GAUSS SUMS OVER QUASI-FROBENIUS RINGS

PHILIPPE LANGEVIN AND PATRICK SOLÉ

Abstract. Quasi-Frobenius (QF) rings comprise finite fields, Galois rings and enjoy remarkable character-theoretic properties. We define two types of Gauss sums over QF rings : complete and incomplete. We compute the modulus of the former. We estimate the modulus of the latter from above and give an application to Gauss sums over Galois rings. We derive the analogue of Stickelberger theorem for Gauss sums indexed by the Teichmüller set.

1. Gauss sums over finite fields

Let K be a finite field of characteristic p and order q = p^{f}. Let χ be a multiplicative character of K and let ψ be an additive character of K. The Gauss sum related to the pair (χ, ψ) is defined by :

G_{K}(χ, ψ) = ∑_{x∈K^{×}} χ(x)ψ(x).

If both χ are ψ not trivial, one uses the orthogonality relations of characters to establish that G_{K}(χ, ψ) has absolute value \sqrt{q} and that

G_{K}(1, 1) = q − 1, G_{K}(χ, 1) = 0, G_{K}(1, ψ) = −1.

Let us denote respectively by ζ and ξ the principal p-root and principal (q − 1)-root of the unity in the complex numbers field. Let ℘ be any prime ideal above p lying in the cyclotomic ring Z[ζ, ξ]. The identification between Z[ζ, ξ]/℘ and the field K suggests the definition of two special characters : an additive canonical character μ_{K} which sends x on μ_{K}(x) = ζ^{tr_{K/Fp}(x)} and a multiplicative Teichmüller character which sends the class of ξ on ξ character of K, denoted by ω_{K}, it has order q − 1. For any positive integer a < q − 1, the ℘-adic valuation of the Gauss sum G_{K}(ω̄^{a}, μ) is given by the Stickelberger ’s congruences [18] :

G_{K}(ω̄_{K}^{a}, μ_{K}) ≡ −\frac{(ζ − 1)^{S(a)}}{R(a)} (mod ℘^{S(a)+1}) (1)

where S(a) is the sum of the p-digit of a and R(a) the product of their factorial. The factorization of the principal ideal generated by G_{K}(ω̄_{K}^{a}, μ_{K}) follows, see e.g. [10].

2. Gauss sums over finite rings

Let A be finite commutative ring. The multiplicative characters group is denoted by Â^{×} and the additive one by Â^{+}. As in the case of fields, we

A commutative local ring of characteristic p^{ℓ}
M maximal ideal of A
K residue field, A/M of order q
A^{×} units
S strong units, 1 + ann(M)
T Teichmüller units, ∼ K^{×}
U principal units, 1 + M, note that A^{×} ∼ T × U
μ generating character when A is a quasi-Frobenius ring

Figure 1. notations for a local commutative ring

define the Gauss sum related to the characters χ ∈ A^{×} and ψ ∈ A^{+} in the same manner :

G_{A}(χ, ψ) = ∑_{x∈A^{×}} χ(x)ψ(x)

It is a complete Gauss sum. For any subgroup X of A^{×}, we also define the incomplete Gauss sum

G_{X}(χ, ψ) = ∑_{x∈X} χ(x)ψ(x)

In this paper, we study the elementary properties of these Gauss sums. The quadratic Gauss sums have been studied by [9]. Some works about the non-commutative case can be found in [6, 7], see also the references pages 55–56 of [1].

All along this paper, we will assume commutativity. A commutative ring decomposes as a direct sum of local ring and we can focus our attention on Gauss sum over commutative, local ring. In order to get general results, it is necessary to put restrictions on the ring A. We will work with quasi-Frobenius rings which is a general framework since most of the usual finite rings are quasi-Frobenius : finite fields, the quotient rings Z/mZ, and the Galois rings are quasi-Frobenius. Moreover the class of quasi-Frobenius rings is stable with the classical constructors : a product of quasi-Frobenius rings, group rings and matrix rings based over quasi-Frobenius ring are quasi-Frobenius rings...

Up to the proposition 4.2, the results of the first sections are known, thought not readily avaible, see for example [8, 20]. For completion, important results on Gauss sums are collected in sections 5-6, with an application to the trivial incomplete Gauss sums. A generalization of the Stickelberger’s congruences is proposed in the last section.

3. Gauss sums over local rings

In this section, we settle notations and definitions summarized by fig.1. For details on the basic properties of finite rings see the monography of McDonald [12]. A finite commutative ring A of characteristic γ decomposes as a direct sum of rings

A = ⨁_{\begin{matrix}p|γ \\ p, prime \end{matrix}} A_{p}

where A_{p} is the p-ring {a ∈ A | ∃n, p^{n}a = 0}. The p-ring A_{p} decomposes as a direct sum of local rings of characteristic a power of p. It follows that the Gauss sum G_{A}(χ, ψ) is a product of Gauss sums over finite local rings, without a loss of generality, we may assume that A is local of characteristic p^{m}.

Let M the maximal ideal of A. The residue field K := A/M is a finite extension of degree f of the prime field F_{p}, of order q = p^{f}. For any ideal I ≠ A of A the set 1 + I is a subgroup of A^{×}, we will distinguish the group U := 1 + M and the group of strong units S := 1 + ann(M) assuming A is not a field ! Since M is a p-group, we can use the exact sequence coming from the reduction map :

1 ⟶ U ⟶ A^{×} ⟶ K^{×} ⟶ 1

to decompose A^{×} as a direct product of T × U where T is a cyclic group of order q − 1 that we call the Teichmüller units of A. The monoïd T ∪ {0} is isomorphic to (K, ×). We call T ∪{0} the Teichmüller set of A. Each a ∈ A admits a unique decomposition as a sum :

a = t + m, t ∈ T ∪ {0}, m ∈ M, (2)

where t is the Teichmüller representative of a. For any x, y in T ∪ {0}, we write x ⊕ y the Teichmüller representative of x + y. The Teichmüller set equipped with the laws × and ⊕ becomes a field isomorphic to K, of course.

The exact values of complete or incomplete Gauss sums are not easy to evaluate. We begin by an elementary case i.e. the case of complete Gauss sum associated to a multiplicative character trivial on the group U.

Proposition 1. Let ψ be an additive character and let χ be a multiplicative character. If χ is trivial on U then

G_{A}(χ, ψ) = \begin{cases}G_{K}(χ, ψ) |M|, & ψ trivial over M; \\ 0, & else. \end{cases}

Proof. Indeed,

G_{A}(χ, ψ) = ∑_{t∈T} ∑_{m∈M} χ(t + m)ψ(t + m) = ∑_{t∈T} χ(t)ψ(t) ∑_{m∈M} ψ(m).

4. Quasi-Frobenius rings

Let A a finite ring, may be not local. The group Â^{+} becomes a Amodule with the external law (a, ψ) ↦ ψ_{a}, where ψ_{a}(x) = ψ(ax). It is an isomorphism if and only if there exists a generating character μ i.e. a character such that the kernel does not contain a proper ideal. In this case, the ring is called quasi-Frobenius or equivalentely self-injective. Note that a direct sum of quasi-Frobenius rings is quasi-Frobenius and conversely. Of course, finite fields are quasi-Frobenius. The group rings and the matrix rings based over a quasi-Frobenius ring are quasi-Frobenius. Indeed, it is easy to construct a generating character for these rings starting from the one of the basic ring.

Theorem 1. Let A be a finite ring. The following assertions are equivalent :

(1) The ring A is quasi-Frobenius.

(2) The A-module A is injective.

(3) The map I ↦ ann(I) is an involution of the set of ideals of A.

Proof. For the the equivalence between (1) and (2) see [12], for those between (2) and (3) see [3].

Proposition 2. Let A be a local ring of maximal ideal M and residual field K. The ring A is quasi-Frobenius if and only if it satisfies one of the two following properties :

(1) there is only one minimal ideal.

(2) the order of ann(M) is equal to the order of K.

For example, the ring A = F_{2}[X, Y]/(X^{2}, XY, Y^{2}) is not quasi-Frobenius. Indeed, let x be the class of X, and let y be the class of Y. The ring A is local of maximal ideal M = (x, y). The annihilator of M is M of cardinality 4, and the residual field has only two elements. En passant, remark that :

ann(ann(A.x)) = ann(M) = M ≠ A.x

Proof. First point. Let I be the minimal ideal of A, and consider a nontrivial character over I. We can extend it to a character of A, see e.g. [16] which is a generating character. Conversely, let I be a minimal ideal of A. We have IM = I or IM = (0) and from Nakayama lemma [10], we deduce IM = (0) and thus ann(I) = M since the ring is quasi-Frobenius I = ann(M).

Second point. ann(M) is a K-vector space. If its dimension is not one 1 then it contains a subspace of dimension 1 which is necessary an ideal. Conversely, if I is a minimal ideal then it is included in ann(M) but that last one is a K-vector space of dimension 1.

Proposition 3. Let ψ be a generating character of A and let χ be a multiplicative character. The modulus of a complete Gauss sum is completely determined :

|G_{A}(χ, ψ)|^{2} = \begin{cases}|A|, & χ ⊥̸ S \\ 0, & χ ⊥ S \end{cases}

Proof.

\begin{aligned}|G_{A}(χ, ψ)|^{2} & = ∑_{x∈A^{×}} ∑_{y∈A^{×}} χ(x/y)ψ(x − y) \\ & = ∑_{z∈A^{×}} ∑_{y∈A^{×}} χ(z)ψ((z − 1)y) \\ & = |A| − |M| ∑_{z∈S} χ(z) \end{aligned} (3)

From this result, we deduce an estimation of for incomplete Gauss sum.

Proposition 4. Let X be a subgroup of A^{×}, let χ be a multiplicative character and let ψ be an generating character of A. If χ⊥S that is if χ is trivial over S then

|G_{X}(χ, ψ)| ≤ \frac{q − ϵ}{q}\sqrt{|A|}

where ϵ is the order of the intersection of X and S.

Proof.

G_{X}(χ, ψ) = \frac{1}{|X^{⊥}|} ∑_{θ∈X^{⊥}} G_{A}(χθ, ψ)

= \frac{1}{|X^{⊥}|} ∑_{θ⊥X,χθ⊥̸S} G_{A}(χθ, ψ)

≤ \frac{1}{|X^{⊥}|}|\{θ ∈ X^{⊥} | θ ⊥̸ XS\}|\sqrt{|A|}

The order of SX is \frac{q|S|}{ϵ}, whence

|X^{⊥} ∩ S^{⊥}| = |(XS)^{⊥}| = \frac{|A^{×}|}{|XS|} = \frac{|X^{⊥}||X ∩ S|}{|S|} = \frac{ϵ|X^{⊥}|}{q}

5. Galois rings

Let ℓ be a positive integer. Up to isomorphism , there exists one and only one Galois extension of degree f of the ring Z/p^{ℓ}Z. It is the Galois ring GR(p^{ℓ}, f) :

A := GR(p^{ℓ}, f) ∼ Z[X]/(g(X), p^{ℓ})

where g(X) is an unitary polynomial of degree f in Z[X] dividing X^{p^{f}} −X modulo p^{ℓ} whose the reduction modulo p is irreducible over F_{p}. It is a local ring of characteristic p^{ℓ} whose maximal ideal is principal generated by p. Any element z of A decomposes as :

z = ∑_{0≤i<ℓ} t_{i}p^{i}, (4)

where the t_{i} are in the Teichmüller set. The Galois group of A is cyclic of order f generated by the Frobenius automorphism which sends z to σ(z) = ∑_{0≤i<ℓ} t_{i}^{p}p^{i}. The trace of A sends z on tr_{A}(z) = ∑_{k=1}^{f} σ^{k}(z), and one can check that the character μ_{A}(z) = exp(2iπtr_{A}(z)/p^{ℓ}) is a generating character i.e. the Galois ring A is a quasi-Frobenius ring.

They are three ways to handle the Galois rings : Witt vectors, Hensel lift and extension of p-adic fields. Let us say a few words about these different approaches.

5.1. Witt ’s vectors. One uses the map

x_{0} + px_{1} + ⋯ + x_{ℓ−1}p^{ℓ−1} ↦ (x_{0}, x_{1}^{p}, …, x_{ℓ−1}^{p^{ℓ−1}}) (5)

to carry on K^{f} the structure of A [15]. For any index j, there exists polynomials S_{j} and P_{j} in Z[X_{0}, X_{1}, … X_{j}, Y_{0}, Y_{1}, …, Y_{j}] such that :

\begin{aligned}(x + y)_{j} & = S_{j}(x_{0}, x_{1}, …, x_{j}, y_{0}, y_{1}, …, y_{j}), \\ (xy)_{j} & = P_{j}(x_{0}, x_{1}, …, x_{j}, y_{0}, y_{1}, …, y_{j}). \end{aligned}

The polynomial S_{j} and P_{j} do not depend on ℓ.

S_{0}(x_{0}, y_{0}) = x_{0} + y_{0} S_{1}(x_{0}, x_{1}, y_{0}, y_{1}) = x_{1} + y_{1} + \frac{x_{0}^{p} + y_{0}^{p} − (x_{0} + y_{0})^{p}}{p}
P_{0}(x_{0}, y_{0}) = x_{0}.y_{0} P_{1}(x_{0}, x_{1}, y_{0}, y_{1}) = y_{o}^{p}x_{1} + y_{1}x_{0}^{p}

5.2. Hensel lifts. This approach is based on the following proposition which is a special case of Hensel’s Lemma.

Proposition 5. Let P (X) and Q(X) two monic polynomial of Z[X] such that P (X)Q(X) = X^{q} −X modulo p^{ℓ−1}. There exist two monic polynomials P_{ℓ} and Q_{ℓ} with integer coefficients satisfying the modulo p^{ℓ} congruences :

P_{ℓ}(X) ≡ P (X) Q_{ℓ}(X) ≡ Q(X), P_{ℓ}(X) ≡ Q_{ℓ}(X)

Modulo p^{ℓ}, P_{ℓ} is completely determined, it is the Hensel lift of P. Let F_{ℓ} the polynomial obtained after ℓ − 1 lifts of a unitary polynomial F_{1} whose the reduction modulo p is primitive then the Galois ring A is isomorphic to the quotient ring Z[X]/(p^{ℓ}, F_{ℓ}(X)), moreover the class of X generates the group of Teichmüller units.

5.3. Extensions of Q_{p}. Let α a root of the unity of order q − 1 in an algebraic extension of the field of p-adic numbers. The extension Q_{p}(α) of Q_{p} is unramified of degree f. The ideal generated by p in the ring Z_{p}[α] is maximal, the Galois ring A is isomorphic to

Z_{p}[α]/(p^{ℓ})

6. Character sums over Galois rings

Let F (X) = f_{0}(X) + p^{1}f_{1}(X) + ⋯ + p^{ℓ−1}f_{ℓ−1}(X), where the f_{i}(X)’s are polynomials whose the coefficients are in the Teichmüller set. Let Deg(F) denote the weighted degree of F i.e.

Deg(F) = max{p^{ℓ−i} deg(f_{i}) | 0 ≤ i < ℓ}

For any positive integer s, let A_{s} be a Galois extension of degree s of A. For any additive character ψ ∈ Â^{+} , let ψ^{(s)} ∈ Â_{s}^{+} the lift of ψ by the trace from A_{s} onto A :

ψ^{(s)} = ψ ∘ tr_{A_{s}/A}

In [2], is proved that the set of solutions (z, x) ∈ A × T of the equation σ(z) − z = F (x) deduces from a curve of genus \frac{(Deg(F)−1)(q−1)}{2} obtained by a tower of ℓ Artin-Schreier extensions.

Theorem 2. Let ψ be a non-trivial character of A. There exists Deg(F)−1 algebraic integers ω_{i} of modulus exactly \sqrt{q} such that

− ∑_{x∈T (A_{s})} ψ^{(s)}(F (x)) = ∑_{i=1}^{Deg(F)−1} ω_{i}^{s}

Proof. See [2].

Using a variation of the Stepanov’s method [11], one can prove a analog in the multiplicative case. More precisely, for any multiplicative character χ ∈ Â^{×} let us denote by χ^{(s)} the lift of χ by the norm of A_{s} onto A :

χ^{(s)} = χ ∘ N_{A_{s}/A}

Theorem 3. Let ψ be a non-trivial additive character of A. Let χ be a multiplicative character of A not trivial over T. There exists Deg(F) algebraic integers ω_{1}, ω_{2}, …, ω_{Deg(F)}, of modulus at most \sqrt{q} such that

− ∑_{x∈T (A_{s})} χ^{(s)}(x)ψ^{(s)}(F (x)) = ∑_{i=1}^{D(f)} ω_{i}^{s}

Proof. See [19].

These two theorems give bounds for Gauss sums under Teichmüller units.

|G_{T} (1, μ_{A}) + 1| ≤ (p^{ℓ−1} − 1)\sqrt{q} (6)

and if χ ∈ A^{×} is not trivial over T then

|G_{T} (χ, μ_{A})| ≤ p^{ℓ−1}\sqrt{q} (7)

when f = ℓ = 2 the estimate of the proposition (4) is slightly more precise.

7. Trivial incomplete Gauss sum

Let x be a Teichmüller element of the Galois ring A. Let us consider the p-adic developement of the trace of x :

tr_{A}(x) = ∑_{j=0}^{ℓ−1} t_{j}(x)p^{j}. (8)

where the functions t_{j} can be seen as functions from K into F_{p}. We say that the t_{j} are tracic functions. The first tracic function t_{0} is the trace of the field K. We determine the polynomial expression of the t_{1} by mean of Witt vectors. Let us continue assuming that p = 2. The second tracic function is equal to :

t_{1}(x) = ∑_{0≤i<j≤f−1} x^{2^{i}+2^{j}} (9)

The case of t_{2} is more difficult, but we can use the Hensel lift to get it. Indeed, let β ∈ K, for any integer ℓ, let P_{ℓ}(X) be the ℓ-lifting of P of the polynomial P (X) = ∏_{i=0}^{f−1}(X − β^{p^{i}}). Let us set

P_{ℓ}(X) = ∑_{i=0}^{f} S_{i}^{(ℓ)}(β)X^{f−i}.

Using the Graeffe’s root-squaring method [17], one can see that P_{ℓ+1}(X^{2}) is equal to (−1)^{f}P_{ℓ}(X)P_{ℓ}(−X). It follows

\begin{aligned}S_{1}^{(ℓ+1)}(β) & = 2S_{2}^{(ℓ)}(β) − S_{1}^{(ℓ)}(β)^{2}, \\ S_{2}^{(ℓ+1)}(β) & = 2S_{4}^{(ℓ)}(β) − 2S_{1}^{(ℓ)}(β)S_{3}^{(ℓ)}(β) + S_{2}^{(ℓ)}(β)^{2} \end{aligned}

In particular, we recover t_{1}(β) = ∑_{0≤i<j<f} β^{2^{i}+2^{j}}. Hence, t_{1} is a quadratic form and one can check that t_{2} is a boolean function of degree 4. The tracic function t_{1} is a quadratic form whose associated bilinear form is :

ϕ(x, y) = tr(x)tr(y) + tr(xy)

In particular, t_{1} is a quadratic form with maximal rank. The trivial Gauss sum depends on the Arf invariant ( see e.g [4], [5] ) its hyperplane intersection with t_{0}(x) = 0. For that approach, one has to determine of symplectic basis, it is not easy but done by Netchaev and Kuzmin [13], see also [14]. But, there exists a very simple way based on Theorem (2) which gives :

S_{A}(μ_{A}) = 1 + G_{T} (1, μ_{A}) = −(−1)^{m}(1 + i)^{m}

We finish this section with a nice formula that links the trivial Incomplete Gauss sums G_{T} (1, μ_{θ}) and G_{T} (1, μ_{A}) where θ ∈ A and μ_{θ}: x ↦ μ(θx).

Proposition 6. [21] Let A be a Galois ring of characteristic 4. Let θ = a + 2b ∈ A where a and b are Teichmüller elements then

S_{A}(μ_{θ}) = \begin{cases}μ_{A}(\frac{b}{a})S_{A}(μ_{A}), & a ≠ 0; \\ q, & a = b = 0; \\ 0, & a = 0 ≠ b. \end{cases}

Proof. The proof is based on the well known relation

x ⊕ y = x + y + 2\sqrt{xy} (10)

that gives the Teichmüller representative of the sum two elements of the Teichmüller set. In particular, μ_{A}(x ⊕ y) = μ_{A}(x + y + 2xy), and :

\begin{aligned}1 + G_{T} (1, μ_{θ}) & = ∑_{x∈T ∪\{0\}} μ_{A}((a + 2b)x) = ∑_{x∈T ∪\{0\}} μ_{A}((1 + 2\frac{b}{a})x) \\ & = ∑_{x∈T ∪\{0\}} μ_{A}(x + 2\frac{bx}{a}) = μ_{A}(b/a) ∑_{x∈T ∪\{0\}} μ_{A}(x ⊕ b/a) \end{aligned}

8. Stickelberger theorem for incomplete Gauss sums

In this section, we show that Gauss sums over Galois ring of characteristic 4 satisfy analogues of Stickelberger congruences. The notation are unchanged : μ_{A} and μ_{K} are the canonical additive characters of A := GR(4, f) and its residue field is denoted by K. We start by a formula linking Gauss sums with the character sum

H(χ, μ) = ∑_{s∈T} μ_{A}(s)χ(s/s + 1),

we get the relation

G_{T} (χ, μ_{A})^{2} = G_{K}(χ, μ_{K})[1 + G_{T} (1, μ_{A}) + H(χ, μ_{A})]

Proof. This is a straightforward calculation. Let χ be a multiplicative character and let us square G_{T} (χ, μ_{A}).

G_{T} (χ, μ_{A})^{2} = ∑_{x∈T} ∑_{y∈T} χ(xy)μ_{A}(x + y)

= ∑_{x∈T,y∈T} χ(xy)μ_{A}(x ⊕ y + 2xy) by 10,

= ∑_{\begin{matrix}x⊕y=u \\ xy=v \end{matrix}} χ(v)μ_{A}(u + 2v)

= ∑_{x∈K^{×}} χ(x^{2})μ_{K}(x^{2}) + 2 ∑_{\begin{matrix}u∈T,v∈T \\ tr_{K/F_{2}}(v/u^{2})=0 \end{matrix}} χ(v)μ_{A}(u + 2v)

= G_{K^{×}}(χ, μ_{K}) + 2 ∑_{\begin{matrix}u∈T,w∈T \\ tr_{K/F_{2}}(w)=0 \end{matrix}} χ(u^{2}w)μ_{A}(u + 2u^{2}w)

changing u by u^{2},

= G_{K^{×}}(χ, μ_{K}) + 2 ∑_{\begin{matrix}u∈T,w∈T \\ tr_{K/F_{2}}(w)=0 \end{matrix}} χ(uw)μ_{A}(u + 2uw)

= G_{K^{×}}(χ, μ_{K}) + 2 ∑_{u∈T} μ_{A}(u)χ(u) ∑_{\begin{matrix}w∈T \\ tr_{K/F_{2}}(w)=0 \end{matrix}} χ(w)μ_{K}(uw).

One can check, for example with the Poisson formula, that

∑_{tr_{K/F_{2}}(w)=0} χ(w)μ_{K}(uw) = \frac{1}{2}[G_{K}(χ, μ_{K}_{u+1}) + G_{K}(χ, μ_{K}_{u})]

= \frac{1}{2} \begin{cases}[χ̄(u + 1) + χ̄(u)]G_{K}(χ, μ_{K}), & u ≠ 1; \\ G_{K}(χ, μ_{K}), & u = 1. \end{cases}

and we can continue,

G_{T} (χ, μ_{A})^{2} = G_{K^{×}}(χ, μ_{K}) + μ_{A}(1)G_{K}(χ, μ_{K})

+ G_{K}(χ, μ_{K}) ∑_{u∉{0,1}} μ_{A}(u)χ(u)[χ̄(u + 1) + χ̄(u)]

= G_{K}(χ, μ_{K})[1 + G_{T} (1, μ_{A}) + H(χ, μ_{A})]

Proposition 7. Let ℘ be a prime ideal above 2 in the ring Z[i, ξ]. Let ω be the Teichmüller character of the Galois rings that occurs in the Stickelberger congruences of the residue field. Then

val_{℘}(G_{T} (ω^{a}, μ_{A})) = S(a).

Proof. It suffices to show that the sum H(χ) is invertible modulo ℘.

∑_{u∉{0,1}} ω^{a}(u + 1)ω̄^{a}(u)μ(u) ≡ ∑_{u∉{0,1}} ω^{a}(u + 1)ω̄^{a}(u) (mod P)

≡ ∑_{u∉{0,1}}(u + 1)^{a}u^{q−1−a} ≡ −1

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Philippe Langevin, GRIM, université de Toulon, B.P. 132, 83957 La Garde France

E-mail address: langevin@univ-tln.fr, Patrick.Sole@agat.univ-lille1.fr