(F_{∞}); see [22], 1.6.

1. Ω AND T 3

are multiplicative: ∥fg∥_{v} = ∥f∥_{v}∥g∥_{v}.

For A an abelian group, let \underline{H}(T , A) denote the group of maps φ : E⃗(T ) → A satisfying φ(\overline{e}) = −φ(e) and ∑_{t(e)=v} φ(e) = 0; such maps are called harmonic cochains (or cocycles).

Define a family of seminorms ∥.∥_{T_{n}} on \underline{H}(T , ℤ) as

∥φ∥_{T_{n}} := max{φ(e) | e^{∗} ∈ E(T_{n})}.

The following result is due (mostly) to van der Put.

Theorem 1. The map

r : O_{Ω}^{∗}(Ω) → \underline{H}(T , ℤ), r(f)(e) := log_{q} \frac{∥f∥_{t(e)}}{∥f∥_{o(e)}}

is a surjective continuous group homomorphism with kernel C_{∞}^{∗}. (Here log_{q} is just the usual logarithm in base q.)

Proofs are given in [12], I.8.9 and V.2, in [38], th. 2.1, and in [19], th. 32. Continuity is not mentioned in the references, but is obvious: for, if ∥f −g∥_{v} < ∥g∥_{v} ∀ v ∈ T_{n}, then (from multiplicativity of ∥.∥_{v}) r(f) = r(g) on T_{n}. So r is a continuous homomorphism between “Fréchet spaces over ℤ” (i.e., projective limit of topological spaces of functions, where the algebraic structure is just of abelian group).

Theorem 1 is based on a non-archimedean version of the residue theorem ([12] I.3.3, [23] II,§3 and III.1.18.4); for the relation between r and the residue map on holomorphic differentials on Ω, cf. [22] 1.8.

Given a, b ∈ ℙ^{1}(F_{∞}) = E_{∞}(T ), let A(a, b) denote the axis in T joining them (oriented from a to b). More generally, for v, w ∈ V(T ), the notation A(v, w) will be used for the geodesic (i.e., path without backtracking) from v to w.

Index the edges of A(∞, 0) as {e_{n}}_{n∈ℤ}, with t(e_{n}) = o(e_{n+1}) and t(e_{0}) = v_{0}. Put also v_{i} := t(e_{i}). Let S_{i} be the maximal subtree intersecting A(∞, 0) only in v_{i}: it consists of all those edges which can be joined to v_{i} by a path not passing through the principal axis A(∞, 0).

Lemma 2. Let v be a vertex in V(S_{i}): then |z| ≡ q^{−i} on λ^{−1}(v).

The proof is an easy consequence of the way the building map was defined; see e.g. [16], 1.8.

By the continuity of |.|, it follows that

ν(t) = i for all t ∈ U(e_{i}) − U(e_{i+1}).

In particular, O_{∞}^{∗} = U(e_{0}) − U(e_{1}).

Given an axis A(a, b), define its “characteristic function” χ_{A(a,b)} ∈ \underline{H}(T , ℤ) as

χ_{A(a,b)}(e) := \begin{cases}1 & if e ∈ A(a, b) \\ −1 & if \overline{e} ∈ A(a, b) \\ 0 & otherwise. \end{cases}

The following lemma comes from [16], 1.14.

Lemma 3. Let a, b ∈ ℙ^{1}(F_{∞}): then r (\frac{z−a}{z−b}) = χ_{A(a,b)}.

Proof. Since ∥f ∘ γ∥_{v} = ∥f∥_{γv} for γ ∈ GL_{2}(F_{∞}) and this group acts transitively on the set of geodesics in T (that is, the set of ordered pairs (a, b), a and b distinct points of ℙ^{1}(F_{∞})), one is reduced to check that r(z) = χ_{A(0,∞)}; this is trivial from the fact that ∥z∥_{v} changes only along the axis A(∞, 0), as follows from lemma 2. □

The reader is reminded that γ ∈ GL_{2}(F_{∞}) is said to be hyperbolic if its eigenvalues λ_{1}, λ_{2} are distinct in absolute value. Such an element has two fixed points a, b in ∂Ω and it acts on T by shifting of ν(λ_{1}) − ν(λ_{2}) along the axis A(a, b).

The prototype is (\begin{matrix}π & 0 \\ 0 & 1 \end{matrix}) (which, by abuse of notation, will be

called simply π). On Ω, π acts as a homothety, decreasing the absolute value of a factor q^{−1}. On T , πv_{i} = v_{i+1} for vertices in A(∞, 0); this completely determines the effect of π, since GL_{2}(F_{∞}) acts on the tree by isometries.

For more detail, see e.g. [34], 3, 6.1.

Define the level of a vertex v as its distance from the principal axis A(∞, 0); the level of an edge or of a path is then the minimum level of its vertices. By abuse of notation, for z, z′ ∈ λ^{−1}(V(T )), A(z, z′) will denote the path from λ(z) to λ(z′).

Lemma 4. Assume that A(z_{1}, z_{2}) has level n > 0: then

|\frac{z_{1}}{z_{2}} − 1| ≤ |π^{n}|.

Proof (sketch). Recall that |z| is constant on λ^{−1}(S_{i}). Therefore, shifting by a power of π, one can assume that A(z_{1}, z_{2}) is contained in S_{0}. In these conditions, |z_{1} − z_{2}| ≤ |π^{n}|, as can be seen from the same reasoning as in lemma 2. □

2. Explicit analytic uniformization of E

The main reference for this section is [22]; see also [18], [17].

2. EXPLICIT ANALYTIC UNIFORMIZATION OF E 5

The Drinfeld modular curve M_{0}(𝔫) is an affine algebraic curve whose set of C_{∞}-valued points can be described (in the case A = 𝔽_{q}[T]) as the quotient of Ω by the action of Γ. The addition of a finite set of cusps yields a “compactified” curve \overline{M}_{0}(𝔫).

In more general terms, M_{0}(𝔫) is a (possibly coarse) moduli scheme representing the functor of rank 2 Drinfeld modules with 𝔫-level structure. These curves were introduced by Drinfeld in [11], where he proved (among other things) that an analogue of the Shimura-Taniyama-Weil conjecture holds for elliptic curves over global function fields.

Let E/F be an elliptic curve of conductor 𝔫∞ and with split multiplicative reduction at ∞: by Drinfeld’s theorems (together with results of Weil, Grothendieck, Jacquet-Langlands, Deligne and Zarhin), there is a non-constant morphism \overline{M}_{0}(𝔫) → E defined over F. By means of rigid analytic geometry, Gekeler and Reversat ([22]) have explicitly described this map.

For simplicity, assume that E is the strong Weil curve in its isogeny class.

The presence of cusps for M_{0}(𝔫) translates, in terms of the tree T , into the fact that Γ\T is the union of a finite graph with a finite set of half-lines.

Let \underline{H}_{!}(T , ℤ)^{Γ} be the group of cuspidal ℤ-valued harmonic cocycles for Γ, i.e.,

\underline{H}_{!}(T , ℤ)^{Γ} := \{φ ∈ \underline{H}(T , ℤ) | Γ-invariant finitely supported mod Γ\}.

Elements in \underline{H}_{!}(T , ℤ)^{Γ} correspond to cusp forms for Γ ([22], section 4): finiteness of support means precisely that these cocycles come from forms vanishing at the cusps.

The action of Hecke operators on modular forms translates into an action on harmonic cocycles. By Drinfeld results in [11], the isogeny class of E corresponds to a cocycle φ_{E} ∈ \underline{H}_{!}(T , ℤ)^{Γ}, determined (up to a scalar) by being an eigenform for the Hecke operators, with the eigenvalues associated to E.

Let Γ̃ := Γ/Γ ∩ Z(GL_{2}(F)) (Z(G) denotes the center of the group G) and \overline{Γ} := Γ^{ab}/tor(Γ^{ab}); choose any vertex v of T . One has a homomorphism j : \overline{Γ} → \underline{H}_{!}(T , ℤ)^{Γ}, where j(α)(e) counts the number of times γe lies in path(v, αv) as γ varies in Γ̃ ([22], section 3); this number is independent of v. In the case A = 𝔽_{q}[T], j is an isomorphism ([22], 6.4.5): put γ_{E} := j^{−1}φ_{E}.

Let w be any point in Ω. To γ ∈ Γ one associates a function u_{γ} : Ω → C_{∞} defined by

u_{γ}(z) := ∏_{ε∈Γ̃} \frac{z − εw}{z − εγw}.

Theorem 5. The product u_{γ} converges to a holomorphic theta function on Ω; the value u_{γ}(z) is independent of the choice of w. The automorphy factor c_{γ}(α) := \frac{u_{γ}(αz)}{u_{γ}(z)} induces a group homomorphism \overline{c} : \overline{Γ} → Hom(\overline{Γ}, C_{∞}^{∗}) by γ ↦ c_{γ}. The map \overline{Γ} × \overline{Γ} → C_{∞}^{∗}, defined by (α, β) ↦ c_{α}(β), is a symmetric bilinear pairing.

All of this (and more) is proved in th. 5.4.1 of [22].

Theorem 6. (Gekeler-Reversat) The analytic uniformization of E is given by the commutative diagram

1 ⟶ \overline{Γ} ⟶_{\overline{c}} Hom(\overline{Γ}, C_{∞}^{∗}) ⟶ J_{0}(𝔫)(C_{∞}) ⟶ 0
↓  ↓^{ev_{E}}  ↓
1 ⟶ Λ_{E} ⟶  C_{∞}^{∗}  ⟶^{Φ_{Tate}}  E  ⟶ 0

Here ev_{E} is just the evaluation map at γ_{E} and Λ_{E} := ev_{E}(\overline{c}(\overline{Γ})). In the case A = 𝔽_{q}[T], it is proved in [17] that Λ_{E} is equal to Q^{ℤ} for some Tate period Q.

To establish \overline{Γ}\Hom(\overline{Γ}, C_{∞}^{∗}) ≅ J_{0}(𝔫)(C_{∞}) one constructs, by theta functions, an explicit Abel-Jacobi map

Φ_{AJ} : Div^{0}(Ω) → J_{0}(𝔫)(C_{∞}), Φ_{AJ}((a) − (b))(γ) = \frac{u_{γ}(a)}{u_{γ}(b)}

([22], section 7). The commutative diagram

Div^{0}(Ω) ⟶ Hom(\overline{Γ}, C_{∞}^{∗}) ⟶ C_{∞}^{∗}
↓  ↓  ↓^{Φ_{Tate}}
Div^{0}(\overline{M}_{0}(𝔫)) ⟶ J_{0}(𝔫)(C_{∞}) ⟶ E

yields that the image in E of a divisor (a) − (b) on Ω is

Φ_{Tate}Φ_{AJ}((a) − (b))(γ_{E}) = Φ_{Tate} (\frac{u_{γ_{E}}(a)}{u_{γ_{E}}(b)}) .

3. Integration

Let X be a topological space such that the compact open subsets form a basis for the topology. Define the group M(X, ℤ) of ℤ-valued distributions on X as consisting of the finitely additive functions on the compact open subsets of X. In the case X is compact, distributions are measures and M_{0}(X, ℤ) is the kernel of μ ↦ μ(X).

Let C be a complete topological abelian group such that a basic system of neighbourhoods of the identity consists of open closed subgroups (e.g., C = C_{∞}^{∗}). The group law on C will be written multiplicatively.

The set of continuous functions C(X, C) is a topological group with the compact open topology: a fundamental system of open subgroups is {f | f(𝔎) ⊂ V }, where 𝔎 varies among compact subsets of X and V runs through a fundamental system of open subgroups of C.

From now on, assume that X is compact.

Definition. Given a continuous function f : X → C, its integral is defined as:

(2)  ∫_{X} f(t)dμ(t) := lim ∏_{U_{n}∈C_{α}} f(t_{n})^{μ(U_{n})},

where the limit is taken over the direct system of finite covers C_{α} of X by compact open subsets and t_{n} is an arbitrary point in U_{n}.

Proposition 7. The limit in (2) exists and is independent of the choice of the t_{n}’s. Besides,

∫_{X} .dμ : C(X, C) → C

is a continuous homomorphism.

Proof (sketch). Let V be an open subgroup of C: the sets f^{−1}(cV), c ∈ C, form an open covering of X, which can be refined to a covering by compact open subsets. Take a finite subcover C_{α(V)}: then for any α ≥ α(V), for any U_{n} ∈ C_{α}, having t and t′ both in U_{n} implies that f(t)f(t′)^{−1} belongs to V. The existence of the limit follows.

It is immediate to see that the integral defines a group homomorphism C(X, C) → C. To check its continuity, just observe that if V is an open closed subgroup of C, integration over X sends {f | f(X) ⊂ V } into V.

□

Caveat. Essentially all integrals considered will be multiplicative: in the case C = C_{∞}^{∗}, the reader is cautioned not to confuse ∫_{X} fdμ with the usual integral taking values in the additive group C_{∞}, which will be denoted by ∫_{X} fd^{+}μ.

The open basis {U_{e}}_{e∈E⃗(T )} for the topology of ∂Ω = ℙ^{1}(F_{∞}) is used to establish a bijection between the set of harmonic cocycles on T , \underline{H}(T , ℤ), and M_{0}(∂Ω, ℤ): to a cocycle φ one associates the measure μ_{φ} by μ_{φ}(U_{e}) := −φ(e). The negative sign is necessary to make theorem 8 consistent with Teitelbaum’s results in [47], where the opposite orientation on ends of T seems to be used.

To compute integrals on ∂Ω, it is convenient to fix a vertex v_{0} in T and, for e ∈ E⃗(T ) pointing away from v_{0}, let l(e) denote the distance between v_{0} and the origin of e. Then it is clear that

∫_{∂Ω} f(t)dμ(t) = lim_{n→∞} ∏_{l(e)=n} f(t_{e})^{μ(U_{e})}

independently of the choice of v_{0}.

Let μ_{E} be the measure coming from φ_{E}.

Notice that GL_{2}(F_{∞}) acts naturally on M(ℙ^{1}(F_{∞}), ℤ), as well as on \underline{H}(T , ℤ), by γ ∗ μ(U) := μ(γ^{−1}U) (respectively γ ∗ φ(e) = φ(γ^{−1}e)). It is obvious that ∫_{γU} fd(γ ∗ μ) = ∫_{U}(f ∘ γ)dμ.

4. The Poisson formula

As in [2], one expects to recover u_{γ_{E}} as a multiplicative integral on ∂Ω. Recall van der Put’s map r : O_{Ω}^{∗}(Ω) → \underline{H}(T , ℤ). Let Θ_{h}(Γ) denote the group of holomorphic theta functions for Γ: then r induces an injection Θ_{h}(Γ)/C_{∞}^{∗} → \underline{H}_{!}(T , ℤ)^{Γ}. (As a matter of fact, this is an isomorphism: cf. [22], 6.4.3.)

Theorem 8. Let u ∈ Θ_{h}(Γ) and define φ = r(u). Choose z_{0} ∈ Ω: then for z ∈ Ω

u(z) = u(z_{0}) ∫_{∂Ω} \frac{z − t}{z_{0} − t}dμ_{φ}(t).

Proof. The partial products P_{N}(z) = ∏_{l(e)=N}(\frac{z−t_{e}}{z_{0}−t_{e}})^{φ(e)} converge uniformly on each U_{n}: hence the integral on the right-hand side defines a nowhere vanishing rigid analytic function of z. To prove that it is u (up to a scalar c ∈ C_{∞}^{∗}), it is enough to check the identity φ = r (∫_{∂Ω} \frac{z−t}{z_{0}−t}dμ_{φ}(t)). It is then immediate to see that c = u(z_{0}).

By the continuity of r,

r (∫_{∂Ω} \frac{z − t}{z_{0} − t}dμ_{φ}(t)) = lim_{N→∞} ∑_{l(e)=N} φ(e)r (\frac{z − t_{e}}{z_{0} − t_{e}} ) .

By lemma 3, r(z − t_{e}) = χ_{A(te,∞)}. Now compute this cocycle on e′ ∈ E⃗(T ): without loss of generality, it can be assumed that e′ points away from ∞. Then \overline{e}′ belongs to A(t_{e}, ∞) if and only if t_{e} is in U(e′): thus

r (∫_{∂Ω} \frac{z − t}{z_{0} − t}dμ_{φ}(t)) (e′) = lim_{N→∞} ∑_{l(e)=N} φ(e)χ_{A(te,∞)}(e′) =

lim_{N→∞} ∑_{\begin{matrix}t_{e}∈U(e′) \\ l(e)=N \end{matrix}} −φ(e) = −φ(e′).

The last passage is a direct consequence of the harmonicity of φ. □

Remark. The above is a multiplicative version of Teitelbaum’s result ([47]) that ∫_{∂Ω} \frac{1}{z−t}d^{+}μ_{φ}(t) is the modular form corresponding to φ. This can be seen applying to both sides the differential logarithm operator, d log : f ↦ \frac{f′}{f}, which can be brought inside the integral by continuity. Note, however, that, as remarked in [22], in the characteristic p context the operator d log tends to be quite forgetful: hence theorem 8 says actually something more than in [47].

Besides, [47] takes a different approach - and proves much more - than the one here, suggested by [2]. It might be an interesting exercise to develop this multiplicative machinery to recover all of [47], thm. 11 (where the Poisson formula is proven for cusp forms of any weight n ≥ 2) and maybe obtain a simpler proof of theorem 16 (the bijection between cusp forms and harmonic cocycles).

The following corollary is the form of theorem 8 to be used in future applications.

Corollary 9. Let u, φ be as in theorem 8: then for z, z′ ∈ Ω

\frac{u(z)}{u(z′)} = ∫_{∂Ω} \frac{z − t}{z′ − t}dμ_{φ}(t).

Proof. Immediate, remembering that one deals with a multiplicative integral.

□

10 1. PROLEGOMENA

Corollary 10. For z ∈ Ω, one has

u_{γ_{E}}(z) = u_{γ_{E}}(z_{0}) ∫_{∂Ω} \frac{z − t}{z_{0} − t}dμ_{E}(t).

Proof. Just use r(u_{γ}) = j(γ) ([22], thm. 5.6.1).

□

CHAPTER 2 De integralibus quibusdam

In the previous sections, a p-adic measure on ∂Ω was associated to E and used to recover geometric information about the elliptic curve (i.e., u_{γ} and hence the uniformizing map). In this chapter, the action of GL_{2}(F_{∞}) on ∂Ω will be exploited to transfer the measure to the profinite group G defined in the next section. It will be then proven that integration on G yields information about the arithmetic of E.

1. Embeddings

Let R be the A-algebra generated by Γ in M_{2}(F) and K/F a separable quadratic extension, inert or split at ∞.

Lemma 11. Assume that (disc(K), 𝔫) = 1. Then there is an F - algebras homomorphism Ψ : K → M_{2}(F) such that B is sent into R if and only if all the primes dividing 𝔫 in F are split in K. The classes of these isomorphisms modulo conjugation by Γ are in bijection with Pic(B).

Proof. An embedding Ψ : B → M_{2}(A) induces an action of B on A ⊕ A, which so becomes a B-module M, projective (since it is torsion free and finitely generated) and of rank 1. The fact that Ψ(B) takes values in R means that N = A ⊕ 𝔫 is a B-submodule of M. Thus the existence of Ψ implies the existence of a torsion B-module M/N ≃ A/𝔫. Clearly M/N is cyclic, and its annihilator is contained in 𝔫B: that is, M/N ≃ B/𝔫̃, with 𝔫̃ a divisor of 𝔫B. Let 𝔫 = ∏ 𝔭^{ν_{p}(𝔫)} be the prime decomposition: localizing, one is reduced to look for Bmodules isomorphic to A/𝔭^{ν_{p}(𝔫)}. Let 𝔓 be a prime in B above 𝔭: if 𝔭 splits, then A/𝔭^{i} is isomorphic to B/𝔓^{i} for all i, while if 𝔭 is inert this never happens.

On the other side, projective B-modules of rank 1 are isomorphic to the fractional ideals of B. If all primes dividing 𝔫 split, let 𝔞 be any ideal of B: then (since A is PID) 𝔞 ≃ A⊕A, and, after choosing a basis, the natural B-action becomes an embedding B → M_{2}(A). Let 𝔫_{1} ⊂ B be an ideal of norm 𝔫, so that A/𝔫 ≃ B/𝔫_{1}. Choosing 𝔞 coprime to 𝔫_{1}, one gets a surjection 𝔞 ⊕ 𝔫_{1} → B, from which 𝔞/𝔞𝔫_{1} ≃ A/𝔫: hence coordinates can be chosen so that 𝔞𝔫_{1} = A ⊕ 𝔫, yielding an embedding of B in R. □

It is still possible to find Ψ when 𝔫 has ramified prime factors 𝔭 with ν_{𝔭}(𝔫) = 1: for also in this case (but not with higher powers) A/𝔭 ≃ B/𝔓.

Choose an embedding Ψ as in lemma 11 and extend it to a map K ⊗_{F} F_{∞} → M_{2}(F_{∞}): this induces an action of (K ⊗ F_{∞})^{∗}/F_{∞}^{∗} on ∂Ω. Since K/F is separable, Ψ(a) has two distinct eigenvalues ∀ a ∈ K − F and so there are 2 or 0 fixed points, according to whether ∞ is split or not in K.

Notice that measures on ∂Ω associated to Γ-invariant cocycles, such as μ_{E}, are invariant under the action of B^{∗}.

By the S-unit theorem (which in this case follows from the finiteness of the Jacobian of the curve corresponding to K - see e.g. [39]), B^{∗} is a finite group if ∞ is inert or ramified, while it is of rank one modulo torsion in the split case. If B^{∗} is finite, then it is contained in the constant field of K; that is, B^{∗} = A^{∗} unless K is the constant field extension.

For the rest of the chapter, assume that the infinite prime splits in K. Set

G := (K ⊗_{F} F_{∞})^{∗}/F_{∞}^{∗}B^{∗}.

Fix u a generator of the free part of B^{∗}: then Ψ(u) is hyperbolic. Let i_{k} be the immersions of K into F_{∞}, with ν(i_{1}(u)) > 0; put

w := \frac{i_{1}(u)^{2}}{N_{K/F} (u)} ∈ F_{∞}^{∗}.

K ⊗_{F} F_{∞} is isomorphic to F_{∞} × F_{∞} by α ⊗ β ↦ (i_{1}(α)β, i_{2}(α)β); then (K ⊗ F_{∞})^{∗} ≃ F_{∞}^{∗} × F_{∞}^{∗} → F_{∞}^{∗}, (x, y) ↦ xy^{−1}. The image of B^{∗} is w^{ℤ}. This yields an isomorphism between G and F_{∞}^{∗}/B^{∗}: let ϱ : F_{∞}^{∗} → G be the corresponding projection.

Stabilizers in M_{2}(F_{∞}) of pair of points of ℙ^{1}(F_{∞}) are 2-dimensional split F_{∞}-algebras: therefore Ψ is determined (up to Galois conjugation in K/F) by its fixed points; denote them as P_{Ψ}^{−}, P_{Ψ}^{+} according to whether repulsive or attractive for Ψ(u). Then Ψ(B^{∗}) acts on T as shift by h := 2ν(i_{1}(u)) along A(P_{Ψ}^{−}, P_{Ψ}^{+}). This allows one to define a “fundamental domain” F_{Ψ} for G: choose e ∈ A(P_{Ψ}^{−}, P_{Ψ}^{+}) and take F_{Ψ} = U(e) − U(Ψ(u)e) ⊂ ℙ^{1}(F_{∞}) (a more precise choice will be made later on).

The choice of a base point P ∈ ∂Ω−{P_{Ψ}^{−}, P_{Ψ}^{+}} gives an identification of (K ⊗_{F} F_{∞})^{∗}/F_{∞}^{∗} and ∂Ω − {P_{Ψ}^{−}, P_{Ψ}^{+}} by a ↦ Ψ(a)P. This yields an isomorphism of topological spaces between F_{Ψ} and G: thus the restriction of a measure μ ∈ M(ℙ^{1}(F_{∞}), ℤ) to F_{Ψ} induces a measure (still denoted by μ) on G.

Lemma 12. For all measures μ ∈ M_{0}(∂Ω, ℤ)^{Γ}, μ(F_{Ψ}) = 0.

Proof. This follows immediately from Ψ(B^{∗}) ⊂ Γ: for, if φ is the cocycle corresponding to μ, μ(F_{Ψ}) = φ(e) − φ(Ψ(u)e). □

2. Galois interpretation

The goal of this section is to identify (K ⊗_{F} F_{∞})^{∗}/F_{∞}^{∗}B^{∗} with a Galois group.

Let S be the set of places of K lying above ∞. Denote by 𝕁_{K} the group of idéles on K; K^{∗}, O_{v}^{∗}, K_{v}^{∗} are embedded in 𝕁_{K} as usual. Put

O_{S}^{∗} := ∏_{v∉S} O_{v}^{∗} and K_{S}^{∗} := ∏_{v∈S} K_{v}^{∗}

and define 𝕁_{S} := O_{S}^{∗}K_{S}^{∗}: then 𝕁_{S} ∩ K^{∗} = B^{∗}. Let K_{ab} be the maximal abelian extension of K, K′_{ab} the maximal abelian extension of K unramified at places outside S; following [40], define the Hilbert class field with respect to S, H, as the maximal unramified abelian extension in which all primes in S split completely. By class field theory (see e.g. [45]), Artin reciprocity identifies 𝕁_{K}/K^{∗} with a dense subgroup of Gal(K_{ab}/K): K′_{ab} is the fixed field of O_{S}^{∗}, while Gal(K_{ab}/H) is the completion of K^{∗}𝕁_{S}/K^{∗} ≃ 𝕁_{S}/B^{∗}. Therefore Gal(K′_{ab}/H) contains a dense subgroup isomorphic to 𝕁_{S}/O_{S}^{∗}B^{∗} ≃ K_{S}^{∗}/B^{∗}. Besides, Gal(H/K) ≃ 𝕁_{K}/K^{∗}𝕁_{S} is canonically isomorphic to Pic(B), by sending an idéle into the corresponding divisor.

Note that K′_{ab}/F and H/F are still Galois. For, in a separable closure of F, K′_{ab} is the fixed field of (the closure of) the commutator subgroup of Gal(F_{sep}/K) and the group generated by inertia at places v |̸ ∞; this group is normal, since conjugation by σ ∈ Gal(F_{sep}/F) just permutes the inertia groups not above ∞ and leaves invariant the commutator part. Analogously for H/F, adding to the picture the decomposition groups for primes above ∞.

Let τ be the generator of Gal(K/F); extend it to τ̃ ∈ Gal(K′_{ab}/F). Since Gal(K′_{ab}/K) is abelian, conjugation by τ̃ is independent of the choice of the extension τ̃: hence get an action (obviously continuous) of Gal(K/F) on Gal(K′_{ab}/K) and on Gal(K′_{ab}/H). On the other hand, τ acts naturally on 𝕁_{K}; the two actions are compatible with the reciprocity map: rec(τa) = τ̃ rec(a) τ̃^{−1}. This is clear if rec(a) is the Frobenius for some 𝔭; by Chebotarev, such are dense.

Define the “anticyclotomic” extension K_{ac}/H as the fixed field, in K′_{ab}, of the subgroup of Gal(K′_{ab}/H) corresponding to F_{∞}^{∗}. Then Gal(K_{ac}/H) is isomorphic to K_{S}^{∗}/F_{∞}^{∗}B^{∗} and τ acts on it as inversion: for rec(a)rec(τa) = rec(N_{K/F} a) ∈ rec(F_{∞}^{∗}). It is clear that K_{ac}/H is the maximal subextension of K′_{ab}/H enjoying this property (which justifies the name anticyclotomic): for F_{∞}^{∗} = N_{K/F} (K_{S}^{∗}) has to act trivially on any such subfield. Note, however, that H/K is not anticyclotomic when Pic(A) → Pic(B) is not trivial.

It is clear from the above that Gal(K_{ac}/H) ≃ G; if K/F splits at ∞, the two groups are isomorphic to F_{∞}^{∗}/w^{ℤ}, as explained in the previous section.

3. Main results

To simplify future computations, change coordinates on ∂Ω = ℙ^{1}(F_{∞}) by α ∈ P GL_{2}(F_{∞}), determined by 0 = αP_{Ψ}^{+}, ∞ = αP_{Ψ}^{−} and αP = 1. Then one can identify (K ⊗ F_{∞})^{∗}/F_{∞}^{∗} with F_{∞}^{∗}, orbit of 1 under multiplication. In particular, γ_{w} := αΨ(u)α^{−1} acts as multiplication by w on ℙ^{1}(C_{∞}) and as shift by ν(w) along the geodesic A(∞, 0) on T . Since ν(w) = h > 0, F_{∞}^{∗} is a disjoint union of cosets π^{i}w^{j}O_{∞}^{∗}, i ∈ {0, ..., h−1}, j ∈ ℤ: hence a “fundamental domain” mod w is

F = ⋃_{i=0}^{h−1} π^{i}O_{∞}^{∗} = U(e_{0}) − U(e_{h}).

To fix ideas, define F_{Ψ} := α^{−1}F.

Proposition 13. Let μ ∈ M(∂Ω, ℤ) be γ_{w}-invariant: then, for any z ∈ Ω,

∫_{ℙ^{1}(F_{∞})} \frac{wz − t}{z − t}dμ(t) = w^{m_{0}} ∫_{F} tdμ(t),

where m_{0} := μ(U(e_{0})).

Cf. prop. 3.3 of [9].

Proof. First of all, notice that γ_{w}-invariance of μ implies that μ(F) = 0 (same proof as for lemma 12).

One can decompose

ℙ^{1}(F_{∞}) = U(\overline{e}_{−Nh}) ∪ U(e_{Nh}) ∪ ⋃_{n=−N}^{N} w^{n}F

and it follows, by definition of integral, that

∫_{ℙ^{1}(F_{∞})} \frac{wz − t}{z − t}dμ(t) =

lim_{N→∞} (\frac{wz − 0}{z − 0})^{m_{0}} (\frac{wz − ∞}{z − ∞})^{−m_{0}} ∏_{n=−N}^{N} ∫_{w^{n}F} \frac{wz − t}{z − t}dμ =

lim_{N→∞} w^{m_{0}} ∏_{n=−N}^{N} ∫_{F} (\frac{wz − w^{n}t}{z − w^{n}t}) dμ(t) =

lim_{N→∞} w^{m_{0}} ∫_{F} ( ∏_{n=−N}^{N} \frac{w(z − w^{n−1}t)}{z − w^{n}t}) dμ(t) =

lim_{N→∞} w^{m_{0}} ∫_{F} (w^{N} \frac{w^{N+1}z − t}{z − w^{N}t}) dμ(t) =

lim_{N→∞} w^{m_{0}} ∫_{F} w^{N}dμ(t) ∫_{F} (\frac{w^{N+1}z − t}{z − w^{N}t}) dμ(t) =

w^{m_{0}} ∫_{F} ( lim_{N→∞} \frac{t − w^{N+1}z}{w^{N}t − z} ) dμ(t) = w^{m_{0}} \frac{∫_{F} tdμ(t)}{∫_{F}(−z)dμ(t)} = w^{m_{0}} ∫_{F} tdμ(t)

using throughout the proof the facts that ∫ is a continuous homomorphism and that ∫_{F} c = 1 for c ∈ C_{∞}^{∗}.

□

It is immediate to check that α ∗ μ_{E} satisfies the conditions of proposition 13:

γ_{w} ∗ α ∗ μ_{E} = αΨ(u)α^{−1} ∗ α ∗ μ_{E} = α ∗ Ψ(u) ∗ μ_{E} = α ∗ μ_{E}.

Following [9], define the winding number

W_{Ψ} := − ∑_{i=0}^{h−1} φ_{E}(e_{i}(Ψ)),

where e_{i}(Ψ) = α^{−1}(e_{i}).

It follows from section 2 of chapter 1 that c_{Ψ(u)}(γ_{E}) = Q^{M} for some M ∈ ℤ.

Theorem 14. Let M be the index of γ_{w} in Λ_{E}: then M = \frac{W_{Ψ}}{ν(Q)} and ∫_{G} tdμ_{E}(t) = ϱ(Q)^{M}, Q the period of the Tate curve E.

Notice the formal similarity of the term Q^{W_{Ψ}/ν(Q)} with the L-invariant of equation (1).

Proof. Exploiting the symmetry of the pairing c_{.}(.),

Q^{M} = c_{Ψ(u)}(γ_{E}) = c_{γ_{E}}(Ψ(u)) = \frac{u_{γ_{E}}(Ψ(u)z)}{u_{γ_{E}}(z)},

so that, by theorem 8 and its corollaries,

Q^{M} = ∫_{ℙ^{1}(F_{∞})} \frac{Ψ(u)z − t}{z − t}dμ_{E}(t).

Put a = αz and observe that

\frac{Ψ(u)z − t}{z − t} = c\frac{γ_{w}a − αt}{a − αt}

for some constant c ∈ C_{∞}^{∗} (both sides have the same divisor). Then

∫_{ℙ^{1}(F_{∞})} \frac{Ψ(u)z − t}{z − t}dμ_{E}(t) = ∫_{ℙ^{1}(F_{∞})} \frac{wa − t}{a − t}d(α ∗ μ_{E})(t)

and, using proposition 13 and ∫_{F} td(α ∗ μ_{E}) = ∫_{F_{Ψ}} tdμ_{E}, get

Q^{M} = w^{−α∗φ_{E}(e_{0})} ∫_{F} td(α ∗ μ_{E})(t) = w^{−φ_{E}(α^{−1}e_{0})} ∫_{F_{Ψ}} tdμ_{E},

so that

ϱ(Q^{M}) = ∫_{G} tdμ_{E}(t)

(where, by abuse of notation, μ_{E} denotes the pushforward of the measure from F_{Ψ} to G).

It is left to compute M. In general, since ν(t) = i for t in the set U(e_{i}) − U(e_{i+1}),

ν (∫_{F} tdμ) = ∑_{i=0}^{h−1} i(μU(e_{i}) − μU(e_{i+1})) = − ∑_{i=1}^{h−1} φ(e_{i}) + (h − 1)φ(e_{h})

(where φ is the cocycle associated to the measure μ). Then, replacing the above in

Mν(Q) = −(α ∗ φ_{E})(e_{0})ν(w) + ν (∫_{F} td(α ∗ μ_{E}))

it follows that M = \frac{W_{Ψ}}{ν(Q)}. □

The result is independent of the choice of base point P: changing it just shifts the fundamental domain along axis A(P_{Ψ}^{−}, P_{Ψ}^{+}) and replaces α with cα, c ∈ C_{∞}^{∗}, which doesn’t affect integration on sets of measure zero.

Dependence on u: the choice of a different fundamental unit might reverse the orientation on the axis of fixed points and therefore change the sign of μ and W_{Ψ}.

As for Ψ, notice that the right hand side of formula in theorem 14 depends on Ψ only through W_{Ψ}. GL_{2}(F) acts on embeddings by conjugation, (γ ∗ Ψ)(a) := γΨ(a)γ^{−1}. Assume Ψ′ = γ ∗ Ψ, γ ∈ Γ: then P_{Ψ′}^{+} = γP_{Ψ}^{+}, etc., μ_{Ψ′} = γ_{Ψ}∗μ and in particular W_{Ψ′} = ∑ φ_{E}(γe_{i}(Ψ)) = W_{Ψ}.

CHAPTER 3 Hypotheses finguntur

1. Iwasawa algebras

Let R be a commutative ring, G = lim_{←} G_{a} a profinite group, G_{a} = G/H_{a}. Let M(G, R) := M(G, ℤ) ⊗ R be the set of R-valued distributions on G: it becomes an R-algebra defining the product as convolution. It is trivial to check that M(G, R) is the inverse limit of the algebras M(G_{a}, R): therefore if R is a topological ring M(G, R) is naturally endowed with a topology.

There is a canonical isomorphism of topological R-algebras between M(G, R) = lim_{←} M(G_{a}, R) and the Iwasawa algebra R[[G]] := lim_{←} R[G_{a}], given by

μ ↦ α_{μ} := lim( ∑_{g∈G_{a}} μ(gH_{a})g)

(see [35], §7.2, for more detail).

Let I_{a} be the augmentation ideal of R[G_{a}], i.e.

I_{a} := ker(R[G_{a}] → R, g ↦ 1)

Define I := lim_{←} I_{a} be the augmentation ideal of R[[G]] ; then μ belongs to M_{0}(G, R) if and only if ∑_{g∈G_{a}} μ(gH_{a}) = 0 ∀ a, that is, if and only if α_{μ} ∈ I.

Apply the above to R = ℤ and G the group of previous chapter. For μ ∈ M_{0}(G, ℤ), define α_{μ} to be the associated ∞-adic L-function; its derivative at 1, “L′(μ, 1)”, is the image of α_{μ} in I/I^{2}. Since G is abelian, I/I^{2} is canonically isomorphic to G by [g − 1] ↦ g; the two will be identified in the rest of this section. Let

L′(E, 1) := L′(μ_{E}, 1).

Proposition 15. L′(E, 1) = ∫_{G} gdμ_{E}(g).

Proof. This is just a matter of working out definitions: in the map I → G,

lim_{a} ∑_{g∈G_{a}} μ(gH_{a})g ↦ lim_{a} ∏_{g∈G_{a}} g^{μ(gH_{a})} = ∫_{G} gdμ(g).

□

Together with theorem 14, this yields an analogue of formula (1).

Remark. In the number field case, the group G one deals with is (essentially) ℤ_{p}. Fix g_{0} a topological generator of such a G: then, thanks to the isomorphism ℤ_{p}[[G]] ≃ ℤ_{p}[[x]], g_{0} ↦ x + 1 (under which I corresponds to the ideal generated by x), α_{μ} can be identified with a power series F_{μ} and the image of α_{μ} in I/I^{2} ≃ ℤ_{p} is c_{1}, the degree one coefficient of F_{μ}. Furthermore, in this case, the dual group G^{∨} := Hom_{cont}(G, ℂ_{p}^{∗}) can be identified with an open neighbourhood of 1 in ℂ_{p} by χ ↦ χ(g_{0}) ^{1}. Taking as p-adic L-function the additive integral of characters, L_{p}(χ) := ∫_{G} χd^{+}μ, one can take derivatives in the usual sense and the formula ∫_{G} χd^{+}μ = F_{μ}(χ(g_{0}) − 1) shows that L′_{p}(1) = c_{1} (cf. [35]).

Besides, in characteristic 0 one can define a p-adic logarithm log : G → ℂ_{p}. Thus, in [2] and [6], ℤ_{p} is embedded in G^{∨} by s ↦ χ_{s}, χ_{s}(g) := exp(s log(g)). Then L_{p}(s) := ∫_{G} χ_{s−1}(t)dμ(t) is analytic and its derivative is \frac{d}{ds}F_{μ}(χ_{s−1}(g_{0}) − 1) = log(g_{0})c_{1} = log(g_{0}^{c_{1}}).

In the next section, an attempt (still very naive and tentative) will be made to develop a similar interpretation of L′_{∞} as derivative of a function on ℤ_{p}.

2. Towards a p-adic L-function

Let G be, as usual, the group (K ⊗F_{∞})^{∗}/F_{∞}^{∗}. Define its dual group as

G^{∨}:= Hom_{cont}(G, C_{∞}^{∗}).

Let 𝔲_{1}, 𝔘_{1} be the groups of 1-units in F_{∞}, C_{∞} respectively; i.e. 𝔘_{1} := {z ∈ C_{∞} | |z − 1| < 1}, 𝔲_{1} := 𝔘_{1} ∩ F_{∞} = 1 + 𝔪_{∞}. The following result is lemma 8.2.2 in [25].

Lemma 16. 𝔘_{1} is a ℚ_{p}-vector space.

1A necessary and sufficient condition for t = χ(g_{0}) is lim_{n→∞} t^{p^{n}} = 1; therefore any character is of the form χ = χ_{t}, χ_{t}(g_{0}^{a}) = t^{a} := exp(a log_{p}(t)), t in the open ball B(1, 1).

2. TOWARDS A p-ADIC L-FUNCTION 21

Proof. For s = ∑_{i=−N}^{∞} a_{i}p^{i} ∈ ℤ_{p} and t = 1 + m ∈ 𝔘_{1}, set

s · t := t^{s} = ∏_{i=−N}^{∞}(1 + m^{p^{i}})^{a_{i}}.

This is well defined, since p-th roots uniquely exist in C_{∞}. The vector space axioms are trivially verified for those s expressed as a finite sum; continuity takes care of the rest. □

Let G_{1} be the image of 𝔲_{1} in G: G_{1} is a pro-p-group, sitting in G with finite index h(q − 1). Since C_{∞}^{∗} ≃ ℚ × \overline{𝔽}_{p}^{∗} × 𝔘_{1} and G is compact, any character χ ∈ G^{∨} takes values in the product of 𝔘_{1} with some finite subgroup of \overline{𝔽}_{p}^{∗}. Since the latter has order coprime to p, χ(G_{1}) is trivial in \overline{𝔽}_{p}^{∗}: therefore G^{∨} decomposes as G_{t}^{∨} ×G_{w}^{∨}, where G_{t}^{∨} is isomorphic to Hom(G/G_{1}, C), C some finite cyclic group, and G_{w}^{∨} := Hom_{cont}(G, 𝔘_{1}) (of course the subscripts t, w stay for tame, wild).

It follows from lemma 16 that 𝔘_{1} is uniquely divisible: in particular, any homomorphism from G_{1} to 𝔘_{1} extends uniquely to all of G, yielding an isomorphism G_{w}^{∨} ≃ Hom_{cont}(𝔲_{1}, 𝔘_{1}).

Define a tentative p-adic L-function L_{tent}(E, χ) : G^{∨} → C_{∞}^{∗} as

L_{tent}(E, χ) := ∫_{G} χdμ_{E}.

In particular, one gets a p-adic, C_{∞}^{∗}-valued L-function by choosing an embedding of ℚ_{p} in the ℚ_{p}-vector space G_{w}^{∨}. The most natural thing to do seems to be the following.

Let χ_{1} ∈ G_{w}^{∨} be the character determined by χ_{1} ∘ϱ|_{𝔲_{1}} = id_{𝔲_{1}}. Define g^{s} := χ_{s}(g) := χ_{1}(g)^{s}. This gives a map ℚ_{p} → G_{w}^{∨} by s ↦ χ_{s}. Then the function

L_{tent}(E, s) := ∫_{G} g^{s−1}dμ = χ_{s−1} (∫_{G} gdμ)

is ℚ_{p}-linear and has constant derivative

\frac{d}{ds}L_{tent}(E, s)|_{s_{0}} = lim_{s→s_{0}} ∫_{G} g^{\frac{s−s_{0}}{s−s_{0}}} dμ = ∫_{G} gdμ.

This definition was naively made by looking at the usual Mellin transform, both classical and p-adic^{2}; however, in those situations, the integrals taken into consideration were additive. In our case, it would be probably better to replace the exponent s − 1 by some polynomial of higher degree in ℤ_{p}[s], in order to avoid having a constant derivative.

2For the latter, in a context very close to ours, see [26].

Observe that χ_{1} seems to play role of exponential and ϱ of logarithm. There is of course some loss in taking just χ_{1} instead of multiplying it with some tame character, but this seems unavoidable.

In terms of the augmentation ideal, the above can be interpreted in the following way. Integration on G gives a natural map

S : M(G, ℤ_{p}) → Hom(G_{w}^{∨}, 𝔘_{1}),

by μ ↦ L_{μ}, where

L_{μ}(χ) := ∫_{G} χdμ.

This map is a homomorphism on the additive group of measures: it is immediate that L_{μ_{1}+μ_{2}} = L_{μ_{1}}L_{μ_{2}}. As before, M(G, ℤ_{p}) is identified with ℤ_{p}[[G]]. Notice that G is contained in ℤ_{p}[[G]] as a set of topological generators {g}, corresponding to the Dirac deltas {δ_{g}} in M(G, ℤ_{p}). An immediate computation yields

S(g)(χ) = S(δ_{g})(χ) = χ(g).

Since ℤ_{p}[[G]] = ℤ_{p}⊕I, the equality S(g−1)(χ) = χ(g) gives S(ℤ_{p}[[G]]) = S(I). Besides S factors through I^{2}, since

S((g − 1)(h − 1))(χ) = χ(gh)χ(g)^{−1}χ(h)^{−1} = 1.

Therefore S(ℤ_{p}[[G]]) = S(I/I^{2}) is the image of G in Hom(G_{w}^{∨}, 𝔘_{1}).

Goss’ theory of L-functions. It seems natural to try and look at our tentative L-function in the framework of Goss’ L-series ([25]).

To define p-adic L-functions in characteristic p, Goss introduces a “complex plane” S_{∞} := C_{∞}^{∗} × ℤ_{p} ([25], 8.1): the idea is that F_{∞}^{∗} ≃ ℤ × 𝔽_{q}^{∗} × 𝔲_{1}, hence its dual is C_{∞}^{∗} × Hom(𝔽_{q}^{∗}, C_{∞}^{∗}) × Hom_{cont}(𝔲_{1}, 𝔘_{1}), which contains S_{∞}.

For s = (x, y) ∈ S_{∞}, α ∈ F_{∞}^{∗}, he defines α^{s} := x^{deg α}⟨α⟩^{y}. The subgroup 𝔘_{1} × ℤ_{p} of G_{w}^{∨} is contained in S_{∞}: one can extend L_{tent} to a function defined on all of S_{∞} by L_{tent}(E, s) := ∫_{G} g^{s−(0,1)}dμ (notice that for α ∈ 𝔘_{1}, deg α = 0, and χ_{0}(G) ⊂ 𝔘_{1}, ergo g^{s} = χ_{0}(g)^{y}).

A function f : S_{∞} → C_{∞} is defined to be entire if the corresponding map ℤ_{p} → O_{C_{∞}^{∗}}(C_{∞}^{∗}) is continuous^{3} (this definition is slightly stronger than 8.5.1 in [25]); this condition is clearly satisfied by L_{tent}(E, s).

3The topology on O_{C_{∞}^{∗}}(C_{∞}^{∗}) is given by seminorms ∥f∥_{A_{n}} = sup{|f(z)| | z ∈ A_{n}}, where A_{n} is the annulus {z | q^{−n} ≤ |z| ≥ q^{n}}.

CHAPTER 4 Paralipomena

The techniques introduced in the previous chapters can be easily extended to the case of an extension inert at ∞: this will be sketched in section 1. In section 2, a few words will be said on what can be done replacing the field K by a split quadratic algebra over F.

1. The inert case

Assume that K/F is inert at ∞: then K ⊗_{F} F_{∞} =: K_{∞} is a field. The action of Gal(K_{∞}/F_{∞}) will be denoted by x ↦ \overline{x}.

1.1. Heegner points. Recall that points of the Drinfeld modular curve M_{0}(𝔫) classify rank 2 Drinfeld A-modules with level-𝔫 structure.

Definition. P ∈ M_{0}(𝔫) is said to be a Heegner point if the corresponding Drinfeld module P has complex multiplication by an order R in an imaginary quadratic extension of F; the action of R on P is requested to preserve the level structure.

In the conditions of the definition above, one says that P is associated to the order R.

Heegner points enjoy rationality properties (as can be seen, e.g., by means of Hayes’ explicit class field theory [31], §8; see also [13], §4).

The reader is reminded of the equivalence of categories between Drinfeld A-modules over C_{∞} and homothety classes of A-lattices of the same rank in C_{∞}. In particular, an endomorphism of the module Φ corresponds to a scalar c ∈ C_{∞} such that cΛ ⊂ Λ, where Λ is the lattice associated to Φ.

Lemma 17. There is a bijection between the set of Heegner points of M_{0}(𝔫) associated to the order B and embeddings Ψ : B → R: lattices generated by fixed points of Ψ(B) correspond to Drinfeld modules with complex multiplication by B.

The proof works like in the case of elliptic curves over ℂ ([43], 4.4).

Proof (sketch). Let P be a Heegner point, Λ = Aw_{1} + Aw_{2} and Λ_{1} = Aw_{1} + 𝔫w_{2} the corresponding pair of lattices. For b ∈ B, the relation bw_{i} ∈ Λ yield an embedding Ψ : B → R; it is immediate to check that z_{Ψ} := \frac{w_{1}}{w_{2}} is a fixed point of Ψ(B).

Vice versa, if z_{Ψ} is a fixed point of Ψ, then (z_{Ψ}, 1) is an eigenvector for Ψ(B): that is, the lattice Az_{Ψ} + A has complex multiplication by B. □

1.2. The theorem. Let K_{∞,1}^{∗} := \{x ∈ K_{∞}^{∗} | x\overline{x} = 1\} be the group of elements of norm 1 in K_{∞}^{∗}.

If K/F is not the constant field extension F ⊗_{𝔽_{q}} 𝔽_{q^{2}}, the map x ↦ x\overline{x}^{−1} establishes an isomorphism between G = K_{∞}^{∗}/F_{∞}^{∗} and K_{∞,1}^{∗}. Thus a convenient way to transfer measures from ∂Ω to G will be by

means of the Möbius transformation α(t) = \frac{t − z_{Ψ}}{t − \overline{z}_{Ψ}} , which yields a bi-

jection α : ∂Ω → K_{∞,1}^{∗}. Notice that in this case K_{∞,1}^{∗} is contained in 𝔘_{1}.

In the case K = F ⊗_{𝔽_{q}} 𝔽_{q^{2}}, to get G one has to factor out the image of the roots of unity: that is, G ≃ K_{∞,1}^{∗} ∩ 𝔘_{1}. So in all cases there is an embedding ι : G → 𝔘_{1}.

As in the split case, define L′(E, 1) := ι(∫_{G} gdμ).

Theorem 18. For K ≠ F ⊗_{𝔽_{q}} 𝔽_{q^{2}}, Φ_{Tate}(L′(E, 1)) is a global point of E.

Proof (sketch). Let z_{Ψ}, \overline{z}_{Ψ} be the fixed points corresponding to the embedding Ψ. Remember the Abel-Jacobi map from section 2. One has

Φ_{AJ}((z_{Ψ}) − (\overline{z}_{Ψ}))(γ_{E}) = \frac{u_{γ_{E}}(z_{Ψ})}{u_{γ_{E}}(\overline{z}_{Ψ})} = ∫_{∂Ω} \frac{t − z_{Ψ}}{t − \overline{z}_{Ψ}} dμ_{E}(t) =

∫_{K_{∞,1}^{∗}} td(α_{∗}μ_{E})(t) = ι ∫_{G} gdμ_{E}.

Now use the rationality properties of Heegner points and of the Abel-Jacobi map. □

In the case of the constant field extension, K_{∞,1}^{∗} is the union of q+1 copies of ι(G), corresponding to the action of B^{∗}. Since the measure μ_{E} is invariant with respect to this action, in this case one gets

Φ_{AJ}((z_{Ψ}) − (\overline{z}_{Ψ}))(γ_{E}) = (ι ∫_{G} gdμ_{E})^{\frac{1}{q+1}} .

As in the previous chapter, also in the inert case one can use the action of ℤ_{p} on 𝔘_{1} to define a p-adic L-function.

The inert case has been studied in Pál’s thesis ([37]): his approach is somewhat different, but the special value L′(E|K) he gets seems to be the same as here^{1}. In [37], as in [4], no mention is made of integration, which makes some computations more cumbersome. On the other side, Pál proves interpolation formulae which show links between ∞-adic and classical L-functions and could give fruitful hints for the construction of a p-adic L-function as attempted in chapter 3.

2. The cyclotomic case

In this section, the field K is replaced by the split quadratic algebra K̃ := F × F, into which F is embedded on the diagonal; the integral closure of A in K̃ is B̃ = A × A.

In this situation the group G̃ := (K̃ ⊗_{F} F_{∞})^{∗}/F_{∞}^{∗}B̃^{∗} ≃ F_{∞}^{∗}/𝔽_{q}^{∗} = π^{ℤ} × 𝔲_{1} is not compact any more and so it cannot be identified with a Galois group. Class field theory shows that G̃ corresponds to a dense subgroup of a “cyclotomic” Galois group (see e.g. [30]); in particular, the uniformizer π is sent by reciprocity into the arithmetic Frobenius, corresponding to extensions of the the constant subfield of F. the role previously played by G will be now given to the geometric cyclotomic group G_{0} ≃ 𝔲_{1}.

The set of fixed points of embeddings K̃ ↪ GL_{2}(F) consists precisely of ℙ^{1}(F).

2.1. Cusps. Let \overline{Ω} := Ω∪ℙ^{1}(F). As said in section 2 of chapter 1, the affine curve M_{0}(𝔫) can be completed to a projective curve \overline{M}_{0}(𝔫), whose C_{∞}-points are identified with Γ\\overline{Ω}. As in the classical situation of modular curves over ℂ, a rigid analytic structure at the cusps s ∈ ℙ^{1}(F) is defined by means of horocyclic neighbourhoods: for s = ∞ one takes Ω_{c} := {z ∈ Ω | |z|_{i} ≥ c}, quotiented by the action of Γ (for more detail, see [22], §2.6-2.7 and [15] V.2).

The theta functions studied in [22] can be evaluated also at cusps of Γ.

Proposition 19. Let s ∈ ℙ^{1}(F): then u_{γ}(s) = ∏_{ε∈Γ̃} \frac{s−εw}{s−εγw} converges to a value in C_{∞}^{∗}; moreover, u_{γ} is a holomorphic function on all of \overline{Ω}.

1It’s not completely clear to the author from the still incomplete draft of [37] he could see.

For the proof, see [22], lemmata 5.3.9 and 5.3.10.

The cusps Γ\ℙ^{1}(F) correspond to the set of ends of Γ\T : recall that this graph consists of the union of a finite graph and a finite number of half-lines (see [42], II.2.3). In particular, this means that, given a cusp s, one can find an end e = e(s) (pointing to s) such that the conical subtree C_{s,e} ⊂ T collapses to a the half-line of s in Γ\T . Here

C_{s,e} := A(o(e), s) ∪ {A(v, w) | v ∈ A(t(e), s) and d(v, w) ≤ d(v, t(e)},

where d(v, w) is the distance between the vertices. Then, for φ ∈ \underline{H}_{!}(T , ℤ)^{Γ}, the cuspidality condition yields that φ is identically 0 on C_{s,e}.

Finally, an analogue of the classical Manin-Drinfeld theorem on cusps holds.

Theorem 20. The group generated by cusps in J_{0}(𝔫) is finite.

A proof is given in [21] (for the case F = 𝔽_{q}(T), a different proof can be found in [14]).

2.2. The theorem. Consider the diagonal embedding of K̃ in GL_{2}(F): the fixed points on ∂Ω are 0 and ∞. As in chapter 2, the choice of a base point allows to identify (K̃ ⊗_{F} F_{∞})^{∗}/F_{∞}^{∗} with ∂Ω − {0, ∞}; the measure μ_{E} is then pulled back to G̃. To get a measure on G_{0} ≃ 𝔲_{1}, recall the notion of level of an edge/vertex, l(e)/l(v), introduced in chapter 1, section 1, and define μ̃_{E} on O_{∞}^{∗} by

μ̃(U_{e}) := − ∑_{l(e′)=l(e)} φ(e′).

Notice that this sum is finite, because under the action of Γ the cones C_{0,e(0)} and C_{∞,e(∞)} collapse to half-lines in Γ\T . The restriction of μ̃_{E} to 𝔲_{1} gives the measure on G_{0}.

For r, s ∈ ℙ^{1}(F) and φ ∈ \underline{H}_{!}(T , ℤ)^{Γ}, define

[r, s] · φ := ∑_{e∈A(r,s)} φ_{E}(e).

This is the pairing between modular symbols and cuspidal harmonic cocycles considerd by Teitelbaum in [48], definition 8.

Let e_{v} be the unique edge with origin v and pointing to 0. For v ∈ V(S_{0}) choose a representative s(v) = u_{v} ∈ U(\overline{e}_{v}). Extend this map to all of V(T ) by letting s(π^{i}v) = π^{i}u_{v}.

Theorem 21. One has

u_{γ_{E}}(0) = π^{[0,∞]·φ_{E}} (∫_{G_{0}} gdμ̃_{E})^{q−1} .

Proof. Choose a vertex w_{0} and let d_{1}, ..., d_{r} be the edges of the path A(w_{0}, γw_{0}). Put t(d_{i}) = o(d_{i+1}) = w_{i} and take z_{0}, ..., z_{r−1} in Ω such that λ(z_{i}) = w_{i}; let z_{r} := γ_{E}z_{0}. Define

P_{n} := ∏_{i=0}^{r−1} ∏_{\begin{matrix}ε∈Γ̃ \\ l(εd_{i})≤n \end{matrix}} \frac{εz_{i}}{εz_{i+1}} .

Then

u_{γ}(0) = ∏_{ε∈Γ̃} \frac{εz_{0}}{εγz_{0}} = ∏_{ε∈Γ̃} ∏_{i=0}^{r−1} \frac{εz_{i}}{εz_{i+1}} = lim_{n→∞} P_{n}

by lemma 4 (the fact that one can rearrange terms in the infinite product follows from the estimates in [22], §5.3).

If l(λ(εz_{i})) ≤ n, each time εz_{i} appears in P_{n} as terminus of an edge εd_{i} in the Γ̃-orbit of A(z_{0}, γz_{0}), it also appears as origin of the next edge.^{2} Thus the only contributions to P_{n} which are not cancelled are those corresponding to vertices of level n + 1. Furthermore, lemma 4 implies that \frac{εz_{i}}{s(λ(εz_{i}))} → 1 uniformly as l(λ(εz_{i})) → ∞. Hence

lim_{n→∞} P_{n} = lim_{n→∞} ∏_{l(v)=n+1} s(v)^{X_{v}}.

To compute the exponent X_{v}, observe that s(v) appears in the numerator exactly the number of times e_{v} falls in Γ̃A(w_{0}, γ_{E}w_{0}); analogously, for the denominator one computes the occurrences of \overline{e}_{v}. But ([22], §3.3)

φ_{E}(e) = ♯\{ε ∈ Γ̃ | e ∈ εA(w, γ_{E}w)\} − ♯\{ε ∈ Γ̃ | \overline{e} ∈ εA(w, γ_{E}w)\}.

That is,

u_{γ}(0) = lim_{n→∞} ∏_{l(v)=n} s(v)^{φ_{E}(e_{v})} = lim_{n→∞} ∏_{l(v)=n} π^{i(v)φ_{E}(e_{v})} ∏_{\begin{matrix}v∈S_{0} \\ l(v)=n \end{matrix}} u_{v}^{−μ̃U(e_{v})}.

The product of units converges precisely to ∫_{O_{∞}^{∗}} udμ̃_{E}.^{3} To pass to an integral over G_{0}, just recall that μ̃_{E} is invariant under B^{∗} ⊂ Γ and that G_{0} ≃ 𝔲_{1} = O_{∞}^{∗}/𝔽_{q}^{∗}.

2Recall that t(d_{r}) = o(γd_{1}), o(d_{1}) = t(γ^{−1}d_{r}).

3Don’t forget that e_{v} is oriented in the “wrong” way: O_{∞}^{∗} is covered by the U(\overline{e}_{v})’s.

As for the product of π’s, recall that i(v) = i for all v ∈ V(S_{i}); besides, it follows from the harmonicity of φ_{E} that

∑_{\begin{matrix}v∈S_{i} \\ l(v)=n \end{matrix}} φ_{E}(e_{v}) = φ_{E}(e_{i+1}) − φ_{E}(e_{i}).

Therefore, the valuation of P_{n} is constant, given by

ν(P_{n}) = ∑_{i=N_{∞}}^{N_{0}} i(φ_{E}(e_{i+1}) − φ_{E}(e_{i})) = [0, ∞] · φ_{E}.

□

According to Gekeler, [21], th. 1.6 (see also [20]), it follows from theorem 20 that u_{γ_{E}}(0) = \frac{u_{γ_{E}} (0)}{u_{γ_{E}} (∞)} is mapped by the Tate map to a torsion point on E and it is therefore a rational power of the Tate period. Theorem 21 should thus be regarded as the function field analogue of the Greenberg-Stevens theorem about the cyclotomic L-invariant.

Conclusion

It has been proven that integration on the anticyclotomic Galois group G with respect to measures obtained from explicit uniformization of elliptic curves can yield arithmetically meaningful information also in characteristic p, thereby generalizing what was done in [2] in the case of number fields. More precisely, in the split case an analogue of the L-invariant of [36] was obtained by integration on G; while in the inert case, we have indicated how this technique allows to recover Heegner points.

An attempt has been made to interpret the values of these integrals as special values of a still hypothetical p-adic L-function L(E, s) : ℤ_{p} → C_{∞}^{∗}. The results obtained are not yet very satisfactory (the function L_{tent} takes the right value at the special point 1, but it has constant derivative). This clearly indicates a direction for further research: to define, by some sort of Mellin transform, a p-adic L-function which encodes a large enough amount of the arithmetic of E; such a function should be related (e.g., by interpolation of special values) to the classical complex L-function L(E, s). The fact that the latter is a polynomial in q^{−s} suggests that the conjectural L_{p} should not be very complicated.

A further goal (still far from the author’s scope) would be to formulate analogues of the Birch and Swinnerton-Dyer conjectures for these L-functions.

Another possible direction of research is the generalization to the characteristic p case of the double integrals considered in [9].

Sed fugit interea, fugit inreparabile tempus, singula dum capti circumvectamur amore.^{4}

And so, for the moment, we stop here our investigation.

A last remark: throughout the paper it has been made use of the simplifying assumption that F and A are respectively 𝔽_{q}(T) and 𝔽_{q}[T].

4[49], III, 284-285.

It should be an easy step to generalize our results to any F, A. Among the things one should take care of, there are:

• if A has class number > 1, Γ\Ω is only one of the irreducible components of the moduli space M_{0}(𝔫);

• for A a generic Drinfeld ring, j is injective with finite cokernel, but it is not known whether it is an isomorphism (cf. [22], remark 6.4.5);

• in general Q^{ℤ} is contained in Λ_{E} with finite index ([22] 9.5.1);

• the classification of embeddings B → R depended on the fact that A was a principal ideal domain.

Bibliography

[1] The arithmetic of function fields (D. Goss et al., eds.) Columbus 1991. Walter de Gruyter 1992

[2] M. Bertolini: CRM-minicourse (September 1998). To appear in the proceedings of the theme year in number theory and arithmetic geometry.

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