Valentin Petit
email: valentin.petit@univ-fcomte.fr

EXPLICIT COMPUTATION OF THE MODULAR PARAMETRIZATION OF ELLIPTIC CURVES OVER FUNCTION FIELDS BY DRINFELD MODULAR CURVES

Keywords: 1991 Mathematics Subject Classication. 11G18, Secondary 11G09, 11F12, 14H52 Elliptic curves, Function Fields, Modular parametrization, Harmonic cochains, Theta functions, Drinfeld modular curves

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Introduction and motivation

Let q be a prime power and E be an elliptic curve dened over F q (T ) given by a Weierstrass model

E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , a i ∈ F q [T ].
We assume that E is non-isotrivial and has split multiplicative reduction at a place, say ∞ := 1/T . By the work of Drinfeld, Grothendieck, Jacquet and Langlands ( [START_REF] Drinfel'd | Elliptic modules[END_REF][START_REF] Jacquet | Automorphic forms on GL(2)[END_REF]), there exists a "modular parametrization" Φ : M Γ0(n) → E, where M Γ0(n) is the Drinfeld modular curve over F q (T ) associated to the Hecke congruence subgroup

Γ 0 (n) = a b c d ∈ GL 2 (F q [T ]) : c ≡ 0 mod n ,
and n is the nite part of the conductor of E. The map Φ has been studied and described by several authors, especially Gekeler-Reversat [START_REF] Gekeler | Jacobians of Drinfeld modular curves[END_REF] and Gekeler [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF][START_REF] Gekeler | Jacquet-Langlands theory over K and relations with elliptic curves[END_REF]. The main goal of this article is to explain how this modular parametrization can be explicitly computed and to deduce some arithmetical properties from this. In particular, we describe a method to compute exactly the images Φ(c) ∈ E of the cusps c ∈ M Γ0(n) .

The situation above has to be compared with the classical one in characteristic zero. In that case, if F is an elliptic curve dened over Q with conductor N , by a series of works culminating with [START_REF] Taylor | Ring-theoretic properties of certain Hecke algebras[END_REF][START_REF] Breuil | On the modularity of elliptic curves over Q: wild 3-adic exercises[END_REF], there exists a modular parametrization given by the composition ϕ : X 0 (N )

φ → C/Λ ℘ → F (C)
, where Λ is the period lattice of F , the map φ is roughly speaking given by the dierential form of the weight-2 modular form f (τ ) = n≥1 a n e 2inπτ associated to F , and the isomorphism ℘ is given by the Weierstrass function and its derivative. In the literature there are many investigations on the explicit description of the map φ and the computation of its arithmetical invariants: its degree ( [START_REF] Zagier | Modular parametrizations of elliptic curves[END_REF][START_REF] Cremona | Computing the degree of the modular parametrization of a modular elliptic curve[END_REF][START_REF] Delaunay | Computing modular degrees using L-functions[END_REF][START_REF] Watkins | Computing the modular degree of an elliptic curve[END_REF]), its critical points ( [START_REF] Chen | Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves[END_REF][START_REF] Delaunay | Critical and ramication points of the modular parametrization of an elliptic curve[END_REF]), its evaluation at cusps ( [START_REF] Brunault | On the ramication of modular parametrizations at the cusps[END_REF][START_REF] Wuthrich | Numerical modular symbols for elliptic curves[END_REF]), its explicit evaluation and applications to Heegner points ( [START_REF] Watkins | Some remarks on Heegner point computations[END_REF]).

In the case of Drinfeld modular curves, the fact that E has split multiplicative reduction at ∞ implies the existence of a Tate parameter t ∈ C * ∞ and an isomorphism E(C ∞ ) C * ∞ /t Z where C ∞ is a completion of an algebraic closure of F q ((1/T )). Hence on the one hand, the modular parametrization is given as in the classical case by a map Φ : M Γ0(n) → C * ∞ /t Z where Φ will be dened in Section 7. On the other hand, unlike the characteristic zero situation, the degree of Φ is easily computable using a formula of Gekeler [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF] recalled in Theorem 25. The denition of Φ also requires a lot of theoretical material that we will recall for the self-containedness of this text.

The organization of the paper is as follows. In the notations and in Sections 2 to 4, we give a survey of classical theoretical results on Drinfeld modular curves and related objects which are used for the construction of Φ: the Bruhat-Tits tree, harmonic cochains, theta functions. Section 5 is devoted to Theorem 13, which states that the expression of the map Φ using a theta function, which originally holds outside of the cusps, may also be evaluated at the cusps. This is our rst key step for evaluating explicitly Φ at a cusp. As far as we know, there is no similar result in the characteristic zero setting (however numerical experimentations suggest that it should be the case). In Section 6, we recall the action of Hecke operators on dierent objects involved in our study. In Section 7, we use Hecke operators to give an explicit upper bound for the order of the image of a cusp by Φ (Theorem 26): this is reminiscent of the Manin-Drinfeld theorem, whose analogue in our context was proved by Gekeler [START_REF] Gekeler | On the cuspidal divisor class group of a Drinfeld modular curve[END_REF], but our bound is explicit and it is the second key step in our method. Finally in Section 8 we illustrate our explicit description of the modular parametrization by two examples in characteristic 2 and 3 with small conductors (note that explicit examples in characteristic > 2 seem to be rare in the literature).

Acknowledgements. The author is a member of the Laboratoire de mathématiques de Besançon (LmB) which receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002).

Notations. Let p be a prime number, q a power of p, and F q a nite eld with q elements. We denote by A = F q [T ] the ring of polynomials with coecients in

F q and K = F q (T ) the fraction eld of A. Let K ∞ = F q (( 1 T )) be the completion of K with respect to the norm |P | = q deg(P ) . The element π = 1 T ∈ K is a uniformizer of K ∞ and let O ∞ = F q [[π]] be the ring of integers of K ∞ . We denote by C ∞ a completion of K ∞ where K ∞ is an algebraic closure of K ∞ . Finally let Ω = C ∞ -K ∞ be the Drinfeld upper half-plane. The imaginary part map on Ω, denoted | • | i , is dened by |z| i = inf{|z -x|, x ∈ K ∞ }.
Recall that the group GL 2 (K ∞ ) acts on Ω by fractional linear transformations:

a b c d z = az + b cz + d .
The imaginary part satises the relation

|γz| i = |z| i |cz + d| 2 z ∈ Ω, γ = a b c d ∈ GL 2 (K ∞ ) .
Any arithmetic subgroup Γ ⊂ GL 2 (A) acts on Ω with nite stabilizers: it implies that the quotient Γ\Ω has a structure of an analytic space over K ∞ . There exists a smooth irreducible ane algebraic curve M Γ dened over K ∞ whose underlying analytic space is canonically isomorphic to Γ\Ω. Let M Γ be the smooth projective model of M Γ . The algebraic curve M Γ is called the Drinfeld modular curve associated to Γ.

The Bruhat-Tits tree

For Sections 2, 3, and 4, we mainly use the references [START_REF] Gekeler | Jacobians of Drinfeld modular curves[END_REF], [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF], and [START_REF] Gekeler | Fundamental domains of some arithmetic groups over function elds[END_REF]. We denote by Z the center of GL 2 (K ∞ ) and I ∞ the Iwahori subgroup dened by

I ∞ = a b c d ∈ GL 2 (O ∞ ), c ≡ 0 mod π .
We recall a description of the Bruhat-Tits tree in terms of the group GL 2 (K ∞ ) as in [START_REF] Serre | Trees. Springer Monographs in Mathematics[END_REF]II,1.3], [START_REF] Gekeler | Fundamental domains of some arithmetic groups over function elds[END_REF].

Denition 1. The Bruhat-Tits tree T is the combinatorial graph with set of vertices

X(T ) = GL 2 (K ∞ )/ZGL 2 (O ∞ ) and set of oriented edges Y (T ) = GL 2 (K ∞ )/ZI ∞ .
We associate to T the following maps

o : Y (T ) -→ X(T ) M mod ZI ∞ -→ M mod ZGL 2 (O ∞ )
and

• : Y (T ) -→ Y (T ) e -→ e = e 0 1 π 0 .
The map o is the canonical map that associates to an edge its origin. The map e → e associates to an edge e its opposite edge denoted by e.

The graph T is a tree in the sense of Serre ([22, I.2]) and it is a (q + 1)-regular graph. The group GL 2 (K ∞ ) acts on the left on T in a natural way. Any arithmetic subgroup Γ ⊂ GL 2 (A) acts on T without inversion. Hence we can consider the quotient graph Γ\T whose set of vertices is X(Γ\T ) = Γ\X(T ) and set of edges is Y (Γ\T ) = Γ\Y (T ). By the work of Serre ([22, II.1.2 , II.1.3]), the graph Γ\T is the disjoint union of a nite graph denoted by (Γ\T ) 0 and a nite number of half-lines called the ends of Γ\T . In fact we have the following statement (see [START_REF] Gekeler | Jacquet-Langlands theory over K and relations with elliptic curves[END_REF]). Proposition 2. There are canonical bijections between the following sets:

(1) the ends of Γ\T , (2) the cusps of M Γ , (3) the orbits Γ\P 1 (K) of P 1 (K) under Γ.

Let (v k ) k∈N be the vertex of T corresponding to the classes of the matrices π -k 0 0 1 k∈N and let e k be the edge from

v k to v k+1 . The end ∞ = (v 0 , v 1 , v 2 , ...)
of T denes an orientation on T i.e. a decomposition

Y (T ) = Y + (T ) Y -(T ) with Y + (T ) = Y -(T ).
An edge e is called positive if it points towards the end ∞. The following statement is well-known and can be found for instance in [11, p. 371].

Lemma 3. The set of matrices π j y 0 1

, with j ∈ Z and y ∈ K ∞ mod π j O ∞ , is a system of representatives for X(T ) and Y + (T ).

Harmonic cochains

Denition 4. Let B be an abelian group. A map ϕ : Y (T ) → B is said to be a (B-valued) harmonic cochain if it satises the two following conditions:

(1) for all e ∈ Y (T ), ϕ(e) = -ϕ(e),

(2) for all v ∈ X(T ),

o(e)=v
ϕ(e) = 0.

Furthermore if for all γ ∈ Γ and e ∈ Y (T ) we have ϕ(γe) = ϕ(e), we say that ϕ is

Γ-invariant.
The additive group of B-valued harmonic cochains is denoted by H(T , B) and its subgroup of Γ-invariant harmonic cochains by H(T , B) Γ .

Elements of H(T , B) Γ can be considered as maps dened on the edges of the quotient graph Γ\T . Let v ∈ X(T ) be a vertex and v its equivalence class modulo Γ. The stabilizer of v, which is denoted by Γ v , acts on the set {e ∈ Y (T ), o(e) = v}.

For e ∈ Y (T ), let Γ e be its stabilizer. The length of the orbit of e is m(e) = [Γ v : Γ e ] and this number only depends on the image ẽ of e in Γ\T . Viewing ϕ ∈ H(T , B) Γ as a map on Y (Γ\T ), the sum condition in Denition 4 translates into

ẽ∈Y (Γ\T ) o(ẽ)=ṽ m(ẽ)ϕ(ẽ) = 0.
Denition 5. A harmonic cochain ϕ ∈ H(T , B) Γ is said to be cuspidal if it has a nite support modulo Γ.

We denote by H ! (T , B) Γ the subgroup of cuspidal harmonic cochains. On H ! (T , C) Γ , the Petersson inner product is dened by [START_REF] Breuil | On the modularity of elliptic curves over Q: wild 3-adic exercises[END_REF] ϕ, ψ =

ẽ∈Y (Γ\T ) q -1 2 |Γ e | -1 ϕ(ẽ)ψ(ẽ) (ϕ, ψ ∈ H ! (T , Z) Γ ).
Let n ∈ A. If n ∈ A is a polynomial dividing n then for each monic divisor a of n/n , we have an embedding i a,n :

H ! (T , B) Γ0(n ) → H ! (T , B) Γ0(n) given by i a,n (ϕ)(e) = ϕ a 0 0 1 e (ϕ ∈ H ! (T , B) Γ0(n ) , e ∈ Y (T )). Denition 6. Let H new ! (T , Q) Γ0(n) be the orthogonal complement in H ! (T , Q),
with respect to the Petersson inner product, of the images of all the i a,n where n runs through the proper divisors of n.

Let H new ! (T , Z) Γ0(n) = H ! (T , Z) Γ0(n) ∩ H new ! (T , Q) Γ0(n) . A harmonic cochain ϕ ∈ H ! (T , Z) Γ0(n) is called a newform if ϕ ∈ H new ! (T , Z) Γ0(n) .
We now introduce a map j which will be used to construct the modular parametrization. Let v, w ∈ X(T ) be two vertices. We denote by g(v, w) the unique geodesic going from v to w. For v ∈ X(T ), e ∈ Y (T ) and γ, α ∈ Γ, we let

ι(e, α, γ, v) =      1 if γe ∈ g(v, αv), -1 if γe ∈ g(αv, v), 0 otherwise.
Since the map γ → ι(e, α, γ, v) has nite support, the quantity

ϕ α,v (e) = 1 |Γ ∩ Z| γ∈Γ ι(e, α, γ, v)
is well-dened and it is also Z-valued. We let Γ = Γ ab / Tor(Γ ab ), where Γ ab is the abelianization of Γ and Tor(Γ ab ) is the torsion subgroup of Γ ab . Lemma 7 ([16,Lemma 3.3.3]). The functions ϕ α,v : Y (T ) → Z have the following properties:

(1) For all v ∈ X(T ) and all α ∈ Γ,

ϕ α,v ∈ H ! (T , Z) Γ . (2) The function ϕ α = ϕ α,v is independent of the choice of v ∈ X(T ). ( 3
) The map α → ϕ α induces a group homomorphism j from Γ to H ! (T , Z) Γ . ( 4
)
The map j : Γ → H ! (T , Z) Γ is one-to-one and with nite cokernel.

Moreover as proved by Gekeler-Nonnengardt in [15, Theorem 3.3], we have in the case of the subgroup Γ 0 (n):

Theorem 8. Let n ∈ A be a non constant polynomial. If Γ = Γ 0 (n) then the map j : Γ → H ! (T , Z) Γ is an isomorphism.
Let g(Γ) be the genus of the modular curve M Γ . For an abelian group B, let denote by rk Z (B) its rank. We have the following equalities (see [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF]):

rk Z (Γ) = rk Z (H ! (T , Z) Γ ) = g(Γ).

Theta functions for Γ

Denition 9 ([16, Section 5]). A holomorphic theta function (resp. meromorphic theta function) for Γ is a holomorphic function on Ω without zeros and poles on Ω and at the cusps (resp. without zeros and poles at cusps), and satisfying

f (γz) = c f (γ)f (z), for all z ∈ Ω and γ ∈ Γ, with c f (γ) ∈ C * ∞ independent of z. The map c f : Γ → C * ∞ is called the multiplier of f . Let m a positive integer. We set U m = {z ∈ Ω, |z| ≤ q m , |z| i ≥ q -m }. The set Ω is a rigid analytic subspace in P 1 (C ∞ ) and Ω = m≥1 U m is an admissible covering.
For the rest of this section, we x two elements ω, η ∈ Ω.

Denition 10. We put

Γ = Γ/(Γ ∩ Z) and (2) θ(ω, η, z) = γ∈ Γ z -γω z -γη .
Convergence and other properties of these theta functions are summarized in the following statement. (1) The product [START_REF] Brunault | On the ramication of modular parametrizations at the cusps[END_REF] converges locally uniformly on Ω and denes a meromorphic theta function for Γ. Moreover, it has only zeros and poles at Γω, Γη if Γω = Γη.

(2) The function θ(ω, η, .) satises θ(ω, η, γz) = c(ω, η, γ)θ(ω, η, z) for any γ ∈ Γ, z ∈ Ω, with c(ω, η, γ) ∈ C * ∞ . Moreover, c(ω, η, .) factors over Γ.

(3) Given α ∈ Γ, the holomorphic function

(3) u α (z) = θ(ω, αω, z) = γ∈ Γ z -γω z -γαω
is well-dened and independent of ω ∈ Ω. It only depends on the class of α ∈ Γ.

(4) The multiplier c(ω, η, .) satises for all α ∈ Γ, c(ω, η, α) =

u α (η) u α (ω)
and in particular, is holomorphic in ω and η. (5) Let c α (.) = c(ω, αω, .) be the multiplier of

u α . The map (α, β) → c α (β) is a symmetric bilinear map on Γ × Γ which takes values in K * ∞ .
We recall that the multiplier c(ω, η, .) satises (see [16, 5.4.9])

(4) c(ω, η, α) =      1 if α∞ = ∞, γ∈ Γ α∞ -γω α∞ -γη otherwise.
Since for all α ∈ Γ, the map

γ ∈ Γ → c α (γ) ∈ C * ∞ factors over Γ, the map c : Γ -→ Hom(Γ, C * ∞ ) α -→ c α (.)
is well-dened. Furthermore the map c factors over Γ and gives a map c : Γ → Hom(Γ, C * ∞ ).

Theorem 12 ([13, Proposition 2.6]). Let ω, η be elements of Ω. The function θ(ω, η, .) has a meromorphic continuation to the boundary P 1 (K) of Ω. Moreover θ(ω, η, .) is holomorphic without any zeros at cusps.

Remark. The value of the meromorphic continuation of θ(ω, η, .) at the cusp ∞ is 1 (this can be derived from [16, Lemma 5.3.10]).

5. Formula for the value of theta functions at a cusp A consequence of Theorem 12 is that the holomorphic function u α has a holomorphic continuation to Ω = Ω P 1 (K) hence it is a holomorphic theta function. Furthermore, as proved in [16, Lemma 5.3.9], for all s ∈ K and all ω, η ∈ Ω, the product γ∈ Γ s -γω s -γη converges. Note that it does not necessarily imply that the image of u α at a cusp s ∈ K is given by the product formula (3). However we prove now that it is indeed the case.

Theorem 13. Let α be an element of Γ and s ∈ P 1 (K) -{∞}. The value of u α at s is given by

u α (s) = γ∈ Γ s -γω s -γαω .
The rest of this section is devoted to the proof of Theorem 13. Let s ∈ P 1 (K) -{∞}. Since the product u α (z) is independent of the choice of ω ∈ Ω, we can choose ω ∈ Ω such that |ω| ∈ q Q-Z . This implies that |ω| = |ω| i . We have

(5) γ∈ Γ s -γω s -γαω - γ∈ Γ z -γω z -γαω = |u α (z)| γ∈ Γ (s -γω)(z -γαω) (s -γαω)(z -γω) -1 . We let F γ,s (z) = (s -γω)(z -γαω) (s -γαω)(z -γω) . Lemma 14. For γ = a b c d ∈ Γ, we have F γ,s (z) = 1 + (z -s)(αω -ω) det(γ) (s -γαω)(z -γω)(cω + d)(cαω + d) . Proof. Let γ = a b c d ∈ Γ and z ∈ Ω. We rst note that (6) γαω -γω = (αω -ω) det(γ) (cω + d)(cαω + d) .
Furthermore we have

(s -γω)(z -γαω) (s -γαω)(z -γω) = sz -s(γαω) -z(γω) + (γω)(γαω) (s -γαω)(z -γω) = 1 + (z -s)(γαω -γω) (s -γαω)(z -γω) = 1 + (z -s)(αω -ω) det(γ) (s -γαω)(z -γω)(cω + d)(cαω + d)
and the lemma follows.

We know that |s -γαω| ≥ κ s,αω |cαω + d| (see [16, proof of Lemma 5.3.9]) where κ s,αω is a positive real number only depending on s, α and ω. It yields

(7) |F γ,s (z) -1| ≤ |z -s||αω -ω|κ -1 s,αω |z -γω||cω + d| ≤ |z -s|C ω,α,s |z -γω||cω + d| where C ω,α,s is independent of γ ∈ Γ and z ∈ Ω.
The goal is to prove that the limit of ( 5) is 0 as z tends to s. From the meromorphic continuation of u α at s, we deduce that u α (z)

→ z→s u α (s). It is then sucient to nd a sequence (z n ) n∈N ⊂ Ω such that lim n→∞ z n = s and such that (8) lim n→∞ u α (z n ) = γ∈ Γ s -γω s -γαω .
We have

(9) |u α (z n )| ≤ max(|u α (s)|, |u α (s) -u α (z n )|) < C
where C > 0 only depends on s and α since u α (z n ) → u α (s). Now to prove (8), we need to prove that for all ε > 0, there exists n ε > 0 such that for all n ≥ n ε , |F γ,s (z n ) -1| < ε. We choose ζ ∈ Ω -Γω such that |ζ| = |ζ| i = 1 and we let

z n = s + T -n ζ. Remark that |z n | i = q -n
for all non-negative integers n.

V. PETIT Lemma 15. Let ε > 0 be a real number. There exists a non-negative integer n 0 such that for all n ≥ n 0 , if

γ satises |z n -s| i > |γω| i then |F γ,s (z n ) -1| ≤ ε.
Proof. Let ε > 0 be xed and n ≥ 0. First note that |cω

+ d| ≥ |cω| i = |c||ω| i . Hence, if |c| > |ω| -1 i q n or if |d| > max |cω|, |ω| -1 i q n 1 2
, q n , we have

|γω| i = |ω| i |cω + d| 2 < |z n -s| i = q -n .
For all the pairs (c, d) satisfying at least one of the two conditions on |c| or |d|, we have |cω + d| > q n and since

|γω| i < |z n -s| i we have |z n -γω| ≥ |z n -s -γω| i = |z n -s| i . So we obtain |F γ,s (z n ) -1| ≤ |z n -s|C ω,α,s |z n -s| i |cω + d| by (7) ≤ C ω,α,s |cω + d| ≤ C ω,α,s q -n
which is less than or equal to ε when n is large enough.

Lemma 16. Let ε > 0 be a real number. There exists n 1 ≥ 0 such that for all

n ≥ n 1 , if γ ∈ Γ satises |γω| i > |z n -s| i then |F γ,s (z n ) -1| ≤ ε.
Proof. Let ε > 0 be xed and n ≥ 0 be large enough. ). Then by [START_REF] Delaunay | Critical and ramication points of the modular parametrization of an elliptic curve[END_REF], we obtain |F γ,s (z n ) -1| ≤ |z n -s|C ω,α,s κ -1 ω,s , which is less than or equal to ε if n is large enough.

If γ ∈ Γ satises |γω| i > |z n -s| i then |s -γω| ≥ |γω| i > |z n -s| i = |z n | i = |z n -s|.
Corollary 17. Let ε > 0. There exists n 2 ≥ 0 such that for all n ≥ n 2 and all

γ ∈ Γ, we have |F γ,s (z n ) -1| ≤ ε.
Proof. Let ε > 0. We let n 2 = max(n 0 , n 1 ) where n 0 and n 1 are taken as in Lemma 15 and 16. We derive

|F γ,s (z n ) -1| < ε if |γω| i = |z n -s| i . Moreover since |ω| = |ω| i ∈ q Q-Z , for all γ = a b c d ∈ Γ we have |cω + d| = max(|cω|, |d|). Hence we get |γω| i ∈ q Q-Z for all γ ∈ Γ, therefore |γω| i = |z n -s| i = q -n .
From Corollary 17, there exists n 2 such that for all n ≥ n 2 ,

γ∈ Γ s -γω s -γαω - γ∈ Γ z n -γω z n -γαω ≤ Cε,
where the constant C is as in [START_REF] Gekeler | Drinfeld modular curves[END_REF]. This concludes the proof of Theorem 13.

Hecke operators

From now on, the arithmetic subgroup Γ will be Γ 0 (n) with n ∈ A non constant and monic. Let ϕ ∈ H(T , B) Γ where B = Z, Q, R or C. We consider ϕ as a map on GL 2 (K ∞ ). Let m = (m) ⊂ A be an ideal coprime to (n). We set Proposition 18. The Hecke operators have the following properties:

T m (ϕ)(e) =
(1) Let p ⊂ A be a prime ideal of A and k be a nonnegative integer. Then T p k is a polynomial in T p . (2) Let m, m be two coprime ideal of A, then T mm = T m T m .

(3) Hecke operators commute with each other.

It is also possible to dene a Hecke action on Γ (see [16, 9.3]). If p = (P ) is an ideal of A coprime to n, we let τ P = P 0 0 1 ∈ GL 2 (K ∞ ) and we dene

∆ P = Γ ∩ τ P Γτ -1 P .
For α ∈ Γ, we dene T p (α) as [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF] T p (α) = τ -1

P αi∈∆ P \Γ α i αα -1 σ(i) τ P
where α i runs through a system of representatives of ∆ P \Γ and where σ is the permutation of ∆ P \Γ such that α i αα -1 σ(i) ∈ ∆ P .

Lemma 19 ([16, Lemma 9.3.2]). Let a = (a) be an ideal of A coprime to n and α ∈ Γ. We have j(T a (α)) = T a (j(α)). In other words, the action of Hecke operators commutes with the isomorphism j.

Proposition 20 [START_REF] Gekeler | Jacobians of Drinfeld modular curves[END_REF]Lemma 9.3.3]). The Hecke operators on Γ dened in [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF] are self-adjoint with respect to the bilinear map (α, β) → c α (β).

Let J Γ be the Jacobian of M Γ . It is an abelian variety over K of dimension g(Γ) where g(Γ) is the genus of the modular curve M Γ .

Denition 21 ([9, VIII,1]). Let p = (P ) be a prime ideal of A and ∆ P , τ P as above. The p-th Hecke operator on M Γ is given by the correspondence [9, Chapter VIII] for more details). The action of Hecke operators on J Γ (C ∞ ) induces the following action on C * ∞ /t Z E(C ∞ ) (see [16, 9.

T p ω = α∈∆ P \Γ τ -1 P αω, ω ∈ M Γ (C ∞ ).

The action of Hecke operators on M Γ induces an action on J

Γ . Indeed if D = a i (ω i ) ∈ J Γ (C ∞ ) then T p (D) = a i (T p (ω i )) (see

Modular parametrization

Proposition 22 ([16, 7.3.3]). The analytic group variety Hom(Γ, C * ∞ )/c(Γ) carries the structure of an abelian variety K ∞ -isomorphic to J Γ .

In other words, we have the exact sequence

(12) 1 → Γ c → Hom(Γ, C * ∞ ) → J Γ (C ∞ ) → 0.
The Hecke operators act on each of its terms and the map c : Γ → C ∞ is compatible with their action (see [16, 9.3.3]). The same holds for the projection Hom(Γ,

C * ∞ ) → J Γ (C ∞ ).
Let E/K be an elliptic curve with split multiplicative reduction at the place ∞ = 1/T of K. Equivalently, E has Tate parametrization

(13) E(K ∞ ) K * ∞ /t Z , for some t ∈ K * ∞ with |t| < 1.
For each prime p, we set

λ p =          q deg(p) + 1 -card(E(F p )) if
ϕ ∈ H ! (T , Z) Γ which is it is not divisible on H ! (T , Z) Γ and such that    c(ϕ, 1) = 1 T m (ϕ) = q deg(m) c(ϕ, m)ϕ for all ideal m ⊂ A, c(ϕ, p) = q -deg(p) λ p for all prime p ⊂ A
where deg(m) = log q (card(A/m)).

For m = (m), the coecient c(ϕ, m) is the Fourier coecient of ϕ given by (see [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF]Section 3])

c(ϕ, m) = q -1-deg(m) y∈F * q \πO∞/π 2+deg(m) O∞ ϕ π 2+deg(m) y 0 1 ν(my)
where ν : K ∞ → Z is the character that maps the element i∈Z a i π i to -1 if a 1 = 0 and to q -1 otherwise. Let Γ ∞ be the subgroup

Γ ∞ = a b 0 d ∈ Γ, a, b ∈ F * q , d ∈ A .
Recall that ϕ can be seen as a map on Y + (Γ ∞ \T ) and its Fourier expansion is given by the following expression, for all j ∈ Z and y ∈ K ∞ mod π j O ∞ (see [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF] or [START_REF] Gekeler | Improper Eisenstein series on Bruhat-Tits trees[END_REF]), ( 14)

ϕ π j y 0 1 = j-2 k=0 q k+2-j f ∈A, f monic deg(f )=k c(ϕ, (f ))ν(f y).
Identifying H ! (T , Z) Γ with Γ by means of the isomorphism j of Theorem 8, we may consider ϕ as an element of Γ. Let Λ = {c ϕ (γ), γ ∈ Γ} = t Z . We have the following commutative diagram

(15) 1 / / Γ cϕ(.) c / / Hom(Γ, C * ∞ ) evϕ / / J Γ (C ∞ ) / / 0 1 / / Λ / / C * ∞ / / C * ∞ /Λ / / 0
Denition 24. The modular parametrization (also called Weil uniformization) is dened as

Φ : M Γ (C ∞ ) -→ C * ∞ /Λ E ϕ (C ∞ ) z -→ u ϕ (z),
where E ϕ is called the Weil curve associated to the harmonic cochain ϕ. We note that E ϕ is a strong Weil curve in the sense that Φ can not be factorized through another Weil uniformization Corollary 3.19]). Let us recall also his formula for the degree of the modular parametrization.

Φ : M Γ (C ∞ ) Φ → E (C ∞ ) → E ϕ (C ∞ ), where E is a Weil curve. We set µ = inf{ ϕ, ψ > 0, ψ ∈ H ! (T , Z) Γ } = inf{log q |c ϕ (γ)| > 0, γ ∈ Γ}. Gekeler has proved that µ = v ∞ (t) = -v ∞ (j E ) where j E is the j-invariant of E ϕ ([10, Theorem 3.2,
Theorem 25 [START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF]Proposition 3.8]). The degree of modular parametrization Φ is given by

deg(Φ) = ϕ, ϕ µ .
We want to compute the values taken by the modular parametrization Φ at the cusps of M Γ . By the analogue of the Manin-Drinfeld theorem, proved in this setting by Gekeler [START_REF] Gekeler | A note on the niteness of certain cuspidal divisor class groups[END_REF], the subgroup of the Jacobian of M Γ generated by the cusps is nite. As a consequence, the image of the cusps by Φ are torsion points of the elliptic curve. However Gekeler's statement provides no explicit bound for the size of the cuspidal subgroup in general. For our purpose, we prove an explicit bound on the order of such torsion points: the bound depends on the elliptic curve and is proved using Hecke operators and the same proof principle as Manin-Drinfeld.

Theorem 26. Let s ∈ Γ\P 1 (K) be a cusp of M Γ . Then Φ(s) is a torsion point of E and its order divides card(E(F p )) for all p = (P ) with P ≡ 1 mod n.

Proof. Let z ∈ M Γ (C ∞ ) and let p = (P ) be a prime ideal of A such that P ≡ 1 mod n. By the analogue of Dirichlet's theorem ([20, Theorem 4.7]), such a polynomial P exists. We have

T p u ϕ (z) = u ϕ (P z) j∈A deg(j)<deg(P ) u ϕ z + j P .
Since T p , as any other Hecke operator, commutes with the map f ∈ Hom(Γ, C ∞ ) → f (ϕ) ∈ C * ∞ , we have by ( 11), ( 16)

u ϕ (z) λp = u ϕ (P z) j∈A deg(j)<deg(P ) u ϕ z + j P .
Let s ∈ Γ\P 1 (K) be a cusp. We let S P = {j ∈ A, deg(j) < deg(P )}. Since P ≡ 1 mod n and by arguments similar to [5, Proposition 2.2.3], there exist Q P and (Q j ) j∈S P ⊂ Γ such that Q P s = P s and Q j s = s + j P for all j ∈ S P . We take z = s in ( 16), divide by u ϕ (s) q deg(P )+1 and obtain

u ϕ (s) λp-(q deg(P ) +1) = u ϕ (P s) u ϕ (s) j∈A deg(j)<deg(P ) u ϕ s+j P u ϕ (s) , = u ϕ (Q P s) u ϕ (s) j∈A deg(j)<deg(P ) u ϕ (Q j s) u ϕ (s) ∈ Λ. So Φ(s) is a torsion point of E(C ∞ ) whose order divides card(E(F p )).
Remark. In specic cases, explicit bounds for the size of the cuspidal subgroup of the Jacobian are known, for example when n is irreducible by Pál ([18]) or deg(n) = 3 by Papikian and Wei ( [START_REF] Papikian | On the Eisenstein ideal over function elds[END_REF]). These bounds, which in contrast with Theorem 26 depend only on n, also are upper bounds for the order of torsion points on the elliptic curve coming from evaluating Φ at cusps. However Theorem 26 requires no assumption on n.

Examples

8.1. Example in F 2 (T ) with n = T 3 . We consider the elliptic curve E/F 2 (T ) dened by

y 2 + T xy = x 3 + T 2 .
Its conductor is n = T 3 ∞. The elliptic curve E is isomorphic over F 2 (T ) to the strong Weil curve given in [21, Theorem 2.1 (a)]. We set n = T 3 and Γ = Γ 0 (n).

The genus of M Γ is 1, thus the abelian group H ! (T , Z) Γ has rank one. Let ϕ E ∈ H ! (T , Z) Γ be the harmonic cochain attached to E as described in Section 7.

We can identify ϕ E with its Fourier expansion: indeed since ϕ E has nite support modulo Γ, it is sucient to know its values on the nite number of edges in (Γ\T ) 0 . Furthermore we can compute the inverse image of ϕ E by the isomorphism j of Theorem 8. For this purpose, we compute the inverse image of a Z-basis of H ! (T , Z) Γ with the maximal subtree method in Γ\T as in [START_REF] Gekeler | Jacobians of Drinfeld modular curves[END_REF]proof of Lemma 3.3.3]. The element α ∈ Γ such that j(α) = ϕ E that we obtain is then

α = T 2 + T + 1 1 T 3 1 + T . Note that α generates Γ since ϕ E generates H ! (T , Z) Γ , hence Λ = {c α (γ), γ ∈ Γ} = c α (α)
. In order to determine the lattice Λ, we just need to compute c α (α). The coecient c α (α) is given by (see ( 4))

(17) c α (α) = γ∈ Γ α∞ -γω α∞ -γαω ,
and this product is independent of ω ∈ Ω. We choose ω = T -2 ρ with ρ ∈ C * ∞ satisfying ρ 2 +ρ+1 = 0. We obtain a suitable approximation of c α (α) by restricting the product [START_REF] Jacquet | Automorphic forms on GL(2)[END_REF] to the set

γ ∈ Γ, α∞ -γω α∞ -γαω -1 ≥ 1 16 .
This set is then determined by similar arguments as those in the proof of Proposition 27. We obtain c α (α) = π -4 +δ cα(α) where

δ cα(α) ∈ K * ∞ is such that |δ cα(α) | < 1. We deduce that a generator of Λ is t = c α (α) -1 = π 4 + δ tα with |δ tα | < 2 -8 .
The modular curve M Γ has four cusps: ∞, 0, 1 T , 1 T 2 . By Theorem 25, the degree of the modular parametrization is 1 and Φ is therefore an isomorphism. For a cusp c ∈ {∞, 0, 1 T , 1 T 2 }, we want to evaluate u α (c) modulo Λ. Using Theorem 26, we compute card(E(F p )) for all primes p = (P ) with deg(P ) ≤ 15 and obtain that the image of c is a torsion point in E of order dividing 16. Note that in this example, the bound given by Papikan-Wei in [START_REF] Papikian | On the Eisenstein ideal over function elds[END_REF] is 4, which is better than ours.

Proposition 27. We have the following approximations:

u α (0) = π + υ, with |υ| < 2 -5 , u α 1 T = π -1 + δ 1/T , with |δ 1/T | < 2 -3 , u α 1 T 2 = π 2 + δ 1/T 2 , with |δ 1/T 2 | < 2 -6 .
In particular, the points Φ(0) and Φ( 1 T ) have order 4 in E(K ∞ ), and the point Φ( 1T 2 ) has order 2 in E(K ∞ ). Proof. We only give details for the computation of u α at s = 0. Recall that by Theorem 13, the value u α (0) is given by

u α (0) = γ∈ Γ 0 -γω 0 -γαω .
As above, we choose ω = T -2 ρ with ρ 2 + ρ + 1 = 0. We then observe that

|ω| = |ω| i = 2 -2 , |αω| = 2 -1 and |αω| i = 2 -4 . First let γ = a b c d be an element of Γ. We have -γω -γαω = 1 + γαω -γω -γαω ≤ max 1, γαω -γω -γαω ,
and by ( 6), [START_REF] Pál | On the Eisenstein ideal of Drinfeld modular curves[END_REF] γαω -γω

-γαω = |αω -ω| | -γαω||cω + d||cαω + d| = 2 -1 |aαω + b||cω + d| .
But we have

|aαω + b| = |(a(1 + T + T 2 ) + bT 3 )ω + b(1 + T ) + a| |T 3 ω + 1 + T | ≥ 2 -1 (a(1 + T + T 2 ) + bT 3 )ω + b(1 + T ) + a ≥ 2 -1 . Hence if c = 0 γαω -γω -γαω ≤ 1 |cω + d| ≤ 2 2 |c| ≤ 2 -1 .
If c = 0, then a = d = 1 and |cω + d| = 1 and we have

γαω -γω -γαω = 2 -1 |αω + b| ≤ 1.
For all subsets S ⊂ Γ, we deduce

(19) γ∈S -γω -γαω ≤ 1.
By this inequality, we see that to compute u α (0) at a precision at most ε, it is sucient to compute the product

γ∈ Γ |Gγ,ω(0)-1|≥ε G γ,ω (0), where G γ,ω (z) = z -γω z -γαω . If |c| > 2 2 ε ,
|G γ,ω (0) -1| ≤ 2 -2 |aαω + b| < 2 2 |a| < ε if |a| > 2 2 ε .
And if c = 0 (so that a = d = 1), then if |b| > 2 -1 ε , we obtain where

|G γ,ω (0) -1| = 2 -1 |αω + b| ≤ 2 -1 |b| < ε.
Γ ε = γ = a b c d ∈ Γ, c = 0, |c| ≤ 2 2 ε , |d| ≤ 2 ε , |a| ≤ 2 2 ε ∪ γ ∈ Γ, c = 0, |b| ≤ 1 2ε .
The computation of the nite product γ∈ Γε G γ,ω (0) for ε = 2 -5 has been done

with the software Pari-GP [START_REF]PARI/GP version 2[END_REF]. We obtain u α (0) = π + υ with υ ∈ K * ∞ satisfying |υ| < 2 -5 , thus u α (0) 4 = t + υ 4 . We conclude that Φ(0) has order 4.

For s = 1 T , we obtain u α (

T ) = π -1 + δ 1/T with δ 1/T ∈ K * ∞ satisfying |δ 1/T | < 2 -3 1
: we deduce that the point Φ(1/T ) has order 4 on E. Finally, we have u α 1

T 2 = π 2 + δ 1/T 2 with |δ 1/T 2 | ≤ 2 -6 , hence the point Φ 1
T 2 has order 2.

Remark.

In this example we can also give an explicit bound on the cardinal of

the set Γ ε . The set S = {c ∈ F 2 [T ] : c ≡ 0 mod T 3 , deg(c) ≤ 2 + log 2 ( 1 ε ) } has cardinal 2 -log 2 (ε) . Let c ∈ S. Consider the set S c of polynomials d ∈ F 2 [T ] such that there exists a matrix a b c d ∈ Γ: an upper bound on its cardinal is card({d ∈ F 2 [T ] : deg(d) ≤ 1 -log 2 (ε) , gcd(d, T ) = 1}) = 2 2-log 2 (ε) .
Then for c ∈ S and d ∈ S c , the number of all possible polynomials a such that

there exists b ∈ A with a b c d ∈ Γ is bounded by card({a ∈ F 2 [T ], gcd(a, T ) = 1 and deg(a) ≤ 2 -log 2 (ε)}) = 2 2-log 2 (ε) .
Hence we get card( Γ ε ) ≤ 2 4-3 log 2 (ε) .

8.2. Example in F 3 (T ) with n = T 3 -T 2 . We consider the elliptic curve E/F 3 (T )

dened by E : y 2 = x 3 + T (T + 1)x 2 + T 2 x. Its conductor is ∞(T 3 -T 2 ). Again the curve E is a strong Weil curve, see [START_REF] Schweizer | Strong Weil curves over F q (T ) with small conductor[END_REF]Theorem 2.4 (b)]. Let n = T 3 -T 2 be the nite part of conductor of E and let Γ = Γ 0 (T 3 -T 2 ). Here the genus of the curve M Γ is 2 hence the abelian group H ! (T , Z) Γ has rank two. Let ϕ E be the harmonic cochain associated to E. Let α the inverse image of ϕ E by j. By the same method as in the rst example, we nd

α = T 3 + 2T 2 + 2T + 2 2T + 1 T 4 + T 3 + T 2 2T 2 + 2T + 1 .
We compute a generator t ∈ K * ∞ of the lattice Λ = {c α (γ), γ ∈ Γ} = t Z . We obtain the following approximation for t:

t = π 4 + 2π 5 + O(π 6 ). The modular curve M Γ has six cusps: ∞, 0, 1 T , 1 T 2 , 1 T -1 , 1 T 2 -T . The Tate period of E is given by (see [23, Section 2 of Appendix A]) t E = n≥1 d(n) 1 j n E = π 4 + 2π 5 + π 7 + 2π 8 + δ t E ,
where (d(n)) n≥1 are the coecients of the series for the reciprocal function of j, and

δ t E ∈ K * ∞ with |δ t E | < 3 -10
. We deduce that t E = t. The degree of the modular parametrization is 2. As before, we compute card(E(F p )) for prime ideals p = (P ) such that deg(P ) ≤ 10 and P ≡ 1 mod n. By Theorem 26, we obtain that the order V. PETIT of image of a cusp divides 8. This bound is sharper than the one of Papikian-Wei [START_REF] Papikian | On the Eisenstein ideal over function elds[END_REF] which is 24 in this example. Proposition 28. We have the following approximations:

u α (0) = 2π 2 + 2π 3 + ϑ 0 , with |ϑ 0 | < 3 -3 , u α 1 T = π -1 + 1 + π + ϑ 1/T , with |ϑ 1/T | < 3 -1 , u α 1 T 2 = π 4 + 2π 5 + ϑ 1/T 2 with |ϑ 1/T 2 | < 3 -5 , u α 1 T -1 = π -2 + π -1 + ϑ 1/(T -1) ,
with |ϑ 1/(T -1) | < 1,

u α 1 T 2 -T = π 3 + ϑ 1/(T 2 -T ) ,
with |ϑ 1/(T 2 -T ) | < 3 -5 .

In particular, we get (1) the points Φ 1 T and Φ Hence we obtain by [START_REF] Delaunay | Computing modular degrees using L-functions[END_REF] This proves the statement.

Remark. We can also provide an upper bound on the cardinal of the set S

Theorem 11 (

 11 [START_REF] Gekeler | Jacobians of Drinfeld modular curves[END_REF] Proposition 5.2.3, Theorem 5.4.1 and Proposition 5.4.12]).

  By the ultrametric inequality, we have |z n -γω| = |(z n -s) + s -γω| = |s -γω| ≥ κ ω,s |cω + d| where κ ω,s is independent of γ ∈ Γ (see [16, proof of Lemma 5.3.9]

  Then we have T m (ϕ) ∈ H(T , B) Γ and the operator T m is called the m-th Hecke operator. The following classical statements may be found for instance in[START_REF] Gekeler | Jacquet-Langlands theory over K and relations with elliptic curves[END_REF] Section 7].

  T m u α (z) = a,b,d∈A a monic deg(d)<deg(b) ad=m,(a,n)=1 u α az + b d .

  Note that, except for the case c = 0, for xed a, c, d there exists at most one value of b ∈ A such that a b c d ∈ Γ. Hence, to approximate u α (0) at precision at most ε we must compute the product γ∈ Γε G γ,ω (0),

1 T 2 1 T - 1 has order 2

 2112 -T have order 4 in E(K ∞ ), (2) the point Φ 1 T 2 has order 1 in E(K ∞ ), (3) the point Φ(0) and Φ on E(K ∞ ).Proof. As before for s ∈ Γ\P 1 (K) and γ ∈ Γ, we let G γ,ω (s) = s -γω s -γαω . As a system of representatives for Γ, we choose the matrices a b c d ∈ Γ with monic a.For ε ≤ 1, let S ε = {γ ∈ Γ, |G γ,ω (s) -1| ≥ ε}.We haveγ∈Sε G γ,ω (s) ≤ γ∈S1 G γ,ω (s) and γ∈ Γ G γ,ω (s) -γ∈Sε G γ,ω (s) < γ∈S1 G γ,ω (s) ε. Indeed if g γ,ω (s) = G γ,ω (s) -1, we have γ∈ Γ-Sε G γ,ω (s) -1 = γ∈ Γ-Sε (1 + g γ,ω (s)) -1and we can easily see thatγ∈ Γ-Sε (1 + g γ,ω (s)) = 1 + δ ω,ε (s) with |δ ω,ε (s)| < ε.We give details only for the cusp s = 1 T . We choose ω = T -2 ρ with ρ 2 + 1 = 0. Note that |ω| = |ω| i = 3 -2 , |αω| = 3 -1 and |αω| i = 3 -4 . Let γ = a b c d ∈ Γ. We have |s -γαω| = |(c -aT )αω + (d -bT )| |T ||cαω + d| . Moreover we have |(c -aT )αω + d -bT | ≥ 3 -2 .

If c = 0 , 3 G 1 T, u α 1 T 2 =

 03112 we have a = 1 and d ∈ F * 3 , so we get if b = 0, |s -γαω| = |s -αω d -b d | = |b|. Hence |G γ,ω (s) -1| = |αω -ω| |b||cω + d| ≤ γ,ω (s) has norm 3. So if we want to approximate the product γ∈ ΓG γ,ω (s) with precision at most ε, we need to consider the product restricted to the set S ε/3 . For ε = 3 -1 , the computation of u α at precision at most 3 -1 gives u α 1T = π -1 + 1 + π + ϑ 1/T , with ϑ 1/T ∈ K * ∞ satisfying |ϑ 1/T | < 3 -1 .For the other cusps we getu α (0) = 2π 2 + 2π 3 + ϑ 0 , with |ϑ 0 | < 3 -3 π 4 + 2π 5 + ϑ 1/T 2 with |ϑ 1/T 2 | < 3 -5 , u α 1 T -1 = π -2 + π -1 + ϑ 1/(T -1) , with |ϑ 1/(T -1) | < 1, u α 1 T 2 -T = π 3 + ϑ 1/(T 2 -T ) , with |ϑ 1/(T 2 -T ) | < 3 -5 .

1 / 9 needed to compute u α 1 T

 191 at precision at most ε = 3 -1 . We have card({c ∈ F 3 [T ], deg(c) ≤ 7 and (T 3 -T 2 ) | c}) = 3 4 , card {d ∈ F 3 [T ], deg(d) ≤ 4 and gcd(d, T 3 -T 2 ) ∈ F * 3 } ≤ 3 4 .

  Proposition 23[START_REF] Gekeler | Jacobians of Drinfeld modular curves[END_REF] Proposition 9.5.1],[START_REF] Gekeler | Analytical construction of Weil curves over function elds[END_REF] Theorem 3.2]). Let ev ϕ be the evaluation map at ϕ from Hom(Γ, C * ∞ ) to C * ∞ . The subgroup ev ϕ (c(Γ)) is isomorphic to Z and generated by a unique t ∈ K * ∞ with |t| < 1.

  If |c| ≤ 2 2 ε and if |d| > 2 ε , we have |G γ,ω (0) -1| < ε. Now for the pairs (c, d) not satisfying the condition (|c| > 2 2 ε or |d| > 2 ε ) then, if c = 0 we have |cω + d| ≥ |c||ω| i ≥ 2 1 , and then

			we have
	|G γ,ω (0) -1| =	2 -1 |aαω + b||cω + d|	(by (18))
	≤	2 2 |c|	< ε.

  that if |c| > 3 4 Furthermore, if |c| ≤ 3 4 ε and |d| > 3 2 ε , we have |cω + d| = |d| and we get If (c, d) does not satisfy the condition (|c| > 3 4 ε or |d| > 3 2 ε ) and if c = 0, we note that when |a| > 3 5 ε , we have |aαω + b| > |cαω + d||s|. We then obtain Since |cω + d| ≥ |c||ω| i ≥ 3 1 , we get by (6)

			ε
	|G γ,ω (s) -1| ≤	3 3 |αω -ω| |cω + d|	≤	3 2 |c||ω| i	≤	3 4 |c|	< ε.
		|G γ(s),ω -1| =	3 2 |d|		< ε.
	|s -γαω| = s -	aαω + b cαω + d		=	|aαω + b| |cαω + d|	.
	|G γ,ω (s) -1| ≤	|αω -ω| |aαω + b||cω + d|	≤	3 2 |a|	< ε,	if |a| >	3 5 ε	.

V. PETIT

So the number of pairs (c, d) such that (c, d) can be lifted to a matrix in Γ is bounded by 3 8 . We also have

We conclude that card(S 1/9 ) ≤ 3 15 .


Visualizing the Margins of Gulf Cities M. Benchetrit, R. Stadnicki

This work is the product of a two-months field mission in 2011 across the urban outskirts of Sana'a (Yemen), Doha (Qatar), Muscat and Sohar (Oman), Dubai, Sharjah, Ajman and Ras Al-Khaymah (UAE). Photographer Manuel Benchetrit and geographer Roman Stadnicki collaborated on this geo-photographic project that explores the emerging urban territories of the Arabian Peninsula and is the result of a dialogue between a scientific viewpoint and an artistic perception.

This work looks at idealized urban forms that are either imitations of regional models or exaggerated copies of foreign dreamlands. A fast-paced process alternates construction and demolition, sometimes resulting in the visual strata that our work intends to show. Public enthusiasm for novelty coincides with a growing interest in reinvented heritage and national identity, which contrast with more versatile wider historic and social realities. This contrast contributes to a dissonance in urban visual environments, with billboards advertising brand-new heritage villages for the middle class, while labor migrants appropriate traditional houses. A similar contrast is offered by the landscape of an urban mega-project, in front of which one sees the buses that shuttle immigrant construction workers every day. Deluxe compounds grew up to house the white-collar expat work force, and are as geographically isolated and compartmentalized as the labor camps built by the authorities in the suburbs for the largest construction projects' workers

In this context of dissonance and contrast, urban practices and projects meet at the intersection of dreams and playfulness, through house ornaments, street games, gleaming mall interiors, a minimalist but richly lit Ferris wheels, concrete wind towers, and more.

All photos in this series were initially shown at the NYUAD New York University Institute in Abu Dhabi as pigment prints on Arches paper, mounted on aluminum, 40 x 60cm.

WADI MUSHAYRIB, DOHA (QATAR)

This luxury hotel is separated from its surroundings and offers a variety of services in a self-sufficient environment. One block away, the area of Mushayrib used to host trade shops managed by working class foreigners. Mushayrib, with its open and porous nature, did not resist the assault of the hotel's comfort and amenities. The destruction of Mushayrib to the benefit of Dohaland, a high standard urban area for housing, shops and offices, is now almost complete; but the extent of the destruction is invisible from the street. Awaiting their turn, the facades block the perspective on a vast field of debris. Some of the residents of Mushayrib were invited to move to Al-Barwa Village (see photo 6).

AL RIFAH, SHARJAH (UAE)

This is a construction site at the border between Ajman and Sharjah. The grounds of a former building, probably a sport facility, are bound to be buried like a soon-to-be archeological relic. A little further, the financial crisis seems to have interrupted a real estate project. This temporary wavering between covering, destroying, and building again looks like a geological cross-section and reflects the everlasting stratification of urban projects.

One observes here the blended effects of urbanization's sudden acceleration and slowing down.

THE PEARL, DOHA (QATAR)

Workers' buses are parked in front of the Pearl, an artificial island on the northern outskirts of town that supports a gigantic tourist and residential complex. These vehicles choreograph the dance of the laborers who are jolted along at fixed hours from their labor camp to the building site. This flow, of which the buses are a mechanical part, seems like a pendulum movement or a clockwork pulse.

BARWA VILLAGE, DOHA (QATAR)

This brand-new village in the southeast of the city was built to house some of the shopkeepers of the now demolished Wadi Mushayrib neighborhood, in the center of the city (see photo 1). The relocation operation seems bound to fail. In this area, rents are higher, customers fewer and official accommodation is reserved only for families. Single tradersthe majority of workers in the areaare left with restricted housing options, lack of access to amenities, services and shops. residents flock to malls because they feel they are unfit for the outside environment, its heat, and its cultural differences from their culture origin? Do they come to enjoy the dream of a tri-dimensional postcard that is even more controlled and more reassuring than the original? In May 2012, the Villagio mall burned down and 19 people died in the fire.

  7. AL WAAB, DOHA (QATAR)This gated community in the western suburb of Doha is a closed compound. The residents' exclusive way of life expresses itself in fences, self-segregation and the use of private vehicles and exclusive amenities. Doha is probably one of the most spatially segregated cities in the world. Even interpenetrations between the various compartmentalized communities are regulated. For instance, the construction laborers have to wait until middle class residents leave their compound to start working on new constructions in the mornings. 8. THE VILLAGIO MALL, DOHA (QATAR) Venetian canals were first imitated by Coney Island's Dreamland in 1904. The model has influenced other developments, from Las Vegas to Dubai and to Doha's Villagio. Several malls in the region are characterized by signature themes such as ancient Egypt, the travels of Ibn Batuta, or Marco Polo's hometown. The Villagio is strategically located among upper class gated communities, and begs the following questions: do middle class

AL BIDDA, DOHA (QATAR)

The construction of heritage villages oddly accompanies the race for novelty. In Doha, spaces that glorify the national heritage emulate forms from the past but use modern materials. The recreation of ancient models seems to indicate a desire to reduce the overriding presence of foreign influences in the public space. The march towards novelty and conservation is part of a larger philosophical dialectic between imitation and model, juxtaposition and replacement.

Manuel Benchetrit is an independent photographer, based in Paris. Email: manuelbenchetrit@yahoo.fr