Thierry Hargé

Some remarks on the complex heat kernel on ν in the scalar potential case * †

Keywords: heat kernel, asymptotic expansion, Wigner-Kirkwood expansion, complex variables, Borel summation; A.M.S. subject classification: 30E15, 32W30, 35K05, 35K08, 35C20

In previous works, we used a so-called deformation formula in order to study, in particular, the Borel summability of the heat kernel of some operators. A goal of this paper is to collect miscellaneous remarks related to these works. Here the complex setting plays an important role. Moreover, the deformation formula provides a solution of the heat equation in "unusual" cases. We also give a uniqueness statement concerning these cases.

Introduction

In previous works [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF][START_REF] Hargé | Borel summation of the heat kernel with a vector potential case[END_REF], we used a so-called deformation formula in order to study the Borel summability of the heat kernel, p(t, x, y), associated to a ν-dimensional partial differential operator (ν ∈ AE * ). This formula is extended to the non-autonomous case [START_REF] Hargé | A deformation formula for the heat kernel[END_REF]. The natural setting for this formula is a complex one (t ∈ , Ret 0 and x, y ∈ ν ). For instance this formula is valid for operators as

P := ∂ 2 x + λx 2 + c(x) := ∂ 2 x1 + • • • + ∂ 2 xν + λ(x 2 1 + • • • + x 2 ν ) + c(x 1 , . . . , x ν ) (1.1)
where λ ∈ Ê and the function c is the Fourier transform of a suitable Borel measure. The aim of this paper is threefold:

1. Defining in a unique way the heat kernel associated to the operator P is a well known procedure if λ 0. If λ > 0, one can use the commutator theorem [Re-Si] for instance (see Remark 3.5). However, we look for a statement adapted to a full complex setting and covering the non-autonomous case: P = P 0 + c where P 0 is defined by (3.1). Here is the purpose of Proposition 3.2. The statement and the proof of this proposition are standard. We assume that the coefficients defining P 0 satisfy a reality-preserving property; see (3.2). This implies that the operator P 0 | iÊ is symmetric with respect to the L2 inner product. The unicity is therefore a consequence of the conservation of the L 2 -norm for the solutions of the time dependent Schrödinger equation associated to P 0 . Let us make the following remarks.

-In our setting, the Schrödinger kernel is viewed as the boundary value of the heat kernel for imaginary1 values of t.

-Our goal is not to solve the heat equation but to study the heat kernel as a function defined on some subset of √ • × 2ν ( √ • denotes the Riemann 2 surface of the square root function).

About the existence problem, we use [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF] and [START_REF] Hargé | A deformation formula for the heat kernel[END_REF].

2. Considering a complex setting allows one to make some remarks.

We reformulate Proposition 3.2 by using the analytic dilation given by (t, x, y) → (e iǫ t, e iǫ/2 x, e iǫ/2 y) , ǫ ∈ Ê/4π

(1.2) (see Corollary 4.1). As a consequence, we see that the deformation formula provides a solution for the Schrödinger equation

i -1 ∂ t p = ∂ 2 x + λx 2 + c(x) p , p| t=0 = δ x=y , (t ∈ Ê, |t| ≪ 1, x, y ∈ Ê ν )
with fast growing potentials such as, for instance, c(x) = e x 2 (see Corollary 4.3 and Remark 4.4). We do not state uniqueness in this case (this can be done by keeping the complex point of view for the space variables). This equation, in the free case (λ = 0), was considered by Kuna, Streit and Westerkamp [K-S-W]. In this paper, the authors build a Feynman integral dealing with such potentials. A version of the deformation formula can also be found there [K-S-W, Remark 19].

Then we give a simple assumption on the potential c providing the existence of the heat kernel on a conical neighboorhood of Ê + but with an aperture larger than π/2 (see Proposition 4.5).

3. We consider the Borel summability of the small time expansion of the conjugate of the heat kernel in the free case (λ = 0): we reformulate a statement given in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF] by using the analytic dilation given by (1.2). As in Proposition 4.5, we consider two cases. We consider a simple class of potentials for which Borel-Nevalinna summability (see Section 5 for the definition) holds in an arbitrary direction. Then we give assumptions on the potential c implying Borel-Watson summability of this small time expansion instead of Borel-Nevalinna summability as in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]. Borel-Nevalinna summability uses that c is the Fourier transform of some Borel measure µ defined on Ê ν with a suitable convergence assumption. Borel-Watson summability uses that c is the Fourier transform of an analytic function defined on a conical neighbourhood of Ê ν , with a similar convergence assumption. Let us remark that Borel-Watson summability of a series is a central tool when the critical time is a non-trivial power of the variable [Bals], [Ma-Ra].

We assume in this paper that the potential c is -valued. Our statements also hold if this potential is matrix-valued as in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF] and [START_REF] Hargé | A deformation formula for the heat kernel[END_REF].

Notation

For z ∈ , we denote sh

z := 1 2 (e z -e -z ), ch z := 1 2 (e z + e -z ). Let ν 1. For λ, µ ∈ ν , we denote λ • µ := λ 1 µ 1 + • • • + λ ν µ ν , λ 2 := λ • λ. These notations are extended to operators as ∂ x := (∂ x1 , . . . , ∂ xν ). We also denote λ := ( λ1 , . . . , λν ), |λ| := (λ • λ) 1/2 (if λ ∈ Ê ν , |λ| = √ λ 2
). We denote by π the canonical projection from Ê/4π onto Ê/2π . For θ ∈ Ê/4π , we denote by e iθ the element of √ • , the Riemann surface of the square root function, with argument θ and modulus 1. Then √

• := {z = re iθ |r > 0, θ ∈ Ê/4π }. For z = re iθ ∈ √ • , we denote z 1/2 := r 1/2 e iθ/2
. If z ∈ , we also denote by z 1/2 the square root of z which is defined up to a sign. Let m 1 and θ ∈ Ê/2π . For every subset A of m , we denote e iθ A := {e iθ z|z ∈ A}. We also use this

notation if θ ∈ Ê/4π and A ⊂ √ • . If α ∈ Ê/2kπ (k ∈ AE * ) and r ∈ Ê + , we denote by ]α -r, α + r[ 2kπ
the open interval3 of Ê/2kπ with end points α -r and α + r. We denote

+ := {z ∈ |Re(z) > 0} , + := {z ∈ |Re(z) 0}
and, if T > 0,

D T := {z ∈ ||z| < T } , D + T := D T ∩ + , D+ T := D T ∩ + .
Let B denote the collection of all Borel sets on Ê m . An -valued measure µ on Ê m is an -valued function on B satisfying the classical countable additivity property (see [Ru]). We denote by |µ| the positive measure defined by

|µ|(E) = sup ∞ j=1 |µ(E j )|(E ∈ B),
the supremum being taken over all partitions {E j } of E. In particular, |µ|(Ê m ) < ∞.

We denote by D(e iθ Ê m ) the space of smooth functions with compact support defined on e iθ Ê m . If Ω is an open domain in m , we denote by A(Ω) the space of -valued analytic functions on Ω. These spaces are equipped with their standard Frechet structure (the semi-norms are indexed by compact sets and eventually differentiation order). If U = e iθ ] -T, T [ or e iθ D+ T or + and F is a Frechet space, C ∞ U, F denotes the standard Frechet space of smooth functions defined on U with values in F . For instance, if the topology of F is defined by a family of semi-norms

(| • | j ) j∈J , the Frechet structure of C ∞ U, F is defined by the semi-norms | • | α,j (|f | α,j = sup t∈U |∂ α t f (t)| j ) if U = e iθ ] -T, T [ or e iθ D+
T . In the case U = + , suprema defining the semi-norms are taken over compact sets + ∩ D(0, R) where R > 0. We now define global spaces (with respect to the space variable). We denote by S(Ê m ) the space of Schwartz functions:

f ∈ S(Ê m ) ⇔ ∀(α, β) ∈ AE m × AE m , sup z∈Ê m |z α ∂ β z f (z)| < ∞.
We consider the following spaces of smooth functions

• if I =] -T, T [ or I = i] -T, T [ f ∈ C ∞ b,1 (I × Ê m ) ⇔ ∀(α, β) ∈ AE × AE m , ∃C > 0, ∀(t, z) ∈ I × Ê m , |∂ α t ∂ β z f (t, z)| C(1 + |z|) α . • if I = Ê or I = iÊ f ∈ C ∞ b (I × Ê m ) ⇔ ∀(α, β) ∈ AE × AE m , sup (t,z)∈I×Ê m |∂ α t ∂ β z f (t, z)| < ∞.
3 The heat kernel viewed from the positive direction

The setting

Let U be a complex open neighbourhood of 0 ∈ . Let P 0 be the operator acting on A(U × ν ), defined by

P 0 = A(t) • (∂ x + B(t)x) ⊗ (∂ x + B(t)x) -C(t) • x ⊗ x (3.1)
where A, B and C are ν × ν complex matrix-valued analytic functions on U . We assume that the matrices A and C are symmetric and that the matrix A(0) is real positive definite. We assume that the functions A| iÊ , (iB)| iÊ and C| iÊ are real-valued near 0.

(3.2)

Then the equation

∂ t u = P 0 u
with the boundary condition

u| t=0 + = δ x=y (3.3)
admits an explicit solution

p 0 := k(t) (4π∆t) ν/2 e -1 4t A -1 (0)•(x-y) 2 +Qt(x,y) . (3.4)
Here ∆ := det A j,k (0) 1 j,k ν , the function k is analytic near 0 and Q t denotes a polynomial of total degree at most 2 in x, y whose coefficients are analytic near 0. Moreover these coefficients take their values in iÊ if x, y ∈ Ê ν and t ∈ iÊ, |t| small enough (see [START_REF] Hargé | A deformation formula for the heat kernel[END_REF] and in particular Lemma 3.4, assertion 3). By (3.3)

we mean that for every

ϕ ∈ D(Ê ν ), x ∈ Ê ν and θ ∈ [-π/2, π/2] Ê ν u(re iθ , x, y)ϕ(y)dy -→ r→0 + ϕ(x).
Let us consider two particular cases. Let λ ∈ Ê and let ω := 2(-λ) 1/2 . Let us denote

p harm := 4π sh(ωt) ω -ν/2 exp - 1 4 ω sh(ωt) (ch(ωt)(x 2 + y 2 ) -2x • y) and p free = (4πt) -ν/2 e -(x-y) 2 /4t .
Then p harm (respectively p free ) satisfies

∂ t u = ∂ 2 x + λx 2 u (respectively ∂ t u = ∂ 2 x u)
with the boundary condition (3.3).

An existence and uniqueness statement

In this section, our aim is to give an existence and uniqueness statement concerning the equation

∂ t u = P 0 + c(t, x) u (3.5)
We also use the following definition in this section.

Definition 3.1 Let T b > 0. Let f be a measurable -valued function on D T b ×
Ê ν , analytic with respect to the first variable and let µ * be a positive measure on Ê ν such that for every R > 0

Ê ν exp(R|ξ|) sup |t|<T b |f (t, ξ)|dµ * (ξ) < ∞.
We denote by c the function belonging to

A(D T b × ν ) defined by c(t, x) = exp(ix • ξ)f (t, ξ)dµ * (ξ).
Proposition 3.2 Let P 0 be as above and let T b > 0. There exists T > 0 such that, for every f and µ * as in Definition 3.1, the following assertions hold.

1. For every ϕ ∈ S(Ê ν ) there exists a unique ψ ∈ C

∞ i] -T, T [, S(Ê ν ) such that    ∂ t ψ = P 0 + c(t, x) ψ ψ| t=0 = ϕ (3.6)
and the mapping ϕ

-→ ψ is continuous from S(Ê ν ) onto C ∞ i]-T, T [, S(Ê ν ) .
2. Let p = p(t, x, y) be the kernel of the operator

P t : S(Ê ν ) → S(Ê ν ) defined by P t (ϕ) = ψ(t, •) for t ∈ i] -T, T [. Then p can be uniquely extended as a function in C ∞ D+ T -{0}, A( 2ν ) ∩ A (D + T -{0}) × 2ν . This function satisfies (3.5) on ( D+ T -{0}) × 2ν and (3.3). Moreover p = p 0 × p conj where p conj = p conj (t, x, y) ∈ A(D + T × 2ν ) ∩ C ∞ D+ T , A( 2ν ) ∩ C ∞ b,1 (i] -T, T [×Ê 2ν ).
3. Let us assume that P 0 = ∂ 2 x (free case). Then

p conj ∈ A( + × 2ν ) ∩ C ∞ + , A( 2ν ) ∩ C ∞ b (iÊ × Ê 2ν ). Remark 3.3 The spaces C ∞ D+ T , A( 2ν ) and A(D + T × 2ν
) are local in the following meaning: the semi-norms are defined by taking suprema over compact sets. The spaces S(Ê ν ) and C ∞ b,1 (i] -T, T [×Ê 2ν ) are global, which is useful for the uniqueness statement. By (3.4)

p 0 = p free × p 1
where p 1 ∈ A(D T × 2ν ) for some T > 0. However one can not replace p 0 by p free in the statement of Proposition 3.

2 since p 1 ∈ C ∞ b,1 (i] -T, T [×Ê 2ν
) in the general case.

Remark 3.4 The reality-preserving property (3.2) is useful for the uniqueness statement. This assumption implies that the operator P 0 | iÊ is symmetric with respect to the L 2 inner product.

Remark 3.5 Let us assume that P 0 = ∂ 2 x + λx 2 where λ ∈ Ê, that the potential c does not depend on t. Since c ∈ L ∞ (Ê ν ), by [Re-Si, Th. X.36], the operator P 0 +c is self-adjoint on a domain containing the domain of the operator -∂ 2

x +x 2 . See also [Bo-Ca-Hä-Mi]. Therefore one can define the Schrödinger operator for the operator P 0 . By [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF], its kernel admits a (unique) analytic continuation on ( D+ T -{0}) × 2ν if T is small enough. This yields an alternative formulation of Proposition 3.2 in the harmonic case.

Proof of Proposition 3.2

We need the following lemma.

Lemma 3.6 Let T > 0 and let B be a ν × ν real positive definite symmetric matrix. Let P t be a polynomial with respect to x, y ∈ Ê ν , of degree at most 2, with coefficients belonging to C ∞ (] -T, T [, Ê). There exists T 2 > 0 such that,

for every u ∈ C ∞ b,1 (] -T, T [×Ê 2ν
) and every ϕ ∈ S(Ê ν ), the function ψ defined by

ψ(t, x) := Ê ν det B 4πit ν/2 e -B•(x-y) 2 /4it e iPt(x,y) u(t, x, y)ϕ(y)dy (3.7) belongs to C ∞ ] -T 2 , T 2 [, S(Ê ν ) . The mapping ϕ -→ ψ is continuous from S(Ê ν ) into C ∞ ] -T 2 , T 2 [, S(Ê ν ) . Moreover, if e iP0(y,y) × u(0, y, y) = 1, then ψ| t=0 = ϕ.
Such a result is standard. The proof is given in the Appendix. Let us prove Proposition 3.2. The assertion 1, of course, is a well-known statement. However, for the convenience of the reader and the completeness of the paper, we give its proof.

-1-In view of the uniqueness statement in assertion 1, let us consider ψ ∈ C

∞ i] -T, T [, S(Ê ν ) satisfying (3.6) with ϕ = 0. Let E(t) := Ê ν ψ(it, x) ψ(it, x)dx, R(t) := -2 Ê ν Im c(it, x) ψ(it, x) ψ(it, x)dx.
Then by (3.2), for every t ∈] -T, T [, ∂ t E = R(t). Since the function |c| is bounded, there exists K > 0 such that |∂ t E| KE. Since ψ ψ is bounded by C(1 + |x|) -ν-1 for (t, x) ∈] -T, T [×Ê ν , one gets E| t=0 = 0 by the dominated convergence theorem. Therefore E = 0 and ψ| i]-T,T [×Ê ν ≡ 0.

-2-Let us prove the existence statement in assertion 1. Let p conj be as in [START_REF] Hargé | A deformation formula for the heat kernel[END_REF]Theorem 2

.1]. Let ϕ ∈ S(Ê ν ). Let ψ(t, x) := Ê ν (p 0 × p conj )(t, x, y)ϕ(y)dy. By (3.4) (p 0 × p conj )(it, x, y) = (4πi∆t) -ν/2 e -A -1 (0)•(x-y) 2 /4it × e iPt × k(it)p conj (it, x, y)
where the polynomial P t satisfies the assumptions of Lemma 3.6. Then, by Lemma 3.6, there exists

T 2 ∈]0, T ] such that ψ ∈ C ∞ i]-T 2 , T 2 [, S(Ê ν ) . More- over the mapping ϕ -→ ψ is continuous from S(Ê ν ) onto C ∞ i]-T, T [, S(Ê ν ) .
Since p 0 × p conj satisfies (3.5) and p conj | t=0 = 1, (3.6) holds.

-3-Let us prove assertion 2. By the regularity properties of p conj [Ha6, Theorem 2.1], one gets a suitable extension of the kernel of the operator P t . We claim that this extension is unique. Let p 1 and p 2 be two extensions. Then

p = p 1 -p 2 ∈ A (D + T -{0}) × 2ν ∩ C ∞ D+ T -{0}, A(

The heat kernel viewed from an arbitrary direction

We must take into account the ramification of the heat kernel at t = 0 in our statements. The ramification is only due to the term t -ν/2 in (3.4). Let ǫ ∈ Ê/4π . Then the free heat kernel p free is invariant, up to a multiplicative constant, under the change of variables (t, x, y) → (e iǫ t, e iǫ/2 x, e iǫ/2 y) (4.1) and the free heat equation is invariant under the change of variables (t, x) → (e iǫ t, e iǫ/2 x). This elementary remark allows a reformulation of Proposition 3.2 (we only consider the harmonic case for the sake of simplicity). We denote 4

p harm ǫ := 4πe -iǫ t -ν/2 × sh(ωe -iǫ t) ωe -iǫ t -ν/2 exp - 1 4 e -iπ(ǫ) ω sh(ωe -iǫ t) (ch(ωe -iǫ t)(x 2 + y 2 ) -2x • y) , |t| ≪ 1 , p free ǫ := 4πe -iǫ t -ν/2 exp - 1 4t (x -y) 2 .
Corollary 4.1 Let λ ∈ Ê and ǫ ∈ Ê/4π . There exists T > 0 such that the following statement holds. Let µ be a complex measure on Ê ν such that for every

R > 0 Ê ν exp(R|ξ|)d|µ|(ξ) < ∞. (4.2) Let c(x) = Ê ν exp(ie -iǫ/2 x • ξ)dµ(ξ).
Then the following assertions hold.

4 Only the first factor of the product defining p harm ǫ is concerned by the ramification.

1. For every ϕ ∈ S(e iǫ/2 Ê ν ) there exists a unique ψ ∈ C ∞ ie iǫ ]-T, T [, S(e iǫ/2 Ê ν )

such that    ∂ t ψ = ∂ 2 x + λe -2iπ(ǫ) x 2 + c(x) ψ ψ| t=0 = ϕ
and the mapping ϕ -→ ψ is continuous from S(e iǫ/2 Ê ν ) onto C ∞ ie iǫ ] -T, T [, S(e iǫ/2 Ê ν ) . 2. Let p = p(t, x, y) be the kernel of the operator P t : S(e iǫ/2 Ê ν ) → S(e iǫ/2 Ê ν ) defined by P t (ϕ) = ψ(t, •) for t ∈ ie iǫ ] -T, T [. Then p can be uniquely continued as a function belonging to

A e iǫ (D + T -{0}) × 2ν ∩ C ∞ e iǫ ( D+ T -{0}), A( 2ν ) . Moreover p = p harm ǫ × p conj where p conj = p conj (t, x, y) ∈ A(e iπ(ǫ) D + T × 2ν ) ∩ C ∞ e iπ(ǫ) D+ T , A( 2ν ) ∩ C ∞ b,1 (ie iπ(ǫ) ] -T, T [×Ê 2ν ).
3. Let us assume that λ = 0. Then Here dy = e iϑν dm(y) where m denotes the standard (non negative) Lebesgue measure on e iϑ Ê ν . Then the kernel p satisfies on e iǫ (D

p conj ∈ A(e iπ(ǫ) + × 2ν ) ∩ C ∞ e iπ(ǫ) + , A( 2ν ) ∩ C ∞ b (ie iπ(ǫ) Ê× Ê 2ν ).
+ T -{0}) × 2ν    ∂ t p = ∂ 2 x + λe -2iπ(ǫ) x 2 + c(x) p p| t=e iǫ 0 + = δ x=y (4.3)
since, for every smooth function g = g(t, x, y) on e iπ(ǫ) D+ T × ν × ν such that g| t=0 = 1, the function p free ǫ × g goes to δ x=y in the direction e iǫ .

Let us choose ǫ = π. Then we get a solution p such that p conj is defined on

t = |t|e iθ ∈ |θ ∈ [π/2, 3π/2] 4π , |t| < T × 2ν or t ∈ |θ ∈ [π/2, 3π/2] 4π × 2ν if λ = 0.
In particular, by considering values of t such that arg t = π/2, 3π/2, we obtain the following result about the standard Schrödinger equation.

p conj = p conj (t, x, y) ∈ C ∞ ] -T, T [, A( 2ν ) such that p = p harm ǫ × p conj satisfies    1 i ∂ t p = ∂ 2 x + λx 2 + c(x) p , x ∈ Ê ν , t ∈] -T, T [ p| t=0 = δ x=y , y ∈ Ê ν . (4.5) If λ = 0, p conj ∈ C ∞ Ê, A( 2ν ) .
Remark 4.4

The assumption (4.2), since c is given by (4.4), allows potentials such as V (x) = λx 2 ± e x 2 , V (x) = λx 2 ± e x1 , . . .

In the case λ = 0, this fact was noticed by Kuna, Streit and Westerkamp [K-S-W]. In particular the function x -→ e x 2 is viewed as a perturbation of the operator ∂ 2 x + λx 2 (the deformation formula is used to deal with this part of the potential V ) whereas the function x -→ λx 2 is not viewed as a perturbation of the operator ∂ 2

x in our method. Using a complex point of view with respect to the space variables perhaps explains this "paradox". A related remark can be done for the uniqueness problem: we do not claim that (4.5) has a unique solution in its natural real setting. However it does, if we consider complex values for the space variables (x, y ∈ e iπ/2 Ê ν ), by taking advantage of the uniqueness statement of Corollary 4.1.

Notice that the dilation given by (t, x, y) → (e iπ t, ix, iy) (ǫ = π in 4.1), which allows one to view Corollary 4.3 as a consequence of Corollary 4.1, "reverses" the direction of t.

Another viewpoint is formally related to the previous proposition. For θ > 0, let

Ê +,ν ≺,θ := {e iϕ x ∈ ν |x ∈ (Ê + ) ν , ϕ ∈] -θ, θ[ 2π }, Ê ν ≺,θ := {e iϕ x ∈ ν |x ∈ Ê ν , ϕ ∈] -θ, θ[ 2π }, Ê + ≺,θ := {re iϕ ∈ |r > 0, ϕ ∈] -θ, θ[ 2π }. One has Proposition 4.5 Let θ, α ∈]0, π/4[. Let µ be a -valued Borel measure on ν .
Let us assume (case 1) that Ê +,ν ≺,θ contains the support of µ and that ∀R > 0, ν exp(R|ξ|)d|µ|(ξ) < ∞ or (case 2) that dµ(ξ) = ĉ(ξ)dξ where ĉ denotes an analytic function on Ê ν

≺,α satisfying ∀R > 0, ∃K > 0, ∀ξ ∈ Ê ν ≺,α , |ĉ(ξ)| Ke -R|ξ| . Let F = ν (case 1) or F = Ê ν (case 2). Let c(x) := F exp(ix • ξ)dµ(ξ).
Then, by the deformation formula, -case 1: The heat equation associated to the operator ∂ 2

x +c(x) has a solution

p defined on Ê + ≺,π/2-2θ -{0} × 2ν satisfying the following boundary condition. For every ϕ ∈ D(Ê ν ), x ∈ Ê ν and α ∈] -π/2 + 2θ, π/2 -2θ[ Ê ν p(re iα , x, y)ϕ(y)dy -→ r→0 + ϕ(x).
-case 2: The heat kernel of the operator ∂ 2 x + c(x), which is defined on

Ê + × Ê 2ν
, admits an analytic continuation on Ê + ≺,π/2+2α -{0} × 2ν .

Ê + ≺, π 2 -2θ Re t Im t π 2 -2θ case 1 Ê + ≺, π 2 +2α
Re t

Im t π 2 + 2α case 2 Proof Let p conj := n 0 v n where v n := t n 0<s1<•••<sn<1 F e i y+s(x-y) •ξ exp -ts(1 -s) • n ξ ⊗ ξ d νn µ ⊗ (ξ)d n s, y + s(x -y) • ξ := y + s 1 (x -y) • ξ 1 + • • • + y + s n (x -y) • ξ n , s(1 -s) • n ξ ⊗ ξ := n j,k=1 s j∧k (1 -s j∨k )ξ j • ξ k , d νn µ ⊗ (ξ) := dµ(ξ n ) • • • dµ(ξ 1 ).
We first check that the series defining p conj is convergent.

(case 1) We claim that

p conj = p conj (t, x, y) ∈ A(Ê + ≺,π/2-2θ × 2ν ). One has |x|, |y| R ⇒ exp i y + s(x -y) • ξ e R|ξ1| × • • • × e R|ξn| . Since 2θ < π/2, Ê + ≺,2θ is a convex cone. Then t ∈ Ê + ≺,π/2-2θ , ξ ∈ supp(µ ⊗ ) ⇒ Re ts(1 -s) • n ξ ⊗ ξ 0.
This implies the convergence of the series defining p conj and the analyticity of

p conj . (case 2) We claim that p conj ∈ A(Ê + ≺,π/2+2α × 2ν ). Let β ∈]0, α[. Since
the function ĉ is analytic on Ê ν ≺,α , one gets by a deformation of the integration contour,

v n := e -iνnβ t n 0<s1<•••<sn<1 Ê νn e ie -iβ y+s(x-y) •ξ exp -te -2iβ s(1 -s) • n ξ ⊗ ξ ĉ(e -iβ ξ 1 ) • • • ĉ(e -iβ ξ n )d νn ξd n s.
Therefore the convergence of the series defining p conj and the analyticity of p conj hold for Re e -2iβ t > 0 and x, y ∈ ν . Since β is arbitrary, one gets that p conj ∈ A(Ê + ≺,π/2+2α × 2ν ). By proceeding as in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF], one can show that p = p free × p conj satisfies the heat equation. Moreover the boundary condition is satisfied.

Example 4.6 Let θ 1 , . . . , θ q ∈] -π/4, π/4[ and λ 1 , . . . , λ q ∈ (Ê + ) ν . Let c(x) = exp ie iθ1 λ 1 • x + • • • + exp ie iθq λ q • x .
Then the function c satisfies the assumptions of Proposition 4.5 (case 1). We do not attempt to give a uniqueness statement in this case.

Example 4.7 Let c(x) = e -x 2 . By Proposition 4.5 (case 2) the heat kernel is well defined on ( -] -∞, 0]) × 2ν .

Remark 4.8 One can generalize Proposition 4.5 (case 2) in the harmonic case. The proof needs a modification of [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]Lemma 4.2].

Borel summability of the conjugate of the heat kernel in an arbitrary direction

For the sake of simplicity, we only consider the free case in this section. Let κ, T > 0. Let

Sκ := z ∈ |d(z, [0, +∞[) < κ , DT := z ∈ |Re 1 z > 1 T .
DT is the open disk of center T 2 and radius T 2 .

Definition 5.1 Let ǫ ∈ Ê/2π . Let a 1 , . . . , a r , . . . ∈ . The formal power series f = r 0 a r t r is called Borel-Nevalinna (respectively Borel-Watson) summable in the direction e i ǫ if

• the radius of convergence of the Borel transform of f , f (τ ) := ∞ r=0 ar r! τ r , does not vanish

• there exist κ > 0 (respectively θ > 0) such that the Borel transform can be analytically continued on e i ǫ Sκ (respectively e i ǫÊ + ≺,θ )

• there exist K, T > 0 such that for every τ ∈ e i ǫ Sκ (respectively

e i ǫÊ + ≺,θ ) | f (τ )| Ke |τ |/T .
If the power series f is Borel-Nevalinna or Borel-Watson summable, the Laplace transform of f f (t) :

= +∞ 0 f (τ )e -τ t dτ t
is called the Borel sum of f .

e i ǫ DT

Re t

Im t e i ǫÊ + ǫ θ e i ǫÊ + ≺,θ ∩ D T Remark 5.2 If a power series f is Borel-Nevalinna (respectively Borel-Watson) summable in the direction e i ǫ, then there exist T > 0 and θ > π/2 such that its Borel sum is well defined for t ∈ e i ǫ DT (respectively e i ǫÊ + ≺,θ ∩ D T ). See [So]. The change of variables (4.1) allows us to give the following corollary of Theorem 3.1 [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF].

Corollary 5.3 Let ε > 0 and ǫ ∈ Ê/4π . Let µ be a -valued measure on Ê ν verifying Ê ν exp(εξ 2 )d|µ|(ξ) < ∞.

(5.1)

Let c(x) = exp(ie -iǫ/2 x • ξ)dµ(ξ) (5.2)
and let u be the solution of (4.3) where λ = 0. Let p conj be defined by u=p free p conj . Then p conj admits a Borel transform p conj (with respect to t) which is analytic on 1+2ν . Let κ, R > 0 and let

C := 2 exp 2κ ε + ε 2 ξ 2 + R|ξ| d|µ|(ξ) 1/2 .
Then, for every (τ, x, y) ∈ e iπ(ǫ) Sκ × 2ν such that Im(e -iǫ/2 x) < R and

Im(e -iǫ/2 y) < R,

p conj (τ, x, y) exp C|τ | 1/2 .
(5.3) Remark 5.4 By the estimate (5.3), the small time expansion of the conjugate heat kernel is Borel-Nevalinna summable in the direction e iπ(ǫ) and its Borel sum is equal to p conj .

We now illustrate Corollary 5.3 by simple examples.

Example 5.5 Let ǫ ∈ Ê/4π , ξ 0 ∈ e -iǫ/2 Ê ν -{0} and c(x) = exp(ix•ξ 0 ). The function c satisfies the assumptions of Corollary 5.3 hence the small time expansion of the conjugate heat kernel is Borel-Nevalinna summable in the direction e iπ(ǫ) . For ǫ ′ ∈ Ê/4π , π(ǫ ′ ) = π(ǫ), the function c is not bounded on e iǫ ′ /2 Ê ν and therefore does not satisfy the assumptions of Corollary 5.3 in the direction e iε ′ : Corollary 5.3 can not be used to study the Borel-Watson summability of this expansion in the direction e iπ(ǫ) .

Example 5.6 Let c(x) = exp ix 1 + ie iπ/8 x 2 and let p be the solution given by Proposition 4.5 (case 1). Then Corollary 5.3 can not help us to study the Borel summability of the small time expansion of p.

Let us now consider c(x) = exp ix 1 + exp ie iπ/8 x 2 . Then by separation of variables, the solution given by Proposition 4.5 is the product of two Borel-Nevalinna summable expansions but in different directions.

Example 5.7 Let K, A > 0 and α ∈]0, π/2[. Let ĉ be an analytical function on

Ê ν ≺,α satisfying, for every ξ ∈ Ê ν ≺,α , |ĉ(ξ)| Ke -A|ξ| 2 . Let c(x) := Ê ν exp(ix • ξ)ĉ(ξ)dξ.
Let ǫ ∈ I :=] -2α, 2α[ 4π and ε < A. Then there exists a measure µ satisfying (5.1) such that the function c is also defined by (5.2) (see also Proposition 4.5 case 2). Therefore the small time expansion of the conjugate heat kernel is Borel-Nevalinna or Borel-Watson summable in every direction belonging to π(I). Functions like c = e -γx 2 , γ > 0, satisfy such a property.

Appendix

Here is a proof of Lemma 3.6.

For the sake of simplicity, we assume B = Id. For δ ∈ AE ν , we denote

|δ| = δ 1 + • • • + δ ν . For m, k ∈ Ê and p ∈ AE, we denote by S m,k p (] -T, T [×Ê ν × Ê ν ) the set of smooth functions f = f (t, x, y) on ] -T, T [×Ê ν × Ê ν such that ∀(q, r) ∈ AE 2 , ∃C > 0, ∀(α, β, γ) ∈ AE × AE ν × AE ν , ∀(t, x, y) ∈] -T, T [×Ê ν × Ê, α p, |β| q, |γ| r ⇒ |∂ α t ∂ β x ∂ γ y f | C(1 + |x|) m+α (1 + |y|) k+α .
For such a function, we denote by |f | m,k,p,q,r the best constant C satisfying the previous inequality. For f ∈ S m,k p (k < -ν), let us denote F f (t, x) := Ê ν (4πit) -ν/2 e -(x-y) 2 /4it e iPt(x,y) f (t, x, y)dy.

We first establish some useful properties of this transform. Let f ∈ S m,k r . Let j = 1, . . . , ν.

• Using the symmetry of the free Schrödinger kernel and integration by parts, one gets

∂ xj F f = F f (6.1)
where

f := e -iPt(x,y) (∂ xj + ∂ yj ) e iPt(x,y) f (t, x, y) ∈ S m+1,k+1 r-1
.

Moreover there exists c 2 > 0, which only depends on the coefficients of P t (x, y), such that

| f | m+1,k+1,p,q-1,r-1 c 2 |f | m,k,p,q,r . • Since 1 i ∂ t (4πit) -ν/2 e -(x-y) 2 /4it = ∂ 2 y (4πit) -ν/2 e -(x-y) 2 /4it
and by integrations by parts, one gets

∂ t F f = F f (6.2)
where f := e -iPt(x,y) (i∂ 2 y + ∂ t ) e iPt(x,y) f (t, x, y) ∈ S m+2,k+2 r-2

.

Moreover there exists c 1 > 0 such that

| f | m+2,k+2,p-1,q,r-2 c 1 |f | m,k,p,q,r .
• We shall need to estimate x j F f . For this, we express the multiplication operator by x j in a convenient way. Let us denote φ := (x -y) 2 4t + P t (x, y).

Let ς = 1, . . . , ν. Then

∂ yς φ = y ς -x ς 2t + 1 2 c(t) + 1 2 ν ς ′ =1 a ς,ς ′ (t)x ς ′ + b ς,ς ′ (t)y ς ′ where a ς,ς ′ , b ς,ς ′ , c are smooth Ê-valued functions defined on ] -T, T [. Then 1-ta ς,ς (t) x ς -t ν ς ′ = 1 ς ′ = ς a ς,ς ′ (t)x ς ′ = y ς +tc(t)-2t∂ yς φ+t ν ς ′ =1 b ς,ς ′ (t)y ς ′ .
Let us consider the above equations as a system of ν equations where the unknowns are x 1 , . . . , x ν . Then there exists T

2 ∈]0, T [ such that, for t ∈] -T 2 , T 2 [, x j = u(t) • ∂ y φ + v(t) • y + w(t)
where u, v (respectively w) are smooth Ê ν -valued (respectively Ê-valued) functions defined on ] -T 2 , T 2 [. These functions and T 2 only depend on the coefficients of the polynomial P t . Then, by integration by parts, Then, by (6.1), (6.2) and (6.3), Hence, for t ∈] -T 2 , T 2 [-{0} and x ∈ Ê ν ,

x δ ∂ α t ∂ β x ψ = F f where f ∈ S
x δ ∂ α t ∂ β x ψ C 1 |4πt| -ν/2 (1 + |x|) 2α+|β| Ê ν (1 + |y|) -ν-1 dy × ϕ k,r
C 2 |t| -ν/2 (1 + |x|) 2α+|β| × ϕ k,r .

-Let p, q, k ′ 0. Let us choose k ∈ Ê and r ∈ AE such that

   k + 2p + q + k ′ < -ν -1 r -2p -q -k ′ 0 .
Then, if α p, |β| q and |δ| k ′ , (6.4) is satisfied and

∀t ∈] -T 2 , T 2 [-{0}, ∀x ∈ Ê ν , x δ ∂ α t ∂ β x ψ C 3 |t| -ν/2 (1 + |x|) 2p+q
where C 3 is a positive number. Let k ∈ Ê. where c only depends on κ and T 2 . Therefore, for every n ∈ AE, ∂ n t ψ(t, •) ∈ S(Ê ν ) and for every (k ′ , q) ∈ Ê × AE there exist (k, r) ∈ Ê × AE and C > 0 such that sup t∈]-T2,T2[ ∂ n t ψ(t, •) k ′ ,q C ϕ k,r .

i.e. the mapping ϕ -→ ψ is continuous.

-Let us consider the assertion on ψ| t=0 . Let γ ∈ D(Ê ν ) be such that γ(z) = 1 if |z| 1. Let x ∈ Ê ν . Then ϕ = ϕ 1 + ϕ 2 where ϕ 1 (y) = γ(y -x)ϕ(y)

and ϕ 2 (y) = 1 -γ(y -x) ϕ(y). Since both functions belong to the Schwartz space, it suffices to check the claim for the corresponding ψ 1 and ψ 2 . Since ϕ 2 vanishes on a neighbourhood of x, one gets ψ 2 (t, x) = O(t ∞ ) by integrations by parts. Since the support of the function ϕ 1 is compact, ψ 2 (•, x)| t=0 = ϕ 2 (x). This proves ψ| t=0 = ϕ.

  2ν ) and p| (i]-T,T [-{0})×Ê ν = 0. Let (x, y) ∈ Ê 2ν and let p be the function on D T -{0} defined by p(t) := 1 Ret 0 p(t, x, y). By regularity properties of p and Cauchy-Riemann equations with respect to t, p is smooth near iT /2 and satisfies Cauchy-Riemann equations. Therefore the function p is analytic near iT /2, vanishes near iT /2 and actually on D + T -{0}. Then by analytic continuation with respect to the space variables the function p vanishes on (D + T -{0}) × 2ν . -4-Assertion 3 can also be checked by considering the deformation formula in the free case.

Figure

  Figure 4.1:

Corollary 4. 3

 3 Let µ as in Corollary 4.1. Let c(x) = Ê ν exp(x • ξ)dµ(ξ).

  Let λ ∈ Ê. Then there exist T > 0 and

Figure

  Figure 4.2:

Figure

  Figure 5.1:

1 .

 1 iu(t) • ∂ y f + v(t) • y + w(t) f ∈ S m,k+1 r-Moreover there exists c 3 > 0 such that | f | m,k+1,p,q,r-1 c 3 |f | m,k,p,q,r . -Let k ∈ Ê and r ∈ AE. Let ϕ ∈ S(Ê ν ) and u ∈ C ∞ b,1 (] -T, T [×Ê 2ν ). Let us denote ϕ k,r := sup |γ| r,y∈Ê ν (1 + |y|) -k |∂ γ y ϕ|.By Leibniz formula, the function f : (t, x, y) -→ u(t, x, y)ϕ(y) belongs to S 0,k r and for every p, q ∈ AE, there exists C > 0 such that|f | 0,k,p,q,r C ϕ k,r(C depends on the function u and the numbers k, p, q, r). Let ψ be defined by(3.7). Then ψ = F f . Let (α, β, δ) ∈ AE × AE ν × AE ν . Let us assume that    k + 2α + |β| + |δ| < -ν -1p -α 0 , q -|β| 0 , r -2α -|β| -|δ| 0 . (6.4)

  2α+|β|,k+2α+|β|+|δ|,p-α,q-|β|,r-2α-|β|-|δ| c α

  Then there exists C 4 > 0, such that,for t ∈] -T 2 , T 2 [-{0} and x ∈ Ê ν , α p, |β| q ⇒ ∂ α t ∂ β x ψ C 3 |t| -ν/2 (1 + |x|) -k .If κ > 0 and g is a smooth function on ] -T 2 , T 2 [-{0} satisfying, for everyN ∈ AE, max n N |∂ n t g(t)| C N |t| -κ , then g is smooth on ] -T 2 , T 2 [ and

	sup t∈]-T2,T2[,n N	|∂ n t g(t)| cC N +[κ]+2

With our notation, a physical interpretation of the parameter t is the reciprocal of the temperature which is a macroscopic variable. We denote by t the "physical" time.

A ramification occurs in our statements if the dimension ν is odd: this is only due to the existence of the factor t -ν/2 in the expression of the free heat kernel.

r > kπ ⇒ ]αr, α + r[ 2kπ = Ê/2kπ .


Anthony Acou

Pauline Boury

Pascal Laffaille

Alain Jean Crivelli

Eric Feunteun

Towards a standardized characterization of the potentially migrating silver European eel

Keywords: yellow eel, silver eel, discrimination, multicriteria approach, silvering process, PIT-tags

Towards a standardized characterization of the potentially migrating silver European eel (Anguilla anguilla, L.)

Introduction

In the context of the major decline of Atlantic eel stocks throughout their distribution range observed since the 1980 s (e. g. [START_REF] Moriarty | Management of European eel fisheries[END_REF] for European eel Anguilla anguilla and Casselman 2003 for American eel Anguilla rostrata), restoration programs and local management plans have been conducted in order to restore abundance for all its inland water stages (ICES1998).Theregulationoffisheriesforglasseelsand/orsilvereels [START_REF] Moriarty | Management of European eel fisheries[END_REF], the management of obstructions (chemical or physical) to migration, in particular fish passes (e. g. [START_REF] Legault | Etude de quelques facteurs de sélectivité des passes à anguilles[END_REF][START_REF] Knights | Enhancing immigration and recruitment of eels: the use of passes and associated trapping systems[END_REF], Feunteun et al. 1998), and restocking [START_REF] Moriarty | Management of European eel fisheries[END_REF] are the most widely used management options [START_REF] Feunteun | Management and restoration of European eel population (Anguilla anguilla): An impossible bargain[END_REF].

Standardized abundance indices, based on monitoring involving many fishery managers and researchers, will be needed to assess the efficiency of such restoration plans on eel stocks. Recently, the monitoring of silver eel escapement has been designated as a major priority at national and European scales [START_REF] Lambert | Compte rendu des journées anguilles de Paimpont du 23 au 25 Septembre 1998. -Rapport GRISAM[END_REF] since it implies an assessment of the inland water dynamics of a sub-population of eels [START_REF] Feunteun | Management and restoration of European eel population (Anguilla anguilla): An impossible bargain[END_REF]. In this context, it seems essential to identify the number and features of candidate eels ready to undertake their seaward migration in the following year, based on the analysis of a component of the sedentary population [START_REF] Feunteun | European eel (Anguilla anguilla): prediction of spawner escapement from continental population parameters. -Can[END_REF].

The silvering process involves a whole series of morphological and physiological transformations that mark the transition between a sedentary growth phase (yellow eel) and a catadromous migration phase (silver eel). Numerous experiments and studies have investigated the changes that occur during the transformation from a resident yellow eel to actively migrating silver eel. They usually showed that the silvering process is a gradual phenomenon that starts in the yellow stage and that in neuro-endocrine terms consists of a succession of stages that take place from spring to autumn [START_REF] Fontaine | L'argenture de l'anguille: métamorphose, anticipation, adaptation[END_REF][START_REF] Durif | Analysis of morphological and physiological parameters during the silvering process of the European eel (Anguilla anguilla) in the Lake of Grand-Lieu (France)[END_REF]. These changes are generally thought to be related to the onset of migration behaviour and gonadal development (Pankhurst 1984). Some of the metamorphic changes that have been documented in anguillid eels include increased eye diameter [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF][START_REF] Beullens | Sex differentiation, changes in length, weight and eye size before and after metamorphosis of the European eel (Anguilla anguilla L.) maintained in captivity[END_REF], changes in integument structure and colour [START_REF] Pankhurst | Changes in vision and olfaction during sexual maturation in the European eel Anguilla anguilla (L.)[END_REF][START_REF] Fontaine | L'argenture de l'anguille: métamorphose, anticipation, adaptation[END_REF], a shift in retinal pigment sensitivity [START_REF] Pankhurst | Changes in vision and olfaction during sexual maturation in the European eel Anguilla anguilla (L.)[END_REF], Pankhurst 1984), a differentiation of the lateral line [START_REF] Zacchei | Lateral line system during the life cycle of Anguilla anguilla L. -Boll[END_REF], etc. Some of these characteristics can be used to assess the migratory status of individuals eels examining one or more of these parameters [START_REF] Cottrill | An examination of utilizing external measures to identify sexually maturing female American eels, Anguilla rostrata, in the St. Lawrence River[END_REF]. However, despite the widespread use of morphometric variables to discriminate resident eels from migrating eels, no methodology is yet available to precisely assess the gradual evolution of silvering. Such a method is lacking since many unclassifiable individuals that are neither completely 'yellow' nor 'silver' can be observed in the field. Thus, the fate of these eels, as candidates to seaward migration, can be questioned and is difficult to determine with a single character of silvering [START_REF] Feunteun | European eel (Anguilla anguilla): prediction of spawner escapement from continental population parameters. -Can[END_REF].

The objective of our study was to define a standardized, non-destructive and rapid method adapted to field investigations to (i) specify the state of the silvering process and (ii) identify silver eels that are candidates for the seaward migration in the following season. The state of advance of metamorphosis of eels was described using a combination of three criteria representative of the sequence of observed changes during metamorphosis. Indeed, the increase of the eye diameter correlated with the higher gonad weight was the earliest change, whereas the typical pigmentation of silver eel occurred generally at the end of the process [START_REF] Durif | Analysis of morphological and physiological parameters during the silvering process of the European eel (Anguilla anguilla) in the Lake of Grand-Lieu (France)[END_REF][START_REF] Han | Synchronous changes of morphology and gonadal development of silvering Japanese eel Anguilla japonica[END_REF]. Thus, a quantitative criterion that expresses the ocular hypertrophy using the Pankurst Ocular Index "OI" (1982) and two qualitative criteria that specify the state of the lateral line (differentiated or not) and the colour contrast (present or not) were used. The identification of candidates to migration was based on the comparison of the combination of these criteria observed between sedentary versus migrating eels in three different sites and was validated by a mark-recapture survey using Passive Integrated Transponder (PIT) tags. Moreover, in order to show that the combination of three silvering criteria was essential for silvering eel identification, consequences of the use of a single criterion (lateral line or colour or OI) on the determination of stages were investigated.

Material and methods

Study sites

Eels were sampled in three rivers in western France: the Frémur, the Oir and the Loire. The Frémur (17 km-long) and Oir (21 km-long) are similar to a wide range of small catchments in western France as their drainage areas cover 60 km 2 and 87 km 2 respectively. The other site is located in the downstream area of the Loire (1012 km-long, the largest river in France) catchment (117500 km 2 ).

Sedentary fish sampling

Eels were sampled in the Frémur and Oir drainages in 2002 and 2003 by electrofishing using a 'Heron' apparatus (100 -500 V D. C. and 0.8-10 A; [START_REF] Lamarque | Comparaison des appareils de pêche à l'électricité EPMC et héron[END_REF]. Sampling was conducted during the low water level period (September in the Frémur and October in the Oir), i. e. after the beginning of the metamorphosis [START_REF] Fontaine | L'argenture de l'anguille: métamorphose, anticipation, adaptation[END_REF] and before the effective downstream migrations of silver eels that typically begin from November in both sites with high water discharges [START_REF] Acou | Catadromous migration dynamics of European Eel (Anguilla anguilla L.) in a dammed catchment[END_REF], Feunteun et al. 2000).

PIT-Tagging

In Frémur, all eels > 200 mm captured by electrofishing in September 2002 were marked individually using Trovan  Passive Integrated Transponder (PIT) tags (dimensions: 2.12 × 11.5 mm) injected with a syringe into the abdominal cavity [START_REF] Prentice | Electronic tag. Feasibility of using implantable passive integrated transponder (PIT) tags in salmonids[END_REF]. Fourteen percent of the tags were rejected one hour of after the injection [START_REF] Acou | L'anguille européenne, dynamique et déterminisme de la migration catadrome dans un bassin versant anthropisé de Bretagne (Frémur)[END_REF]. Therefore, eels were kept at least an hour after tagging before they were released in the river.

Downstream migrant fish sampling

In the Frémur, a Wolf trap (6 km from the estuary) was designed to capture every descending eel > 200 mm under all flow conditions. Daily monitoring was conducted between September 2002 and September 2004. In order to identify marked eels, we used a Trovan  apparatus (LID500 Portable reader).

In the Loire, 13 commercial fishermen, located between 100 km and 130 km from the mouth concentrate their fishing on emigrant silver eels during river floods using a boat that supports a specific net called "guideau". Daily monitoring was conducted during river floods in December 2002 in order to characterize the migrant eels. This net is on average 25 meters long and 8 meters wide. It is supported by two horizontal metal bars, one placed at the bottom and the other one at the surface during the fishing, and ends in a double pocket with meshing of approximately 20 mm compared to 320 mm at the entry. Fishing takes place preferably at night and lasts from 30 minutes to several hours depending on the number of captures and the obstruction of the guideau. No eel sampling was conducted in the Oir during the seaward migration period.

Each fish collected at the three different sites (Frémur, Oir, Loire) was anaesthetized with Benzocaine at a concentration of 0.15 g l -1 , and the total body length (TL) was measured at the nearest mm with an ichthyometer. Sex was determined by macroscopic observation of gonads, using the criteria described by [START_REF] Colombo | Gonad differentiation and body growth in Anguilla anguilla[END_REF] on a subsample of 45 silver eels caught in the Frémur's downstream trap. Male lengths ranged between 300 mm and 450 mm whereas all eels > 450 mm were only females as shown in other studies [START_REF] Tesch | The eel. Biology and management of anguillid eels[END_REF][START_REF] Rossi | Some observations on age, sex, growth of silver eels (Anguilla anguilla L.) in north Adriatic Lagoons. -Rapp. P.-v. Réun[END_REF].

Combination of silvering criteria

Qualitative criteria

For all fish, the lateral line consists of surficial receptors (i. e. neuromasts) highly sensitive to hydrodynamic disturbances [START_REF] Montgomery | The lateral line can mediate rheotaxis in fish[END_REF]. In yellow eels, these neuromasts located on the surface of the skin constitute the lateral line, and are grouped into white points of small diameter ( < 1 mm) OI value OI < 6.5 0 6.5 ≤ OI < 8.0 1 OI ≥ 8.0 2 1 The OI value of 6.5 corresponds to the minimal silvering threshold value of [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF] defined for female eels. The OI value of 8.0 corresponds to silvering threshold value defined by [START_REF] Marchelidon | Etude des variations de paramètres anatomiques et endocriniens chez l'anguille européenne (Anguilla anguilla) femelle, sédentaire et d'avalaison: Application à la caractérisation du stade argenté[END_REF] and [START_REF] Acou | Silvering of females eels (Anguilla anguilla) in two sub-populations of the Rhône Delta. -Bull[END_REF] resulting from the analysis of female eels.

along the body. During silvering, the lateral line is clearly differentiated [START_REF] Zacchei | Lateral line system during the life cycle of Anguilla anguilla L. -Boll[END_REF]. Visual observations in the field indicate that the white points, which are gradually surrounded by a black circle, increase in diameter to finally become black bodies of a diameter varying from 1 to 2 mm (Fig. 1). They form a distinct and continuous line along the body from the base of the pectoral fin to the tail [START_REF] Voigt | Response of anterior lateral line afferent neurones to water flow[END_REF]. They are located 1 to 2 cm apart, and their numbers vary between 5 and 30 according to the size and the maturity of the eel (A. Acou, personal observation).

The typical silver eel has a dark dorsal surface and a white ventral surface with metallic reflections, both zones being generally delimited by the lateral line (Fig. 1). The evaluation of the colour of an eel is subjective [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF] and does not constitute a relevant criterion for determining the different stages of eels [START_REF] Pankhurst | Changes in vision and olfaction during sexual maturation in the European eel Anguilla anguilla (L.)[END_REF]. The changes in the colour of eel skin are often variable, since they are not only dependent on the degree of maturity but also on growth habitat and adaptation of eel to its habitat [START_REF] Tesch | The eel. Biology and management of anguillid eels[END_REF]. The relative contrast observed between ventral and dorsal surface, generally well-marked in silver individuals, is a more objective criterion which disregards colorimetry (Table 1). However, both ventral and dorsal surfaces in silver eels can be quite variable in hue and intensity. We therefore considered in this study that the colour contrast used for our classification could be (i) the typical blackhish-brown back/silvery-white belly or (ii) a significant contrast between dorsal and ventral surface whatever the colours (with the exception of yellow and green coloration) (Table 1).

Thus, the clear differentiation of the lateral line, based on the presence of the black corpuscles, and the changes in skin colour, represented by the contrast of colours between the dorsal and ventral surfaces, are two objective qualitative criteria to be tested in our study in order to accurately determine the degree of silvering (Table 1).

Quantitative criterion

The ocular index (OI), defined as the relation between the total length of the eel and the mean size of the two eyes, was calculated using the following formula [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF]:

OI = {((A R + B R )/4) × ((A L + B L )/4) ×π/TL} × 100
where A and B are the horizontal and vertical eye diameters, respectively, TL is the total body length and R and L are the diameters of the right and left eyes. Measurement of horizontal and vertical eye diameters are made from the margins formed by the pigmented area adjacent to the unpigmented conjunctiva (Fig. 1). In this study, vertical and horizontal eye diameters were measured for both eyes in order to limit OI error imprecision. Measurements of total length were made with an ichthyometer (i. e. a ruler fixed in the bottom of a gutter) to the nearest mm. Measurements of eye diameters were made with a standard calliper to the nearest 0.02 mm. Length and OI data presented are means ± Standard Deviation (S. D.).

Three classes of OI defined from the threshold values at silvering of 6.5 [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF]) and 8.0 [START_REF] Marchelidon | Etude des variations de paramètres anatomiques et endocriniens chez l'anguille européenne (Anguilla anguilla) femelle, sédentaire et d'avalaison: Application à la caractérisation du stade argenté[END_REF][START_REF] Acou | Silvering of females eels (Anguilla anguilla) in two sub-populations of the Rhône Delta. -Bull[END_REF] characterize the ocular hypertrophy observed during metamorphosis (Table 1). Thus, each eel was described by the combination of the three categorical criteria in this order: clear differentiation of the lateral line (0 or 1), colour contrast of the skin (0 or 1) and OI class (0, 1 or 2) (Table 1).

Results

Sedentary fraction

214 and 149 eels were sampled by electrofishing in the Frémur and in the Oir, respectively. They were ranked according to the presence of the three silvering criteria (lateral line, skin colour, OI class) (Table 2). Eels that did not have any silvering criterion were the most abundant: 72.9 % (N = 156) and 91.3 % (N = 136) of total catches in the Frémur and Oir, respectively. Eels with the three criteria (combinations 1-1-1 and 1-1-2) represented 12.6 % (N = 27) and 6 % (N = 9) of total catches. In the Oir, no eel had a single criterion and eels with two criteria only represented 2.7 % (N = 4) of all individuals. In the Frémur, Table 2. Description of the combinations of silvering criteria observed for (i) sedentary eels caught by electrofishing in the Frémur and the Oir and (ii) migrant eels (descending) caught during downstream migration peaks in the Frémur and in the Loire. The combinations are the qualitative silvering criteria in the following order (1) clearly differentiated lateral line and (2) contrasting colour between dorsal and ventral surfaces, and (3) a quantitative criterion characterizing ocular hypertrophy compared to the OI threshold values (cf. Table 1 for the coding of the 3 criteria).

Number of silvering criteria observed

Combination

Sedentary Migrant Frémur Oir Frémur Loire 0 0 -0 -0 156 136 8 0 8.9 % (N = 19) and 7.7% (N = 12) of sampled eels had 1 and 2 cumulated criteria, respectively.

1 0-0-1 7 0 0 0 0-0-2 1 0 0 0 0-1-0 3 0 0 0 1-0-0 8 0 0 0 total 19 0 0 0 2 0-1-1 1 0 0 0 0-1-2 2 0 0 0 1-0-1 2 0 0 0 1-0-2 2 0 0 0 1-1-0 5 4 4 2 total 12 4 4 2 3 1-1-

Migrant fraction

General characteristics

373 and 153 individuals were caught in the Frémur trap and in the Loire, respectively (Table 2). In the Frémur, female silver eels represented 55.7% (N = 208) of the total catch. In the Loire, all eels sampled were females. The mean length of female silver eels in the Frémur (589.5 ± 97 mm) was significantly lower than in the Loire (765.8 ± 85 mm) [One-Way ANOVA, F(1,349) = 311.2, p < 0.001].

Combination of criteria

96.8 % (N = 361) and 98.7% (N = 151) of emigrant eels trapped had a differentiated lateral line, a contrasting colour and an OI value higher than the thres- hold value of 6.5 (combinations 1-1-1 and 1-1-2) in the Frémur and the Loire, respectively (Table 2). Among them, 88.4 % (N = 319) and 91.4 % (N = 138) had an OI ≥ 8.0 (combination 1-1-2) in the two sites, respectively. The mean OI of eels with the combinations 1-1-1 and 1-1-2 were 7.42 ± 0.39 and 10.16 ± 1.40 in the Frémur and 7.35 ± 0.42 and 10.30 ± 1.27 in the Loire. The 6.5 OI value appeared to be an accurate threshold value since only 1.1% and 1.3 % of emigrant eels had only two silvering criteria (combination 1-1-0) in the Frémur and Loire, respectively. No migrant eels at either sampling site had only one silvering criterion. Lastly, 2.1% (N = 8) of eels trapped in the Frémur were characterized by the combination 0 -0 -0 with a mean OI of 4.64 ± 0.98, whereas no eel was found with this combination in the Loire.

PIT-tagging survey

111 eels sampled by electrofishing in autumn 2002 in the Frémur were individually tagged (Table 3). Among them, 55 individuals were yellow eels (combination 0 -0 -0). None were recaptured in the following season as emigrant silver eel in the downstream trap, and 29.1 % (N = 16) were recaptured in the catchment in autumn 2003 as yellow eel (N = 11), presilver eel (N = 1) and silver eels (N = 4). Among the 12 eels tagged in autumn 2002 that showed between 1 and 2 silvering criteria, only 1 eel (combination 1-1-0) was recaptured in the downstream trap in 2002 as silver eel (combination 1-1-2). However, the difference of OI variation (coefficient of variation c. v.: 23.7 %) of this individual between the sedentary (combination 1-1-0) and the migrant phase (1-1-2) compared with the mean variation in silver eels during the same period (mean c. v.: 5 %, N = 7) suggested an error measurement of ocular diameters. This individual was therefore a posteriori reclassified as silver eel in the sedentary fraction. Indeed, among the 23 eels tagged that presented three silvering criteria (combinations 1-1-1 and 1-1-2), 8 were found in the downstream trap in 2002 (Table 3). In other words, 34.8 % of the potentially emigrant silver eels present in the catchment actually emigrated during the following migration season. The remaining silver eels were neither recaptured in the catchment in autumn 2003 nor in the trap in 2003.

Comparisons of silvering criteria within both life stages

Emigrant silver eels were characterized by a minimum reference combination corresponding to 1-1-1 (Table 2). If the identification of the stages was based on using only one criterion among the three (differentiated lateral line or contrasting colour or OI value higher than the threshold values of 6.5) this led to large variations in the estimated proportion of potentially silver eels compared to our minimum silver eel reference (combination 1-1-1, Table 4). For eels sampled in the sedentary phase, the variations observed were very large since they varied between -7 % and + 63 % in the Frémur and -22 % and + 44 % in the Oir (Table 4). Using the differentiated lateral line or the contrasting colour as a single silvering criterion led to an average overestimation of 50 % in the proportion of the potentially silver eels in both sites.

In the Frémur, we observed a positive and significant correlation between OI and TL values of eels having between 0 and 2 criteria of silvering (Pearson r = 0.62, p < 0.05, N = 182; Fig. 2). According to our silver reference, the yellow eels (0 criteria present) and/or eels in the process of silvering (1 to 2 criteria present) were not expected to be "emigration candidates" in the following season. However, among these eels in the process of silvering (1 to 2 criteria present), 48.4 % (N = 15) and 16.1 % (N = 5) had an OI higher than the threshold values of 6.5 and 8.0, respectively. As a consequence, the use of OI threshold values as a single criterion for silvering led to poor classification. First, the use of 6.5 threshold value led to overestimation ( + 56 %; Table 4) because all emigration candidates as well as eels in progress of silvering were selected. Second, the use of any other threshold value > 6.5 (7.0, 7.5 for example) lead to underestimation because part of the 'real' emigration candidates were ex- cluded of the classification whereas some eels in the process of silvering could also be selected (-7 % for OI threshold value 8.0 for example; Table 4) because of the positive linear relation between OI and TL in yellow eels and eels in progress of silvering (Fig. 2).

In the Oir, we also observed a positive and significant correlation between TL and OI values in eels showing 0 to 2 criteria of silvering (Pearson r = 0.61, p < 0.05, N = 136; Fig. 2). However, the low number of eels characterized by the presence of 1 and 2 criteria of silvering (N = 4) resulted in there being no difference between silver eels defined by the 1-1-1 combination and the OI threshold value of 6.5. Lastly, the threshold value of 8.0 led to an underestimation in the proportion of silver eels of -22 % (Table 4).

For eels sampled in the migrant phase, we recorded a negligible variation in the proportions of silver eels defined with the 1-1-1 reference combination and the use of the lateral line, colour or OI threshold value silveri ng criteria on their own (≤ 1 % of overestimation) (Table 4; Fig. 3). The use of the OI threshold value of 8.0 led to an underestimation in the proportions of emigrant silver eels of -12 % and -9 % in the Frémur and Loire, respectively (Table 4; Fig. 3).

Discussion

Our method, which aims at quickly identifying in the field a silver eel ready to migrate, in a standard way and without sacrificing fish, is based on the analysis of three standard macroscopic criteria which constitute adaptations to migration in the deep sea. At the term of their continental phase, the silver eel Gonado-Somatic Index does never exceed 2 % [START_REF] Fontaine | L'argenture de l'anguille: métamorphose, anticipation, adaptation[END_REF][START_REF] Marchelidon | Etude des variations de paramètres anatomiques et endocriniens chez l'anguille européenne (Anguilla anguilla) femelle, sédentaire et d'avalaison: Application à la caractérisation du stade argenté[END_REF][START_REF] Acou | Silvering of females eels (Anguilla anguilla) in two sub-populations of the Rhône Delta. -Bull[END_REF] whereas it can attain 80 % under experimental conditions simulating hydrostatic pressure conditions of the Sargasso Sea spawning site [START_REF] Dufour | Neuroendocrinologie de la reproduction de l'anguille: de la recherche fondamentale aux problèmes appliqués[END_REF]. Thus, silver eels are not able to reproduce at the end of their life in fresh waters, and they can be said to be "subadult" individuals that have already undergone a series of adaptations. The significant increase of in eye diameter observed during metamorphosis is accompanied with a number of changes in the retina including a decrease in the density of cones [START_REF] Pankhurst | Changes in vision and olfaction during sexual maturation in the European eel Anguilla anguilla (L.)[END_REF] consistent with the condition of the retinae of deep-sea fish in which cones are often absent. The clear differentiation of the lateral line observed during silvering is also a fundamental adaptation for the transoceanic migration. The lateral line is a mechanosensoral organ sensitive to currents permitting eels to perceive their close environment through predator avoidance [START_REF] Blaxter | Function of the free neuromasts of marine teleost larvae[END_REF], obstacle avoidance [START_REF] Hassan | Hydrodynamic imaging of the surroundings by the lateral line of the blind cave fish Anoptichthys jordani[END_REF]) and rheotaxis [START_REF] Montgomery | The lateral line can mediate rheotaxis in fish[END_REF], Montgomery et al. 2000). During metamorphosis, this organ becomes well-pigmented and very visible as a result of an increase in the number of neuromasts [START_REF] Zacchei | Lateral line system during the life cycle of Anguilla anguilla L. -Boll[END_REF]). In the same way, the capacity to increase the contrast between dorsal and ventral coloration, typical of pelagic organisms, provide an effi-cient camouflage, which increases their chance of survival [START_REF] Denton | A survey of reflectivity in silvery teleosts[END_REF].

Analyses of eels sampled in the sedentary phase in the Frémur and Oir watersheds have shown that the combination of the presence/absence of the three macroscopic criteria of silvering, provided a discrimination of the state of progress of this metamorphosis. We clearly distinguished a gradient of present criteria between yellow eels (0 criterion observed) and silver eels (3 criteria present), whereas a relatively large number of eels had between 1 and 2 silvering criteria. Almost 14.5 % of eels caught in the Frémur showed between 1 (8.9 %) and 2 (5.6 %) silvering criteria. These eels, which can be assumed to be in the process of metamorphosis, only accounted for 2.7% of the total catch in the Oir watershed. Acquisition of ocular hypertrophy (OI ≥ 6.5) as well as the differentiation of the lateral line in the Frémur, were the earliest signs of silvering since they each identified 42 % of eels having a sole silvering criterion, the contrasting colour appearing later. These results confirmed the conclusion of [START_REF] Durif | Analysis of morphological and physiological parameters during the silvering process of the European eel (Anguilla anguilla) in the Lake of Grand-Lieu (France)[END_REF][START_REF] Durif | The silvering process of Anguilla anguilla:a new classification from the yellow resident to the silver migrating stage[END_REF] who emphasized a succession of internal and external changes during the metamorphosis. In particular, they showed that the increase of the ocular diameter, correlated with the higher gonad weight, were the earliest changes, whereas the typical pigmentation of silver eel occurred generally at the end of the process. Therefore, our method allows an eel to be placed in a sequence of observed changes during metamorphosis. Nevertheless, the time required for an eel to complete metamorphosis remains unknown [START_REF] Cottrill | An examination of utilizing external measures to identify sexually maturing female American eels, Anguilla rostrata, in the St. Lawrence River[END_REF]. Such information is fundamental if we want to precisely determine those individuals that will migrate in the following season among eels in the process of silvering. In our study, the sampling was conducted in September/October in both sites, when most silvering eels had already finished their metamorphosis [START_REF] Tesch | The eel. Biology and management of anguillid eels[END_REF][START_REF] Fontaine | L'argenture de l'anguille: métamorphose, anticipation, adaptation[END_REF]. Moreover, PIT-tagging results confirmed that neither yellow nor presilver eels actually emigrated as silver eels in the following season.

Analyses of migrant eels caught in the Frémur and the Loire during the seaward peak migration confirmed that metamorphosis does not proceed further in freshwater. Our results clearly showed that > 97 % of emigrant silver eels were characterized by a differentiated lateral line, a contrasting color and an OI ≥ 6.5 as defined by [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF] (combinations 1-1-1 and 1-1-2).

A very low proportion of eels ( < 1.5 %) observed in the Frémur and Loire samples presented the combination 1-1-0. Considering the sequence of appearance of metamorphic characteristics during silvering, those eels may have an underestimated OI value. Indeed, OI implies the measurement of 5 variables (TL and eyes diameters) in a formula which can be sensitive to measurement error. A comparative study using 13 European eels and 3 replicated measurements by three different observers showed that the estimated amplitude of OI imprecision at 95 % confidence level was 1.8 whatever eel length and observer (Acou, unpubl. data). However, we assumed in this study that the combination 1-1-0 did not characterize a potentially silver eel that would migrate in the following season.

PIT-tagging survey provided evidence that only silver eels identified from their external appearance (1-1-1 or 1-1-2) in the catchment actually emigrated the following season. However, the recapture rate observed between the sedentary and the migratory fraction was rather low (34.8 %) and, therefore, the fate of the remaining identified silver eels (64.2 %) that were not recaptured is unknown. Our sampling effort allowed the recapture of tagged yellow eels (combination 0 -0 -0) in the following year confirming the sedentary lifestyle of this stage in inland freshwaters [START_REF] Laffaille | The yellow European eel (Anguilla anguilla L.) may adopt a sedentary lifestyle in inland freshwaters[END_REF]. In contrast, no identified silver eels tagged in autumn 2002 were recaptured the following year in the catchment suggesting that they at least moved away from their initial tagging site. Several hypotheses can be postulated. First, eels may escape from the downstream trap during migration. Secondly, natural mortality (about 5-10 % year -1 , e. g., Adam 1997), fishing pressure (no professional fishery but many anglers) or predation might have reduced the tagged population. Thirdly, a major obstruction located between the stations sampled by electrofishing and the downstream trap may alter downstream migration dynamics of silver eels. It is 14-m high and creates a reservoir to provide the county with drinking water. Most of the time, it is not full and the flow is controlled by pipes. During the period of downstream migration, the influence of classical environmental parameters (e. g., river discharge, rainfall) that triggered emigration intensity was overruled by dam management. Emigration was impossible until the reservoir was full and overflowing water permitted eels to slide down the weir (see [START_REF] Feunteun | European eel (Anguilla anguilla): prediction of spawner escapement from continental population parameters. -Can[END_REF] for further details) which caused an important delay of migration compared to a non-obstructed river and/or stop of the run [START_REF] Acou | Catadromous migration dynamics of European Eel (Anguilla anguilla L.) in a dammed catchment[END_REF][START_REF] Vollestad | Environmental factors regulating the seaward migration of silver eels (Anguilla anguilla). -Can[END_REF]). Indeed, an eel that has missed the favourable 'environmental windows' can possibly stop its catadromous migration and revert back to the growth phase [START_REF] Durif | Behavioral study of downstream migrating eels by radio-telemetry at a small hydroelectric power plant[END_REF][START_REF] Durif | The silvering process of Anguilla anguilla:a new classification from the yellow resident to the silver migrating stage[END_REF]. An extra sampling campaign in the reservoir should enable us to better understand the fate of these eels.

Our results further showed the need to use a multicriteria approach for estimating potentially silver eels. Indeed, the use of a sole silvering criterion among the three led to great differences in the estimated proportions of silver eels in comparison with our silver eel reference (combination 1-1-1). These variations are related to the fact that the use of a single criterion (i) cannot be used to position an individual in the metamorphic sequence observed during silvering and (ii) emphasizes the inherent skew due to the use of this criterion.

The most meaningful example is indisputably the use of OI the threshold values 6.5 [START_REF] Pankhurst | Relation of visual changes to the onset of sexual maturation in the European eel Anguilla anguilla (L.)[END_REF]) and 8.0 [START_REF] Marchelidon | Etude des variations de paramètres anatomiques et endocriniens chez l'anguille européenne (Anguilla anguilla) femelle, sédentaire et d'avalaison: Application à la caractérisation du stade argenté[END_REF][START_REF] Acou | Silvering of females eels (Anguilla anguilla) in two sub-populations of the Rhône Delta. -Bull[END_REF], which led to an overestimation of 56 % and an underestimation of 22 % of the potentially emigrating eels, respectively. Because of the imprecision in the calculation of OI and the fact that ocular hypertrophy appears to be among the first metamorphic changes [START_REF] Durif | Analysis of morphological and physiological parameters during the silvering process of the European eel (Anguilla anguilla) in the Lake of Grand-Lieu (France)[END_REF][START_REF] Han | Synchronous changes of morphology and gonadal development of silvering Japanese eel Anguilla japonica[END_REF], identifying silvering solely on the basis of OI threshold values thus poses problems since sampled eels have OI values that are close to these boundaries. Moreover, as the increase in OI is a gradual phenomenon that starts at the yellow eel stage [START_REF] Marchelidon | Etude des variations de paramètres anatomiques et endocriniens chez l'anguille européenne (Anguilla anguilla) femelle, sédentaire et d'avalaison: Application à la caractérisation du stade argenté[END_REF], large overlaps may occur between eels at the OI threshold values during sampling of a "sedentary" population fraction in late summer or in autumn, whatever the eel stages are. This was confirmed by [START_REF] Cottrill | An examination of utilizing external measures to identify sexually maturing female American eels, Anguilla rostrata, in the St. Lawrence River[END_REF] for female American silver eel A. rostrata. The relevance of our method depends on the combination of the three silvering criteria. We therefore recommend the use of the criteria defined in this study in order to precisely assess the proportion of potentially emigrating eels among a sedentary population fraction, i. e. we consider that a silver eel ready to migrate in the following season is characterized by a differentiated lateral line, a contrasting colour and an OI ≥ 6.5.

Silvering dynamics probably vary according to the ecological subunits (catchments, lagoon, etc.), habitat index and geographic situation, since such variables are known to influence the sex ratio, population structure, and population size [START_REF] Feunteun | European eel (Anguilla anguilla): prediction of spawner escapement from continental population parameters. -Can[END_REF]. Length and age at silvering for European eel are known to vary greatly between and within ecological units [START_REF] De Leo | A size and age-structured model of the European eel (Anguilla anguilla L.). -Can[END_REF][START_REF] Vollestad | Geographical variation in age and length at metamorphosis of maturing European eels: environmental effects and phenotypic plasticity[END_REF][START_REF] Acou | Silvering of females eels (Anguilla anguilla) in two sub-populations of the Rhône Delta. -Bull[END_REF]. The standardized identification of potentially silver migrant eels constitutes a fundamental step in order to analyse and model the life history traits of the European eel (age and size at silvering notably) at the scale of its distribution range. Moreover, this method should permit also to better understand the migratory dynamics of silver eels according to the degree of river regulation that may alter the natural flow regimes. This approach is thus required to start reliable management of this panmictic species, which is considered outside safe biological limits (ICES 1998).

Fig. 1 .

 1 Fig. 1. (a) Body section of A. anguilla silver eel, showing the left profile with the differentiated lateral line composed by black corpuscules (i. e. neuromasts) and effective colour contrast between the blackish-brown back and the silvery-white belly. (b) Head section of A. anguilla silver eel showing the left eye and 4 differentiated neuromasts.D. f., dorsal fin; V. f., ventral fin; Bl. br. ba., blackhish-brown back; Si. wh. bel., silvery white belly; Nm, neuromast; A, horizontal eye diameter; B, vertical eye diameter; Un. cj., unpigmented conjunctiva.

Fig. 2 .

 2 Fig. 2. Relation between the Ocular Index (OI) and body length (TL) for eels having between 0 and 2 silvering criteria and silver eels (silver combination reference 111) in catchments of (a) Frémur and (b) Oir. Significant correlations were observed for eels (p < 0.001). OI threshold values (TV) are indicated.

Fig. 3 .

 3 Fig. 3. Relation between Ocular Index (OI) and body length (TL) for emigrant silver eels (silver combination reference 1-1-1), for each sex, sampled in downstream migration in catchments of (a) Frémur and (b) Loire. No significant correlations are observed. OI threshold values (TV) are indicated.

Table 1 .

 1 Characteristics and coding of the criteria used to describe the silvering state of eels.

	Criteria	Description	Modalities	Coding
	Lateral line	Presence of at least one black corpuscle on the True	1
		lateral line	False	0
	Colour contrast Typical blackish brown back/silvery white belly
		Or	True	1
		Significant contrast between dorsal and ventral
		surfaces whatever the colours (except yellow or False	0
		green)
	Eyes 1

Table 3 .

 3 Number of eels marked by Passive Integrated Transponder (PIT)-tags following electrofishing (EF) in autumn 2002 in the Frémur and subsequent recaptures either in the downstream trap (trap) or in the catchment (EF) in autumn 2003.

	Tagging				Recapture
	No. of silvering	EF	Trap 2002	EF Autumn 2003: No.	Total
	Criteria	Autumn	3 silvering		of slivering criteria
	observed	2002	criteria	01 o r 2	3
	0	55	0	11	1	4	16
	1 or 2	12	1	0	0	0	1
	32 3	7	0	0	0	7
	Total	111	8	11	1	4	24

Table 4 .

 4 Differences expressed as percentage between numbers of silver eels defined according to the combination of the 3 silvering criteria present (combination reference 111; see Table2for reference number calculation) and those estimated by each criterion taken individually (in %). The number of eels is given in brackets.

	Fractions Sites	Reference Differentiated Contrasted	Ocular Index
			number of lateral line silver eels	colour	≥ 6.5	≥ 8.0
	Migrant	Frémur	361	+ 1 (365)	+ 1 (365)	0 (361)	-12 (319)
		Loire	151	+ 1 (153)	+ 1 (153)	0 (151)	-9 (138)
	Sedentary Frémur	27	+ 63 (44)	+ 41 (38)	+ 56 (42) -7 (25)
		Oir	9	+ 44 (13)	+ 44 (13)	0 (9)	-22 (7)

Acknowledgements

We are grateful to 'Kassin', 'Cuche' and 'Mimile' for their comments on the MS. Sonia and Lukas helped us for the photos. We would like to acknowledge Fredéric Marchand (Institut National de Recherche Agronomique de Rennes) and Richard Delanoë for their assistance during the field study on the Oir. This study was funded by the 'Fédération de Pêche d'Ille et Vilaine', the 'Contrat de Plan Poissons Migrateurs', and various regional and local councils. We thank Bob Britton for the English translation throughout the manuscript.