On Kummer covers with many rational points over finite fields

by

Arnaldo Garcia and Alvaro Garzon^{∗}

1 Introduction

Let C be a nonsingular, projective, geometrically irreducible algebraic curve defined over a finite field 𝔽_{q} with q elements. Let C(𝔽_{q}) denote the set of 𝔽_{q}-rational points on C; i.e., points on the curve C having all coordinates in 𝔽_{q}. A celebrated theorem of A. Weil gives in particular the following upper bound on the cardinality of C(𝔽_{q}):

#C(𝔽_{q}) ≤ q + 1 + 2\sqrt{q} · g(C),

where g(C) stands for the genus of the curve C. If the cardinality q of the finite field is not a square, the upper bound above was improved by J.-P. Serre substituting 2\sqrt{q} by its integer part [2\sqrt{q}]. If the cardinality q is a square (say q = r^{2}) then the curve C is called maximal over 𝔽_{r^{2}} if #C(𝔽_{r^{2}}) attains Weil’s upper bound; i.e.,

#C(𝔽_{r^{2}}) = r^{2} + 1 + 2r · g(C).

The interest on curves over finite fields with many rational points with respect to their genera (i.e., with #C(𝔽_{q}) close to known upper bounds; e.g., see the tables in [6] and [11]) was greatly renewed after Goppa’s construction of linear codes with good parameters from such curves. Other important applications include: lowdiscrepancy sequences, stream ciphers, hash functions, finite geometries, etc... . We refer to the book in [9] for an extensive study on this subject and its connections with Ray Class Fields, Hilbert Class Fields, Drinfeld Modules, etc....

The aim hence is to present an elementary and effective method for the construction of Kummer covers of the projective line over finite fields with many rational points. The key-idea comes from [5] (see also [3]) and that is the construction of rational functions f(x)/r(x) in the rational function field 𝔽_{q}(x) having the value 1 for many elements x = β in 𝔽_{q}.

(∗) This paper was completed while the first author was visiting the Institute for Mathematical Sciences at the National University of Singapore. Both authors acknowledge the support from CNPq/PRONEX # 66.2408/1996-3 (Brazil)

The paper is organized as follows: in Section 2 we present our method for the construction of good curves C and we explain how one computes its genus and its number of rational points; Section 3 is just to single out an important particular case of our general construction; in Section 4 and Section 5 we give several examples based in the two preceding sections, and in Section 6 we consider fibre products of two curves, both obtained by the method in Section 2.

Three things are worth mentioning:

a) Lots of the records on the tables (see [6] and [11]) that we match were obtained by totally different methods; so here we have an unified way to get them.

b) Our construction involves the choice of a divisor m of the cardinality (q −1) of the multiplicative group of the finite field 𝔽_{q} in question, and we have almost always chosen m small enough so that the resulting genus g satisfies g ≤ 50. This is done only in order to compare with the tables in [6] and [11].

c) We can always determine the exact amount of rational points in the case m = q − 1; however if m is a proper divisor of (q − 1) we sometimes used the computer program Mathematica to complete the determination of the rational points (see end of Section 2 here).

In the following tables we summarize the good pairs (g, N) obtained in the examples of sections 4, 5 and 6 here, where g = g(C) and N = #C(𝔽_{q}).

Table 1 (New records)

Example q g N former entry
6.1  16 34 183  161
5.1  81 13 243  224

Table 2 (New entries)

\begin{matrix}Example & q & g & N \\ 6.2 & 8 & 57 & 168 \\ 6.3 & 64 & 19 & 297 \\ 4.10 & 64 & 25 & 335 \\ 5.5 & 27 & 18 & 148 \\ 5.6 & 27 & 30 & 196 \\ 4.9 & 27 & 37 & 236 \\ 5.7 & 27 & 44 & 278 \\ 5.2 & 81 & 11 & 220 \\ 5.3 & 81 & 14 & 278 \\ 5.4 & 81 & 25 & 392 \\ 4.6 & 25 & 7 & 84 \\ 4.3 & 25 & 13 & 104 \\ 4.7 & 25 & 19 & 132 \\ 6.5 & 25 & 19 & 132 \\ 4.3 & 25 & 21 & 152 \\ 4.7 & 25 & 43 & 252 \\ 6.5 & 25 & 43 & 252 \\ 4.3 & 25 & 45 & 296 \end{matrix}

Table 3 (Meets the record)

Example q g  N
6.6 8 25 86
4.4  16 2 maximal
4.2 16 4 45
4.5  16 6 maximal
4.4  16 12  83
4.5  16 20  127
5.8  64 3 maximal
5.9  64 4 maximal
5.10  64 7 maximal
5.11  64 10 maximal
5.12  64 12 maximal
5.13  64 28 maximal
4.3  9 1 maximal
4.8  27 2  48
6.4  27 11  96
4.3  81 16 maximal
4.3  25 4 maximal
4.8  125 9  252

2 The Method

Let f(x) and ℓ(x) be two polynomials in 𝔽_{q}[x] such that deg f(x) ≥ deg ℓ(x). Consider the euclidian division of f(x) by ℓ(x); i.e.,

f(x) = h(x) · ℓ(x) + r(x)  (1)

for some polynomials h(x), r(x) ∈ 𝔽_{q}[x] with deg r(x) < deg ℓ(x). We will always assume that r(x) ≠ 0 (i.e., we will always assume that f(x) is not a multiple of ℓ(x)). The method for the construction is then to consider the nonsingular projective model C of the curve given by the affine Kummer equation:

y^{m} = \frac{f(x)}{r(x)}, with m a divisor of (q − 1).  (2)

As was shown in [7], the genus g(C) can be easily derived from the multiplicities of the zeros and poles of the rational function f(x)/r(x) ∈ 𝔽_{q}(x) (see also [12, Proposition III.7.3]). If x = α ∈ \overline{𝔽}_{q} is a zero or a pole for f(x)/r(x) with multiplicity m_{α}, then we have d_{α} points on the curve C with first coordinate to x = α, where

d_{α} = gcd(m, m_{α}).  (3)

These d_{α} points have ramification index e_{α} := \frac{m}{d_{α}} and they have different exponent equal to (e_{α} − 1), by Dedekind’s Different Theorem. The same holds for the points of the curve C above the point at infinity of the projective line; i.e., for α = ∞ one defines its multiplicity by m_{∞} = (deg f(x) − deg r(x)) and, similarly,

d_{∞} = gcd(m, m_{∞}) and e_{∞} = \frac{m}{d_{∞}} .  (4)

It now follows from Hurwitz’s genus formula that:

g(C) = 1 + m · (−1 + \frac{1}{2} ∑_{α∈\overline{𝔽}_{q}∪\{∞\}} (1 − \frac{d_{α}}{m} )) .  (5)

The sum over α ∈ \overline{𝔽}_{q} ∪ \{∞\} in the formula (5) above is indeed a finite sum over only the zeros and poles of the rational function f(x)/r(x) ∈ 𝔽_{q}(x). In fact, if α is neither a zero nor a pole, then its multiplicity m_{α} is equal to zero and hence d_{α} = m.

From the genus formula (5), we see that the genus g(C) is smaller as the rational function f(x)/r(x) has fewer distinct zeros and poles. So it is desirable to have a product f(x) · r(x) with lots of multiple roots. Typically we take f(x) as a power of a polynomial and we try to get a remainder r(x) with many multiple roots.

Now we turn to the rational points over 𝔽_{q} of the curve C given by Equation (2).

Roots of h(x) · ℓ(x) = 0. If β ∈ 𝔽_{q} satisfies h(β) · ℓ(β) = 0 and f(β) ≠ 0, then the value of the rational function f(x)/r(x) at x = β is equal to 1 (see Formula (1)). This guarantees that we have m rational points of the curve C with first coordinate x = β as above. So in order to have many rational points over 𝔽_{q} we will always take ℓ(x) as a separable polynomial having all roots in the finite field 𝔽_{q}.

Ramification points. If x = α ∈ \overline{𝔽}_{q} ∪ \{∞\} is a zero or a pole for the rational function f(x)/r(x) with multiplicity m_{α}, then we have d_{α} := gcd(m, m_{α}) points on C with first coordinate x = α. If this zero or pole is rational (i.e., if x = α ∈ 𝔽_{q} ∪{∞}), then it may happen that the d_{α} points on C with x = α are again rational points over 𝔽_{q}. Here is how one decides if this is the case: We can assume that α is a zero for f(x)/r(x) (otherwise substitute y by 1/y) and hence we write

y^{m} = k(x) · (x − α)^{m_{α}} or (\frac{y^{m/d_{α}}}{(x − α)^{m_{α}/d_{α}}} )^{d_{α}} = k(x),

where k(x) ∈ 𝔽_{q}(x) with k(α) ≠ 0 and k(α) ≠ ∞. Then the d_{α} points are all rational over 𝔽_{q} if and only if k(α) is a d_{α}-th power of an element of 𝔽_{q}.

Case m = q − 1. In this case we see from Equation (2) that a rational point on C which is not ramified (i.e., not a zero nor a pole for the rational function f(x)/r(x)) must have second coordinate satisfying y ∈ 𝔽_{q}^{∗} and hence y^{q−1} = 1. Hence in this case besides the possible rational ramification points, the other rational points on C must come from x = β ∈ 𝔽_{q} with h(β) · ℓ(β) = 0 and f(β) ≠ 0.

Case m < (q − 1). In this case it is in general harder to determine the exact amount of rational points on C over 𝔽_{q}. Besides the ones obtained as in the case m = q − 1, one has also to consider the following:

For x = γ ∈ 𝔽_{q} which is not a zero or a pole for the function f(x)/r(x), we consider the element f(γ)/r(γ) ∈ 𝔽_{q}^{∗}. This element γ̃ := f(γ)/r(γ) ∈ 𝔽_{q}^{∗} may satisfy

γ̃ ≠ 1 and γ̃ is a m-th power of an element of 𝔽_{q}.  (6)

For each γ ∈ 𝔽_{q} as above with γ̃ satisfying the conditions on (6), we have m rational points on C with first coordinate x = γ. So in pratice (when m is a proper divisor of (q − 1)), after having a good candidate for a curve C with many rational points over 𝔽_{q} with respect to its genus g(C), we sometimes have carried out a computer search to determine the cardinality K of the set of such elements

K := #{γ ∈ 𝔽_{q} | γ is not zero or pole for f(x)/r(x), γ̃ ≠ 1
and γ̃ is m-th power in 𝔽_{q} }.  (7)

This then gives rise to K · m “extra rational points” on the curve C.

3 An important particular case of the method

Lots of the curves C with many rational points over 𝔽_{q} with respect to the genus g(C) that we will construct here are obtained thru this particular case of this section.

Let ℓ(x) ∈ 𝔽_{q}[x] be a separable polynomial having all roots in the finite field 𝔽_{q}. Let g(x) ∈ 𝔽_{q}[x] be another polynomial which is relatively prime to the polynomial ℓ(x). Then there are polynomials h(x) and v(x) in 𝔽_{q}[x] such that:

v(x) · g(x) = h(x) · ℓ(x) + 1.  (8)

We then consider the nonsingular projective model C of the curve given by

y^{m} = v(x) · g(x), with m a divisor of (q − 1).  (9)

As in Section 2 we usually take g(x) highly inseparable (e.g., a power of a polynomial) and we try to get the polynomial v(x) ∈ 𝔽_{q}[x] from Formula (8) also highly inseparable. As before this is done in order to get a low genus for the constructed curve C, since from Formula (5) we see that the genus depends essentially on the number of distinct roots of the polynomial f(x) := v(x) · g(x).

One should think of the polynomial g(x) ∈ 𝔽_{q}[x] as an auxiliary polynomial that is used to construct the polynomial f(x) := v(x) · g(x) whose remainder in the division by ℓ(x) satisfies r(x) = 1 (compare Formula (8) with Formula (1)).

4 Examples based on Section 2

Suppose that q = r^{n} with n ≥ 2. An irreducible polynomial in 𝔽_{r}[x] of degree n has all roots in 𝔽_{q}. A natural choice for the polynomial ℓ(x) is then to take certain products of such irreducible polynomials.

Example 4.1. (n = 2). Let m be a divisor of (r^{2}−1) and denote d := gcd(m, r−1). Let w ∈ 𝔽_{r^{2}} be such that w^{d} = −1. Then the curve C given by

y^{m/d} = w · (x^{r} − x)^{(r−1)/d}

is a maximal curve over 𝔽_{r^{2}} with g(C) = (r − 1) · (m − d)/2d.

Proof. The point here is to obtain the equation defining the curve C using our method. For that we consider

ℓ(x) = \frac{x^{r^{2}} − x}{x^{r} − x} = (x^{r} − x)^{r−1} + 1;

i.e., ℓ(x) is the product of all irreducible polynomials in 𝔽_{r}[x] of degree 2. Then taking f(x) = (x^{r} − x)^{r−1}, we have

f(x) = ℓ(x) − 1; i.e., we get r(x) = −1.

We are then led by our method to consider the equation

y^{m} = −(x^{r} − x)^{r−1}.

We then extract the d-th root of both sides above to obtain the defining equation for the curve. The assertions about the genus and the number of rational points of the curve C are easily verified.

As a variation of Example 4.1 we have the next example where we take the same polynomial ℓ(x) but a different polynomial for f(x).

Example 4.2. (n = 2 and r = 4). This is an example of a curve C over 𝔽_{16} with g(C) = 4 and #C(𝔽_{16}) = 45. This is the best value known (see the table in [6]). This curve C over 𝔽_{16} can be given by

y^{3} = (x^{3} + x^{2} + 1)^{4}/x^{3}(x + 1)^{3}(x^{3} + x + 1).

Taking f(x) = (x^{3} + x^{2} + 1)^{4} and ℓ(x) = \frac{x^{16}−x}{x^{4}−x}, we have after some computation that r(x) = x^{3}(x+1)^{3}(x^{3} +x+1). This shows that the equation for C is obtained by our method. That the genus is equal to 4 follows from the fact that we have exactly 6 ramified points and they are totally ramified (corresponding to the solutions of (x^{3} + x^{2} + 1) · (x^{3} + x + 1) = 0). These points are not rational over 𝔽_{16}. Besides the 12 roots of ℓ(x) = 0, one can check easily that x = 0, x = 1 and x = ∞ also give rise to rational points of C over 𝔽_{16}. Hence #C(𝔽_{16}) ≥ (12 + 3) · 3 = 45. The other possible rational points must have first coordinate satisfying x = γ ∈ 𝔽_{4} \ 𝔽_{2}. A direct computation shows

\frac{(γ^{3} + γ^{2} + 1)^{4}}{γ^{3}(γ + 1)^{3}(γ^{3} + γ + 1)} = (\frac{1}{γ})^{2},

and this shows that the points on C with x = γ are not rational over 𝔽_{16}. Remark. a) Example 4.2 is an instance where m = 3 is a proper divisor of (q−1) = 15 and we have determined the exact number of rational points over 𝔽_{q} without performing a computer search (i.e., we have shown directly that K = 0 with notation as in Case m < q − 1 in Section 2).

b) The curve C in Example 4.2 can also be given by the somewhat simpler equation

z^{3} = \frac{x^{3} + x^{2} + 1}{x^{3} + x + 1}.

In fact just take z = y x(x + 1)/(x^{3} + x^{2} + 1). In several other examples still to come one can perform similar simplifications for the defining equations of the curves C, but we have prefered not to perform such simplifications in order to make more evident that the equations are really obtained thru the method in Section 2.

Example 4.3. Let r be a power of an odd prime number and let m be a divisor of (r^{2} − 1). The curve C over 𝔽_{r^{2}} defined by the affine Kummer equation

y^{m} = x^{t}(x^{t} + 1)^{r}/(x^{t} − 1), with t = (r − 1)/2,

has genus satisfying 2g(C) = (m − 1)(r − 3) + 2(m − d), where d = gcd(m, t), and moreover #C(𝔽_{r^{2}}) ≥ m · (r^{2} − 1)/2 + (r − 1) + 2d.

Proof. Taking ℓ(x) = (x^{t(r+1)} + 1) and f(x) = x^{t}(x^{t} + 1)^{r}, we have

f(x) = ℓ(x) + r(x) with r(x) = x^{t} − 1.

This explains the equation for the curve. We have 2t = (r − 1) totally ramified points and they are rational points over 𝔽_{r^{2}} ( the points corresponding to the roots of (x^{t} + 1) · (x^{t} − 1) = 0). There are d points lying over x = 0 and also d points lying over x = ∞; all these 2d points are rational over 𝔽_{r^{2}} and all have ramification index e = m/d. This gives us the genus formula. Adding to the above rational points the ones coming from the roots of ℓ(x) = 0, we get the inequality for #C(𝔽_{r^{2}}).

Remark. The curves considered in Example 4.3 give in the particular case r = 5 the following good curves (they are new entries for the table in [11]):

• For m = 8, a curve with g = 13 and at least 104 rational points over 𝔽_{25}.

• For m = 12, a curve with g = 21 and at least 152 rational points over 𝔽_{25}.

• For m = 24, a curve with g = 45 and at least 296 rational points over 𝔽_{25}. One can check that in all cases m = 8, m = 12 and m = 24 the exact amount of

rational points over 𝔽_{25} is 104, 152 and 296, respectively. In particular this shows that in all such cases we have K = 0 (notation as in Case m < q − 1 in Section 2). Remark. Some of the curves in Example 4.3 are maximal curves over 𝔽_{r^{2}}. For instante this is the case if we have:

• r = 3 and m = 2; this gives a maximal curve C_{1} over 𝔽_{9} of genus 1.

• r = 9 and m = 5; this gives a maximal curve C_{2} over 𝔽_{81} of genus 16.

• r = 5 and m = 3; this gives a maximal curve C_{3} over 𝔽_{25} of genus 4.

The equation for the curve C_{2} over 𝔽_{81} above is

y^{5} = x^{4}(x^{4} + 1)^{9}/(x^{4} − 1).

It follows from [2] that this curve C_{2} can also be represented by the simpler equation z^{9} + z = w^{5}. One can see this here directly: take a ∈ 𝔽_{81}^{∗} with a^{40} = −1 and let

z = a^{5}x and w = (−a)x(x^{4} + 1)^{2}/y.

Similarly our curve C_{3} over 𝔽_{25} above admits the representation z^{5} + z = w^{3}. In fact let a ∈ 𝔽_{25}^{∗} be a primitive element and then set

z = a^{3}x and w = (−a)(x^{2} + 1)^{2}x/y.

In the next two examples we are going to construct four curves over 𝔽_{16} whose numbers of rational points attain the best known values (see the table in [6]).

Example 4.4. We will construct here two curves C_{1} and C_{2} over 𝔽_{16} satisfying: • g(C_{1}) = 2 and #C_{1}(𝔽_{16}) = 33; i.e., C_{1} is maximal.

• g(C_{2}) = 12 and #C_{2}(𝔽_{16}) = 83.

Consider the curve C over 𝔽_{16} defined by

y^{m} = (x^{3} + 1)^{2}/(x + 1), with m a divisor of 15 = r^{2} − 1.

The curve C_{1} (resp. C_{2}) corresponds to the choice m = 5 (resp. m = 15).

Taking ℓ(x) = \frac{x^{5}−1}{x−1} (i.e., the roots of ℓ(x) are exactly the primitive 5-th roots of unity inside 𝔽_{16}^{∗}) and f(x) = (x^{3} + 1)^{2}, one has f(x) = (x^{2} + x)ℓ(x) + (x + 1). This explains the equation for the curve C here. It is easy to see that

2g(C) = (m − 1) + (m − d), with d = gcd(m, 5).

Besides the 4 roots of ℓ(x) = 0 one has that x = β = 0 also gives m rational points on C (the ones with first coordinate equal to zero). Then #C(𝔽_{16}) ≥ 5m+d+3. The number d of rational points above comes from the points on C with x = ∞ and the number 3 above comes from the 3 totally ramified points (corresponding to x = 1 and to the roots of x^{2} + x + 1 = 0). One can verify directly (without a computer search) that in any case we have #C(𝔽_{16}) = 5m + d + 3.

Example 4.5. We will now construct two curves C_{1} and C_{2} over 𝔽_{16} satisfying: • g(C_{1}) = 6 and #C_{1}(𝔽_{16}) = 65; i.e., C_{1} is maximal.

• g(C_{2}) = 20 and #C_{2}(𝔽_{16}) = 127.

Consider the curve C over 𝔽_{16} given by

y^{m} = (x^{2} + x + 1)^{4}/x^{6}(x + 1), where m is a divisor of 15.

The curve C_{1} (resp. C_{2}) corresponds to the choice m = 5 (resp. m = 15).

Taking ℓ(x) = (x^{4} + x + 1) · (x^{5} − 1)/(x − 1) and f(x) = (x^{2} + x + 1)^{4}, we get

f(x) = ℓ(x) + r(x), with r(x) = x^{6}(x + 1).

This explains the equation for the curve C here. One can show that

• 2g(C) = 2(m − 1) + (m − d), with d = gcd(m, 3).

• #C(𝔽_{16}) = \begin{cases}12m + d + 4 & if m = 5 \\ 8m + d + 4 & if m ≠ 5. \end{cases}

It follows from [10] that the curve C_{1} above admits the simpler model z^{4}+z = w^{5}. One can see it directly by taking z = x and xyw = (x^{2} + x + 1).

Before considering curves over finite fields 𝔽_{q} with q = r^{n} and n ≥ 3 we will give two more examples leading to good curves over 𝔽_{25}. They provide now entries for the table in [11]. In these two examples we will use the following irreducible polynomials of degree 2 over 𝔽_{5}:

ℓ_{1}(x) = x^{2} + 2; ℓ_{2}(x) = x^{2} − 2; ℓ_{3}(x) = x^{2} − x + 1;
ℓ_{4}(x) = x^{2} + 2x − 2; ℓ_{5}(x) = x^{2} + 2x − 1; ℓ_{6}(x) = x^{2} − 2x − 2.

Example 4.6. The curve C over 𝔽_{25} given by y^{8} = x(x − 1)^{3}(x + 2) has g(C) = 7 and #C(𝔽_{25}) = 84.

As in Example 4.1, the point here is to obtain the equation defining the curve C using our method. For that we consider

ℓ(x) = ℓ_{1}(x) · ℓ_{2}(x) · ℓ_{4}(x) · ℓ_{6}(x) and f(x) = x^{3}(x + 2)^{3}(x − 1)^{9}.

We then have f(x) = h(x) · ℓ(x) + 1, with h(x) = (x + 1)(x − 2)^{2}(x^{4} + 2x^{2} − 2). We are then led to consider the equation y^{24} = x^{3}(x + 2)^{3}(x − 1)^{9}. Taking the third root of the equation above we get the desired equation for the curve. The genus is easily obtained since there are four totally ramified points.

For the rational points on C over 𝔽_{25} we have:

• The eight roots of ℓ(x) plus the two roots of h(x) in 𝔽_{25} give (8 + 2) × 8 = 80 rational points.

• The four ramification points (i.e., x = 0, 1, −2, ∞) are clearly rational.

Example 4.7. We are going to construct curves C_{1}, C_{2} and C_{3} over 𝔽_{25} with: • g(C_{1}) = 3 and #C_{1}(𝔽_{25}) = 52.

• g(C_{2}) = 19 and #C_{2}(𝔽_{25}) = 132.

• g(C_{3}) = 43 and #C_{3}(𝔽_{25}) = 252.

Consider the curve C over 𝔽_{25} defined by

y^{m} = \frac{x^{5}(x + 1)^{5}}{(x − 2)(x^{2} + 2x − 2)^{4}} , with m a divisor of 24.

This equation comes from ℓ(x) = ℓ_{1}(x)ℓ_{2}(x)ℓ_{3}(x)ℓ_{5}(x)ℓ_{6}(x) and f(x) = (x^{2} + x)^{5}. The curve C_{1} (resp. C_{2} and C_{3}) corresponds to m = 4 (resp. m = 12 and m = 24).

This curve C satisfies

g(C) = (m − 1) + (m − d), with d = gcd(m, 4), and #C(𝔽_{25}) ≥ 10m + 2d + 4.

The 2d rational points appearing in the inequality for #C(𝔽_{25}) above are the ones with first coordinate satisfying x^{2} + 2x − 2 = 0. We explain why they are rational points, but we are not going to give similar details in other examples. Since d is a divisor of 4 we can write the equation of C as below

(\frac{y^{m/d}(x^{2} + 2x − 2)^{4/d}}{x^{4/d} · (x + 1)^{4/d}} )^{d} = \frac{x^{2} + x}{x − 2}.

Let α ∈ 𝔽_{25} be such that α^{2} + 2α − 2 = 0, then α^{2} + α = −(α − 2) and hence (α^{2} + α)/(α − 2) = −1. The rationality of the points with x = α now follows since (−1) is a d-th power in 𝔽_{25}^{∗}; in fact, d divides 4 and hence 2d divides 24.

It is easy to verify directly that (10m + 2d + 4) is the exact amount of rational points in the cases m = 12 and m = 24. Thru a computer search one sees that the same holds also for m = 4.

It’s worth to point out that one knows the existence of a maximal curve over 𝔽_{25} with genus g = 3; i.e., its number of rational points is equal to 56 (see [4]).

Now we will give some examples of good curves over 𝔽_{q} with q = r^{n} and n ≥ 3.

Example 4.8. Let r be a power of a prime number p and let m be a divisor of (r^{3} − 1). The curve C over 𝔽_{r^{3}} defined by the affine equation

y^{m} = − \frac{(x^{r} − x)^{r^{2}−1}}{(x^{r} − x)^{r−1} + 1}

has the following properties (with d := gcd(m, r − 1)):

• 2g(C) = (r + 1) · [(r − 2)(m − 1) + (m − d)]

• #C(𝔽_{r^{3}}) = \begin{cases}m(r^{3} − r) + d(r + 1) & if p · (r − 1)/d is even \\ m(r^{3} − r) & if p · (r − 1)/d is odd. \end{cases}

Proof. We are just going to show that the equation for the curve C is obtained thru over method. Taking ℓ(x) = \frac{x^{r^{3}}−x}{x^{r}−x} (i.e., ℓ(x) is the product of all irreducible polynomials in 𝔽_{r}[x] of degree 3) and f(x) = (x^{r} − x)^{r^{2}−1}, we have:

f(x) = ℓ(x) + r(x), with r(x) = −(x^{r} − x)^{r−1} − 1.

This explains the equation for the curve C. Since m divides (r^{3} − 1) we have gcd(m, r^{2} − 1) = gcd(m, r − 1). We also have that there are d(r + 1) points on C with x = α ∈ 𝔽_{r} ∪ {∞}.

Remark. As an illustration of Example 4.8 we have:

• For m = 2 and r = 3, we get a curve over 𝔽_{27} with genus g = 2 and 48 rational points; i.e., this curve attains Serre’s bound (see the table in [6]).

• For m = 2 and r = 5, we get a curve over 𝔽_{125} with genus g = 9 and 252 rational points; this is the best value known (see the table in [11]).

Example 4.9. Let r ≠ 2 be a power of a prime number p and let m be a divisor of (r^{3} − 1). The curve C over 𝔽_{r^{3}} defined by the Kummer equation

y^{m} = − \frac{x^{r}(x^{r−1} + 1)^{r}}{x − 1}

has the following properties (with d := gcd(m, r − 1)):

•  2g(C) = \begin{cases}(m − 1)(r − 3) + 2(m − d) & if p = 2 \\ (m − 1)(r − 1) + (m − d) & if p ≠ 2. \end{cases}

•  #C(𝔽_{r^{3}}) ≥ \begin{cases}m(r^{2} − 1) + 2d + (r − 1) & if p = 2 \\ mr^{2} + d + 2 & if p ≠ 2 and (r − 1)/d even \\ mr^{2} + 2 & if p ≠ 2 and (r − 1)/d odd. \end{cases}

Proof. Here one takes ℓ(x) = x^{r^{2}} + x^{r} + x − 1 and f(x) = x^{r^{2}} + x^{r}.

Remark. As an application of Example 4.9 we get a curve C over 𝔽_{27} with genus 37 and with 236 rational points; this is a new entry for the table in [6]. Indeed just take r = 3 and m = 26 in Example 4.9. Notice that m = 26 is the case m = q − 1 in Section 2, and hence 236 is really the exact amount of rational points on C.

Example 4.10. The following affine equation gives a curve C with g(C) = 25 and #C(𝔽_{64}) = 335:

y^{9} = \frac{(x + 1)^{30}}{x^{4}(x^{2} + x + 1)^{3}(x^{4} + x^{3} + x^{2} + x + 1)^{4}} .

This example provides a new entry in the table in [6]. In this construction we will take the polynomial ℓ(x) as a product of five irreducible polynomials in 𝔽_{2}[x], each one having degree equal to 6. Take then

ℓ_{1}(x) = (x^{6} + x + 1); ℓ_{2}(x) = (x^{6} + x^{3} + 1); ℓ_{3}(x) = (x^{6} + x^{5} + 1);
ℓ_{4}(x) = (x^{6} + x^{4} + x^{2} + x + 1); ℓ_{5}(x) = (x^{6} + x^{5} + x^{4} + x^{2} + 1).

Taking now ℓ(x) = ∏_{i=1}^{5} ℓ_{i}(x) and f(x) = (x + 1)^{30}, we get the equation. The determination of its genus is routine and for the number of rational points one needs to perform a computer search to show that K = 6 (see Formula (7) Section 2).

The next remark shows that our method can be applied to prime fields. Remark. a) Let q = 5, ℓ(x) = x^{2} + x and f(x) = x^{3} − x^{2} + x − 1. We then consider the curve C over 𝔽_{5} given by y^{4} = 2(x − 1)(x + 2). This curve C satisfies g(C) = 1 and #C(𝔽_{5}) = 10; i.e., it attains Serre’s bound.

b) Let q = 5, ℓ(x) = x^{5} − x and f(x) = (x^{2} + x + 1)^{3}. We then have

f(x) = (x + 3)ℓ(x) + r(x), with r(x) = (x^{4} + 2x^{3} + 2x^{2} + x + 1).

We then consider the curve C over 𝔽_{5} given by y^{4} = (x^{2} + x + 1)^{3}/r(x). This curve C satisfies g(C) = 7 and #C(𝔽_{5}) = 22; this is the best known value (see [11]).

5 Examples based on Section 3

In the preceding section we have given only two new entries for the table in [6] (see Remark after Example 4.9 and Example 4.10) and six new entries in characteristic p = 5 for the table in [11] (see Remark after Example 4.3, Example 4.6 and Example 4.7). All the other examples were either maximal curves or curves attaining the existing record for their genera. That is not so bad since it represents an unified way to reach those records. In this section our performance will be better, especially in characteristic p = 3. On the other hand we will have to perform here several computer searches to determine the quantity K (see Formula (7) in Section 2). In the next four examples we construct curves over 𝔽_{81} taking the polynomial ℓ(x) as a product of two irreducible polynomials in 𝔽_{3}[x] of degree 4. All these examples give new entries in [6]. We use the following polynomials:

ℓ_{1}(x) = (x^{4} + x^{3} + x^{2} + 1); ℓ_{2}(x) = (x^{4} + x^{2} − x + 1); ℓ_{3}(x) = (x^{4} + x − 1);

ℓ_{4}(x) = (x^{4} + x^{2} + x + 1); ℓ_{5}(x) = (x^{4} − x^{3} + x + 1); ℓ_{6}(x) = (x^{4} − x^{3} − 1).

We start with a new record:

Example 5.1. The curve C over 𝔽_{81} given by

y^{10} = (x − 1)^{4}(x + 1)^{3}(x^{3} − x^{2} + x + 1)

has g(C) = 13 and #C(𝔽_{81}) = 243. The former best value was 224 rational points.

To see that the equation for C above comes from Section 3, one takes ℓ(x) = ℓ_{1}(x) · ℓ_{2}(x) and g(x) = (x − 1)^{3}. Notice that the two factors in ℓ(x) are irreducible in 𝔽_{3}[x] and hence ℓ(x) has all roots in 𝔽_{81}. We have v(x) · g(x) = h(x) · ℓ(x) + 1, with h(x) = x^{2} and v(x) = (x − 1)(x + 1)^{3}(x^{3} − x^{2} + x + 1). This explains the equation for the curve. The determination of the genus is routine. The nine roots of h(x) · ℓ(x) = 0 give rise to 90 rational points. Surprisingly enough a computer search gives K = 14 (see the notation of Case m < q − 1 in Section 2). The point x = ∞ has 10 points above it and they are rational points (notice that the equation defining C has degree 10 on both sides). The unique point above x = −1 and the two points above x = 1 are also rational. This then gives

#C(𝔽_{81}) = 9 × 10 + 14 × 10 + 10 + 1 + 2 = 243.

Example 5.2. The curve C given by the equation y^{10} = x^{5}(x − 1)^{4}(x^{2} + x − 1) has g(C) = 11 and #C(𝔽_{81}) = 220.

Here one takes ℓ(x) = ℓ_{3}(x) · ℓ_{4}(x) and g(x) = x(x − 1)^{4}. Then

v(x) · g(x) = h(x) · ℓ(x) + 1,

with h(x) = (x + 1)^{3} and v(x) = x^{4}(x^{2} + x − 1). Besides the nine distinct roots of h(x)ℓ(x) = 0, one has here K = 12. Also one has 10 rational ramification points and then #C(𝔽_{81}) = 9 × 10 + 12 × 10 + 10 = 220.

Example 5.3. The curve C defined by y^{10} = −x^{3}(x − 1)^{6}(x^{3} + x^{2} + x − 1)^{2} has genus g(C) = 14 and #C(𝔽_{81}) = 278.

Here one takes ℓ(x) = ℓ_{1}(x) · ℓ_{5}(x) and g(x) = x^{3}(x − 1)^{5}; then

v(x) · g(x) = h(x) · ℓ(x) + 1

with h(x) = −(x + 1)(x^{2} + 1) (\frac{x^{5}−1}{x−1}) and v(x) = −(x − 1) · (x^{3} + x^{2} + x − 1)^{2}. There are 15 distinct roots of h(x) · ℓ(x) = 0 in this case, and one computes that K = 12. Adding the 8 rational points that are ramification points, we get

#C(𝔽_{81}) = 15 × 10 + 12 × 10 + 8 = 278.

Example 5.4. The curve C over 𝔽_{81} given by y^{20} = −x^{4}(x−1)^{9}(x^{2} −x−1) satisfies g(C) = 25 and #C(𝔽_{81}) = 392.

Here we take ℓ(x) = ℓ_{1}(x) · ℓ_{6}(x) and g(x) = x(x − 1)^{7}. We omit the details.

Continuing in characteristic p = 3, we will give now three examples of curves over 𝔽_{27} and they all give new entries for the table in [6]. Here the polynomial ℓ(x) will be a product of degree 3 irreducible polynomials in 𝔽_{3}[x]. We will use:

ℓ_{1}(x) = (x^{3} − x^{2} − x − 1); ℓ_{2}(x) = (x^{3} − x^{2} + x + 1); ℓ_{3}(x) = (x^{3} + x^{2} + x − 1);

ℓ_{4}(x) = (x^{3}+x^{2}−x+1); ℓ_{5}(x) = (x^{3}−x^{2}+1); ℓ_{6}(x) = (x^{3}−x+1); ℓ_{7}(x) = (x^{3}−x−1).

We will just give in each of the three examples below the pair of polynomials ℓ(x) and g(x) ∈ 𝔽_{27}[x] used to get the equation of the curve.

Example 5.5. Taking ℓ(x) = ℓ_{1}(x) · ℓ_{2}(x) and g(x) = (x − 1)^{6}, we get the curve C over 𝔽_{27} given by the equation y^{13} = −(x−1)^{7}(x^{3} +x^{2} −1). We have that g(C) = 18 and #C(𝔽_{27}) = 148.

Example 5.6. Taking ℓ(x) = ℓ_{1}(x) · ℓ_{3}(x) · ℓ_{4}(x) · ℓ_{5}(x) and g(x) = (x^{2} + x − 1)^{3}, we get the curve C defined by y^{13} = (x^{2} + x − 1)^{4} · (x^{4} − x^{3} − x^{2} + x − 1). This curve has g(C) = 30 and #C(𝔽_{27}) = 196.

Example 5.7. Taking ℓ(x) = ℓ_{5}(x) · ℓ_{6}(x) · ℓ_{7}(x) and g(x) = x^{4}(x + 1), we get the curve C given by the affine equation y^{26} = x^{7}(x + 1)^{3}(x^{3} + x^{2} + x − 1). This curve has g(C) = 44 and #C(𝔽_{27}) = 278.

Now we move to characteristic p = 2. We are going to give 6 maximal curves over 𝔽_{64} having genus g = 3, 4, 7, 10, 12 and 28. As before we will just give the pair of polynomials ℓ(x) and g(x) used to get the equation of the curve. We will need in this construction the following 10 irreducible polynomials in 𝔽_{2}[x]:

ℓ_{1}(x) = (x^{6} + x^{5} + x^{3} + x^{2} + 1); ℓ_{2}(x) = (x^{6} + x^{5} + x^{2} + x + 1);

ℓ_{3}(x) = (x^{6} + x^{4} + x^{3} + x + 1); ℓ_{4}(x) = (x^{3} + x^{2} + 1); ℓ_{5}(x) = (x^{3} + x + 1);

ℓ_{6}(x) = (x^{6} + x + 1); ℓ_{7}(x) = (x^{6} + x^{5} + x^{4} + x + 1); ℓ_{8}(x) = (x^{6} + x^{3} + 1);

ℓ_{9}(x) = (x^{6} + x^{5} + x^{4} + x^{2} + 1); ℓ_{10}(x) = (x^{6} + x^{4} + x^{2} + x + 1).

Example 5.8. Taking ℓ(x) = ℓ_{1}(x) · ℓ_{2}(x) · ℓ_{3}(x) and g(x) = x^{9}(x + 1)^{10}, we obtain the equation y^{9} = x^{9}(x + 1)^{15}(x^{3} + x + 1)^{3}. Taking the third root of this equation, we get the curve C given by y^{3} = x^{3}(x + 1)^{5}(x^{3} + x + 1). It is a maximal curve over 𝔽_{64} with g(C) = 3.

Example 5.9. Taking ℓ(x) = ℓ_{1}(x) · ℓ_{4}(x) · ℓ_{5}(x) and g(x) = (x^{2} + x + 1)^{9}, we get the curve C given by the affine equation y^{9} = (x^{2} + x + 1)^{14}, which is a maximal curve over 𝔽_{64} with g(C) = 4.

Example 5.10. Taking ℓ(x) = ℓ_{6}(x) · ℓ_{7}(x) and g(x) = (x^{3} + x)^{6}, we get the curve C given by the affine equation y^{9} = x^{10}(x + 1)^{15}(x^{2} + x + 1), which is a maximal curve over 𝔽_{64} with g(C) = 7.

Example 5.11. Taking ℓ(x) = ℓ_{3}(x) · ℓ_{6}(x) and g(x) = (x^{4} + x)^{6}, we get the curve C defined by the affine equation y^{9} = x^{8}(x + 1)^{6}(x^{2} + x + 1)^{8}, which is a maximal curve over 𝔽_{64} with g(C) = 10.

Let r be a power of p = 2. The largest genus g_{1} for a maximal curve over 𝔽_{r^{2}} is g_{1} = r(r − 1)/2 and the second largest genus g_{2} is given by g_{2} = r(r − 2)/4. When r = 8 one gets g_{2} = 12 and g_{1} = 28.

Example 5.12. Taking ℓ(x) = ℓ_{1}(x) · ℓ_{8}(x) · ℓ_{9}(x) and g(x) = (x + 1)^{10}, we get the curve C_{2} given by the equation y^{9} = x^{5}(x + 1)^{10}(x^{3} + x + 1), which is a maximal curve over 𝔽_{64} with g(C_{2}) = g_{2} = 12.

This curve can also be represented by the much simpler Artin-Schreier plane equation z^{4} + z^{2} + z = w^{9} (see [1] and [8]). One can see it directly here by taking an element a ∈ 𝔽_{8} ⊆ 𝔽_{64} such that a^{4} + a^{2} + a = 1 and then setting

z = (\frac{1}{x} + a) and w = (\frac{y}{x(x + 1)}) .

Example 5.13. Taking ℓ(x) = ℓ_{7}(x) · ℓ_{9}(x) · ℓ_{10}(x) and g(x) = (x + 1)^{10}, we get the curve C_{1} given by the equation y^{9} = x(x + 1)^{18}(x^{2} + x + 1)(x^{6} + x^{4} + x^{3} + x + 1), which is a maximal curve over 𝔽_{64} with g(C_{1}) = g_{1} = 28.

It follows from [10] that this curve can be given by z^{8} + z = w^{9}. One can see it directly here by taking a ∈ 𝔽_{64} such that a^{8} + a = 1 and setting

z = (\frac{1}{x} + a) and w = (\frac{y}{x(x + 1)^{2}} ) .

6 Fibre products and examples

Let C_{1} and C_{2} be two curves over 𝔽_{q} whose equations are obtained thru our method; i.e., for i = 1, 2 we have polynomials ℓ_{i}(x), f_{i}(x) ∈ 𝔽_{q}[x] and the curve C_{i} is then given by an equation

y^{m_{i}} = \frac{f_{i}(x)}{r_{i}(x)}, with m_{i} a divisor of (q − 1),

where r_{i}(x) is the remainder in the division of f_{i}(x) by ℓ_{i}(x). We will consider in this section the fibre product C of the curve C_{1} with the curve C_{2} (in the language of function fields we are then considering the compositum of the function fields corresponding to C_{1} and C_{2}). In order to get a nice genus formula for the product curve C, we will assume that all ramification points on C_{i} (i = 1, 2) are totally ramified; i.e. each multiplicity of a zero or a pole of f_{i}(x)/r_{i}(x) is relatively prime to m_{i} or a multiple of m_{i}. In the later case of multiplicities ≡ 0 mod m_{i}, the points are unramified. We denote the set of totally ramified points on C_{i} by T_{i}; i.e.,

T_{i} = \{α ∈ \overline{𝔽}_{q} ∪ \{∞\} | α is a zero or a pole for f_{i}(x)/r_{i}(x)

with multiplicity ≢ 0 mod m_{i}}.

Also we set t_{i} = #T_{i} and g_{i} = g(C_{i}), for i = 1, 2. Because of the assumption that the points are totally ramified we have

2g_{i} = (m_{i} − 1) · (t_{i} − 2).  (10)

We will also assume that the curve C is a degree m_{1} · m_{2} cover of the projective x-line (for example, this is the case if T_{1} ≠ T_{2}). Applying Abhyankar’s lemma (see [12, Proposition III.8.9]) and Hurwitz’s genus formula, we get the formula for the genus g(C) of the product curve (where t = #(T_{1} ∩ T_{2}) and δ = gcd(m_{1}, m_{2})):

g(C) = (m_{1} − 1)(m_{2} − 1) + m_{1}g_{2} + m_{2}g_{1} − \frac{t}{2}(m_{1}m_{2} − m_{1} − m_{2} + δ).  (11)

Remark. In the above situation Castelnuovo’s inequality says that (see [12, Theorem III.10.3])

g(C) ≤ (m_{1} − 1)(m_{2} − 1) + m_{1}g_{2} + m_{2}g_{1}.

Formula (11) above then gives the correction term in our particular situation.

Let ℓ(x) ∈ 𝔽_{q}[x] be the greatest common divisor of ℓ_{1}(x) and ℓ_{2}(x), and let

λ = #{β ∈ 𝔽_{q} | ℓ(β) = 0 and f_{1}(β) · f_{2}(β) ≠ 0}.  (12)

We then have the following inequality

#C(𝔽_{q}) ≥ λ · m_{1} · m_{2}.  (13)

In order to guarantee that the curve C has many rational points, one usually takes the polynomials ℓ_{1}(x) and ℓ_{2}(x) with many common roots.

In the next five examples we will construct good curves C that are obtained as fibre products as above. We start with a new record:

Example 6.1. Let C be the curve over 𝔽_{16} which is the fibre product of the curves C_{1} and C_{2} given by

y_{1}^{5} = (x^{4} + x)^{3} and y_{2}^{3} = \frac{(x^{2} + x + 1)^{3}(x^{3} + x + 1)^{2}}{x(x + 1)(x^{3} + x^{2} + 1)^{3}} .

This curve C satisfies g(C) = 34 and #C(𝔽_{16}) = 183; the former best known value was 161 rational points.

In this case we have ℓ_{1}(x) = ℓ_{2}(x) = (x^{15} − 1)/(x^{3} − 1); i.e., the roots of ℓ_{i}(x) are the elements of 𝔽_{16} outside 𝔽_{4}. The curves C_{1} and C_{2} are obtained by taking

f_{1}(x) = (x^{4} + x)^{3} and f_{2}(x) = (x^{2} + x + 1)^{3}(x^{3} + x + 1)^{2}.

One can check easily that λ = 12 = deg ℓ(x) (see Formula (12)), and hence

#C(𝔽_{16}) ≥ 12 · 5 · 3 = 180.

We have three more rational points on C corresponding to x = 0, x = 1 and x = ∞. The determination of the genus g(C) is a direct application of Formulas (10) and (11), since we have here t_{1} = 5, t_{2} = 6 and t = 3.

Now example we construct a curve C over 𝔽_{8} with g(C) = 57 and #C(𝔽_{8}) = 168. One cannot find genus 57 curves on the tables, but if one compares with the table in [6] one sees that 168 is certainly a very good entry for such a genus.

Example 6.2. Let C be the curve over 𝔽_{8} which is the fibre product of the curves C_{1} and C_{2} given by

y_{1}^{7} = x(x + 1)^{2} and y_{2}^{7} = (x^{2} + x + 1)^{2}/(x + 1).

This curve C satisfies g(C) = 57 and #C(𝔽_{8}) = 168.

Here one takes ℓ_{1}(x) = ℓ_{2}(x) = (x^{3} + x + 1), f_{1}(x) = x^{4} + x^{2} and f_{2}(x) = (x^{2} + x + 1)^{2}. It is easy to see that λ = 3 and then from Formula (13) we have

#C(𝔽_{8}) ≥ 3 · 7 · 7 = 147.

We think of C as a degree 7 cover of C_{1} and it is easy to see that the unique point on C_{1} with x = 0 has 7 rational preimages in the curve C. The same holds for the unique point on C_{1} with x = 1 or with x = ∞. Also the points with x^{3} + x^{2} + 1 = 0 are not rational and then #C(𝔽_{8}) = 147 + 3 × 7 = 168.

We give the details here that the 7 points on C with x = 1 are indeed rational points. In fact we have that u := (x + 1)/y_{1}^{3} is a local parameter at the unique point on C_{1} with x = 1. We write the equation for the cover C → C_{1} in the form

(uy_{2})^{7} = (x^{2} + x + 1)^{2}/x^{3}.

The value of the right hand side above is one at x = 1, and this shows the rationality of the points. The genus follows from Formulas (10) and (11).

Example 6.3. Let C be the curve over 𝔽_{64} which is the fibre product of the curves C_{1} and C_{2} given by

y_{1}^{3} = \frac{(x + 1)^{4}}{(x^{3} + x + 1)} and y_{2}^{3} = (x^{8} + x)^{7}.

This curve has g(C) = 19 and #C(𝔽_{64}) = 297; this gives a new entry in [6].

The equation for the curve C_{1} is just another way of representing the maximal curve in Example 5.8 and the curve C_{2} is the curve in Example 4.1 with r = 8 and m = 3. Here one takes

ℓ_{1}(x) = (x^{6} + x^{4} + x^{3} + x + 1) · (x^{6} + x^{5} + x^{2} + x + 1) · (x^{6} + x^{5} + x^{3} + x^{2} + 1) ·
· (x^{6} + x^{5} + x^{4} + x + 1),
ℓ_{2}(x) = (x^{64} − x)/(x^{8} − x), f_{1}(x) = (x + 1)^{30} and f_{2}(x) = (x^{8} + x)^{7}.

The genus is again straightforward but the number of rational points requires a computer search. The 24 zeros of ℓ_{1}(x) = 0 certainly give rational points on C_{1}, and there are K = 12 extra-values x = γ such that γ̃ ≠ 1 is a third power in 𝔽_{64}^{∗} (notation as at the end of Section 2). Putting together the 24 zeros and the 12 extra-values, we get the polynomial ℓ̃_{1}(x) ∈ 𝔽_{64}[x] of degree 36 below

ℓ̃_{1}(x) = x^{36} +x^{35} +x^{34} +x^{33} +x^{29} +x^{26} +x^{24} +x^{22} +x^{21} +x^{17} +x^{16} +x^{15} +x^{14} +x^{7} +x,

and again by a computer calculation one has that deg(gcd(ℓ̃_{1}(x), ℓ_{2}(x))) = 30. This gives #C(𝔽_{64}) ≥ 30 × 3 × 3 = 270. Each of the points with x = 1, x = ∞ or x^{3} + x + 1 = 0 has 3 points on C above it and they are all rational over 𝔽_{64} (this computation is similar to the one in Example 6.2 for the points with x = 1). This gives us 15 new rational points and there are 12 others with x · (x^{3} + x^{2} + 1) = 0. Hence #C(𝔽_{64}) = 270 + 15 + 12 = 297.

Example 6.4. Let C be the fibre product of the curves C_{1} and C_{2} over 𝔽_{27}, where

y_{1}^{2} = \frac{x^{8}(x^{2} + 1)^{9}}{x^{8} + 1} and y_{2}^{2} = − \frac{x^{8}(x^{2} − 1)^{8}}{(x^{4} + 1)(x^{2} + 1)}.

This curve C has g(C) = 11 and #C(𝔽_{27}) = 96; this is the best value known.

Here we take ℓ_{1}(x) = (x^{26} − 1); ℓ_{2}(x) = (x^{26} − 1)/(x^{2} − 1); f_{1}(x) = (x^{3} + x)^{9} and f_{2}(x) = (x^{3} − x)^{8}. We then get

gcd(ℓ_{1}(x), ℓ_{2}(x)) = ℓ_{2}(x) and hence #C(𝔽_{27}) ≥ 24 × 2 × 2 = 96.

The determination of the genus is routine and we leave to the reader the checking that there are no further rational points.

Example 6.5. Here we give two new entries over 𝔽_{25} in [11]. They are:

• A curve C̃_{1} with g(C̃_{1}) = 19 and #C̃_{1}(𝔽_{25}) = 132.

• A curve C̃_{2} with g(C̃_{2}) = 43 and #C̃_{2}(𝔽_{25}) = 252.

The curve C̃_{1} (resp. C̃_{2}) is the fibre product of the curve in Example 4.1 with r = 5 and m = 6 (resp. m = 24) with the curve given by the following equation:

y_{2}^{2} = − \frac{x^{4}(x^{4} − a)^{5}}{a^{5}x^{4} − 1} , with a ∈ 𝔽_{25} and a^{6} = −1.

The condition a^{6} ≠ 1 implies that there is no common roots for the numerator and the denominator above, and that these roots are not in 𝔽_{25} (except x = 0). We have 12 rational points with x = α ∈ 𝔽_{5} ∪ {∞}. Indeed we think of C̃_{1} and C̃_{2} as double covers of the corresponding curves from Example 4.1, and we use a^{6} = −1 again to show that the two points above x = 0 and the two points above x = ∞ are rational. The rationality of the 8 points satisfying x = α ∈ 𝔽_{5}^{∗} is easier. Moreover the roots of (x^{25} − x)/(x^{5} − x) give 20 × 2 × 3 = 120 rational points on the curve C̃_{1} and 20 × 2 × 6 = 240 rational points on the curve C̃_{2}. This gives us the assertions about rational points and the genera follow from Formulas (10) and (11).

We finish this paper with the following example. The goal here is just to illustrate that fibre products of Kummer covers with Artin-Schreier covers should also be considered when trying to get good curves over finite fields.

Example 6.6. Consider the curve C over 𝔽_{8} which is the fibre product of

y_{1}^{7} = \frac{x^{3}(x + 1)^{3}}{x^{2} + x + 1} and y_{2}^{2} + y_{2} = \frac{ℓ(x)}{x^{2} + x + 1},

where ℓ(x) = (x^{8} − x)/(x^{2} − x). This curve C satisfies g(C) = 25 and #C(𝔽_{8}) = 86.

This is the best value known (see [6]). Notice that the curve C_{1} is obtained by considering the division of f(x) = x^{3}(x + 1)^{3} by ℓ(x). We omit the details here.

References

[1] M. Abdon, F. Torres, On maximal curves in characteristic two, Manuscripta Math. 99 (1999), 39–53.

[2] R. Fuhrmann, A. Garcia, F. Torres, On maximal curves, J. Number Theory 67 (1997), 29–51.

[3] A. Garcia, L. Quoos, A construction of curves over finite fields, Act Arithmetica 98 (2001), 181–195.

[4] A. Garcia, F. Torres, On maximal curves having classical Weierstrass gaps, Contemporary Math. 245 (1999), 49–59.

[5] G. van der Geer, M. van der Vlugt, Kummer Covers with many rational points, Finite Fields and their Appl. 6 (2000), 327–341.

[6] G. van der Geer, M. van der Vlugt, Tables for the function N_{q}(g), available at www.wins.uva.nl/∼geer.

[7] H. Hasse, Theorie der relativ zyklischen algebraischen Funktionenkörper, J. Reine Angew. Math. 172 (1934), 37–54.

[8] G. Korchmaros, F. Torres, On the genus of a maximal curve, preprint arXiv: math. AG/000802.

[9] H. Niederreiter, C.-P. Xing, Rational Points on Curves over Finite Fields, Cambridge Univ. Press, United Kingdom, 2001.

[10] H.-G. Rück, H. Stichtenoth, A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math. 457 (1994), 185–188.

[11] V. Shabat, Tables of curves with many points, available at www.wins.uva.nl/∼shabat/tables.html.

[12] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, 1993.

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