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Merel's theorem on the boundedness of the torsion of elliptic curves

Marusia Rebolledo

Merel's theorem on the boundedness of the torsion of elliptic curves

Marusia Rebolledo

Introduction

Interest in elliptic curves dates back at least to Fermat, who introduced his fundamental method of infinite descent to prove his "Last Theorem" in degree 4. Poincaré seems to have been the first to conjecture, around 1901, the now famous theorem of Mordell asserting that the group of rational points of an elliptic curve over Q is finitely generated. This result was later generalized by Weil to encompass all abelian varieties over number fields. If E is an elliptic curve over a number field K, it is therefore known that

E(K) ∼ = Z r ⊕ T
as abstract groups, where T = E(K) tors is the finite torsion subgroup of E(K).

The integer r, called the rank, is a subtle invariant about which little is known and which can be rather hard to compute given E and K. The torsion subgroup, in contrast, is readily computed in specific instances, and this makes it realistic to ask more ambitious questions about the variation of E(K) tors with E and K.

A fundamental result in this direction is the theorem of Mazur presented in Chapter 3 of Darmon's lecture in this volume, which gives a uniform bound on E(Q) tors as E varies over all elliptic curves over Q. Kamienny [Kam92] was able to extend Mazur's result to quadratic fields, obtaining a bound on E(K) tors for K quadratic that was even independent of K itself. This led him to formulate the Strong Uniform Boundedness Conjecture, asserting that the cardinality of E(K) tors can be bounded above by a constant which depends only on the degree1 of K/Q. Actually, Demjanenko claimed that this conjecture would have been set out already in the 70's by Shafarevich and proved in 1972 a result in this direction (see [START_REF] Dem Janenko | The boundedness of the torsion of elliptic curves[END_REF] and the mathematical review MR0302654). The Strong Uniform Boundedness Conjecture was proved in 1994 by Merel, building on the methods developped by Mazur and Kamienny.

Theorem 1 (Merel -1994). For all d ∈ Z, d ≥ 1 there exists a constant B(d) ≥ 0 such that for all elliptic curves E over a number field K with [K :

Q] = d then | E(K) tors |≤ B(d).
Merel actually proves the following bound on the prime numbers dividing E(K) tors :

Theorem 2 (Merel -1994). Let E be an elliptic curve over a number field K such that

[K : Q] = d > 1. Let p be a prime number. If E(K) has a p-torsion point then p < d 3d 2 .
It is then sufficient to conclude for the case d > 1: Mazur and Kamienny [START_REF] Kamienny | Rational torsion of prime order in elliptic curves over number fields[END_REF] have indeed showed that, by Faltings and Frey works, Theorem 2 implies Theorem 1. The case d = 1 of Theorem 1 has been proved by Mazur [START_REF] Mazur | Modular curves and the Eisenstein ideal[END_REF][START_REF] Mazur | Rational isogenies of prime degree (with an appendix by D. Goldfeld)[END_REF] We will now give the sketch of the proof of Theorem 2. From now on, we will denote by d ≥ 1 an integer, by p a prime number and by Z = Z[1/p]. Following the traditional approach, Mazur and Kamienny translated the assertion of the theorem into an assertion about rational points of some modular curves.

Mazur's method

To a problem on modular curves

We briefly recall that there exist smooth schemes X 0 (p) and X 1 (p) over Z which classify respectively coarsely and finely the generalized elliptic curves endowed with a subgroup, respectively a point, of order p. We refer for instance to Chapter 3 of [dar] for more details. We denote by Y 0 (p) and Y 1 (p) the respective affine parts of X 0 (p) and X 1 (p). We use the subscript Q for the algebraic curves over Q obtained by taking the generic fiber of X 0 (p) or X 1 (p). We will denote by J 0 (p) the Néron model over Z of the jacobian

J 0 (p) Q of X 0 (p) Q .
Suppose that E is an elliptic curve over a number field K of degree d ≥ 1 over Q, endowed with a K-rational p-torsion point P. Then (E, P ) defines a point x ∈ Y 1 (p)(K). We can map this point to a point x ∈ Y 0 (p)(K) through the usual covering X 1 (p) -→ X 0 (p).

If we denote by v 1 , . . . , v d the embeddings of K into C, we then obtain a point x = (v 1 (x), . . . , v d (x)) ∈ X 0 (p) (d) (Q). Here we denote by X 0 (p) (d) the d-th symmetric power of X 0 (p) that is to say the quotient scheme of X 0 (p) by the action of the permutation group Σ d . It is a smooth scheme over Z.

The Mazur and Kamienny strategy

The strategy is almost the same as in the case d = 1 explained in [dar] Ch.3. Let A Q denote an abelian variety quotient of J 0 (p) Q and A its Néron model over Z. Kamienny's idea is to approach the Uniform Boundedness Conjecture by studying the natural morphism

φ (d) A : X 0 (p) (d) φ (d)
--→ J 0 (p) -→ A defined as follows. Over Q, this morphism is defined as the composition of the Albanese morphism

(Q 1 , . . . , Q d ) → [(Q 1 ) + . . . (Q d ) -d(∞)]
with the surjection of J 0 (p) Q to A Q . It then extends to a morphism from the smooth Z-scheme X 0 (p) (d) to A. For any prime number l = p, we denote by φ

(d) A,F l : X 0 (p) (d)
F l -→ A F l the morphism obtained by taking the special fibers at l. Just as in the case d = 1, we have Theorem 3 (Mazur-Kamienny). Suppose that

1. A(Q) is finite; 2. there exists a prime number l > 2 such that p > (1 + l d/2 ) 2 and φ (d) A,F l is a formal immersion at ∞ (d) F l .
Then Y 1 (p)(K) is empty for all number fields K of degree d over Q, i.e. there does not exist any elliptic curve with a point of order p over any number field of degree d.

Proof. The proof of this theorem is analogous to the one in the case d = 1. The principal ingredients of the proof are explained in [dar] Ch. 3. For a complete proof, the reader can see [START_REF] Mazur | Rational isogenies of prime degree (with an appendix by D. Goldfeld)[END_REF], [START_REF] Kamienny | Torsion points on elliptic curves over fields of higher degree[END_REF] or, for a summary, [START_REF] Edixhoven | Rational torsion points on elliptic curves over number fields (after Kamienny and Mazur)[END_REF]. The idea is the following: suppose that there exists a number field K of degree d and a point of Y 1 (p)(K) and consider the point x ∈ X 0 (p) (d) (Q) obtained as explained in Section 2.1. The condition p > (1+l d/2 ) 2 of Theorem 3 implies that the section s of X 0 (p) (d) corresponding to x crosses ∞ (d) in the fiber at l. Since s = ∞ (d) , the fact that φ

(d) A,F l is a formal immersion at ∞ (d)
F l and Condition 1 will then give a contradiction.

We now need an abelian variety A Q quotient of J 0 (p) Q of rank 0 (see section 3.1) and a formal immersion criterion (see below).

Criterion of formal immersion

Recall first that a morphism φ : X -→ Y of noetherian schemes is a formal immersion at a point x ∈ X which maps to y ∈ Y if the induced morphism on the formal completed local rings φ : O Y,y -→ O X,x is surjective. Equivalently, it follows from Nakayama's lemma that φ is a formal immersion at x if the two following conditions hold :

1. the morphism induced on the residue fields k(y) -→ k(x) is an isomorphism;

2. the morphism

φ * : Cot y (Y ) -→ Cot x (X) is surjective.
The first condition is verified in our situation, so we are now looking for a criterion to have

φ (d) A,F l * : Cot(A F l ) -→ Cot ∞ (d) F l (X 0 (p) (d) F l )
surjective. For this, we will look in more details at φ

(d) A * .
Let R be a Z-algebra. As in [dar], denote by S 2 (Γ 0 (p), R) the regular differentials on X 0 (p) R = X 0 (p) × Z R. For R = C, we obtain the vector space of classical modular forms S 2 (Γ 0 (p), C). The q-expansion principle gives an injective morphism of R-modules

S 2 (Γ 0 (p), R) → R[[q]].
Furthermore, we have an isomorphism between Cot(J 0 (p)(C)) and S 2 (Γ 0 (p), C) coming from the composition of 1. the isomorphism H 0 (J 0 (p)(C), Ω 1 ) -→ Cot(J 0 (p)(C)) which maps a differential form to its evaluation at 0 ;

2. the isomorphism H 0 (J 0 (p)(C), Ω 1 ) φ * -→ H 0 (X 0 (p)(C), Ω 1 ) = S 2 (Γ 0 (p), C) given by Serre duality.
It is a non trivial fact that this isomorphism Cot(J 0 (p)(C)) ∼ = S 2 (Γ 0 (p), C) extends to an isomorphism over Z (and actually even over Z). Indeed, Grothendieck duality can be applied in this setting instead of Serre duality and we then obtain an isomorphism: Cot(J 0 (p)) ∼ = S 2 (Γ 0 (p), Z) (see [START_REF] Mazur | Rational isogenies of prime degree (with an appendix by D. Goldfeld)[END_REF] 2 e)).

On the other hand, we have to understand the cotangent bundle Cot ∞

(d) (X 0 (p) (d) ). Recall that q is a formal local parameter of X 0 (p) at ∞ i.e. O X0(p),∞ ∼ = Z[[q]]. We then have O X 0 (p) (d) ,(∞) (d) ∼ = Z[[q 1 , . . . , q d ]] Σ d = Z[[σ 1 , . . . , σ d ]]
where for i = 1, . . . , d, q i is a local parameter at ∞ on the ith factor of X 0 (p) d and σ

1 = q 1 +• • •+q d , . . . , σ d = q 1 . . . q d are the symmetric functions in q 1 , . . . , q d . Consequently, Cot ∞ (d) (X 0 (p) (d)
) is a free Z-module of rank d with a basis given by the differential forms (dσ 1 , . . . , dσ d ).

We obtain the following diagram:

Cot(J 0 (p)) φ * ∼ / / φ (d) * S 2 (Γ 0 (p), Z) q-exp / / Z[[q]] Cot(X 0 (p) (d) )
Lemma 1. Let ω ∈ Cot(J 0 (p)) be such that φ * (ω) has a q-expansion equal to m≥1 a m q m dq q . Then we have

φ (d) * (ω) = a 1 dσ 1 -a 2 dσ 2 + • • • + (-1) d-1 a d dσ d .
Proof. Denote by π : X 0 (p) d -→ X 0 (p) (d) the canonical map. We have

π * φ (d) * (ω) = d i=1 m≥1 a m q m i dq i q i = m≥1 a m m -1 ds m where s m = d i=1 q m i . Then Newton's formula s m -σ 1 s m-1 + • • • + (-1) m mσ m = 0 gives m -1 ds m = (-1) m dσ m for m ∈ {1, . . . , d}.
We suppose in the sequel that A Q is the quotient of J 0 (p) Q by an ideal I of the Hecke algebra T ⊂ End(J 0 (p) Q ), so that there is an induced action of T on A. The exact sequence

0 → IJ 0 (p) Q → J 0 (p) Q → A Q → 0
induces a reverse exact sequence for the cotangent bundles after scalar extension by

Z[1/2] 0 → Cot(A Z[1/2] ) → Cot(J 0 (p) Z[1/2] ) → Cot(J 0 (p) Z[1/2] )[I] → 0
where we denote by Cot(J 0 (p) Z[1/2] )[I] the differential forms annihilated by I. This is due to a specialization lemma of Raynaud (see [START_REF] Mazur | Rational isogenies of prime degree (with an appendix by D. Goldfeld)[END_REF] Proposition 1.1 and Corollary 1.1).

Let l = 2, p be a prime number. We finally have the following diagram in characteristic l:

Cot(A F l ) / / φ (d) A,F l * ' ' O O O O O O O O O O O Cot(J 0 (p) F l ) φ * F l ∼ / / φ (d) F l * S 2 (Γ 0 (p), F l ) q-exp / / F l [[q]] Cot ∞ (d) F l (X 0 (p) (d) F l )
This diagram and Lemma 1 give a criterion for φ (d)

A,F l to be a formal immersion at ∞ (d) F l (see Theorem 5 below). Historically, Mazur first showed the following result which completes the proof of Mazur's theorem sketched in Section 4 of [dar] using for A Q the Eisenstein quotient.

Theorem 4. Let A Q be the quotient of J 0 (p) Q by an ideal I of T. The morphism φ A,F l is a formal immersion at ∞ F l for all prime numbers l = 2, p.

Proof. There is a non zero element ω ∈ Cot(A F l ) such that φ * F l (ω) ∈ S 2 (Γ 0 (p), F l ) is an eigenform (under the action of the Hecke algebra T). Then by the qexpansion principle and the injectivities in the above diagram, its q-expansion is not identically zero (because if it was, φ * F l (ω) itself would be zero). We deduce that a 1 (ω) = 0: indeed, if it was, since ω is an eigenform, we should have

a m (ω) = a 1 (T m ω) = λ m (ω)a 1 (ω) = 0 for all m ≥ 1, so ω = 0, which is impos- sible. It follows that a 1 (ω) spans Cot ∞ F l (X 0 (p) F l ) ∼ = F l and, by Lemma 1, that φ A,F l is a formal immersion at ∞ F l .
Theorem 5 (Kamienny). The following assertions are equivalent: Proof. The equivalence of 1. and 2. follows directly from Lemma 1 since Cot(A) maps to the forms annihilated by I via the isomorphism φ * . Condition 3. is dual to Condition 2. Indeed, the multiplicity one theorem implies that the pairing

1. φ (d) A,F l is a formal immersion at ∞ (d)
<, > : S 2 (Γ 0 (p), Z) × T -→ Z (f, t) -→ a 1 (tf )
is perfect and then induces an isomorphism of T-modules between S 2 (Γ 0 (p), Z) and the Z-dual of T. For a more detailed proof of this theorem, see [START_REF] Kamienny | Torsion points on elliptic curves over fields of higher degree[END_REF] or [Oes] Sections 3, 4 and 6.

3 Merel's proof

The Winding Quotient

Denote by J e,Q the winding quotient (see [dar] Ch. 3) and J e its Néron model over Z. We just recall that J e,Q is the abelian variety quotient of J 0 (p) Q by the winding ideal I e of T.

Considering Theorem 3, we are now looking for a quotient A Q of J 0 (p) Q by an ideal I ⊂ T such that A(Q) is finite. Mazur and Kamienny have used the Eisenstein quotient which has this property (see [START_REF] Mazur | Modular curves and the Eisenstein ideal[END_REF][START_REF] Kamienny | Torsion points on elliptic curves over fields of higher degree[END_REF]). Merel's fundamental innovation has been to use the winding quotient: this quotient is greater and easier to exploit than the Eisenstein quotient. This has been made possible after the works of Kolyvagin on the Birch and Swinnerton-Dyer conjecture: indeed, it then turned out that J e (Q) is finite by construction (see [START_REF] Merel | Bornes pour la torsion des courbes elliptiques sur les de nombres[END_REF] or [dar] for a summary). Actually, the Birch and Swinnerton-Dyer conjecture predicts that the winding quotient is the greatest quotient of J 0 (p) Q of rank zero.

Finally, to prove Theorem 2, thanks to Theorems 3 and 5, it suffices to determine for which prime numbers p the following is true for a prime number l = 2 such that p > (1 + l d/2 ) 2 : ( l ) the images of T 1 , . . . , T d in T/(lT + I e ) are F l -linearly independent.

Merel's strategy

Suppose now that d ≥ 3. Recall that the Hecke algebra T ⊂ End(J 0 (p)) also acts on the first group of absolute singular homology H 1 (X; Z) of the compact Riemann surface X = X 0 (p)(C) and that I e is the annihilator of the winding element e ∈ H 1 (X; Q) (see the article of Darmon in this volume). Then T.e is a free T/I e -module of rank 1. It follows that ( l ) is equivalent to ( l ) the images of T 1 e, . . . , T d e in Te/lTe are F l -linearly independent.

As before, the characteristic zero analogous condition ( ) T 1 e, . . . , T d e are Z-linearly independent in T.e. is equivalent for φ Q . If ( l ) is true for a prime number l then ( ) is true, while the condition ( ) implies ( l ) for almost all prime number l. Kamienny showed that if ( ) is true then there exists a prime number l < 2(d!) 5/2 (depending on p) such that ( l ) is true (see [START_REF] Kamienny | Torsion points on elliptic curves over fields of higher degree[END_REF] Corollary 3.4 and [START_REF] Edixhoven | Rational torsion points on elliptic curves over number fields (after Kamienny and Mazur)[END_REF] 4.3 for the precise bound). The heart of the Merel's proof for the boundedness of the torsion of elliptic curve is then to prove ( ) for

p > d 3d 2 > 2 d+1 (d!) 5d/2 ≥ (1 + (2(d!) 5/2 ) d/2 ) 2 .
We will now explain the key steps of this proof omitting the details of the calculus. For a completed proof, we will refer to [START_REF] Merel | Bornes pour la torsion des courbes elliptiques sur les de nombres[END_REF].

Consider a fixed prime number p > d 3d 2 for d ≥ 3 an integer. To prove that e, T 2 e, . . . , T d e are linearly independent, it suffices to prove that so do e, t 2 e, . . . , t d e where t r = T r -σ (r) with σ (r) the sum of divisors of r coprime to p. Those slight different Hecke operators t r are more pleasant to work with because they annihilate the "Eisenstein part" of e and we can then work as if e was equal to the modular symbol {0, ∞} (see the section 3.3 for a definition) 2 .

The idea of the proof is to use the intersection product

• : H 1 (X; Z) × H 1 (X; Z) -→ Z.
Suppose indeed that λ 1 e + λ 1 t 2 e + • • • + λ c t c e = 0 for 1 ≤ c ≤ d and some λ 1 , . . . , λ c in Z with λ c = 0. The strategy is then to find

x c ∈ H 1 (X; Z) such that i) t c e • x c = 0 and ii) t r e • x c = 0 (1 ≤ r ≤ c -1).
This will give a contradiction. 3Two key facts make it possible to follow this strategy: first, there is a presentation of H 1 (X; Z) by generators and relations due to Manin [START_REF] Ju | Parabolic points and zeta functions of modular curves[END_REF] (see the section 3.3); secondly, some lemma called lemme des cordes by Merel (Proposition 1 below) enables us to compute the intersection product of two such generators. It suffices then to express t r e in terms of the Manin's generators (see 3.4).

Manin's symbols

Denote by H the Poincaré upper half plane. For α, β ∈ P 1 (Q), consider the image in Γ 0 (p)\H of the geodesic path from α to β in H. Denote by {α, β} its homology class in the homology group H 1 (X, cusps; Z) relative to the set cusps of the cusps of X.

Exercise 1.

1. Show that {α, β} is the sum of classes of type {b/d, a/c} with a, b, c, d ∈ Z such that ad -bc = 1 (hint: use continued fractions).

2. Show that {b/d, a/c} depends only on the coset Γ 0 (p) a b c d . For a solution of this exercise, see [START_REF] Ju | Parabolic points and zeta functions of modular curves[END_REF] for instance.

The preceding results imply that there is a surjective map

ξ : Z[Γ 0 (p)\SL 2 (Z)] -→ H 1 (X, cusps; Z) Γ 0 (p).g -→ {g.0, g.∞} = { b d , a c } g = a b c d ∈ SL 2 (Z).
Since there is moreover an isomorphism

Γ 0 (p)\SL 2 (Z) -→ P 1 (F p ) Γ 0 (p). a b c d → [c : d],
we will simply write ξ(c/d) := ξ( a b c d ). For k ∈ F × p we obtain ξ(k) = {0, 1/k} which is an element of H 1 (X; Z) (seen as a submodule of H 1 (X, cusps; Z)) because 0 and 1/k are conjugated modulo Γ 0 (p). Those elements are generators of H 1 (X; Z). The other generators of H 1 (X, cusps; Z) are ξ(0) and ξ(∞) and they verify ξ(0) = -ξ(∞) = {0, ∞}.

The following proposition, called lemme des cordes by Merel, gives a method to compute the intersection product of two Manin symbols in the absolute homology group. For k ∈ {1, . . . , p -1}, denote by k * the element of {1, . . . , p -1} such that kk * ≡ -1 (mod p).

Proposition 1 (Merel). Let k, k ∈ {1, . . . , p -1}. Denote by C k the corde of the unit circle from e 2iπk * /p to e 2iπk/p and similarly for k . Then

ξ(k) • ξ(k ) = C k ∧ C k where C k ∧ C k is the number of intersection of C k by C k (equal to 1, 0 or -1 according to the trigonometric orientation of the unit circle). k * -1 k k * k Proof. See [Mer96] Lemma 4.

Two useful formulas

Because of their technical aspect, we will not reproduce the proofs of the following formulas which appear in Lemmas 2 and 3 of [START_REF] Merel | Bornes pour la torsion des courbes elliptiques sur les de nombres[END_REF].

We have first a formula for t r e (r > 1) in terms of the Manin symbols ξ(k):

Proposition 2 (Merel). Let r < p be a positive integer. Then

t r e = - ( u v w t )∈Xr ξ(w/t)
where X r is the set of matrices ( u v w t ) of determinant r such that 0 < w < t and u > v ≥ 0.

For r = 1, we can compute directly the intersection of e with a Manin's generator:

Proposition 3 (Merel). For any k ∈ {1, . . . , p -1} we have

(p -1)e • ξ(k) = k * -k p (p -1) -12S(k, p),
where S(k, p) = p-1 h=0 B1 ( h p ) B1 ( kh p ) is the Dedekind sum, B1 the first Bernoulli polynomial made 1-periodic.

Remark 3. Note that in Proposition 2 the ξ(0) and ξ(∞) terms vanish. This is not surprising since t r e lies in the absolute homology group.

Conclusion of the proof

We will now explain substantially how Merel put all the previous ingredients together to obtain the proof of ( ) for p great enough.

Suppose that there are integers λ 1 , . . . , λ d such that

λ 1 e + λ 2 t 2 e + • • • + λ d t d e = 0.
We will show successively that λ i = 0 for all i ∈ {1, . . . , d}, treating the case of λ 1 independently.

Case i = 1. We look for x 1 of the form

x 1 = ξ(k) for some k such that i) e • ξ(k) = 0 and ii) t r e • ξ(k) = 0 (1 < r ≤ d)
Suppose that p > d. By Proposition 2, the condition ii) is equivalent to

( u v w t )∈Xr ξ(w/t) • ξ(k) = 0 (1 ≤ r ≤ d).
It suffices to find k such that ξ(w/t) • ξ(k) = 0 for all ( u v w t ) ∈ X r . That is what Merel does. Let l ∈ {1, . . . , p -1} such that l ≡ wt -1 (mod p) for some ( u v w t ) ∈ X r . Then l * ≡ -tw -1 (mod p). By the Remark 3, we can suppose that neither t nor w are divisible by p.

Exercise 2. Show that l and l * are larger than p-1 d .

(p -1)/d

l * k l 0 k *
Applying the lemme des cordes it suffices to find k such that both the complex numbers e 2iπk/p and e 2iπk * /p are in a portion of the circle where e 2iπl/p can not be, so for instance, by the exercise, such that both k and k * lie in [0, for all k as before.

Case i > 1. Suppose now that

λ 2 t 2 e + • • • + λ c t c e = 0
for some c ≤ d. The method is almost the same as before: we look for

x c = ξ(k) such that i) t c e • ξ(k) = 0 and ii) t r e • ξ(k) = 0 (2 ≤ r < c).
We remark that in the formulas for t r e, r = 2 . . . c of Proposition 2, the Manin symbol ξ(1/c) occurs only in t c e and not in t r e for r < c. So we will look for k

such that ξ(1/c) • ξ(k) = ±1 and ξ(w/t) • ξ(k) = 0 for all ( u v w t ) ∈ X r (r ≤ c) such that w/t = 1/c. q -δ k * q k q + δ q *
Let q and l in {1, . . . , p -1} such that q ≡ 1/c (mod p) and l ≡ w/t = 1/c (mod p) for some ( u v w t ) ∈ X r (r ≤ c).

Exercise 4. Show that |l -q| ≥ δ where δ = p-d 2 d(d-1) .

By the same analytic lemma as before, it is possible to find k ∈]q, q + δ] such that k * ∈ [q -δ, q[ and q * ∈ [q -δ, q + δ] when p is great enough, more precisely when p/ log 4 (p) > Sup(d 8 , 400d 4 ). By the lemme des cordes, it forces then λ c to be zero. This finishes the proof of Theorem 2.

Oesterlé's variant

As we said in Remark 1, Oesterlé improved the Merel's bound for the torsion of elliptic curves. For this, Oesterlé proved directly the formal immersion in positive characteristic: Proposition 4. Suppose that p/ log 4 p ≥ (2d) 6 . Then for all l ≥ 3, the condition ( l ) is true, that is to say φ

(d) A,F l is a formal immersion at ∞ (d) F l .
For d ≥ 33, Theorem 2 with the bound (3 d/2 + 1) 2 then follows directly from Theorem 4, since p > (3 d/2 + 1) 2 implies p/ log 4 p ≥ (2d) 6 in that case. Oesterlé studied the cases d < 37 by computations.

Let us give a sketch of proof of Proposition 4. Let T s defined by T r = s|r T s for all r ≥ 1 and, instead of t r = T r -σ (r), (r ≥ 1), consider the following generators of the Eisenstein ideal I :

I 1 = n p and I r = T r -r if p |r T r if p|r (r ≥ 2),
where we denote by n p the numerator of (p -1)/12. We have t r = s|r,s =1 I s for all r > 1.

Proposition 5. If the images of I 2 e, . . . , I 2d e in Ie/lIe are F l -linearly independent, then T 1 e, . . . , T d e are F l -linearly independent in Te/lTe that is to say ( l ) is true.

Proof. We have T 2 T r = I 2r -2I r if r is odd I 2r -3I r + 2I r/2 if r is even. So if I 2 .e, . . . , I 2r e are linearly independent in Ie/lIe, so do T 2 e, . . . , T 2 T 2r e and, since T 2 e = (T 2 -3)e ∈ Ie, we obtain that T 1 e, . . . , T d e are linearly independent in Te/lTe. But T r = T r + s|r,s<r T s so T 1 e, . . . , T d e are linearly independent in Te/lTe. Moreover, Oesterlé used Proposition 2 and the lemme des cordes to give an explicit formula for t r e • ξ(k) and then for I r e • ξ(k) (which is the unique "r-th term" of t r e • ξ(k)) : Remark 4. As Merel pointed out to me, the result of Proposition 5 is still true replacing I r by t r , (2 ≤ r ≤ 2d). Indeed, a calculus proves that t 2 T i ∈ t 2i + 1≤j≤i ZT j . Using the results of the section 3.5 case i > 1, it follows that when p/ log 4 (p) > Sup(d 8 , 400d 4 ), ( l ) is true for all l ≥ 3. Since p > (3 d/2 + 1) 2 implies p/ log 4 (p) > Sup(d 8 , 400d 4 ) provided that d ≥ 37, it gives the Oesterlé's bound in that case. The other cases have been studied by Oesterlé.

  F l ; 2. there exists d weight-two cusp forms f 1 , . . . , f d annihilated by I such that the vectors (a 1 (f i ), . . . , a d (f i )) i=1,...,d are linearly independent mod l;3. the images of T 1 , . . . , T d in T/(lT + I) are F l -linearly independent.

I

  e to be a formal immersion at ∞ (d)

I

  r e • ξ(k) = rk p -rk * p + v r (k) -v r (k * ) (r ≥ 2, k ∈ {1, . . . , p -1}), (1)where v r (k) = {(a, a , b, b ) ∈ Z, a, a , b, b ≥ 1, aa + bb = r, (a, b) = 1, bk ≡ a (mod p)}.The end of the proof is then mutatis mutandi the same as Merel's one: using Lemma 2, Oesterlé showed that, when p/log 4 (p) > d 6 , it is possible for each r ≥ 2 to find k such that I r e • ξ(k) = 1 and I s e • ξ(k) = 0 for s < r. He deduced that for p/log 4 (p) > d 6 I 2 e, . . . , I d e are linearly independent. Applying this for 2d instead of d and using Proposition 5 gives Proposition 4. This is how one can obtain Oesterlé's bound. As we said in Remark 2, the question of finding a bound growing polynomially in d remains open.

  in 1976 as explained Henri Darmon in his lecture. Mazur gives more precisely a list of all possibilities for the torsion group over Q. It was actually a conjecture of Levi formulated around 1908. We can mention also that the cases 2 ≤ d ≤ 8 and 9 ≤ d ≤ 14 have been treated respectively by Kamienny and Mazur (see[START_REF] Kamienny | Rational torsion of prime order in elliptic curves over number fields[END_REF]), and Abramovich[START_REF] Abramovich | Formal finiteness and the torsion conjecture on elliptic curves. A footnote to a paper[END_REF].The goal of this note is to give the key steps of the proof of Theorem 2.

Remark 1. Oesterlé

[Oes]

later improved the bound of Theorem 2 by (3 d/2 + 1) 2 but we will focus on Merel's original proof (see Section 3.6 concerning the Oesterlé's trick).

Remark 2. Unfortunately, the reduction of Theorem 1 to Theorem 2 is not effective; this explains why the global bound B(d) is not explicit. However, in 1999, Parent

[START_REF] Parent | Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres[END_REF]

gave a bound for the p r -torsion (r ≥ 1, p prime) and thus obtained a global effective bound for the torsion (later improved by Oesterlé). This bound is exponential in d. It is conjectured that B(d) can be made polynomial in d.

  Let p be a prime number and a, b ≥ 1 two real numbers. Let A, B ⊂ {1, . . . , p -1} be two intervals of cardinality p/a and p/b respectively. If p > a 2 b 2 log 4 (p) then there exists k ∈ A such that k * ∈ B.We deduce from the following exercise that the condition i) is also verified assuming that p > d 3d 2 .

	p-1 d [. Merel uses then the following analytic lemma ([Mer96] Lemma 5) to ensure that, provided that p > d 3d 2 , it suffices to choose some k ∈ Z∩] p 10d , p 5d + 1[ and then k * ∈ Z∩] p 2d -1-1 d , p-1 d [ (more precisely, it is already true when p/log 4 (p) > d 4 ). 12(S(k, p) + S(p, k)) = -3 + p k + k p + 1 pk and the inequality |12S(p, k)| ≤ k, show that Lemma 2. Exercise 3. Using the Dedekind's reciprocity formula e • ξ(k) ≥ p 10d -10d -2

The weaker conjecture asserting that the torsion can be bounded uniformly in the field K is presented as beeing " a part of the folklore" by Cassels[START_REF] Cassels | Diophantine equations with special reference to elliptic curves[END_REF] (p.

264).

In the relative homology group, the winding element e differs from {0, ∞} by an element which is an eigenvector for all Tn with system of eigenvalues {σ (n)} n≥1 (up to a constant): this is what I called the Eisenstein part.

Actually, for c = 1 the situation will be slightly different because of the Eisenstein part of e.


How is the LPFC activity related to the activity in MT+ in the task-irrelevant condition? The correlation coefficient between the task-irrelevance-related activity (28) in MT+ and the LPFC was -0.90. This is in accord with the view that when the LPFC is activated, it provides direct or indirect inhibitory control on the activity of MT+.

One might think that the low performance at the 5% coherence ratio was obtained because, despite the instructions to focus on RSVP task performance, the participants may have tried to find a coherent motion direction or to detect whether coherent motion was presented. If these motion tasks are difficult, they may leave fewer resources available for the RSVP task. However, this is not likely. If the participants engaged in the search for motion direction, this task should be hardest at 0% coherence and therefore, the lowest RSVP performance should have occurred at 0% and not at 5% coherence. Second, if the participants engaged in motion detection, this task should be hardest at 5% coherence because indecision may be greatest near the coherent motion threshold, and thus in accordance with the observed RSVP performance result. However, the lowest blood oxygen level-dependent (BOLD) activity was observed at 5% coherence ratio in the LPFC and cannot be directly explained by this possibility.

The results of the present study demonstrate two important points. First, a weak taskirrelevant stimulus feature that is below but near the perceptual threshold more strongly activates the visual area (MT+) that is highly related to the stimulus feature and more greatly disrupts task performance. There was a tendency for activity in the posterior occipitotemporal sulcus (pOTS) (29,30) and the left angular gyrus (31), which are sensitive to letters and words and may be related to the RSVP task, to be lower at the 5% coherence than at the other coherent motion ratios. This contradicts the general view that irrelevant signals that are stronger in stimulus properties have a greater influence on the brain and performance and that the influence of a subthreshold stimulus is smaller than that of a suprathreshold stimulus.

Second, the results may reveal important bidirectional interactions between a cognitive controlling system and the visual system. The LPFC, which has been suggested to provide inhibitory control on task-irrelevant signals (22)(23)(24)(25)(26), may have a higher detection threshold for incoming signals than the visual cortex. Task-irrelevant signals around the threshold level may be sufficiently strong to be processed in the visual system but not strong enough for the LPFC to notice and, therefore, to provide effective inhibitory control on the signals (Fig. 4A). In this case, such signals may remain uninhibited, take more resources for a taskirrelevant distractor, leave fewer resources for a given task (32,33), and disrupt task performance more than suprathreshold signals. On the other hand, suprathreshold coherent motion may be noticed, may be given successful inhibitory control by the LPFC, and may leave more resources for a task (Fig. 4B) (22)(23)(24)(25)(26). This mechanism may underlie the present paradoxical finding that subthreshold task-irrelevant stimuli activate the visual area strongly and disrupt task performance more than some suprathreshold stimuli. It could also be one of the reasons why subthreshold stimuli often lead to relatively robust effects (2,11,14). 1 Ilgam Khalilov, 1 Marat Minlebaev, 1 Christian A. Hübner, 2 Alfonso Represa, 1 Yehezkel Ben-Ari, 1 * Rustem Khazipov 1 We report a signaling mechanism in rats between mother and fetus aimed at preparing fetal neurons for delivery. In immature neurons, g-aminobutyric acid (GABA) is the primary excitatory neurotransmitter. We found that, shortly before delivery, there is a transient reduction in the intracellular chloride concentration and an excitatory-to-inhibitory switch of GABA actions. These events were triggered by oxytocin, an essential maternal hormone for labor. In vivo administration of an oxytocin receptor antagonist before delivery prevented the switch of GABA actions in fetal neurons and aggravated the severity of anoxic episodes. Thus, maternal oxytocin inhibits fetal neurons and increases their resistance to insults during delivery.

D

elivery is a stressful event associated with high risks to the fetal brain (1); however, whether the fetal brain prepares for delivery remains largely unknown. We addressed this issue by studying g-aminobutyric acid (GABA)-mediated (GABAergic) signaling in the rat hippocampus. GABA is the principal inhibitory neurotransmitter in the adult brain. However, during fetal and postnatal periods GABA has a depolarizing action (2-4) and provides the major, and often the only, excitatory synaptic input to immature neurons (5). Excitatory action of GABA is due to elevated intracellular chloride and depolarized value of the GABA type A (GABA A ) reversal potential (E GABA ) (6-10). Excitatory GABA is pivotal in control of neuronal firing, generation of the primitive patterns of activity, intracellular calcium signaling, and neuronal development (2,3). GABA signaling is therefore expected to be an essential element in physiological adaptation to stress during delivery.

To characterize the properties of GABA signaling during the perinatal period, we used cellattached recordings from fetal and neonatal rat hippocampal slices [from embryonic day (E) 18 to postnatal day (P) 5; term is E21] (11). During fetal (E18 and E19) and postnatal periods (P1 to P5), activation of GABA A receptors increased the firing of action potentials in the majority of CA3 pyramidal cells [Fig. 1A; also (6,7,12)]. However, during a brief period extending from E20 to day of birth (P0), the proportion of cells excited by GABA sharply decreased. The loss of the excitatory effect of GABA peaked at E21 [~1 to 2 hours before delivery (Fig. 1A)].

The action of GABA depends on the direction of the transmembrane current elicited by GABA, its driving force (DF GABA ) being the difference between E GABA and resting membrane potential (E m ). Positive values of DF GABA determine excitatory actions of GABA in immature neurons (2,3,6,7). To estimate DF GABA , we used cellattached recordings of single GABA A channels (10) (Fig. 1, B to D), which affect neither E GABA nor E m . With 1 mM GABA in the pipette solution, GABA A channels were observed in ~90% of patches (n = 553) but not in the presence of the GABA A receptor antagonist picrotoxin [100 mM; n = 25 (Fig. 1B)]. We found that DF GABA is strongly depolarizing in fetal and postnatal periods (Fig. 1D). However, DF GABA negatively shifted during a brief near-term period (from E20 to P0), switching to a hyperpolarizing value of -8.4 ± 2.5 mV at term [mean ± SEM, n = 34 (Fig. 1D)]. Thus, the disappearance of GABA-mediated excitation at term (Fig. 1A) coincides with a switch in polarity of GABA signals.

The perinatal changes in DF GABA could be due to a negative shift of E GABA or to a depolarizing shift of E m . By using cell-attached recordings of single N-methyl-D-aspartate (NMDA) channels as voltage sensors (12,13), we found that during early fetal and postnatal periods E m is -78.7 ± 1.9 mV (n = 42; pooled data for E18 and E19 and P1 to P5) [also (13,14)], but near term there is a small hyperpolarizing shift to -85.4 ± 1.5 mV [n = 14 at birth (Fig. 1E)]. Knowing DF GABA and E m , we calculated E GABA (E GABA = DF GABA + E m ) and found that E GABA switches from -40 mVat E18 to -92 mV at E21 and then returns to depolarizing values shortly after birth (Fig. 1F). This corresponds to a decrease in [Cl -] i (intracellular chloride concentration) from 18 mM to 4 mM (Fig. 1F).

Because GABA-induced depolarization raises [Ca 2+ ] i (12,(15)(16)(17)(18), we used a multibeam two-photon microscope to monitor [Ca 2+ ] i changes in hundreds of CA3 neurons in slices loaded with the calcium indicator fura 2-AM (Fig. 2). The GABA A receptor agonist isoguvacine (10 mM) elicited a robust increase of [Ca 2+ ] i in the majority of neurons at E18 and at P2 to P4 [average of 56 ± 3%, n values of 7 movies and 931 cells ( events in many neurons near term [9 ± 2% (Fig. 2)]. Thus, during delivery, the ability of GABA to increase [Ca 2+ ] i is significantly reduced.

What underlies the near-term switch in GABA actions? Because this phenomenon is not observed in fetal neurons grown in culture (8,9), we hypothesized that it is related to parturition and in particular to maternal hormones released during delivery. Parturition is initiated by a massive release of oxytocin (19). In addition to its pivotal role in parturition, there are also indications that oxytocin exerts multiple effects in the adult central nervous system (19)(20)(21)(22)(23)(24)(25). Maternal oxytocin easily crosses the placenta to reach the fetus (26), suggesting that oxytocin may be responsible for the hyperpolarizing switch in action of GABA in fetal neurons during delivery. With selective antibodies, we found a high density of oxytocin receptor immunoreactivity in the hippocampus and neocortex during the perinatal period (Fig. 3A and figs. S1 and S2). During cell-attached recordings from slices at E18 and P2, applications of oxytocin (1 mM) induced a negative shift in DF GABA (Fig. 3, C andD) and suppressed GABA-mediated excitation (Fig. 3B). The effects of oxytocin were completely prevented by bath application of the selective oxytocin receptor antagonist atosiban (AT, 1 to 5 mM) (Fig. 3D). At term, application of oxytocin did not cause a significant effect on the hyperpolarizing DF GABA (Fig. 3D), suggesting that the effects of exogenous oxytocin are occluded by the endogenous hormone. In keeping with this hypothesis, we found that atosiban switched DF GABA from hyperpolarizing to depolarizing at term but not in fetal and postnatal neurons (Fig. 3D). Moreover, intracardial perfusion of E21 fetuses with artificial cerebrospinal fluid (ACSF, to wash out endogenous hormone) produced a similar shift of DF GABA from hyperpolarizing to depolarizing (fig. S3). Addition of 1 mM oxytocin at the end of the perfusion restored hyperpolarizing values of DF GABA (fig. S3). Therefore, endogenous oxytocin is present in slices at term and provides tonic activation of oxytocin receptors, leading to the shift of DF GABA . We also found that the negative shift in DF GABA from 38.4 ± 6.2 mV (n = 8) to -7.2 ± 2.2 mV (n = 18) occurred in E18 fetuses obtained from mothers treated with oxytocin (50 mg/kg) during 2 hours before cesarean delivery.

To directly measure the effects of oxytocin on [Cl -] i , we used two-photon chloride imaging in slices loaded with a chloride-sensitive dye, MQAE (Fig. 3, E andF) (27). In E18 and P4 slices, oxytocin (1 mM) produced a significant increase in the baseline fluorescence signal in half of the imaged cells, indicating a strong reduction of [Cl -] i [average of 43 ± 4%, n values of 10 movies and 1792 cells (Fig. 3F)], and the effect was prevented by atosiban (5 mM; reduction of [Cl -] i in 5 ± 3% cells, n values of 2 movies and 435 cells). Furthermore, incubation of P4 slices with oxytocin reversed the effect of isoguvacine on spontaneous calcium events [59 ± 17% imaged cells not activated by isoguvacine, n values of 3 movies and 228 cells (fig. S4)]. Thus, the effects of oxytocin fully matched the changes in GABA signaling occurring at term. This result indicates that oxytocin is sufficient to trigger changes of [Cl -] i .

To determine whether endogenous oxytocin is necessary for the near-term switch in the action of GABA, we treated pregnant rats orally with the selective oxytocin receptor antagonist SSR126768A (28) (1 mg/kg, starting from E20) and measured the consequences at term. Cell-attached recordings revealed major differences between treated and age-matched control pups, including strongly depolarizing values of DF GABA (Fig. 1D, red circle) and a high proportion of cells excited by GABA (Fig. 1A, red circle). Similar effects were also observed in neocortical neurons, in which DF GABA was -9.4 ± 3.4 mV in control rat pups (n = 8) and 28.3 ± 3.7 mV (n = 9) in pups of mothers pretreated with SSR126768A (Fig. 1D). Thus, maternal oxytocin is necessary and sufficient to trigger the near-term switch in GABA action in the hippocampus and neocortex.

What are the mechanisms underlying the oxytocin-mediated reduction of [Cl -] i at term? Because the NKCC1 chloride inward cotransporter is responsible for elevated [Cl -] i in immature neurons (4,7,29,30), we tested the effects of the selective NKCC1 antagonist bumetanide. We found that bumetanide (10 mM) produces a robust negative shift of DF GABA and occludes the effects of oxytocin (fig. S5). The short latency of oxytocin actions (Fig. 3F) and the observation that blockade of oxytocin receptors does not alter chloride transporters gene expression (fig. S6) suggest that the hormone down-regulates NKCC1 activity.

Hypoxic-ischemic brain damage is a principal cause of newborn death and neurological impairment (1). We hypothesized that the excitatory-to- inhibitory switch in GABA signaling reduces neuronal activity and metabolic demand, thus helping to protect fetal neurons from hypoxic insults (31). Episodes of anoxia and aglycemia were induced by superfusion with a solution in which oxygen was substituted for nitrogen and glucose was substituted for sucrose. The onset of anoxic depolarization (AD) that is an electrophysiological marker of neuronal death (31,32) was measured by using extracellular field potential recordings in the intact hippocampi of E21 rats in vitro after various treatments in vivo to the mother and/or to the fetuses intracardially before the in vitro experiment (Fig 4). In control fetuses, AD occurred after 55.6 ± 1.4 min (n = 30) of perfusion with the anoxicaglycemic solution (Fig. 4). In the hippocampi prepared from the fetuses intracardially perfused with atosiban (5 mM; n = 15) or from the fetuses whose mothers received SSR126768A (1 mg/kg; n = 19), AD onset was significantly accelerated to 44.1 ± 1.1 min (n = 34; P < 0.01). Bumetanide (10 mM) applied in the presence of the oxytocin receptor antagonists delayed AD to the control values (53.4 ± 1.7 min; n = 18; P < 0.01). Bumetanide (10 mM) occluded the effects of the oxytocin receptor antagonists and restored the control delays (53.4 ± 1.7 min, n = 18; P < 0.01). These data indicate that maternal oxytocin exerts a neuroprotective action on fetal neurons during parturition and that this action is likely due to a reduction of [Cl -] i .

Our results suggest that oxytocin, in addition to its well-established role in labor and lactation and its multiple effects in the adult central nervous system (19)(20)(21)(22)(23)(24)(25), also exerts a powerful action on fetal neurons. This mechanism adds a previously unknown facet to the plasticity of GABA signaling via modulation of [Cl -] i (4,33). The dual action produced by a single messenger in the mother and fetus enables a perfect timing for adaptation of fetal neurons to delivery.

ATP Release Guides Neutrophil Chemotaxis via P2Y2 and A3 Receptors

Yu Chen,1 * Ross Corriden, 1,2 * Yoshiaki Inoue, 1 Linda Yip, 1 Naoyuki Hashiguchi, 1 Annelies Zinkernagel, 4 Victor Nizet, 4 Paul A. Insel, 2,3 Wolfgang G. Junger 1 † Cells must amplify external signals to orient and migrate in chemotactic gradient fields. We find that human neutrophils release adenosine triphosphate (ATP) from the leading edge of the cell surface to amplify chemotactic signals and direct cell orientation by feedback through P2Y2 nucleotide receptors. Neutrophils rapidly hydrolyze released ATP to adenosine that then acts via A3-type adenosine receptors, which are recruited to the leading edge, to promote cell migration. Thus, ATP release and autocrine feedback through P2Y2 and A3 receptors provide signal amplification, controlling gradient sensing and migration of neutrophils. N eutrophils are primary phagocytic cells with important roles in host defense and tissue repair. However, activated neutrophils damage host tissues and contribute to chronic inflammatory diseases, including rheumatoid arthritis, inflammatory bowel disease, and asthma (1). A key feature of neutrophils is their ability to detect and migrate to compromised tissues by following a concentration gradient of chemotactic substances released from microbial pathogens or injured cells.

Neutrophils can respond to chemoattractant gradients that differ in concentration by as little as 1% across the length of the cell body (2). Chemotaxis must involve signal amplification because a strongly polarized distribution of intracellular signaltransduction components is observed even in shallow gradients. The mechanisms of signal amplification are unclear (3,4). We identified the polarized release of adenosine triphosphate (ATP), the activation of P2Y2 receptors, and the translocation and activation of A3 adenosine receptors as key mechanisms of signal amplification that control cell orientation and direct the migration of neutrophils.

Membrane deformation caused by mechanical or osmotic stress induces the release of cellular ATP from mammalian cells; however, detailed information on the underlying mechanisms is lacking (5,6). Because cell migration also involves membrane deformation, we tested whether the stimulation of human neutrophils with the chemoattractant N-formyl-Met-Leu-Phe (FMLP) causes ATP release. Treatment of cells (10 7 cells in 250 ml of solution) with 100 nM FMLP rapidly tripled extracellular ATP concentrations in bulk media (Fig. 1, A andB) by inducing the release of ~0.5% of their ATP pool. Concentrations of extracellular ATP and its hydrolytic products adenosine monophosphate (AMP) and adenosine peaked 5 min after FMLP stimulation; but while the concentration of ATP returned to basal levels after 15 min, AMP and adenosine concentrations remained >5-fold above baseline (Fig. 1B), which is consistent with the presence of ecto-adenosine triphosphatases (ecto-ATPases) (7,8). Neutrophils completely hydrolyzed exogenous ATP (5 mM) within 2 min after ATP addition, which suggests that they have potent ecto-ATPase activity (Fig. 1C).

Because membrane deformation occurs predominantly at the leading edge closest to the chemoattractant source, we hypothesized this region to be the principal site of ATP release. Fluorescent microscopy that was used to visualize ATP release [based on conversion of nicotinamide adenine dinucleotide phosphate (NADP + ) to its reduced form NADPH] revealed that neutrophils discharge ATP within seconds after FMLP stimulation (Fig. 1D and movies S1 to S3), with ATP release highest near the cell membrane with the greatest degree of protrusion (Fig. 1D, inset).

Extracellular ATP and adenosine modulate neutrophil functions, including chemotaxis (8,9). We tested the effect of apyrase, which hydrolyzes ATP, on chemotaxis in a trans-well system composed of upper wells with neutrophils and lower wells with 1 nM FMLP separated by a filter with 3-mm pores. Addition of apyrase to the upper wells reduced chemotaxis by nearly 100% (Fig. 2A).

Fig. 1 .

 1 Fig. 1. Transient perinatal loss of the GABA Amediated excitation. (A) Responses of CA3 pyramidal cells recorded in cell-attached mode to the GABA A agonist isoguvacine. Below, summary plot of the proportion of cells excited by isoguvacine during the perinatal period. There is a transient loss of the excitatory effect of isoguvacine near term. Red corresponds to the fetuses whose mothers received SSR126768A. [E21 corresponds to the early phase of delivery (1 to 2 hours before birth); P0 is the day of birth; pooled data from 146 neurons.] (B) Cellattached recordings of single GABA A channels with 1 mM of GABA in patch pipette (top trace); the channels were not observed in the presence of the GABA A antagonist picrotoxin (100 mM; bottom trace). (C) I-V relationships of the currents through GABA A channels in two cells at E21 and E18; their reversal potential corresponds to DF GABA . (D) Summary plot of the age dependence of DF GABA inferred from single GABA A channels recordings [mean ± SEM; 209 CA3 pyramidal cells ( ○ ) and 17 neocortical pyramidal cells ( □ ); 6 to 24 patches for each point]. Red indicates pretreatment with SSR126768A (n values are 25 hippocampal and 9 neocortical patches). (E) Age dependence of the resting membrane potential (E m ) of CA3 pyramidal cells inferred from the reversal of single NMDA channels recorded in cell-attached mode (n = 84 cells; 4 to 12 patches for each point). (F) Age dependence of the GABA A reversal potential (E GABA = E m + DF GABA ). There is a transient hyperpolarizing shift of E GABA near birth.

Fig. 2 .

 2 Fig. 2. Perinatal effects of GABA A receptors activation on [Ca 2+ ] i in CA3 neurons. (A) Twophoton [Ca 2+ ] i fluorescence image of the CA3 region from a P0 hippocampal slice loaded with fura 2-AM; scale bar indicates 100 mm. (Right) Automatically detected contours of the cells. (B) Three types of [Ca 2+ ] i responses (DF/F) to bath application of the GABA A agonist isoguvacine (10 mM, 1 min): excited (red), inhibited (blue), or not affected (black) by isoguvacine (Isog). (C) Contour maps representing the distribution of different cellular responses to isoguvacine at E18, P0, and P5: excited (red), inhibited (blue), and not affected (open contours). Bottom histograms show the percentage of cells that are detected as being active on each movie frame corresponding to the above contour plots (140 ms per frame). Isoguvacine application (10 mM) produces a significant increase in the fraction of active neurons at E18 and P5, whereas it slightly decreased the activity at birth. (D) Histograms of the averaged fraction of imaged neurons excited (top) or inhibited or nonaffected (bottom) by isoguvacine at different ages (for P0, n values of 10 movies and 1546 cells; for E18, n values of 2 movies and 375 cells; and for P5, n values of 3 movies and 328 cells; *P < 0.01). Error bars indicate SEM.

Fig. 3 .

 3 Fig. 3. Oxytocin causes a switch in GABA A signaling from depolarizing to hyperpolarizing. (A) Oxytocin-receptor immunostaining of a P0 rat hippocampus; (inset) CA3 pyramidal cells layer. Bars, 200 mm and (inset) 20 mm. (B) Histograms of the proportion of cells excited by brief application of isoguvacine (Isog) in control and in the presence of oxytocin (1 mM) at E18 and P2 (n = 57 cells). (C) I-V relationships of the currents through single GABA A channels recorded from two CA3 pyramidal cells at E18 in control and after addition of oxytocin (1 mM). (D) Histograms of DF GABA measured at E18, E21, and P2 in control conditions and in the presence of oxytocin (1 mM), antagonist of oxytocin receptors atosiban (1 to 5 mM), and oxytocin plus atosiban (n = 187 cells). (E) Twophoton imaging of [Cl -] i in P4 hippocampal slice loaded with Cl -indicator MQAE. Scale bar, 100 mm. (Right) [Cl -] i fluorescence change produced by oxytocin application. The intensity of the fluorescence signal increases, and [Cl -] i decreases, in the illustrated region. (F) Automatically detected contours of the cells imaged in (E), indicating the distribution of cells in which oxytocin application produced a significant decrease in [Cl -] i (blue filled contours). On the right, chloride fluorescence changes (DF/F) in four representative neurons. Time resolution is 100 ms per frame.

Fig. 4 .

 4 Fig. 4. Blockade of oxytocin receptors decreases fetal brain resistance to anoxia-aglycemia at birth. (A) Representative traces to illustrate the effects of anoxic-aglycemic solution on extracellular field potential recordings from E21 intact hippocampi. Arrows indicate terminal AD that marks neuronal death. AD occurs earlier in the presence of SSR126768A (middle trace). Addition of bumetanide (10 mM) occludes the effects of the antagonist and restores the initial delay (bottom trace). (B) Summary plot of the onset of AD in control, in the presence of the oxytocin receptors antagonists atosiban (AT, 5 mM, fetal intracardial perfusion and SSR126768A (1 mg/kg to the mother), and after further addition of bumetanide (10 mM). Each circle corresponds to one hippocampus (n = 82 intact hippocampi at E21). Error bars indicate SEM.

DECEMBER 2006 VOL 314 SCIENCE www.sciencemag.org

on January 16, 2009 www.sciencemag.org Downloaded from

Institut de Neurobiologie de la Méditerranée, INSERM U29, Université de la Méditerranée, Campus Scientifique de Luminy, Boite Postale 13, 13273 Marseille Cedex 09, France.

Institut für Humangenetik, Universitätsklinikum Hamburg-Eppendorf, 20251 Hamburg, Germany.*To whom correspondence should be addressed. E-mail: ben-ari@inmed.univ-mrs.fr

Department of Surgery, University of California San Diego, San Diego, CA 92103, USA.

Department of Pharmacology, University of California San Diego, La Jolla, CA 92093, USA.

Department of Medicine, University of California San Diego, La Jolla, CA 92093, USA.

Department of Pediatrics, University of California San Diego, La Jolla, CA 92093, USA. *These authors contributed equally to this work. †To whom correspondence should be addressed. E-mail: wjunger@ucsd.edu 15 DECEMBER 2006 VOL 314 SCIENCE www.sciencemag.org