Curves over Finite Fields with Many Rational Points Obtained by Ray Class Field Extensions

Roland Auer

Rijksuniversiteit Groningen, Vakgroep Wiskunde, Blauwborgje 3, NL-9747 AC Groningen, The Netherlands

                             auer@math.rug.nl

Abstract. A general type of ray class fields of global function fields is investigated. The computation of their genera is reduced to the determination of the degrees of these extensions, which turns out to be the main difficulty. While in two special situations explicit formulas for the degrees are known, the general problem is solved algorithmically. The systematic application of the methods described yields several new examples of algebraic curves over 𝔽_{2}, 𝔽_{3}, 𝔽_{4}, 𝔽_{5} and 𝔽_{7} with comparatively many rational points.

1 Introduction

The maximum number of 𝔽_{q}-rational points on a (smooth, projective, absolutely irreducible algebraic) curve X|𝔽_{q} of genus g(X) = g defined over the finite field 𝔽_{q} is usually denoted by N_{q}(g). In the early eighties, Serre [20–22] has written down formulas for N_{q}(1) and N_{q}(2).

Since the precise value of N_{q}(g) is quite difficult to determine in general, the work of many mathematicians has instead led to large tables, such as [6], giving an interval for this quantity. The lower bounds are usually realized by abelian coverings of small genus curves, which are either given by explicit equations (Hansen and Stichtenoth [7], [8], [24], van der Geer and van der Vlugt [3], [4], [5], Niederreiter and Xing [12], [13], [15], Shabat [23], and others) or obtained by class field theory or an equivalent construction (Schoof [19], Lauter [11], Niederreiter and Xing [25], [13], [14], [16]). (Please note that these references are far from being complete.) The present paper, which adds to the second category, summarizes the author’s thesis [1], where all results stated here are proved in detail.

Since we employ ray class field extensions, to the curve X|𝔽_{q} we associate the global function field K = 𝔽_{q}(X). Its genus g_{K} equals g(X), and coverings of X correspond to field extensions of K, the degree of the covering being the degree of the extension. By construction, 𝔽_{q} is the full constant field, i.e. is algebraically closed in K, and we express this instance by writing K|𝔽_{q}.

A place of K, by which we mean the maximal ideal 𝔭 in some discrete valuation ring of K, with (residue field) degree d = deg 𝔭, corresponds to (a Galois conjugacy class of) d points in X(𝔽_{q^{d}}), and each point on X having 𝔽_{q^{d}} as its minimal field of definition over 𝔽_{q} lies in such a conjugacy class. In particular K|𝔽_{q} has N_{K} = |X(𝔽_{q})| rational places, i.e. places of degree 1. Throughout this paper, K|𝔽_{q} is a global function field, and all algebraic extensions of K are assumed to lie in some fixed algebraic closure K̄ of K.

2 Ray Class Fields

We fix a non-empty set S of places of K, and denote the greatest common divisor of the degrees of its elements by d := gcd{deg 𝔭 | 𝔭 ∈ S}. Let 𝔪 be an S-cycle, i.e. an effective divisor of K with support away from S. We consider the S-ray class field mod 𝔪, denoted K_{S}^{𝔪}, which is defined as the largest abelian extension L|K of conductor at most 𝔪 such that every place of S splits completely in L. These extensions occur e.g. in Perret [17].

In the special case of 𝔪 = 𝔬 (the zero element in the divisor group), K_{S}^{𝔬} is also known as the S-Hilbert class field (cf. Rosen [18]). We recall that the Galois group G(K_{S}^{𝔬}|K) is isomorphic to the (ideal) class group Cℓ(O_{S}) of the Dedekind ring O_{S} consisting of all functions with poles only in S.

Since S is non-empty, by class field theory K_{S}^{𝔪} is a finite (algebraic) extension of K. In fact, using Čebotarev’s Density Theorem, any finite abelian L|K is seen to be equal to some K_{S}^{𝔪}. Here for 𝔪 we can take the conductor of L|K, and S can always be chosen finite. Furthermore, the ray class fields satisfy the following properties.

Proposition 1. Let S, d and 𝔪 be as above, T another non-empty set of places of K and 𝔫 a T -cycle.

(a) The full constant field of K_{S}^{𝔪} has degree d over 𝔽_{q}, thus K_{S}^{𝔪}|𝔽_{q^{d}}.

(b) If S ⊇ T and 𝔪 ≤ 𝔫, then K_{S}^{𝔪} ⊆ K_{T}^{𝔫}.

(c) K_{S}^{𝔪} ∩ K_{T}^{𝔫} = K_{S∪T}^{min{𝔪,𝔫}}, where the minimum is taken coefficientwise.

In terms of the ray class fields of K, we can write down the genus for any abelian extension of K.

Theorem 1. Let S, d, 𝔪 = ∑_{p} m_{𝔭}𝔭 be as above, and L an intermediate field of K_{S}^{𝔪}|K. Then the genera g_{K} and g_{L} of K and L satisfy

d · (g_{L} − 1) = [L : K](g_{K} − 1 + \frac{deg 𝔪}{2}) − \frac{1}{2} ∑_{𝔭} ∑_{n=1}^{m_{𝔭}}[L ∩ K_{S}^{𝔪−n𝔭} : K] deg 𝔭 .

This formula can be proved either by applying Möbius inversion to the Conductor Discriminant Product Formula, as done by Cohen et al. [2] in the number field case, or by using Hilbert’s Different Formula, the Hasse-Arf Theorem and the connection between upper ramification groups and higher unit groups known from local class field theory (see [1]).

We observe that computation of the genus of ray class field extensions amounts to determining their degrees. This is easily done if S consists of just one place, but becomes much more intricate if we require more places to split. In the following section we indicate an algorithmic solution of this problem.

3 Computation of Degrees

For simplicity we shall restrict to the case of ramification at only one place 𝔭 of K|𝔽_{q} (outside S), i.e. to 𝔪 = m𝔭 with m ∈ ℕ_{0}. Recall that, by definition, the residue field 𝔽_{𝔭} of 𝔭 satisfies [𝔽_{𝔭} : 𝔽_{q}] = deg 𝔭. We determine the degrees [K_{S}^{m𝔭} : K] in three steps.

First of all, from what has been said about the Hilbert class field, [K_{S}^{𝔬} : K] equals the S-class number h_{S} := |Cℓ(O_{S})|. Its computation is connected with the problem of finding generators for the group O_{S}^{∗} of S-units, which in turn are needed for the other two steps.

Indeed, by class field theory, the Galois group G(K_{S}^{𝔭}|K_{S}^{𝔬}) is isomorphic to the cokernel of the canonical group homomorphism O_{S}^{∗} → 𝔽_{𝔭}^{∗}. Since 𝔽_{q}^{∗} ⊆ O_{S}^{∗},

it follows that [K_{S}^{𝔭} : K_{S}^{𝔬}] divides \frac{q^{deg 𝔭}−1}{q−1}.

Similarly, the cokernel of O_{S}^{∗} ∩ (1 + 𝔭) → (1 + 𝔭)/(1 + 𝔭^{m}) is isomorphic to G(K_{S}^{m𝔭}|K_{S}^{𝔭}). According to the following theorem, its order can be determined for all m ∈ ℕ simultaneously. Let p be the characteristic of K, and define

⌈a⌉_{p} := min{p^{l} | a ≤ p^{l}, l ∈ ℕ_{0}}

for any real a > 0.

Theorem 2. There are s := |S| − 1 positive integers n_{1}, …, n_{s} depending only on S and 𝔭 such that

[K_{S}^{m𝔭} : K_{S}^{𝔭}] = q^{(m−1) deg 𝔭}/ ∏_{i=1}^{s} ⌈\frac{m}{n_{i}} ⌉_{p}

for all m ∈ ℕ.

The proof given in [1, p. 43] provides an algorithm for the computation of the numbers n_{1}, …, n_{s}. Since the order of these numbers is irrelevant, the behaviour of the degrees can be summarized in the polynomial

δ_{S,𝔭} := ∑_{i=1}^{s} t^{n_{i}},

which is uniquely determined by S and 𝔭. Unfortunately we have no explicit formula for δ_{S,𝔭} except in some particular cases, which are treated in the next section.

4 Rational Function Field

Here we want to draw the attention to two special situations where the polynomial δ_{S,𝔭} can be given explicitly.

Theorem 3. Let K be the rational function field over the prime field 𝔽_{p}, S a non-empty set of rational places of K and 𝔭 a rational place of K not occurring in S; thus 0 ≤ s := |S| − 1 < p. Then δ_{S,𝔭} = ∑_{n=1}^{s} t^{n}.

In particular K_{S}^{(s+1)𝔭} = K, and for m ≥ s + 2 it follows that K_{S}^{m𝔭} has exactly

N_{K_{S}^{m𝔭}} = 1 + [K_{S}^{m𝔭} : K](s + 1)

rational places. The genus can be calculated by means of Theorem 1. As an example we have carried out these computations for p ∈ {5, 7} and different values of s and m. The results are displayed in the tables 1 and 2.

Table 1. Ray class fields over 𝔽_{5}

\begin{matrix}s & 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 \\ m & 3 & 4 & 5-6 & 7 & 4 & 5-6 & 7 & 8 & 5-6 & 7 & 8 & 9 \\ [K_{S}^{m𝔭} : K] & 5 & 5 & 5 & 5 & 25 & 25 & 25 & 25 & 125 & 125 & 125 & 125 \\ g_{K_{S}^{m𝔭}} & 2 & 4 & 6 & 10 & 22 & 34 & 56 & 70 & 172 & 284 & 356 & 420 \\ N_{K_{S}^{m𝔭}} & 11 & 16 & 21 & 26 & 51 & 76 & 101 & 126 & 251 & 376 & 501 & 626 \end{matrix}

Table 2. Ray class fields over 𝔽_{7}

\begin{matrix}s & 1 & 2 & 3 & 4 & 5 & 6 & 1 & 2 & 3 & 4 & 5 & 6 \\ m & 3 & 4 & 5 & 6 & 7-8 & 9 & 4 & 5 & 6 & 7-8 & 9 & 10 \\ [K_{S}^{m𝔭} : K] & 7 & 7 & 7 & 7 & 7 & 7 & 49 & 49 & 49 & 49 & 49 & 49 \\ g_{K_{S}^{m𝔭}} & 3 & 6 & 9 & 12 & 15 & 21 & 45 & 69 & 93 & 117 & 162 & 189 \\ N_{K_{S}^{m𝔭}} & 15 & 22 & 29 & 36 & 43 & 50 & 99 & 148 & 197 & 246 & 295 & 344 \end{matrix}

Now let q = p^{e} with e ∈ ℕ be an arbitrary power of the characteristic p again. Take Q := {1, …, q − 1} ⊆ ℤ as a set of representatives for the cyclic group ℤ/(q − 1)ℤ ≃ 𝔽_{q}^{∗}. Via this latter isomorphism, the group G := G(𝔽_{q}|𝔽_{p}) acts on Q. Clearly, two elements n, n′ ∈ Q lie in the same G-orbit Gn = Gn′ iff n′ ≡ p^{l}n mod q − 1 for some l ∈ ℕ_{0}. For n ∈ ℕ we define

e_{n} := \begin{cases}|Gn| & if n ∈ Q and n = min Gn, \\ 0 & otherwise. \end{cases}

The following has been proved by Lauter [11].

Theorem 4. Let K be the rational function field over 𝔽_{q}, 𝔭 a rational place of K, and S the set consisting of the other q rational places. Then δ_{S,𝔭} = ∑_{n∈ℕ} e_{n}t^{n}.

Let us set r := \sqrt{q} or \sqrt{pq} according to whether q is a square or not. By investigating the numbers e_{n}, we see that K_{S}^{(r+1)𝔭} = K. Furthermore we obtain two fields, namely K_{S}^{(r+2)𝔭} and K_{S}^{(2r+2)𝔭} of degree r and rq over K if q is a square, and p − 1 fields K_{S}^{(r+i+1)𝔭} with 1 ≤ i < p of degree q^{i} over K in case q is non-square. Lauter [10] has pointed out that the corresponding curves generalize certain families of Deligne-Lusztig curves. Here we want to write down defining equations for them, which might have been found by J. P. Pedersen before but without publishing.

Proposition 2. Let K = 𝔽_{q}(x) with x an indeterminate over 𝔽_{q} such that 𝔭 is the pole of x, and let S (as in Theorem 4) consist of the remaining q rational places of K.

(a) Assume that r := \sqrt{q} ∈ ℕ and let y, z ∈ K̄ satisfy y^{r} + y = x^{r+1} and z^{q} − z = x^{2r}(x^{q} − x). Then K_{S}^{(r+2)𝔭} = K(y) and K_{S}^{(2r+2)𝔭} = K(y, z).

(b) For r := \sqrt{pq} ∈ ℕ let y_{1}, …, y_{p−1} ∈ K̄ satisfy y_{i}^{q} − y_{i} = x^{ir/p}(x^{q} − x). Then K_{S}^{(r+i+1)𝔭} = K(y_{1}, …, y_{i}) for i ∈ {1, …, p − 1}.

5 Tables

Now we use ray class field extensions of small genus ground fields K|𝔽_{q} to produce curves of higher genus with many rational points over 𝔽_{2}, 𝔽_{3} and 𝔽_{4}. Like in the tables [6] by van der Geer and van der Vlugt, we restrict ourselves to genus g ≤ 50 and give a range (or the precise value) for N_{q}(g). The upper bound is taken from [6]. The lower bound is attained by a field L|𝔽_{q} of genus g_{L} = g with precisely that many rational places, and is set in boldface if our example actually improves the lower bound known before. The field L satisfies K_{S}^{(m−1)𝔭} ⊊ L ⊆ K_{S}^{m𝔭} (thus has conductor m𝔭) with m ∈ ℕ, 𝔭 a place of K of degree d := deg 𝔭, and S a non-empty set of rational places of K not containing 𝔭.

The degrees [K_{S}^{m𝔭} : K] are computed as indicated in Sect. 3. In order to search through a large variety of possibilities for 𝔭 and S, the algorithms have been implemented in KASH/KANT [9]. Then, by Theorem 1, the genus of L is

g_{L} = 1 + [L : K](g_{K} − 1 + \frac{md}{2}) − \frac{d}{2} ∑_{n=0}^{m−1}[K_{S}^{n𝔭} : K] .

Since the inertia degree of 𝔭 in L is h_{S}/h_{S∪{𝔭}}, L has

N_{L} ≥ [L : K] |S| + \begin{cases}h_{S} & if h_{S} = h_{S∪\{𝔭\}} and d = 1 \\ 0 & otherwise \end{cases}

rational places. Here equality holds iff S already contains all rational places of K that split completely in L, which in our examples is always the case. Complete information on the precise construction of each field L occurring in the tables 3–5 is given in [1].

Table 3. Function fields over 𝔽_{2} with many rational places

\begin{matrix}g & N_{2}(g) & [L : K] & |S| & m & d & g_{K} \\ 6 & 10 & 10 & 1 & 2 & 1 & 1 \\ 7 & 10 & 2 & 5 & 2 & 6 & 1 \\ 8 & 11 & 2 & 5 & 10 & 1 & 2 \\ 9 & 12 & 4 & 3 & 4 & 2 & 0 \\ 10 & 13 & 4 & 3 & 4 & 1 & 2 \\ 12 & 14–15 & 7 & 2 & 1 & 6 & 0 \\ 14 & 15–16 & 15 & 1 & 1 & 4 & 0 \\ 15 & 17 & 8 & 2 & 7 & 1 & 0 \\ 16 & 17–18 & 2 & 8 & 14 & 1 & 5 \\ 17 & 17–18 & 16 & 1 & 5 & 1 & 0 \\ 19 & 20 & 4 & 5 & 2 & 6 & 1 \\ 22 & 21–22 & 4 & 5 & 12 & 1 & 2 \end{matrix} \begin{matrix}g & N_{2}(g) & [L : K] & |S| & m & d & g_{K} \\ 27 & 24–25 & 12 & 2 & 3 & 2 & 1 \\ 28 & 25–26 & 8 & 3 & 7 & 1 & 2 \\ 29 & 25–27 & 4 & 6 & 14 & 1 & 3 \\ 30 & 25–27 & 4 & 6 & 12 & 1 & 4 \\ 35 & 29–31 & 4 & 7 & 16 & 1 & 4 \\ 37 & 29–32 & 4 & 7 & 14 & 1 & 5 \\ 39 & 33 & 16 & 2 & 8 & 1 & 0 \\ 41 & 33–35 & 8 & 4 & 6 & 1 & 4 \\ 42 & 33–35 & 8 & 4 & 8 & 1 & 3 \\ 44 & 33–37 & 8 & 4 & 11 & 1 & 2 \\ 49 & 36–40 & 12 & 3 & 1 & 6 & 3 \\ 50 & 40 & 8 & 5 & 2 & 7 & 1 \end{matrix}

Table 4. Function fields over 𝔽_{3} with many rational places

\begin{matrix}g & N_{3}(g) & [L : K] & |S| & m & d & g_{K} \\ 5 & 12–13 & 3 & 4 & 2 & 2 & 1 \\ 7 & 16 & 8 & 2 & 1 & 4 & 0 \\ 9 & 19 & 3 & 6 & 5 & 1 & 2 \\ 10 & 19–21 & 9 & 2 & 5 & 1 & 0 \\ 14 & 24–26 & 3 & 8 & 5 & 2 & 2 \\ 15 & 28 & 9 & 3 & 6 & 1 & 0 \\ 16 & 27–29 & 9 & 3 & 3 & 2 & 0 \\ 17 & 24–30 & 6 & 4 & 5 & 1 & 2 \\ 19 & 28–32 & 3 & 9 & 12 & 1 & 3 \\ 22 & 30–36 & 3 & 10 & 3 & 5 & 3 \\ 24 & 31–38 & 3 & 10 & 14 & 1 & 4 \\ 30 & 37–46 & 9 & 4 & 8 & 1 & 1 \end{matrix} \begin{matrix}g & N_{3}(g) & [L : K] & |S| & m & d & g_{K} \\ 33 & 46–49 & 9 & 5 & 4 & 1 & 3 \\ 34 & 45–50 & 9 & 5 & 3 & 3 & 1 \\ 36 & 46–52 & 9 & 5 & 9 & 1 & 1 \\ 37 & 48–54 & 24 & 2 & 1 & 2 & 2 \\ 39 & 48–56 & 24 & 2 & 2 & 2 & 1 \\ 43 & 55–60 & 9 & 6 & 11 & 1 & 1 \\ 45 & 54–62 & 18 & 3 & 3 & 2 & 1 \\ 46 & 55–63 & 27 & 2 & 6 & 1 & 0 \\ 47 & 54–65 & 18 & 3 & 6 & 1 & 1 \\ 48 & 55–66 & 9 & 6 & 11 & 1 & 2 \\ 49 & 63–67 & 9 & 7 & 3 & 4 & 1 \\ 50 & 56–68 & 28 & 2 & 1 & 2 & 2 \end{matrix}

\begin{matrix}g & N_{4}(g) & [L : K] & |S| & m & d & g_{K} \\ 4 & 15 & 2 & 7 & 6 & 1 & 1 \\ 5 & 17–18 & 4 & 4 & 6 & 1 & 0 \\ 6 & 20 & 4 & 5 & 2 & 3 & 0 \\ 8 & 21–24 & 2 & 10 & 6 & 1 & 3 \\ 9 & 26 & 8 & 3 & 3 & 1 & 1 \\ 10 & 27–28 & 12 & 2 & 2 & 1 & 1 \\ 11 & 26–30 & 2 & 13 & 4 & 3 & 3 \\ 12 & 29–31 & 4 & 7 & 8 & 1 & 1 \\ 13 & 33 & 8 & 4 & 6 & 1 & 0 \\ 14 & 32–35 & 8 & 4 & 2 & 3 & 0 \\ 19 & 37–43 & 4 & 9 & 10 & 1 & 2 \\ 20 & 40–45 & 8 & 5 & 3 & 3 & 0 \\ 21 & 41–47 & 4 & 10 & 8 & 1 & 3 \\ 22 & 41–48 & 4 & 10 & 10 & 1 & 3 \\ 23 & 45–50 & 4 & 11 & 10 & 1 & 3 \end{matrix} \begin{matrix}g & N_{4}(g) & [L : K] & |S| & m & d & g_{K} \\ 24 & 49–52 & 4 & 12 & 10 & 1 & 3 \\ 25 & 51–53 & 12 & 4 & 3 & 1 & 2 \\ 27 & 49–56 & 16 & 3 & 6 & 1 & 0 \\ 28 & 53–58 & 4 & 13 & 14 & 1 & 3 \\ 31 & 60–63 & 15 & 4 & 1 & 3 & 2 \\ 32 & 57–65 & 8 & 7 & 10 & 1 & 1 \\ 33 & 65–66 & 16 & 4 & 7 & 1 & 0 \\ 34 & 57–68 & 8 & 7 & 8 & 1 & 2 \\ 36 & 64–71 & 8 & 8 & 3 & 3 & 2 \\ 41 & 65–78 & 20 & 3 & 3 & 1 & 2 \\ 43 & 72–81 & 24 & 3 & 2 & 3 & 0 \\ 45 & 80–84 & 16 & 5 & 4 & 2 & 0 \\ 47 & 73–87 & 8 & 9 & 12 & 1 & 2 \\ 48 & 80–89 & 16 & 5 & 3 & 3 & 0 \\ 49 & 81–90 & 8 & 10 & 10 & 1 & 3 \end{matrix}

References

1. R. Auer. Ray class fields of global function fields with many rational places. Disser-

tation at the University of Oldenburg, www.bis.uni-oldenburg.de/dissertation
/ediss.html, 1999.

2. H. Cohen, F. Diaz y Diaz, M. Olivier. Computing ray class groups, conductors and discriminants. In: Algorithmic Number Theory. H. Cohen (ed.). Lecture Notes in Computer Science 1122, Springer, Berlin, (1996) 51–59.

3. G. van der Geer, M. van der Vlugt. Curves over finite fields of characteristic 2 with many rational points. C. R. Acad. Sci. Paris 317 (1993) série I, 593–597.

4. G. van der Geer, M. van der Vlugt. How to construct curves over finite fields with many points. In: Arithmetic Geometry (Cortona 1984). F. Catanese (ed.). Cambridge Univ. Press (1997) 169–189.

5. G. van der Geer, M. van der Vlugt. Constructing curves over finite fields with many points by solving linear equations. Preprint 1997.

6. G. van der Geer, M. van der Vlugt. Tables of curves with many points. Math. Comp., to appear.

7. J. P. Hansen. Group codes and algebraic curves. Mathematica Gottingensis, Schriftenreihe SFB Geometrie und Analysis, Heft 9, 1987.

8. J. P. Hansen, H. Stichtenoth. Group codes on certain algebraic curves with many rational points. Appl. Alg. Eng. Commun. Comp. 1 (1990) 67–77.

9. The KANT Group. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, K. Wildanger. KANT V4. J. Symb. Comp. 24/3 (1997) 267–283.

10. K. Lauter. Deligne Lusztig curves as ray class fields. Manuscripta Math. 98/1 (1999) 87–96.

11. K. Lauter. A formula for constructing curves over finite fields with many rational points. J. Number Theory 74/1 (1999) 56–72.

12. H. Niederreiter, C. P. Xing. Explicit global function fields over the binary field with many rational places. Acta Arith. 75/4 (1996) 383–396.

13. H. Niederreiter, C. P. Xing. Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places. Acta Arith. 79/1 (1997) 59–76.

14. H. Niederreiter, C. P. Xing. Algebraic curves over finite fields with many rational points. Number theory (Eger 1996) 423–443, de Gruyter, Berlin, 1998.

15. H. Niederreiter, C. P. Xing. Global function fields with many rational places over the ternary field. Acta Arith. 83/1 (1998) 65–86.

16. H. Niederreiter, C. P. Xing. A general method of constructing global function fields with many rational places. Algorithmic Number Theory (Portland 1998), Lect. Notes in Comp. Sci. 1423 555–566, Springer, Berlin, 1998.

17. M. Perret. Tours ramifiées infinies de corps de classes. J. Number Theory 38 (1991) 300–322.

18. M. Rosen. The Hilbert class field in function fields. Expo. Math. 5 (1987) 365–378.

19. R. Schoof. Algebraic curves and coding theory. UTM 336, Univ. of Trento, 1990.

20. J.-P. Serre. Sur le nombre des points rationelles d’une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris 296 (1983) série I, 397–402.

21. J.-P. Serre. Nombres de points des courbes algébrique sur 𝔽_{q}. Séminaire de Théorie des Nombres de Bordeaux 22 (1982/83).

22. J.-P. Serre. Résumé des cours de 1983–1984. Annuaire du Collège de France (1984) 79–83.

23. V. Shabat. Unpublished manuscript. University of Amsterdam, 1997/98.

24. H. Stichtenoth. Algebraic geometric codes associated to Artin-Schreier extensions of 𝔽_{q}(z). In: Proc. 2nd Int. Workshop on Alg. and Comb. Coding Theory. Leningrad (1990) 203–206.

25. C. P. Xing, H. Niederreiter. Drinfel’d modules of rank 1 and algebraic curves with many rational points. Report Austrian Academy of Sciences, Vienna, 1996.