Yves Aubry
email: yaubry@univ-tln.fr

Philippe Langevin
email: langevin@univ-tln.fr

On The Weights of Binary Irreducible Cyclic Codes

This paper is devoted to the study of the weights of binary irreducible cyclic codes. We start from McEliece's interpretation of these weights by means of Gauss sums. Firstly, a dyadic analysis, using the Stickelberger congruences and the Gross-Koblitz formula, enables us to improve McEliece's divisibility theorem by giving results on the multiplicity of the weights. Secondly, in connection with a Schmidt and White's conjecture, we focus on binary irreducible cyclic codes of index two. We show, assuming the generalized Riemann hypothesis, that there are an infinite of such codes. Furthermore, we consider a subclass of this family of codes satisfying the quadratic residue conditions. The parameters of these codes are related to the class number of some imaginary quadratic number fields. We prove the non existence of such codes which provide us a very elementary proof, without assuming G.R.H, that any two-weight binary irreducible cyclic code c(m, v) of index two with v prime greater that three is semiprimitive.

Introduction

In a recent paper [START_REF] Wolfmann | Are 2-weight projective cyclic codes irreducible[END_REF], Wolfmann has proved that a two-weight binary cyclic code is necessarily irreducible. On the other hand, it is well-known that there exist two infinite classes of irreducible cyclic codes with at most two nonzero weights: the subfield codes and the semiprimitive ones. Apart from these two families, 11 exceptional codes have been found by Langevin (see [START_REF] Ph | A new class of two weight codes, Finite Fields and Their Applications[END_REF]) and, Schmidt and White (see [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF]). It has been conjectured in the later paper that this is the whole story. This question is investigated in this paper in the case of the characteristic two.

In the first part of this article, we recall the McEliece interpretation of the weights of an irreducible cyclic code by means of linear combinations of Gauss sums. McEliece's divisibility theorem plays a significant role in the study of weight distributions of irreducible cyclic codes. In particular, Schmidt and White deduce a necessary and sufficient condition for an irreducible cyclic code to be a two-weight code.

In the second part, we use the Stickelberger congruences and the Gross-Koblitz formula to obtain two new results that improve McEliece's theorem. We study the Boolean functions that appear in the dyadic expansion of the weight of a codeword. The estimation of their algebraic degree leads us to results on the divisibility concerning multiplicities by means of Ax's and Katz's theorems.

In the last part, we are interested in Schmidt and White's conjecture on irreducible cyclic c(m, v) codes. Since they proved that it holds for codes of index two (conditionally on the Generalized Riemann Hypothesis), we focus our attention on this class of codes. We prove that, conditionally on G.R.H., there are an infinity of binary irreducible cyclic c(m, v) codes of index two with v prime. This result can be seen as an analogue of the Artin conjecture on primitive roots. Thus, this family of codes seems to be interesting in view Schmidt and White's result. Then, we use a result of Langevin in [START_REF] Ph | A new class of two weight codes, Finite Fields and Their Applications[END_REF] to prove that there does not exist any two-weight irreducible cyclic c(m, v) code of index two with v > 3 prime and v ≡ 3 (mod 4). This provides us an elementary proof, without assuming G.R.H, of a particular instance of the Schmidt and White conjecture, namely that any two-weight binary irreducible cyclic c(m, v) code of index two with v prime greater that 3 is semiprimitive.

McEliece's theorem

Let L be a finite field of order q = 2 m . Let n be a divisor of q -1 and write v = (q -1)/n. Let ζ be a primitive n-th root of unity in L. Consider the following map Φ:

Φ : L -→ F n 2 a -→ Tr L/F2 (aζ -i ) n-1 i=0
where Tr L/F2 is the trace of the field L over F 2 . The image Φ(L) of L by Φ is an irreducible cyclic code of length n, denoted c(m, v), see [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF] for the material about these codes. Its dimension is equal to the multiplicative order of 2 modulo n, denoted ord n (2). Any binary irreducible cyclic code can be viewed as a c(m, v) code, so let us consider such codes. For an element t of L, let us denote by w(t) the weight of Φ(t). The well-known McEliece formula gives the weight of Φ(t) in term of Gauss sums

w(t) = n 2(q -1) q + χ∈Γ \{1} τ L (χ) χ(t) (1)
where Γ is the subgroup of multiplicative characters of L * that are orthogonal to ζ, see [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF]. The Gauss sum τ L (χ) is implicitly defined with respect to the canonical additive character, say µ L , of L. By definition,

τ L (χ) = - x∈L * χ(x)µ L (x).
Note that a change of additive character produces a permutation of weights. As in [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF], let us denote by θ the greatest integer such that, for all non trivial χ ∈ Γ , 2 θ divides τ L (χ). The famous Stickelberger theorem (see next section) claims

θ = min 0<j<v S(jn)
where S(k) denotes the sum of the bits in the binary expansion of the natural integer k.

Theorem 1 (McEliece). All the weights of the irreducible cyclic code c(m, v) are divisible by 2 θ-1 . Moreover, one of them is not divisible by 2 θ .

Sketch of the proof. It suffices to group together the terms of minimal 2-adic valuation in [START_REF] Ax | Zeroes of polynomial over finite fields[END_REF] to get the first part of the theorem. The second part comes from the independence (modulo 2) of the multiplicative characters of L.

A two-weight code is a code with two nonzero Hamming weights. The McEliece formula appears as the Fourier inversion formula of the map t → f (t) = qz(t) -n, where z(t) denotes the number of zero components of the codeword Φ(t). Moreover if G denotes the group of order n in L * , the map f (t) is defined over the quotient group V = L * /G. Let us set f := ord v (2), and since nv = 2 m -1, f divides m and we set m = f s.

Theorem 2 (Schmidt-White). The irreducible cyclic code c(m, v) is a twoweight code if and only if there exists an integer k satisfying the three conditions

(i) k divides v -1 (ii) k2 sθ ≡ ±1 (mod v) (iii) k(v -k) = (v -1)2 s(f -2θ)
Sketch of the proof. Using Fourier analysis, one can prove that

D = {t ∈ V | 2 θ divides w(t)}
is a difference set of order 2 f -2θ implying (iii). This set or its complementary is a (v, k, λ) difference set satisfying (i) & (ii). Surprisingly, the three conditions are sufficient.

Traditionally, one says that 2 is semiprimitive modulo v when -1 is in the group generated by 2 in (Z/vZ) * . In this case, all the Gauss sums are rationals, equal to √ q whence θ = f /2, and the code c(m, v) is a two-weight code with k = 1. Each of these assertions characterizes the semiprimitivity.

Dyadic weight formula

In this section, we analyse dyadicaly the function

f (t) = 1 =χ∈Γ τ L (χ) χ(t) = 2 θ +∞ i=0 f i (t)2 i (2)
where the f i are Boolean functions i.e. map L into {0, 1}. By definition, see [START_REF] Macwilliams | The Theory of Error-Correcting Codes[END_REF], the degree of a Boolean function f defined over a

F 2 -space E of dimension m is equal to the smallest degree of a polynomial p ∈ F 2 [X 1 , X 2 , . . . , X m ] such that ∀(x 1 , x 2 , . . . , x m ) ∈ F m 2 p(x 1 , x 2 , . . . , x m ) ≡ f (x 1 β 1 + x 2 β 2 + • • • + x m β m ) (mod 2),
where (β 1 , β 2 , . . . , β m ) is any basis of L considered has a vector space over F 2 .

In the first part of this section, we use the Stickelberger's congruences to determine the algebraic degree of f 0 . In the second part, we will use the Gross-Koblitz formula to give an upper bound on the degree of f 1 . For this, we realize the finite field L as the quotient ring Z 2 [ξ]/(2), where ξ is a (q -1)-root of unity in an algebraic extension of Q 2 the field of 2-adic numbers. The Teichmüller character of L, denoted by ω, is the multiplicative character of L defined by the relation ω(ξ (mod 2)) = ξ.

It is important to remark that t → ω(t) (mod 2) is nothing but the identity of L * . The Gross-Koblitz formula below (see [START_REF] Koblitz | p-adic Analysis: a Short Course on Recent Work[END_REF]) claims the existence of an additive character ψ such that, for any residue a modulo q -1, the following holds:

τ L (ω a , ψ) = (-2) S(a) f -1 j=0 Γ 2 1 - 2 j a q -1 (3)
where S(a) = a 0 + a 1 + . . . + a f -1 is the sum of the bits of a = f -1 i=0 a i 2 i , x is the fractional part of x, and Γ p the 2-adic gamma function defined by

∀k ∈ N, Γ 2 (k) = (-1) k j<k, 2 |j j, ∀s ∈ Z 2 , Γ 2 (s) = lim k→s Γ 2 (k).

3.1

The function f 0 .

The first approximation of the 2-adic gamma function gives the famous Stickelberger's congruences τ L (ω a , ψ) ≡ 2 S(a) (mod 2 1+S(a) ).

We introduce the set

J = {j | S(jn) = θ}, so that f 0 (t) ≡ j∈J t jn (mod 2).
Using any F 2 -basis of L, the function f 0 becomes a mapping from F f 2 into F 2 . Since all the exponents jn have a constant 2-ary weight equal to θ, the algebraic degree of f 0 is less or equal to θ. The previous McEliece theorem claims that the weights are divisible by 2 θ-1 . The next result gives precisions concerning the multiplicities of the weights. Let us recall that by Ax's theorem (see [START_REF] Ax | Zeroes of polynomial over finite fields[END_REF]), for any polynomial f ∈ F 2 [X 1 , X 2 , . . . , X m ] of degree k, the number of solutions in F m 2 of the equation :

f (x 1 , x 2 , . . . , x m ) = 0 is divisible by 2 ⌈ m k ⌉-1
where ⌈r⌉ denotes the smallest integer greater or equal to r.

Theorem 3. The number of codewords of weight of dyadic valuation θ -1 is divisible by 2 ⌈f /θ⌉-1 .

Proof. The weight of Φ(t) has valuation θ -1 if and only if f 0 (t) = 1. By Ax's theorem the number of solutions is divisible by 2 ⌈f /θ⌉-1 since the degree of the Boolean function f 0 is less or equal to θ.

Example 1. The weights of the binary [23, 11] (subcode of the Golay code) are : 0, 8, 12 and 16 whence θ = 3 and Theorem 3 claims that the number of codewords of weight 12 is divisible by 2 ⌈11/3⌉-1 = 8. According to [START_REF] Macwilliams | The weight distributions of some minimal cyclics codes[END_REF], this number is 56 × 23 = 8 × 161.

Remark 1. In the case of a two-weight code, the condition (3) of the theorem of Schmidt and White implies a divisibility by a large power of 2. It seems very interesting to study more precisely the function f 0 .

The function f 1

The first values of the 2-adic gamma function are: Γ 2 (0) = 1, Γ 2 (1) = -1, Γ 2 (2) = +1, Γ 2 (3) = -1, and Γ 2 (4) = 3 ≡ -1 (mod 4). In particular,

Γ 2 ( 1 - a q -1 ≡ Γ 2 (1 + a 0 + a 1 2) ≡ (-1) 1+a0+a0a1 (mod 4)
and we get the congruence

τ L (ω a , ψ) ≡ (-1) Q(a) 2 S(a) (mod 2 2+S(a) ) (4)
where Q(a) = f + a 0 a 1 + a 1 a 2 + . . . + a f -1 a 0 . To improve our approximation of f (t), we introduce the set K = {k ∈ N | 1 ≤ k < v, S(kn) = θ + 1} and the partition J ǫ = {j ∈ J | Q(jn) ≡ ǫ (mod 2)}. We have

f 0 (t) + 2f 1 (t) ≡ j∈J0 ω jn (t) - j∈J1 ω jn (t) + 2 k∈K ω kn (t) (mod 4).
The Boolean function f 1 depends on the sets K and J 1 but also of the "carry function" g(t) corresponding to the relation j∈J ω jn (t) ≡ f 0 (t) + 2g(t) (mod 4).

By classical 2-adic tricks, we get:

g(t) = 1 2 j∈J ω jn (t) -( j∈J ω jn (t)) 2 ≡ j<j ′ ω (j+j ′ )n (t) (mod 2).
Reducing modulo 2, gluing all pieces together, we get:

f 1 (t) = j<j ′ t (j+j ′ )n + j∈J1 t jn + k∈K t kn .
Let us recall that Katz's divisibility theorem (see [START_REF] Katz | On a theorem of Ax[END_REF]) implies that for any pair of polynomials f 1 and

f 2 ∈ F 2 [X 1 , X 2 , . . . , X m ] of degree k 1 ≤ k 2 ,
the number of solutions in F m 2 of the system of equations :

f 1 (x 1 , x 2 , . . . , x m ) = 0 f 2 (x 1 , x 2 , . . . , x m ) = 0 is divisible by 2 ⌊ m-k 1 -k 2 k 2
⌋ where ⌊r⌋ denotes the largest integer smaller or equal to r. Theorem 4. Let w 0 be an integer. The number of codewords with weight of the form w 2 θ-1 with w ≡ w 0 (mod 4) is divisible by 2 ⌊ f -3θ 2θ ⌋ . Proof. Let a + 2b + • • • be the 2-adic decomposition of w 0 . The weight of Φ(t) is of the form w 2 θ-1 if and only if t is a solution of the system

f 0 (t) = a, f 1 (t) = b.
The result is a consequence of the above Katz divisibility theorem since the algebraic degrees of f 0 and f 1 are respectively less or equal to θ and 2θ.

Example 2. A sufficient condition to obtain a non trivial result is n > 1 and 5θ < f . The first instance is the [11, 10]-code (v = 93, θ = 2) and the second one is the [6765, 20]-code (n = 6765, v = 155, θ = 4). According to [START_REF] Macwilliams | The weight distributions of some minimal cyclics codes[END_REF], the weight distribution is given by Tab. [START_REF] Ax | Zeroes of polynomial over finite fields[END_REF]. All the weight are divisible by 8, and the number A w of codewords of weight w satisfy: w≡0 (mod 4)

A w = 1 + 25n ≡ 0 mod 2, w≡1 (mod 4)
A w = (5 + 45)n ≡ 0 mod 2, w≡2 (mod 4)

A w = (5 + 20 + 20 + 5)n ≡ 0 mod 2, w≡3 (mod 4)
A w = (4 + 25 + 1)n ≡ 0 mod 2. Is there infinitely many primes v such that 2 generates the squares modulo v ? Before answering this question, recall that the Artin conjecture asserts that 2 is a primitive root for infinitely many primes (the conjecture is proved by Hooley assuming the Generalized Riemann Hypothesis). In other words, there is infinitely many primes v such that the order of 2 modulo v is equal to v -1.

We consider here an analogue question : is there infinitely many primes v such that 2 generates exactly the squares modulo v ? We can give an another formulation of this question : is there infinitely many primes v such that the order of 2 modulo v is equal to v-1 2 or equivalently such that 2 has index 2 modulo v ? Indeed, these problems are equivalent since the group (Z/vZ) * is cyclic and the subgroup of squares has index 2 (v odd).

For a positive integer x, let H(x) be the cardinality of the set

{v ≤ x | v prime and ord v (2) = v -1 2 }.
Murata has proved (see [START_REF] Murata | A problem analogous to Artin's conjecture for primitive roots and its applications[END_REF]) that G.R.H. implies that for every ε > 0,

H(x) = 3 8 δπ(x) + O 2 ε x log log x log 2 x ,
where

δ = ℓ prime 1 - 1 ℓ(ℓ -1)
is the Artin constant.

Then, under G.R.H., we can use the previous result of Murata to conclude positively to our question: there is infinitely many primes v such that 2 has index 2 modulo v.

Recall that a code c(m, v) is said to have index 2 if the multiplicative order of 2 modulo v is equal to ϕ(v)/2, where ϕ is the Euler function. In particular, we have shown that: Proposition 1. Conditionally on G.R.H., there are infinitely many index 2 binary irreducible cyclic codes c(m, v) with v prime. Remark 2. Recall that an index 2 binary irreducible cyclic codes c(m, v) with v prime has at most three different nonzero weights. Thus, these codes are good candidates to be two-weight codes. By the way, we can state that, conditionally on G.R.H., there are infinitely many binary cyclic codes with at most three different nonzero weights.

The residue quadratic case and the semiprimitivity

For the study of a special class of three-weight codes, Langevin in [START_REF] Ph | A new class of two weight codes, Finite Fields and Their Applications[END_REF] introduced more restrictive conditions on our integer v which lead us to the quadratic residue case for v, namely the index 2 case with the additional conditions that v is an odd prime greater than 3 with v ≡ 3 (mod 4). In other words, the integer v satisfies the quadratic residue conditions if:

(i) v is a prime greater than 3, (ii) ord v (2) = v-1 2 , (iii) v ≡ 3 (mod 4).
This case is of particular interest because of an explicit relation between the class number h of the imaginary quadratic number field Q( √ -v) and the Gauss sums (see [START_REF] Ph | A new class of two weight codes, Finite Fields and Their Applications[END_REF]).

Proposition 2. There does not exist a two-weight binary irreducible cyclic code satisfying the quadratic residue conditions. Proof. Let s be the integer introduced in the section [START_REF] Katz | On a theorem of Ax[END_REF]. By theorem 3.3 of [START_REF] Ph | A new class of two weight codes, Finite Fields and Their Applications[END_REF], we know that the code c(m, v) has at most two weights if and only if

v + 1 4 = 2 hs . (5)
The previous relation implies that:

2 hs+2 ≡ 1 (mod v).
This implies that the order of 2 modulo v divides hs + 2. But, by hypothesis, we have ord v (2) = (v -1)/2. Then, taking the logarithm in (5), we have the inequalities:

v -1 2 ≤ hs + 2 = log( v + 1 4 ) + 2 (6)
implying v = 7. But this leads to a code with only one nonzero weight: the proposition follows.

The conjecture of Schmidt and White in even characteristic states that an irreducible cyclic code c(m, v) is a two-weight code if and only if it is a semiprimitive code. They proved it, conditionally on G.R.H. for all index 2 codes. We can now prove it also for all index 2 codes with v prime greater than 3 but without assuming G.R.H.: Theorem 5. A binary irreducible cyclic code c(m, v) of index 2 with v prime greater than 3 is a two-weight code if and only if it is a semiprimitive code.

Proof. The odd prime v is congruent to 1 or 3 modulo 4. The last congruence comes to the quadratic residue case and the previous proposition implies that the code has three weights. The first congruence v ≡ 1 (mod 4) implies that -1 is a square modulo v and thus is a power of 2 modulo v since 2 has index 2 modulo v and then generates the squares. Thus, the code is semiprimitive.

The converse is a well-known result: the semiprimitivity implies that the code has two weights.

Table 1 .

 1 Weight distribution of the [6765, 20] irreducible cyclic code. The number of codewords of weight w is equal to µ × n, w denotes the congruence of w 8 modulo 4.

	w 3272 3280 3320 3352 3376 3392 3400 3408 3448 3504
	w 1	2	3	3	2	0	1	2	3	2
	µ 5	5	4	25 20 25 45 20	1	5
	4 Two-weight Binary Irreducible Cyclic Codes
	4.1 Primes which generate squares and index 2 codes


Yves Aubry
email: yaubry@univ-tln.fr

Philippe Langevin
email: langevin@univ-tln.fr

On the semiprimitivity of cyclic codes

We prove, without assuming the Generalized Riemann Hypothesis, but with at most one exception, that an irreducible cyclic code c(p, m, v) with v prime and p of index 2 modulo v is a two-weight code if and only if it is a semiprimitive code or it is one of the six sporadic known codes. The result is proved without any exception for index-two irreducible cyclic c(p, m, v) codes with v prime not congruent to 3 modulo 8. Finally, we prove that these two results hold true in fact for irreducible cyclic code c(p, m, v) such that there is three p-cyclotomic cosets modulo v.

Introduction

Irreducible cyclic codes are extensively studied in the literature. They can be defined by three parameters p, m and v and are denoted c(p, m, v) (see section 2 for a precise definition). Such codes with only few different (Hamming) weights are highly appreciated, especially those with exactly two non-zero weights, called two-weight codes. The classification of two-weight codes is a classical problem in coding theory (see [START_REF] Calderbank | The geometry of two-weight codes[END_REF]); it is still an open problem but recent progress has been made. An infinite family, namely the semiprimitive codes (i.e. when -1 is a power of p modulo v), and eleven sporadic examples are known. Schmidt and White in [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF] provided evidence to conjecture that this is the whole story: Conjecture 1. An irreducible cyclic code c(p, m, v) is a two-weight code if and only if it is a semiprimitive code or it is one of the eleven sporadic known codes.

They proved their conjecture, conditional on the Generalized Riemann Hypothesis (G.R.H.), for index-two codes, that is when p has index 2 modulo v. Note that semiprimitive codes have two non-zero weights and thus only the "only if" part had to be proved.

We considered in [START_REF] Aubry | On the weights of binary irreducible cyclic codes, Coding and Cryptography[END_REF] the conjecture in the binary case and we proved it in a particular case without assuming G.R.H.. Our main result here is a proof of this conjecture without assuming G.R.H. but with at most one exception in the case where p has index 2 and v is prime. We prove before, using near-primitive root densities and conditionally on G.R.H., that for any prime number p there are infinitely many such codes namely index-two irreducible cyclic codes c(p, m, v) with v prime.

1

We prove the conjecture without any exception (and without assuming G.R.H.) in the case where p has index 2 and v is a prime not congruent to 3 modulo 8. Finally, we remark that the results hold true in fact for irreducible cyclic codes c(p, m, v) with v an integer such that there is three p-cyclotomic cosets modulo v.

Irreducible cyclic codes and McEliece weight-formula

Let us introduce irreducible cyclic codes over a prime finite field (indeed, it is enough for our purpose, namely the classification of twoweight irreducible cyclic codes, to consider such codes over prime fields, as remarked in [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF]).

Let p be a prime number and consider the finite field K with p elements. Let L be the extension of degree m of K, consider a divisor n of p m -1 and write v = (p m -1)/n (thus v and p are coprime). Let ζ be a primitive n-th root of unity in L (i.e. ζ is a generator of the cyclic subgroup of order n of the multiplicative group L * ). We define the c(p, m, v) code to be the image of the following map Φ m :

Φ m : L -→ K n t -→ Tr L/K (tζ -i ) n-1 i=0
where Tr L/K is the trace of the field L over K.

It is a code of length n and dimension ord n (p), the multiplicative order of p modulo n. Every irreducible cyclic code over K can be viewed as a c(p, m, v) code (see [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF]), so we can take c(p, m, v) as the definition of irreducible cyclic codes over K of length n. The c(p, m, v) codes are known to be projective or saturated according to whether gcd(n, p -1) = 1 or gcd(n, p -1) = p -1. As remarked in [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF], we may assume the saturated situation. Now we are interested in the weight w(t) of a codeword Φ m (t) of such a code, for t ∈ L * . Let χ be a character of the multiplicative group L * and let [START_REF] Aubry | On the weights of binary irreducible cyclic codes, Coding and Cryptography[END_REF] τ

L (χ) = - x∈L * χ(x)e 2iπ p Tr L/K (x)
be the Gauss sum associated with χ.

Let V be the subgroup of L * of index v and let Γ be the subgroup of characters of L * which are trivial both on V and K * . Note that the order of Γ is equal to v gcd(n, p -1)/(p -1) which is just equal to v in the saturated situation. We have the following McEliece formula: Proposition 2. For any t ∈ L * , the weight w(t) of the codeword Φ m (t) is given by:

(2) w(t) = p -1 pv p m + χ∈Γ\{1} τ L (χ) χ(t) .
And, conversely by Fourier inversion

(3) τ L (χ) = p p -1 t∈L * /V w(t)χ(t).
One says that p is semiprimitive modulo v when -1 is in the group generated by p in (Z/vZ) * , i.e. when ord v (p) is even. Note that in this case all the Gauss sums are rational and a c(p, m, v) code is a two-weight code. In the paper we investigate the reciprocal: besides some sporadic known examples, is any two-weight irreducible cyclic code semiprimitive ?

The case v small

Before going further let us treat the case where v is small, i.e. v = 2 or 3. We know that a c(p, m, 2) code is a two-weight code, and that the weights can be expressed in term of quadratic Gauss sum (see [START_REF] Berndt | Gauss and Jacobi Sums[END_REF]). In the same way, the weights of a c(p, m, 3) code can be expressed by means of cubic Gauss sums. However, it is hard to give the exact values of the cubic Gauss sums (see [START_REF] Matthews | Gauss sums and elliptic functions I. The Kummer sums[END_REF]), and thus also the weights of such a code. Nevertheless, we have the following characterization:

Proposition 3. A c(p, m, 3

) code has two weights if and only if it is semiprimitive (that is here, if and only if p ≡ 2 mod 3).

Proof. Let χ be a multiplicative character of L of order 3. The number of weights of a c(p, m, 3) code is equal to the number of distincts values taken by the mapping:

L * ∋ t → f (t) = τ L (χ)χ(t) + τ L ( χ) χ(t).
Let 1 = j be a cubic root of unity. Let t be such that

χ(t) = j. It is easy to see that f (1) = f (t) implies τ L ( χ) = jτ L (χ), that f (t) = f (t 2 ) implies τ L ( χ) = τ L (χ) and that f (1) = f (t 2 ) implies τ L ( χ) = j 2 τ L (χ).
Therefore, the code has two weights if and only if there exists a cubic root of unity ω such that

(4) τ L ( χ) = ωτ L (χ).
In particular, since τ L (χ) 3 is an algebraic integer of degree 2 and norm p 3m , we deduce that τ L (χ) 6 = τ L ( χ) 6 = p 6m . Hence the Gauss sums τ L (χ) are pure Gauss sums (see [START_REF] Berndt | Gauss and Jacobi Sums[END_REF] for a definition of a pure Gauss sum). It follows by a theorem of Baumert, Mills and Ward (see Theorem 11.6.4 of [START_REF] Berndt | Gauss and Jacobi Sums[END_REF] for example) that p is semiprimitive modulo 3.

Infinitely many index-two c(p, m, v) codes with v prime

For the study of c(p, m, v) codes with v prime and p of index two modulo v, we are interested in primitive and near-primitive root densities.

In 1927, Emil Artin made the following conjecture (called now the Artin's primitive root conjecture): for any integer α = ±1 not a square, the natural density

lim x→+∞ ♯{v prime | v ≤ x and α generates F * v } ♯{v prime | v ≤ x}
exists and is positive. In 1967, Hooley proved this conjecture under the assumption of G.R.H.. In particular, he proved that if α is neither ±1 nor a perfect square, then there are infinitely many primes v for which α is a primitive root modulo v.

If we ask α to generate only the squares of F * v and not the whole group F * v , i.e. to have index 2 and not index 1 modulo v, we come to the notion of near-primitive roots. Precisely, fix α = ±1 not a perfect power and let v be a prime and t be an integer such that v ≡ 1 (mod t). Consider

N α,t (x) = ♯{v prime | v ≤ x and v |α and ind v (α) = t}.
Notice that for t = 1 this quantity is just the previous one studied by Artin and Hooley. In 2000, Moree introduced in [8] a weighting function depending on α and t and gave an estimation of N α,t (x) assuming G.R.H.. In particular, for α = p a prime number and t = 2, he proved that

N p,2 (x) = v odd prime v≤x ϕ( v-1 2 ) v -1 + O( x log log x log 2 x ).
This implies that there exist infinitely many primes v such that p has index 2 modulo v.

In particular, we have:

Proposition 4. Conditionally on G.R.H., for any prime number p there are infinitely many index-two irreducible cyclic codes c(p, m, v) with v prime.

Necessary conditions on two-weight codes

The irreducible cyclic codes c(p, m, v), with v a prime number and with p of index 2 modulo v, range in two families: the first one with v ≡ 1 (mod 4) and the second one with v ≡ 3 (mod 4). If v ≡ 1 (mod 4), then -1 is a square modulo v and since p generates the squares modulo v, we are reduced to the semiprimitive case. This lead us to consider the second case, where -1 is not a square modulo v. Moreover, in view of Proposition 3, we can suppose that v is greater than 3.

Hence, let us consider a prime number p and an integer v satisfying the following (♯) conditions:

(a) v is a prime greater than 3, (b) ord v (p) = (v -1)/2 i.e. p has index 2 modulo v, (c) v ≡ 3 (mod 4).

Let f be the multiplicative order of p modulo v. Note that f divides m, and we set s = m/f . It is shown in [START_REF] Ph | A new class of two weight codes, Finite fields and their applications[END_REF] that if a c(p, m, v) code with v satisfying the (♯) conditions has two weights then: [START_REF] Louboutin | Simple proofs of the Siegel-Tatuzawa and Brauer-Siegel theorems[END_REF] v + 1 4 = p hs .

We give, now, a more precise result:

Theorem 5. If a c(p, m, v) code with v satisfying the (♯) conditions is a two-weight code then we have:

m = ord v (p).
Proof. Since p has index 2 modulo v, then p is a square modulo v, and

(p) = P P ′ splits in the extension Q( √ -v)/Q. We have that the norm

N Q( √ -v)/Q (P ) = p and that P h = (α) is a principal ideal (since h is the ideal class number of Q( √ -v)), with α = (a + b √ -v)/2 (with a, b ∈ Z) an algebraic integer of Q( √ -v).
Taking norms, we obtain p h = (a 2 + vb 2 )/4 and since a and b cannot be zero in this situation, we conclude that v + 1 4 ≤ p h .

But by (5) a c(p, m, v) code with v satisfying the (♯) conditions has two weights if and only if [START_REF] Matthews | Gauss sums and elliptic functions I. The Kummer sums[END_REF] v + 1 4 = p hs .

Thus, p hs ≤ p h and s = 1.

Then, the previously defined parameter s appearing in [START_REF] Ph | A new class of two weight codes, Finite fields and their applications[END_REF] and [START_REF] Schmidt | All two-weight irreducible cyclic codes ?[END_REF] is equal to 1 under the (♯) conditions. In particular, we have: Corollary 6. If a c(p, m, v) code with v satisfying the (♯) conditions is a two-weight code then

(7) v + 1 4 = p h .
where h is the class number of the imaginary quadratic number field

Q( √ -v).
In particular, such a code is completely defined by the parameter v.

Furthermore, we have the following necessary condition on p for twoweight c(p, m, v) code with v satisfying the (♯) conditions: Corollary 7. If a c(p, m, v) code with v satisfying the (♯) conditions has two weights, then p is the least prime which totally splits in the extension Q( √ -v)/Q, i.e. p is the least prime which is a square modulo v.

Proof. Indeed, if ℓ is a prime which totaly splits in Q( √ -v)/Q, then the previous proof implies that ℓ h ≥ v+1 4 = p h which gives ℓ ≥ p.

Main results

Using the previous section, we can state the following result which can also be derived from the proof of lemma 4.1. of [START_REF] Ph | A new class of two weight codes, Finite fields and their applications[END_REF].

Theorem 8. There is no two-weight c(p, m, v) code with v satisfying the (♯) conditions and with v ≡ 7 (mod 8). Hence, Conjecture 1 holds true for index-two irreducible cyclic c(p, m, v) codes with v a prime not congruent to 3 modulo 8.

Proof. Since v ≡ 7 (mod 8), it follows that 2 is a square modulo v, and the ideal (2) splits in the extension Q( √ -v)/Q. By Corollary 7, we conclude that p = 2. But we proved in [START_REF] Aubry | On the weights of binary irreducible cyclic codes, Coding and Cryptography[END_REF] that there exists no two-weight binary c(p, m, v) code with v satisfying the (♯) conditions, so we get the non-existence of such codes. Hence, this proves the conjecture since the case v ≡ 1 (mod 4) is trivial, as quoted in the previous section, and the last case v ≡ 3 (mod 4) is divided in two subcases : when v ≡ 7 (mod 8), which is now solved, and when v ≡ 3 (mod 8) which is the remainder case.

Actually, we will consider now a more general approach using the identity of Corollary 6 but with at most one exception.

If a c(p, m, v) code with v satisfying the (♯) conditions has two weights then we have the following relation

v + 1 4 = p h ,
where h is the class number of the imaginary quadratic number field Q( √ -v) (see Corollary 6). In 1935, Siegel gave a non-effective lower bound on the residue at s = 1 of the L-function L(s, χ v ) associated to the primitive odd Dirichlet character χ v of Q( √ -v). Tatuzawa, in 1951, proved an effective lower bound of L(1, χ v ) but with at most one exception (see [START_REF] Tatuzawa | On a theorem of Siegel[END_REF] and see also [START_REF] Louboutin | Simple proofs of the Siegel-Tatuzawa and Brauer-Siegel theorems[END_REF] for a simple proof): if 0 < ε < 1/2 and v ≥ max(e 1/ε , e 11.2 ), then

L(1, χ v ) ≥ 0.655εv -ε . Since the class number h of Q( √ -v) with -v ≡ 1 (mod 4
) is linked to L(1, χ v ) by the following Dirichlet class number formula:

L(1, χ v ) = πh √ v ,
we can use Tatuzawa theorem to get an upper bound on v.

Proposition 9. There exists at most one two-weight c(p, m, v) code with v ≥ 10 8 satisfying the (♯) conditions. Proof. Let ε = 1/ log(10 8 ) ∈ (0, 1/2). For v ≥ max(e 1/ε , e 11.2 ) = 10 8 , we have, with at most one exception:

L(1, χ v ) ≥ 0.655εv -ε = 0.035v -0.054 . Now, v+1 4 = p h ≥ 2 h implies that log v+1 4 ≥ h log 2.
By the Dirichlet class number formula, we get:

log v + 1 4 ≥ √ vL(1, χ v ) π log 2.
But, for v ≥ 10 8 , we have on one hand log v+1

4

≥ 17.03 and on the other hand √ vL(1,χv) π log 2 > 28.55 by Tatuzawa theorem. Thus, there exists no v ≥ 10 8 such that v+1 4 = p h , with at most one exception. Now, we make an exhaustive research of the primes v ≤ 10 8 such that (v + 1)/4 is a power of a prime p. Then, for such primes v, we check whether (v + 1)/4 = p h(v) holds true or not, with h(v) the class number of Q( √ -v). Actually, we recover the following sporadic known examples of Table 1.

Thus, we have proved the following theorem:

Cyclotomic cosets

Let p be a prime. For any integer v prime to p, consider on the ring Z/vZ the equivalence relation given by: for a, b ∈ Z/vZ, we set a ∼ b if and only if there exists t ∈ Z such that a = bp t . The equivalence classes for this equivalence relation are the so-called p-cyclotomic cosets modulo v.

Recall that the order ord v (g) of an element g of the multiplicative group (Z/vZ) * divides the order ϕ(v) of this group, where ϕ is the Euler function. We denote by ind v (g) the index of g modulo v i.e. ind v (g) = ϕ(v) ord v (g) .

Then ind v (g) = [(Z/vZ) * : g ] where g denotes the subgroup of (Z/vZ) * generated by g. But the number γ(p, v) of p-cyclotomic cosets modulo v is also equal to the number of irreducibles polynomials in the decomposition of the polynomial X v -1 over F p , thus it is equal to [START_REF] Moree | Asymptotically exact heuristics for (near) primitive roots[END_REF] γ(p, v)

= d|v ϕ(d) ord d (p) = d|v ind d (p)
with the convention that ind 1 (p) = 1. For example, the condition Proof. The result is in fact much general: the number of weights is less or equal than the number of cyclotomic cosets. It follows from the fact that the weight of a codeword of a c(p, m, v) code is invariant under t → tζ and under t → t p ; see Theorem 2.5 of [START_REF] Fitzgerald | Sums of Gauss sums and weights of irreducible codes[END_REF] for a detailed proof.

γ(p, v) = 2 is equivalent to ind v (p) = 1,
The case (ii) of Proposition 11 falls into the semiprimitive case since p generates the whole group (Z/vZ) * and thus contains -1.

Finally, we have proved the following result:

Theorem 13. If v is an integer such that there is three p-cyclotomic cosets modulo v then any two-weight irreducible cyclic code c(p, m, v) which is not one of the six sporadic examples of Table 1 is semiprimitive, with at most one exception. Hence, Conjecture 1 holds true, with at most one exception, for all c(p, m, v) codes with v an integer such that there is three p-cyclotomic cosets modulo v.

Proof. If a binary irreducible cyclic code with three-cyclotomic cosets has two weights then it is semiprimitive. Indeed, by Proposition 11, an irreducible cyclic code with three-cyclotomic cosets leads to two cases. The first one leads c(p, m, v) codes with v a square of a prime and p of index 1 modulo v which gives a semiprimitive code.

The other case leads to c(p, m, v) codes with v a prime and p of index 2 modulo v (the so-called index-two codes). When v ≡ 1 (mod 4), we saw that we obtain a semiprimitive code. When v ≡ 3 (mod 4), we obtain c(p, m, v) codes with v satisfying the (♯) conditions. In the case where p = 2, i.e. the binary case, we found in [START_REF] Aubry | On the weights of binary irreducible cyclic codes, Coding and Cryptography[END_REF] that there is no two-weight codes. When p = 2, theorem 10 gives the result.

Table 1 .

 1 Sporadic c(p, m, v) codes with v satisfying the (♯) conditions and v ≤ 10 8 .

	v 11 19 67 107 163 499
	p 3 5 17 3	41	5
	h 1 1 1	3	1	3

  that is p is a primitive root modulo v.Proposition 11. Let v be an integer. The ring Z/vZ contains exactly 3 p-cyclotomic cosets if and only if one of the following holds:(i) v is a prime and p has index 2 mod v;(ii) v is the square of a prime and p has index 1 mod v.

Proof. By

[START_REF] Moree | Asymptotically exact heuristics for (near) primitive roots[END_REF]

we have γ(p, v) = 3 if and only if ind v (p) = 2 and v has no proper divisor, or ind v (p) = 1 and v has a unique proper divisor. The proposition is then proved.

Proposition 12. Let v be an integer. If the ring Z/vZ contains exactly three p-cyclotomic cosets then any c(p, m, v) code has at most three nonzero weights.


Yves Aubry

R Égis Blache

ON SOME QUESTIONS RELATED TO THE GAUSS CONJECTURE FOR FUNCTION FIELDS

Keywords: 2000 Mathematics Subject Classification. 11R29, 11R58, 11R11, 14H05 Functions fields, Gauss conjecture, zeta functions, jacobian, hyperelliptic curves, finite fields

We show that, for any finite field F q , there exist infinitely many real quadratic function fields over F q such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field F q , of infinitely many real function fields over F q with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over F q such that the numerator of their zeta function is an irreducible polynomial.

Introduction

According to a conjecture of Gauss, there exist infinitely many real quadratic number fields having a ring of integers which is a principal ideal domain, i.e. with class number one. We shall study a related conjecture for functions fields over a finite field.

Let q be a power of a prime p, and F q a finite field with q elements. A function field over F q is a finite extension K of a field of rational function F q (x) (where x is transcendental over F q ) with full constant field F q . The extension K/F q (x) is said to be quadratic when we have [K : F q (x)] = 2, and real when the infinite prime 1 x of F q (x) splits in K. Otherwise (i.e. if the prime 1 x ramifies or is inert), we say that K/F q (x) is imaginary. By abuse of language, we will say that the function field K is real when one can find a transcendental element z ∈ K such that the extension K/F q (z) is quadratic and real. If S denotes the set of places in K above the place at infinity, we denote by O S the ring of regular functions of K outside S. Then the Gauss conjecture for function fields asserts that, for any finite field F q , there exist infinitely many real quadratic function fields with full constant field F q such that O S has ideal class number one. This is the main conjecture BC discussed by Lachaud and Vladut in [START_REF] Lachaud | Gauss problem for function fields[END_REF], among some weaker ones.

Using Iwasawa theory, Bruno Anglès showed in [START_REF] Anglès | On L-functions of cyclotomic function fields[END_REF] that if K is a real quadratic function field over the finite field F q with q ≥ 5 having a principal ring of integers, then the numerator of its zeta function (called the zeta polynomial) is a polynomial with only simple roots; in other words is a separable polynomial. Thus, Anglès asks the following question which is a necessary condition for the Gauss conjecture to hold true: is there exist infinitely many real quadratic function fields over F q with zeta polynomial being separable ? We give a positive answer to this question considering the action of the Frobenius on the F 2vector space of 2-torsion points of the jacobian of certain hyperelliptic curves. Note that the separability of this polynomial is equivalent for the endomorphism ring of the jacobian to be commutative, see [START_REF] Tate | Endomorphisms of abelian varieties over finite fields[END_REF].

Then we consider if it remains true when we ask the zeta polynomial to be irreducible. This problem is stronger than the previous one, and it has been studied by various authors (see [START_REF] Chavdarov | The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy[END_REF], [START_REF] Kowalski | The large sieve, monodromy, and the zeta function of curves[END_REF]) from a different point of view: it is known that for a family of curves of fixed genus g having large monodromy, the zeta polynomial is "generically" irreducible. More precisely, if the parameter space is d-dimensional, the number of curves having a zeta polynomial whose splitting field is not the biggest possible is O(q d-γ ln q), where γ is a positive constant depending on g. Unfortunately, this result is of no use when q is fixed and g gets large. Thus it can not be used to solve our question. Note that there are also unpublished results of Katz about irreducibility of the L-functions of elliptic curves over function fields, providing similar results in this case.

Here we show that assuming Artin's primitive root conjecture, then for any q > 4 there exist infinitely many (real, quadratic) function fields having an irreducible zeta polynomial. On the other hand, we remark that the existence of infinitely many real quadratic function fields with prime divisor class number (i.e. such that the order of the group of rational divisor classes Pic 0 (K) of K is prime) implies the previous ones, as quoted in the following figure Artin's conj prime order jacobian q>4 q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j % -

T T T T T T T T T T T T T T T T T T T T T T T T T T T T irred zeta pol % - T T T T T T T T T T T T T T T T T T T T T T T T T T T T
O S principal q>4 q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j separable zeta pol The paper is organized as follows: in section 2, we give some notations and we recall some basic facts about ideal class numbers, divisor class numbers and regulators of real quadratic function fields. Then, we recall the result of Anglès (Proposition 4.4 of [START_REF] Anglès | On L-functions of cyclotomic function fields[END_REF]) which is the startingpoint of the paper. Section 3 is devoted to the main result, namely the answer to the question of Anglès. In section 4, we give a conditional answer to the related irreducibility problem, and compare this result with previously known results in the literature.

Structure of rings of S-integers

In this section we recall well-known facts about rings of S-integers in function fields (see [START_REF] Rosen | Number theory in function fields[END_REF] for more details), and we introduce the question of Anglès.

Let F q be a finite field and let us fix F q (x) a one-variable rational function field over F q . A real quadratic extension K over F q is a degree 2 extension of F q (x) such that the infinite place ∞ of F q (x) (corresponding to the pole of x) splits in K: there are two places ∞ 1 and ∞ 2 above ∞ in K.

Recall that for X any (smooth, projective) curve over F q having K as function field, we can define for any k ≥ 1 the number N k (X) of F q k rational points of X, and encode these numbers into the zeta function of

K Z K (T ) = exp k≥1 N k (K) T k k .
Weil's theorem (the Riemann hypothesis over finite fields) says that this function is actually a rational function, having denominator (1 -T )(1 -qT ), and whose numerator is a degree 2g polynomial in

Z[T ],
where g is the genus of K. We shall denote this polynomial by L K , and call it the zeta polynomial of K.

The ring of S-integers O S of K (with respect to ∞) is the integral closure of the ring of polynomials F q [x] in K. It is also the ring of elements of K whose poles are in S = {∞ 1 , ∞ 2 }. It is a Dedekind domain; we denote by Cl(O S ) the ideal class group of O S and by h O S the order of Cl(O S ), called the ideal class number. Then, the ring O S is a principal ideal domain if and only if h O S = 1. We will say abusively in this case that K is principal.

Consider the Picard group Pic 0 (K) of rational divisors of K of degree zero modulo the principal ones. It is well-known that the Picard group is isomorphic to the group of rational points over F q of the jacobian J K of the non singular projective algebraic curve over F q associated to the function field K:

Pic 0 (K) ≃ J K (F q ). The order of this group, called the divisor class number of K, is denoted by h K and is equal to L K (1), the evaluation at 1 of the zeta polynomial of K since it is the number of rational points of the jacobian J K .

Consider now the group Div 0 S (K) of divisors of K of degree zero with support in S. Since the places of S have degree 1, the divisors of Div 0 S (K) are of the form n∞ 1 -n∞ 2 with n an integer. Thus we have an isomorphism Div 0 S (K) ≃ Z. Furthermore, since a rational function of K whose support is contained in S lies in the group of units O * S , it is clear that the group P S (K) of principal divisors of K with support in S is isomorphic to O * S /F * q . The units theorem of Dirichlet tell us that

O * S ≃ F * q × < ε > where ε is a function whose divisor is of the form r O S ∞ 1 -r O S ∞ 2 (ε is a fundamental unit) and r O S is called the regulator of K (with respect to ∞ or S)
. Thus, we have:

R S := Div 0 S (K)/P S (K) ≃ Z/r O S Z.
We have the following well-known exact sequence of groups:

0 -→ R S -→ J K -→ Cl(O S ) -→ 0.

It implies in particular the Schmidt relation for a real quadratic function field

K: r O S h O S = h K .
Moreover, if we suppose that the ring O S is a principal ideal domain, then this implies that J K (F q ) is a cyclic group: Proposition 1. Let K be a real quadratic function field. We have the following (i) K is principal if and only if we have the isomorphism

J K (F q ) ≃ R S . (ii) If K is principal, then J K (F q ) is a cyclic group.
(iii) If the group J K (F q ) has prime order then K is principal.

Proof. Assertion (i) follows from the exact sequence above, whereas assertion (ii) comes from the cyclicity of the group R S .

To show assertion (iii), first remark that we must have r O S > 1. Indeed if r O S = 1, the divisor ∞ 1 -∞ 2 is principal: thus it is the divisor of a rational function ε and since [K : F q (ε)] = deg ε = 1, K must be a rational function field. Hence the Picard group Pic 0 (K) is trivial and J K (F q ) cannot have prime order. By the previous Schmidt relation, we obtain h O S = 1. Bruno Anglès proved in [START_REF] Anglès | On L-functions of cyclotomic function fields[END_REF] (proposition 4.4) the following result: Proposition 2. Let K be a real quadratic extension of F q (x) with q > 4. If K has ideal class number one, then the zeta polynomial of K is separable.

Remark. The hypothesis q > 4 of the previous proposition is necessary. Indeed, we can provide counter-examples for each field F 2 , F 3 and F 4 .

For instance, let us consider the example of the hyperelliptic curve X of genus 5 defined by the equation

y 2 = x 12 + 2x 11 + 2x 6 + x 5 + 2 over F 3 .
The curve X has two rational points over F 3 , corresponding to the places above infinity, thus the function field k(X) of X is a real quadratic function field. Moreover, the group of rational points over F 3 of its jacobian is cyclic of prime order (isomorphic to Z/139Z) and thus k(X) has class number one by (iii) of proposition 1. However the zeta polynomial L X (T ) of X has multiple roots since we have:

L X (T ) = (T 2 -3T + 3) 2 (T 6 + 4T 5 + 12T 4 + 23T 3 + 36T 2 + 36T + 27).

Construction of real quadratic function fields

In this section we shall construct infinitely many hyperelliptic curves in any odd characteristic, having a separable zeta polynomial. Then we show that, up to twisting, the function field of any of these curves can be considered as a real quadratic function field.

It is well-known that the zeta polynomial L K (T ) of a function field K over F q can be seen as the reciprocal polynomial of the characteristic polynomial of the Frobenius endomorphism acting on any of the Tate modules of the jacobian of K. Let q > 4 be the power of an odd prime; the reduction modulo 2 of L K (T ) can then be read by looking at the action of the Frobenius on the F 2 -vector space of 2-torsion points of the jacobian. We begin by showing that for a wide family of curves, the reduction modulo 2 of L K (T ) has only simple roots in an algebraic closure F 2 of F 2 .

Lemma 3. Let g be a positive integer, and f (x) be an irreducible polynomial in F q [x] of degree 2g + 1. Consider the quadratic extension K of F q (x) defined by the equation y 2 = f (x). Then the zeta polynomial of K is separable.

Proof. Let us remark that since f is irreducible, all its roots are simple and the function field K has genus g; hence the zeta polynomial L K of K has degree 2g. It is well-known that if ℓ is a prime number distinct from the characteristic of K, the Tate module T ℓ (J f ) of the jacobian of K is a free Z ℓ -module of rank 2g, on which the Frobenius endomorphism F acts. We have the following relation between the characteristic polynomial χ of this endomorphism and the zeta polynomial L K :

χ(T ) = T 2g L K ( 1 T ).
In order to prove the lemma, it suffices to prove that the polynomial χ has only simple roots. Let us fix ℓ = 2. Then we have:

T 2 (J f )/2T 2 (J f ) ≃ J f [2],
the group of 2-torsion points of J f . If α 1 , . . . , α 2g+1 are the roots of f in an algebraic closure F q of F q , it is easy to show that a basis of the F 2 -vector space J f [START_REF] Chavdarov | The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy[END_REF] is given by the divisor classes

D 1 := [(α 1 , 0) -P ∞ ], . . . , D 2g := [(α 2g , 0) -P ∞ ]
where P ∞ is the unique point over the infinite place ∞ corresponding to the prime ideal 1 x of F q (x) (note that ∞ ramifies in the extension K/F q (x) since deg f is odd).

But, since f is irreducible over F q , we can reorder its roots such that α 2 = α q 1 , . . . , α 1 = α q 2g+1 . Hence for 1 ≤ i ≤ 2g -1, we have [START_REF] Chavdarov | The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy[END_REF]. Thus the matrix of the action of F on J f [START_REF] Chavdarov | The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy[END_REF] .

F (D i ) = [(α q i , 0) -P ∞ ] = [(α i+1 , 0) -P ∞ ] = D i+1 . For i = 2g, since the principal divisor of the function y is (y) = (α 1 , 0) + • • • + (α 2g+1 , 0) -(2g + 1)P ∞ , we have F (D 2g ) = [(α q 2g , 0) -P ∞ ] = [(α 2g+1 , 0) -P ∞ ] and thus F (D 2g ) = [(α 1 , 0) -P ∞ ] + • • • + [(α 2g , 0) -P ∞ ] = D 1 + • • • + D 2g in J f
The characteristic polynomial of this matrix is

T 2g + • • • + T + 1 in F 2 [T ]
and this is the reduction modulo 2 of χ(T ).

Since the roots of T 2g + • • • + T + 1 are pairwise distinct in F 2 (they are the (2g + 1)-th non trivial roots of unity), the polynomial P has only simple roots, and the lemma is proved.

Unfortunately, the functions fields that appear in the previous lemma are not real extensions of F q (x) since the infinite place of this last field is ramified. We have just obtained that there exist infinitely many imaginary quadratic function fields with separable zeta polynomial.

We now show that infinitely many of the function fields constructed above can be seen as real function fields. Lemma 4. Let X be the hyperelliptic curve over F q defined by

y 2 = f (x)
where f (x) is an irreducible polynomial of F q [x]. If ♯X(F q ) > 1 then the function field of X can be viewed as a real quadratic function field, i.e. there exists a transcendental element z in K such that the extension K/F q (z) is quadratic, and real.

Proof. The function field K of X can be viewed as a real quadratic extension of a rational function field F q (z) if there exists a place of F q (x) which splits in K; actually taking z as the inverse of a uniformizing parameter at this place, we see that F q (z) = F q (x) (and thus that [K : F q (z)] = 2), and that the pole of z splits in this extension.

We are reduced to show that there is a place of F q (x) which splits in K; since f (x) is supposed to be irreducible, the Weierstrass points are not rational except the infinite place if f (x) has odd degree. Thus, there is at most one rational ramified place and then if ♯X(F q ) > 1 there exists at least one place which splits totally.

In view of the previous results, it remains to show that there exist infinitely many hyperelliptic curves X defined by y 2 = f (x) where f (x) is an irreducible polynomial of F q [x] of odd degree such that ♯X(F q ) > 1.

If X is the hyperelliptic curve defined by y 2 = f (x) where f (x) is an irreducible polynomial of F q [x] of degree 2g + 1, and if v is a non quadratic residue in F q , we defined the twist X of X (with respect to v) to be the hyperelliptic curve defined by

y 2 = f v (x) with f v (x) = vf ( x v
). We have the following

Lemma 5. (i) The polynomial f (x) is irreducible if and only if f v (x) is irreducible.
(ii) If n is odd the numbers of F q n -rational points of X and X are related by

♯X(F q n ) + ♯ X(F q n ) = 2q n + 2,
and if n is even the relation becomes

♯X(F q n ) = ♯ X(F q n ).
Proof. The first part of the proof is trivial. We come to the second part. It is well-known that if X is a complete smooth model of the curve with affine equation y 2 = f (x), with f separable, and F q n is any extension of F q , then the cardinality of X(F q n ) is given by

♯X(F q n ) = q n + 1 + x∈F q n χ 2,n (f (x)),
where χ 2,n is the multiplicative character of order 2 of F × q n extended to F q n by setting χ 2,n (0) = 0.

Applying this result to X, we get

♯ X(F q n ) = q n + 1 + x∈F q n χ 2,n (f v (x)) = q n + 1 + χ 2,n (v) x∈F q n χ 2,n (f ( x v )) = q n + 1 + χ 2,n (v) x∈F q n χ 2,n (f (x)). Now we have χ 2,n = χ 2 • N F q n /
Fq where N F q n /Fq is the norm. Since v ∈ F q , we get N F q n /Fq (v) = v n , and χ 2,n (v) = χ 2 (v) n = (-1) n since v is not a quadratic residue. This ends the proof.

Corollary 6. (i) If ♯X(F q ) ≤ 1 then ♯ X(F q ) ≥ 2q + 1.
(ii) We have: L X (T ) = L X (-T ). In particular, L X (T ) has only simple roots if and only if it is the case for L X (T ) (and L X (T ) is irreducible if and only if L X (T ) is).

Proof. (i) is trivial. To show (ii), let us look at the zeta function of X:

Z X (T ) = exp n≥1 ♯ X(F q n ) T n n .
Using the formulas in the proof of the previous lemma, and setting A n := x∈F q n χ 2,n (f (x)), we get:

Z X (T ) = exp n≥1 (q n + 1 + (-1) n A n ) T n n = exp n≥1 A n (-T ) n n (1 -T )(1 -qT ) . Thus L X (T ) = exp n≥1 A n (-T ) n n
. A similar calculation for X gives T ). Hence, we have proved the following theorem.

L X (T ) = exp n≥1 A n T n n , that is L X (T ) = L X (-
Theorem 7. For any finite field F q , there exist infinitely many real quadratic function fields over F q with separable zeta polynomial.

Irreducibility of the zeta polynomial

We now consider the related question of the existence, for any finite field F q , of infinitely many real quadratic function fields over F q with irreducible zeta polynomial. We prove a conditional result, and then we briefly survey the known results about this question. Lemma 8. Let q > 4, and X be a projective algebraic curve over F q such that the group of rational points J X (F q ) of the jacobian of X has prime order. Then the zeta polynomial of X is irreducible over Q (and thus has simple roots).

Proof. The value L X (1), which is exactely ♯J X (F q ), is prime by hypothesis and any factor

Q(T ) = i∈I (1 -ω i T ),
with I ⊂ {1, . . . , 2g}, of L X (T ) in Z[T ] would have a value at 1 strictly greater than 1 since

Q(1) = i∈I | 1 -ω i |≥ i∈I ( √ q -1) > 1
by the Riemann Hypothesis and because q > 4.

Thus we have the following conditional result:

Theorem 9. Let q > 4. Assuming that there exist infinitely many real quadratic function fields over F q with prime divisor class number, then there exist infinitely many real quadratic function fields over F q with irreducible zeta polynomial. Furthermore in this case Gauss conjecture for function fields over F q holds true.

Proof. The first assertion follows directly from lemma 8, and the second one from proposition 1 (iii).

Using the previous section, we have another way to consider our problem. Indeed, assume Artin's conjecture on primitive root for the integer 2, i.e. there exist infinitely many primes ℓ such that 2 generates the multiplicative group (Z/ℓZ) * . Hence the polynomial

(x ℓ -1)/(x -1) = x ℓ-1 + • • • + x + 1 is irreducible over F 2 .
From the proof of lemma 3, this polynomial is the reduction modulo 2 of the zeta polynomial of any hyperelliptic curve of the form y 2 = f (x), f irreducible of degree ℓ. Thus we obtain the following other conditional result: Theorem 10. Assuming Artin's primitive root conjecture, there exist infinitely many real quadratic function fields with irreducible zeta polynomial.

Note that, in another direction, namely when the characteristic of the function field is not fixed, Koblitz in [START_REF] Koblitz | Jacobi sums, irreducible zeta-polynomials, and cryptography[END_REF] found conditions under which the zeta polynomial of the curve y 2 + y = x d over F p , where d = 2g + 1 is a prime, d = p, is irreducible over Q. As a corollary, he states the following result: Theorem 11. For any fixed prime d ≥ 3 there are infinitely many primes p such that the zeta polynomial of y 2 + y = x d over F p is irreducible over Q.

For a fixed characteristic, the question is still open for this class of curves, as its following conjecture states: Conjecture 1. For any prime p there are infinitely many primes d such that the zeta polynomial of y 2 + y = x d over F p is irreducible over Q.

On the other hand, the question of the irreducibility of the zeta polynomial of smooth projective curves varying in a family is considered in [START_REF] Chavdarov | The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy[END_REF], [START_REF] Kowalski | The large sieve, monodromy, and the zeta function of curves[END_REF]. Precisely, let U be a geometrically irreducible smooth affine scheme of dimension d ≥ 1, and π : C → U be an algebraic family of smooth projective curves of genus g over U , such that the geometric monodromy group of the integral sheaves R 1 π ! Z ℓ is the full symplectic group Sp(2g) for any prime ℓ large enough. Then the number N π (U/F q ) of u ∈ U (F q ) such that the curve π -1 ({u}) has its zeta polynomial reducible or having splitting field with degree strictly less than 2 g g! (which is the maximal degree imposed by the pairing on its roots), satisfies N π (U/F q ) = O(q d-γ ln q), where γ is some constant depending on g, generally γ = O( 1 g 2 ). There is an other similar estimate for families of hyperelliptic curves of the form y 2 = f (x)(x -u), where the parameter space U is A 1 with the zeroes of f removed: here Kowalski gets N π (U/F q ) = O(q 1-γ ln q) for some γ as above.

¿From Lang-Weil estimates on the number of points of varieties over finite fields, we have ♯U (F q ) = O(q d ), and we see that the bound above is non trivial only when q -γ ln q = o(1), for instance when q is large with respect to g. Thus if we fix q and vary g, this estimate is trivial for g large enough, and cannot be used to answer our question.

More generally, we can ask for any fixed finite field F q , is there exist infinitely many jacobians of hyperelliptic curves with irreducible zeta polynomial. Since an ordinary abelian variety over F q is simple if and only if its Weil polynomial is irreducible, we can look for the existence of simple ordinary jacobian varieties of every dimension. However, Howe and Zhu in [START_REF] Howe | On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field[END_REF] proved that for any finite prime fields there exist absolutely simple ordinary abelian varieties of every dimension. But the question of whether there exist absolutely simple jacobians of every dimension over a given finite field is still open (whereas the answer is yes if the field is the algebraic closure of a finite field, if it has characteristic p but is not algebraic over F p or if it has characteristic zero), see [START_REF] Howe | On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field[END_REF].

  in the basis D 1 , . . . , D 2g is 

Acknowledgments.

The authors thank Everett Howe for the idea of Lemma 3, Christophe Ritzenthaler for the numerical example of the remark of section 2 and Marc Perret for a lightning discussion.