Duursma's reduced polynomial

The weight distribution of a linear code C is put in an explicit bijective correspondence with Duursma's reduced polynomial of C. We prove that the Riemann Hypothesis Analogue for a linear code C requires the formal self-duality of C and imposes an upper bound on the cardinality q of the basic field, depending on the dimension and the minimum distance of C. Duursma's reduced polynomial of the function field of a curve X of genus g over the field with q elements is shown to provide a generating function for the numbers of the effective divisors of non-negative degree degree of a virtual function field of a curve of genus g-1 over the same finite field.

genus g over F q is shown to provide a generating function Let F q = ∪ ∞ m=1 F q m be the algebraic closure of a finite field F q and X/F q ⊂ P N (F q ) be a smooth irreducible projective curve of genus g, defined over F q . Denote by F = F q (X) the function field of X over F q and choose n different F q -rational points P 1 , . . . , P n ∈ X(F q ) := X ∩ P N (F q ). Suppose that G is an effective divisor of F of degree 2g − 2 < deg G = m < n, whose support is disjoint from the support of D = P 1 + . . . + P n . The space L(G) := H 0 (X, O X (G)) of the global holomorphic sections of the line bundle, associated with G will be referred to as to the Riemann-Roch space of G. We put l(G) := dim Fq L(G) and observe that the evaluation map E D (f ) = (f (P 1 ), . . . , f (P n )) for ∀f ∈ L(G) is an F q -linear embedding. Its image C := im(E D ) = E D L(G) is known as an algebraic geometry code or Goppa code. The minimum distance of C is d(C) ≥ n − m. For an arbitrary s ∈ N let N s (F ) := |X(F q s )| be the number of the F q s -rational points of X.
Then the formal power series is called the Hasse-Weil zeta function of F . It is well known (cf. Theorem 4.1.11 from [8]) that for a polynomial L F (t) ∈ Z[t] of degree 2g. We refer to L F (t) as to the Hasse-Weil polynomial of F .
In [2], [3] Duursma introduces the genus of a linear code C ⊂ F n q as the deviation g := n + 1 − k − d of its dimension k := dim Fq C and minimum distance d from the equality in Singleton bound. Let W (w) C be the number of the codewords c ∈ C of weight d ≤ w ≤ n. Then W C (x, y) := x n + n w=d(C) W (w) C x n−w y w is called the homogeneous weight enumerator of C. Denote by M n,s (x, y) the homogeneous weight enumerator of an MDS-code of length n and minimum distance s. Put g ⊥ for the genus of the dual code C ⊥ of C and r := g + g ⊥ . Proposition 1. (Duursma [3]) For an arbitrary F q -linear [n, k, d]-code C, which is not contained in a coordinate hyperplane H i := {x ∈ F n q | x i = 0} of F n q , there exist uniquely determined rational numbers a 0 , . . . , a r ∈ Q, such that the homogeneous weight enumerator W C (x, y) = a 0 M n,d (x, y) + a 1 M n,d+1 (x, y) + . . . + a r M n,d+r (x, y) (1) of C is the linear combination of the homogeneous weight enumerators M n,d+i (x, y) of MDS-codes of length n and minimum distance d + i with coefficients a i and The ζ-polynomial P C (t) := r i=0 a i t i of C is uniquely determined by where Coeff t n−d (f (t)) stands for the coefficient of t n−d in a formal power series f (t) ∈ C[[t]].
Proposition 2. (Duursma's considerations from [2]) Let X/F q ⊂ P N (F q ) be a smooth irreducible curve of genus g, defined over F q and G 1 , . . . , G h be a complete list of effective representatives of the linear equivalence classes of the divisors of F = F q (X) of degree 2g − 2 < m < n. Assume that there exist n different F q -rational points P 1 , . . . , P n ∈ X(F q ), such that D = P 1 + . . . + P n ∈ Div(F ) has support Supp(D) ∩ Supp(G i ) = {P 1 , . . . , P n } ∩ Supp( are the Riemann-Roch spaces of G i , E D (f ) = (f (P 1 ), . . . , f (P n )) for ∀f ∈ L(G i ) are the evaluation maps at D and C i := E D L(G i ) are the corresponding Goppa codes with homogeneous weight enumerators W C i (x, y), then for the ζ-polynomial L F (t) of F . In particular, for the ζ-polynomials P C i (t) of C i = E D L(G i ) and the Hasse-Weil polynomial L F (T ) of the function field F .
Proof. Note that (4) is an equality of homogeneous polynomials of x and y of degree n, whose monomials are of degree s ≥ 1 with respect to y. Therefore (4) is equivalent to for ∀s ∈ N. Note that On the other hand, That is why it suffices to verify (6) for s ≥ n − m, s ∈ N.
Note that the number of the codewords c = (f (P 1 ), . . . , f (P n )) ∈ C i , f ∈ L(G i ) of weight s equals the number of the rational functions f ∈ L(G i )\{0}, vanishing at n − s of the points P 1 , . . . , P n . Bearing in mind that the projective space P(L(G i )) = P m−g (F q ) parameterizes the effective divisors, linearly equivalent to G i and two rational functions f, f ′ ∈ F \ {0} have one and a same divisor exactly when they are on one and a same Then e m,s = i e m (i) and it suffices to show that e m (i) = Coeff t s−n+m ((1 − t) s ζ F (t)) for any i, in order to justify (6) and (4).
To this end, observe that E ∈ Div(F ) ≥0 is an effective divisor of degree deg is the generating function for the number A i of the effective divisors of F of degree i. Bearing in mind that D i = ν i 1 + . . . + ν is is a sum of s different places i j of degree deg ν i j = 1, one observes that (1 − t) s ζ F (t) is the generating function for the number of the effective divisors of F of degree i, whose support is disjoint with Supp(D i ). In other words, e m (i) = Coeff t m−n+s ((1 − t) s ζ F (t)).
The equality (5) is an immediate consequence of Proposition 1, (4) and the fact that and, therefore, of genus Proposition 2 motivates Duursma to refer to P C (t) as to the zeta polynomial of an arbitrary linear code C ⊂ F n q . He establishes that P C (t) and W C (x, y) are in a bijective correspondence and Mac Williams identities, relating the weight distributions {W C ⊥ } n w=d ⊥ of a pair (C, C ⊥ ) of mutually dual linear codes are equivalent to the functional equation for the corresponding zeta polynomials P C (t), P C ⊥ (t).
In [2] and [4] Duursma observes the existence of a polynomial D C (t) = defined by the identity of polynomials in t, but does not make use of D C (t) for the study of the homogeneous weight enumerator W C (x, y) of C. He mentions in [4] that the analogue D F (t) of D C (t) for a function field F of one variable accounts for the contribution of the special divisors of F to the zeta function Z F (t). From now on, we refer to D C (t) as to Duursma's reduced polynomial of C.
The present note provides an explicit bijective correspondence between the weight distribution {W The classical Hasse-Weil Theorem establishes that all the roots of the Hasse-Weil polynomial L F (t) ∈ Z[t] of the function field F = F q (X) of a curve X of genus g over F q are on the circle S 1 √ q : z ∈ C |z| = 1 √ q (cf. Theorem 4.2.3 form [8]). Duursma says that a linear code C ⊂ F n q satisfies the Riemann Hypothesis Analogue if all the roots of its zeta polynomial P C (t) = r i=0 a i t i ∈ Q[t] are on the circle S 1 √ q . Let C be an F q -linear code of dimension k and minimum distance d, which satisfies the Riemann Hypothesis Analogue. Proposition 4 shows that C is formally self-dual, while Corollary 5 provides an explicit upper bound on the cardinality q of the basic field, depending on k and d. Let us recall that C is formally self-dual if it has the same weight distribution W C ⊥ , ∀0 ≤ w ≤ n as its dual code C ⊥ ⊂ F n q . In the light of Duursma's results and our Proposition 3, the formal self-duality of C turns to be equivalent to the functional equation P C (t) = P C 1 qt q g t 2g for P C (t) and to the functional equation D C (t) = D C 1 qt q g−1 t 2g−2 for D C (t). Proposition 6 from the present note expresses explicitly the homogeneous weight enumerator W C (x, y) of a formally self-dual code C ⊂ F n q by the lowest half of the coefficients of D C (t) or by the numbers W (d) of the codewords c ∈ C, whose weights are between the minimum distance d of C and the dimension k.
In [1] Dodunekov and Landgev introduce the near-MDS code C ⊂ F n q as the ones with quadratic zeta polynomial P C (t). Kim and Hyun's article [7] provides a necessary and sufficient condition for a near-MDS code to satisfy the Riemann Hypothesis Analogue. Note that the zeta polynomials P C (t) and Duursma's reduced polynomials D C (t) of formally self-dual codes C ⊂ F n q are of even degree. Our Proposition 7 is a necessary and sufficient condition for a formally self-dual code C ⊂ F n q with zeta polynomial P C (T ) of deg P C (t) = 4 to be subject to the Riemann Hypothesis Analogue. Let S ν , ν ∈ N be the uniquely determined logarithmic coefficients of P C (t), defined by the equality of formal power series log P . Adapting Bombieri's proof of the Hasse-Weil Theorem, [5] shows that a linear code C satisfies the Riemann Hypothesis Analogue exactly when the sequence {S ν q − ν 2 } ∞ ν=1 ⊂ C is absolutely bounded. The last, third section is devoted to Duursma's reduced polynomial D F (t) of the function field F = F q (X) of a curve X/F q ⊂ P N (F q ) of genus g over F q . It establishes that D F (t) ∈ Z[t] is determined uniquely by its lowest g coefficients, which equal the numbers A i of the effective divisors of F of degree 0 ≤ i ≤ g − 1. Our Proposition 9 shows that the zeta function associated with D F (t) has the properties of a generating function for the numbers B i of the effective divisors of degree i ≥ 0 of a virtual function field of genus g − 1 over F q . There arises the following Open Problem: To characterize the function fields F = F q (X) of curves X/F q ⊂ P N (F q ) of genus g over F q , for which there are curves Y /F q ⊂ P M (F q ) of genus g − 1, defined over F q with Hasse-Weil zeta function 1 The homogeneous weight enumerator of an arbitrary code is Duursma's reduced polynomial of C and M n,n+1−k (x, y) is the homogeneous weight enumerator of an MDS-code of length n, dimension k and minimum distance n + 1 − k, then the homogeneous weight enumerator of C is More precisely, Duursma's reduced polynomial (10) In particular, The aforementioned formulae imply that W determine uniquely the homogeneous weight enumerator W C (x, y) of C by the formula with explicit polynomials and Proof. In the case of g = 0, note that C is an MDS-code and W C (x, y) = M n,n+1−k (x, y). Form now on, we assume that g > 0 and put r := g + g ⊥ . Making use of d+ g = n + 1− k, let us express Let us introduce c −2 = c −1 = c r−1 = c r = 0 and compare the coefficients of t i from the left and right hand side of (16), in order to obtain Setting j = i − 1, respectively, j = i − 2 in the last two sums, one obtains Let us put and recall that the homogeneous weight enumerator of an MDS-code of length n and minimum distance d + j is Making use of the weight distribution (18) of an MDS-code and introducing Making use of n w Bearing in mind that The equality W n,n−k (x, y) = n k (q − 1)(x − y) k y n−k is exactly the claim (c) of Lemma 1 from Kim and Nyun's work [7]. Plugging in (19) in (17) and bearing in mind that d + g = n + 1 − k, one obtains (8).
In order to prove (9) and (10), let us put Making use of (8), one expresses after changing the summation order. Setting w := n − s, one obtains Then which proves (9), (10).
Towards (11), (12), let us introduce z := x − y and express (8) in the form On the other hand, after changing the summation order. Comparing the coefficients of y d+i z n−d−i in the left and right hand side of (20), one obtains Combining with (18), one justifies (11) and (12). These formulae imply also the fact that The substitution by (11), (12), (18) in (8) yields One exchanges the summation order in the double sums towards Introducing s := d + i, one obtains (13) with (14) and (15).
Comparing the coefficients of x n−d y d in the left and right hand sides of (8), one obtains W (d) We claim that c 0 < 1. To this end, note that for any d-tuple That is due to the fact that the columns H i 1 , . . . , H i d of an arbitrary parity check matrix H of C are of rank d − 1 and there are no words of weight ≤ d − 1 in the right null space If we assume that c 0 = 1 then any d-tuple of columns of H is linearly dependent. Bearing in mind that rkH = n − k, one concludes that d > n − k. Combining with Singleton Bound d ≤ n − k + 1, one obtains d = n − k + 1. That contradicts the assumption that C is not an MDS-code and proves that c 0 < 1 for any F q -linear code C ⊂ F n q of genus g ≥ 1. Note that c 0 can be interpreted as the probability for a d-tuple to support a word of weight d from C.

The Riemann Hypothesis Analogue and the formal selfduality of a linear code
Recall that a linear code C ⊂ F n q with dual code C ⊥ ⊂ F n q is formally self-dual if C and C ⊥ have one and a same number W (w) C ⊥ of codewords of weight 0 ≤ w ≤ n. Let us mention some trivial consequences of the formal self-duality of C. First of all, C and C ⊥ have one and a same minimum distance d = d(C) = d(C ⊥ ) = d ⊥ . Further, C and C ⊥ have one and a same cardinality so that k = dim C = dim C ⊥ = k ⊥ and the length n = k + k ⊥ = 2k is an even integer.
The genera g = k + 1 − d = g ⊥ also coincide. Let P C (t) = 2g i=0 a i t i and P C ⊥ = 2g i=0 a ⊥ i t i be the zeta polynomials of C, respectively, of C ⊥ . The consecutive comparison of the coefficients of x n−d y d , x n−d−1 y d+1 , . . . , x n−d−2g y d+2g from the homogeneous polynomial , y), so that the formal self-duality of C is tantamount to the coincidence P C (t) = P C ⊥ (t) of the zeta polynomials of C and C ⊥ . Duursma has shown that Mac Williams identities for W C ⊥ are equivalent to the functional equation (7) for the zeta polynomials P C (t), P C ⊥ (t) of C, C ⊥ ⊂ F n q with genera g, g ⊥ .
Thus, an F q -linear code C ⊂ F n q is formally self-dual if and only if its zeta polynomial P C (t) satisfies the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g over F q .
Proposition 4. If a linear code C ⊂ F n q satisfies the Riemann Hypothesis Analogue then C is formally self-dual, i.e., the zeta polynomial P C (t) of C is subject to the functional equation (21) of the Hasse-Weil polynomial of the function field of a curve of genus g over F q .
Proof. Let us assume that P C (t) of degree r := g + g ⊥ satisfies the Riemann Hypothesis Analogue, i.e., for some α j ∈ C with |α j | = 1 √ q for all 1 ≤ j ≤ r. If α j is a real root of P C (t) then α j = ε √ q with ε = ±1. We claim that in the case of an even degree r = 2m, the zeta polynomial P C (t) is of the form or of the form while for an odd degree r = 2m + 1 one has for some ε ∈ {±1}. Indeed, if α i ∈ C\R is a complex, non-real root of P C (t) ∈ Q[t] ⊂ R[t] then α i = α i is also a root of P C (t) and P C (t) is divisible by (t − α i )(t − α i ). If P C (t) = 0 has three real roots α 1 , α 2 , α 3 ∈ 1 √ q , − 1 √ q , then at least two of them coincide. For Thus, P C (t) has at most two real roots, which are not complex conjugate (or, equivalently, equal) to each other and P C (t) is of the form (22), (23) or (24).
after multiplying each of the factors 1 Plugging in t = 1, one concludes that q g−m = 1, whereas g = m. As a result, g + g ⊥ = 2m = 2g specifies that g = g ⊥ and (25) yields P C (t) = P C ⊥ (t), which is equivalent to the formal self-duality of C.
If P C (t) is of the form (23) then (7) provides Expressing by Duursma's reduced polynomials D C (t), D C ⊥ (t), one obtains The substitution t = 1 in the last equality of polynomials yields −1 − q g−m = 0, which is an absurd, justifying that a zeta polynomial P C (t), subject to the Riemann Hypothesis Analogue cannot be of the form (23). If P C (t) is of odd degree 2m + 1, then (24) and (7) yield after multiplying 1 qt − ε √ q by − ε √ q qt and each 1 q 2 t 2 − 2Re(α i ) qt + 1 q by qt 2 . Expressing by Duursma's reduced polynomials The substitution t = 1 implies −1 − εq g−m− 1 2 = 0, which is an absurd, as far as q x = 1 if and only if x = 0, while g − m − 1 2 cannot vanish for integers g, m. Thus, none zeta polynomial of odd degree satisfies the Riemann Hypothesis Analogue.

Corollary 5.
If an F q -linear code C of dim Fq C = k and minimum distance d satisfies the Riemann Hypothesis Analogue then the cardinality q of the basic field satisfies the upper bound Proof. By Proposition 4, if C satisfies the Riemann Hypothesis Analogue then for some ϕ j ∈ [0, 2π). The formal self-duality of C is equivalent to the functional equation P C (t) = P C 1 qt q g t 2g of the Hasse-Weil polynomial of a function field of genus g over F q and implies that a 2g = q g a 0 . Comparing the coefficients of x 2k−d y d in the expression of the homogeneous weight enumerator W C (x, y) of C by the homogeneous weight enumerators M 2k,d+i (x, y) of MDS-codes of length 2k and minimum distance d + i, one concludes that W Note that any word c ∈ C of weight d is a solution of a homogeneous linear system of rank d − 1 in d variables, as far as any d − 1 columns of a parity check matrix of C are linearly independent. Thus, there are exactly q − 1 words of weight d from C with the same support as c. If ν is the number of the d-tuples, supporting a word c ∈ C of weight d then W is the probability for a d-tuple to support a word of weight d from C. Altogether, one obtains that In particular, Bearing in mind that cos ϕ j ∈ [−1, 1], one estimates and concludes that As a result, there follows By assumption, C is of minimum distance d, so that ν ≥ 1 and Proposition 6. The following conditions are equivalent for a linear code C ⊂ F n q : (i) C is formally self-dual, i.e., the zeta polynomial P C (t) of C satisfies the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g over F q ; of the Hasse-Weil polynomial of the function field of a curve of genus g − 1 over F q ; (iii) the coefficients of Duursma's reduced polynomial D C (t) = (28) (iv) the dual code C ⊥ ⊂ F n q of C has dimension dim Fq C ⊥ = dim Fq C = k, genus g(C ⊥ ) = g(C) = g and the homogeneous weight enumerator of C is where (v) the dual code C ⊥ ⊂ F n q of C has dimension dim Fq C ⊥ = dim Fq C = k, genus g(C ⊥ ) = g(C) = g and the homogeneous weight enumerator , so that C can be obtained from an MDS-code of the same length 2k and dimension k by removing and adjoining appropriate words, depending explicitly on the numbers W (d) (21), in order to obtain whereas (27).
That proves the equivalence (i) ⇔ (ii). Towards (ii) ⇔ (iii), note that the functional equation of D C (t) reads as Comparing the coefficients of the left-most and the right-most side, one expresses the formal self-duality of C by the relations Let i := g − 1 − j, in order to express the above conditions in the form For any −g + 1 ≤ i ≤ −1 note that c g−1+i = q i c g−1−i is equivalent to c g−1−i = q −i c g−1+i and follows from (34) with 1 ≤ −i ≤ g − 1.
Towards (iii) ⇒ (iv), one introduces a new variable z := x − y and expresses (8) in the form V C (y + z, y) := W C (y + z, y) − M 2k,k+1 (y + z, y) = (q − 1) Let us change the summation index of the first sum to 0 ≤ j := g − 1 − i ≤ g − 1, put 1 ≤ j := i − g + 1 ≤ g − 1 in the second sum and make use of d + g = k + 1, in order to obtain Extracting the term with j = 0 from the first sum, one expresses for an arbitrary F q -linear code C ⊂ F n q . If C is formally self-dual, then plugging in by (28) in (36) and making use of (30), (31), one gets V C (y + z, y) = g−1 j=0 c g−1−j w j (y + z, y).
Substituting z := x−y and V C (x, y) := W C (x, y)−M 2k,k+1 (x, y), one derives the equality (29) for the homogeneous weight enumerator of a formally self-dual linear code C ⊂ F 2k q . In order to justify that (iv) suffices for the formal self-duality of C, we use that (29) with (30) and (31) is equivalent to Comparing the coefficients of y k+j z k−j with 1 ≤ j ≤ g − 1 from (36) and (37), one concludes that c g−1+j = c g−1−j q j for ∀1 ≤ j ≤ g − 1.
These are exactly the relations (28) and imply the formal self-duality of C.
Towards (iv) ⇔ (v), it suffices to put E(x, y) := g−1 j=0 c g−1−j w j (x, y) and to derive that Plugging in by (11) and exchanging the summation order, one gets Introducing s := d + i and extracting W (w) C as coefficients, one obtains Let C ⊂ F n q be an F q -linear code of genus g, whose dual C ⊥ ⊂ F n q is of genus g ⊥ . In [1], Dodunekov and Landgev introduce the near-MDS linear codes C as the ones with zeta polynomial P C (t) ∈ Q[t] of degree deg P C (t) := g + g ⊥ = 2. Thus, C is a near-MDS code if and only if it has constant Duursma's reduced polynomial D C (t) = c 0 ∈ Q. Kim an Hyun prove in [7]) that a near-MDS code C satisfies the Riemann Hypothesis Analogue exactly when The next proposition characterizes the formally-self-dual codes C ⊂ F n q of genus 2, which satisfy the Riemann Hypothesis Analogue. By Proposition 6 (ii), C is a formally self-dual linear code of genus 2 exactly when its Duursma's reduced polynomial is
In other words, the quadratic equation has roots −1 ≤ t 1 = cos(ϕ) ≤ t 2 = cos(ψ) ≤ 1. This, in turn, holds exactly when the discriminant is non-negative, the vertex belongs to the segment [−1, 1] and the values of f (t) at the ends of this segment are non-negative, The equivalence of (41) to (38) is straightforward. Since C is of minimum distance d = k − 1 and W positive rational number and one can multiply (42) by −4 √ qc 0 < 0, add (q + 1)c 0 to all the terms and rewrite it in the form Making use of c 0 > 0, one observes that the above inequalities are tantamount to (39). Finally, can be expressed as (40).

Duursma's reduced polynomial of a function field
Let F = F q (X) be the function field of a curve X of genus g over F q and h g := h(F ) be the class number of F , i.e., the number of the linear equivalence classes of the divisors of F of degree 0. The present section introduces an additive decomposition of the Hasse-Weil polynomial L F (t) ∈ Z[t] of F , which associates to F a sequence {h i } g−1 i=1 of virtual class numbers h i of function fields of curves of genus i over F q . Lemma 8. The following conditions are equivalent for a polynomial L g (t) ∈ Q[t] of degree deg L g (t) = 2g: (i) L g (t) satisfies the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g over F q ; is a polynomial with rational coefficients of degree 2g−2, satisfying the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g − 1 over F q ; Proof. Towards (i) ⇒ (ii), let us note that the polynomial M g (t) := L g (t) − L g (1)t g vanishes at t = 1, so that it is divisible by 1 − t. Further, satisfies the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g over F q . In particular, M g 1 q = M g (1) q g q 2g = 0 and M g (t) is divisible by the linear polynomial q 1 q − t = 1 − qt, which is relatively prime to 1 − t in Q[t]. As a result, is a polynomial of degree deg L g−1 (t) = 2g − 2. Straightforwardly, satisfies the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g − 1 over F q . The implication (ii) ⇒ (i) follows from the functional equation of L g−1 (t), applied to We derive (i) ⇒ (iii) by an induction on g, making use of (ii). More precisely, for g = 1 one has L 0 (t) : In the general case, (ii) provides a polynomial subject to the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g − 1 over F q . By the inductional hypothesis, there exist h ′ i ∈ Q, 0 ≤ i ≤ g − 1 with satisfies the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g over F q .
Proposition 9. Let F = F q (X) be the function field of a smooth irreducible curve X/F q ⊂ P N (F q ) of genus g, defined over F q , with h(F ) linear equivalence classes of divisors of degree 0, A i effective divisors of degree i ≥ 0, Hasse-Weil polynomial L F (t) ∈ Q[t] and Duursma's reduced polynomial D F (t) ∈ Q[t], defined by the equality Then: is a polynomial with integral coefficients, which is uniquely determined by A 0 = 1, A 1 , . . . , A g−1 ; (ii) the equality of formal power series of t holds for (iii) the natural numbers B i , i ≥ 0 from (ii) satisfy the relations (iv) the number h(F ) of the linear equivalence classes of the divisors of F of degree 0 satisfies the inequilities Proof. (i) By Theorem 4.1.6. (ii) and Theorem 4.1.11 from [8], the Hasse-Weil zeta function of F is the generating function . According to Lemma 8 and L F (1) = h(F ), is a polynomial of deg D F (t) = 2g−2, subject to the functional equation of the Hasse-Weil polynomial of the function field of a curve of genus g − 1 over F q . Thus, Let l(G) is the dimension of the space H 0 (X, O X (G)) of the global holomorphic sections of the line bundle O X (G) → X, associated with a divisor G ∈ Div(F ). Riemann-Roch Theorem asserts that for a canonical divisor K X of X. For any j ≥ g − 1, suppose that G 1 , . . . , G h(F ) ∈ Div(F ) is a complete set of representatives of the linear equivalence classes of the divisors of F of degree j. Then for g ≤ j ≤ 2g − 2 and is a complete set of representatives of the linear equivalence classes of the divisors of F of degree 2g − 2 − j, so that Plugging in by (53) in (51), one obtains whereas Putting i := 2g − 2 − j in the second sum and i := j − g in the third sum, one expresses

Summing up the geometric progressions
one derives In particular, D F (t) ∈ Z[t] has integral coefficients.
(ii) Let us expand as sums of geometric progressions and note that Then represent Duursma's reduced polynomial in the form Now, the comparison of the coefficients of t i , i ≥ 0 from the left hand side and the right hand side of (44) provides (45), (46) and The last formula can be expressed in the form A j t j ∈ Z[t] and express In particular, Straightforwardly, That proves (48) for i = g −1.
In the case of g ≤ i ≤ 2g −4 note that 0 ≤ 2g −4−i ≤ g −4 and Changing the summation index of the second sum to s := 2g − 2 − j, one obtains An appropriate grouping of the sums yields That justifies (48). Note that (49) with i ≥ 2g − 2 coincides with (47). In the case of i = 2g − 3, A s (q g−1 − q g−1−s ), after changing the summation index of the second sum to s := 2g − 2 − j. Then which is tantamount to (49) with i = 2g − 3.
(iv) By the Hasse-Weil Theorem, all the roots of L F (t) belong to the circle S 1 √ q = z ∈ C | |z| = 1 √ q . The proof of Proposition 4 specifies that for some ϕ j ∈ [0, 2π). The functional equation L F (t) = L F 1 qt q g t 2g implies that a 2g = q g a 0 . Combining with a 0 = L F (0) = 1, one gets (qt 2 − 2 √ q cos ϕ j t + 1).