Further results on fibre products of Kummer covers and curves with many points over finite fields

We study fibre products of an arbitrary number of Kummer covers of the projective line over $\mathbb{F}_q$ under suitable weak assumptions. If $q-1 = r^a$ for some prime $r$, then we completely determine the number of rational points over a rational point of the projective line. Using this result we obtain explicit examples of fibre products of three Kummer covers supplying new entries for the current table of curves with many points (http://www.manypoints.org, October 31 2015).


Introduction
Let F q be a finite field with q = p n elements, where p is a prime number. Let χ be an absolutely irreducible, nonsingular and projective curve defined over F q , N be the number of F q -rational points of χ and g(χ) be its genus. It is well known that the number N is bounded by the Hasse-Weil bound If the bound in (1) is attained then χ is called a maximal curve. For some improvements in the literature especially when g(χ) is large we refer to [3,4,7,11,12].
Let N q (g) denote the maximum number of F q -rational points among the absolutely irreducible, nonsingular and projective curves of genus g defined over F q . Determining the number N q (g) and constructing explicit curves with many rational points (see [2], and [13] for the current tables) is important since there are many applications of such curves in current research areas such as coding theory, cryptography and quasi-random points etc. (see [4,7,8,11,12]).
In the literature using certain types of fibre products of Kummer covers of the projective line, explicit curves with many points were found (see [1,6,9,10]). It is also well known that the theory of algebraic curves is essentially equivalent to the theory of algebraic function fields, we will use the terminology of function fields [11] and call a degree one place of an algebraic function field as a rational place of the function field throughout this paper.
In particular in [10] we studied the general fibre products of a finite number of Kummer covers of the projective line. More precisely we considered the following fibre product where k ≥ 2 and n 1 , n 2 , . . . , n k ≥ 2 are integers, h 1 (x), h 2 (x), . . . , h k (x) ∈ F q (x), E = F q (x, y 1 , y 2 , . . . , y k ) is the algebraic function field with the system of equations in (2). We assumed that [E : F q (x)] = n 1 n 2 . . . n k and that the full constant field of E is F q . We determined the number of rational places of E over P (where P is a rational place of the rational function field F q (x)) exactly for an arbitrary k ≥ 2 under some strong conditions [10, Theorems 2 and 3]. Moreover, in [10] we determined the number of rational places of E over P when k = 2 exactly under a weak condition [10,Theorem 4].
In this paper we determine the number of rational places of E over P for an arbitrary positive integer k ≥ 3 under the joint condition m 2 | q − 1, m 3 | q − 1, . . . , m k | q − 1. Note that this joint condition is independent from the order (see [10,Remark 2]). The problem seems to be much more challenging for an arbitrary q. Therefore, in this paper we study the case q − 1 = r a for some prime r (see Remark 1 below).
As in the proof of [10,Theorem 4] we determine the number of rational places of E over P using suitable intermediate field extensions of F q (x) ⊆ E depending on P . The main difficulty is to determine the ratios of the number of rational extensions to the number of non-rational extensions in some intermediate field extensions for certain rational places having ramification index 1.
By a detailed study of such intermediate field extensions we observe an important connection to a system of linear equations over the ring Z q−1 (see Equation 6 in the proof of Lemma). The form of this system of linear equations is determined by the equations of E in (2) and the place P as explained in Sections 2 and 3.
We determine the number of rational places of E over P completely for arbitrary k ≥ 3 under the condition that q − 1 = r a for some prime r using local r-adic techniques. These techniques are new in application to this problem in the sense that they were not used for example in [10]. Therefore we obtain the condition C2 in the statement of Theorem 3.1 depending on some inequalities, which correspond to some r-adic valuations. A generalization of the methods of [10] do not extend from k = 2 to arbitrary k ≥ 3. In particular even for k = 2 the conditions in [10,Theorem 4] are very complicated.

Remark 1. We explain what the condition
for some prime r (3) means in this remark. We refer to, for example, [5, Chapter IX, Lemma 2.7] for details. The condition in (3) holds such that F q is a finite field of characteristic p with q = p n if and only if one of the following cases hold: i) p = 2, n is a prime, a = 1 and r is a Mersenne prime. ii) n = 1, a = 2 m , r = 2 and q = p = 2 2 m + 1 is a Fermat prime. iii) q = 9, p = 3, n = 2, r = 2, a = 3.
The paper is organized as follows. In Section 2 we give the notation. Then we present our main result in Section 3, and give some examples in Section 4.

Preliminaries
For an algebraic function field F with a separating element x ∈ F and full constant field F q , if z ∈ F and P is a rational place of F , then we denote the evaluation of z at P by Ev P (z). For an arbitrary u ∈ F q , we denote the rational place of the rational function field F q (x) corresponding to the zero of (x − u) as P u . Similarly the rational place corresponding to the pole of x is denoted as P ∞ . For 1 ≤ i ≤ k, the evaluation of f i (x) ∈ F q (x) at P u is denoted also by f i (u). We denote the multiplicative group F q \ {0} by F * q . We consider the fibre product Let E be the algebraic function field E = F q (x, y 1 , y 2 , . . . , y k ) with the system of equations in (4).
The integer a i and the rational function f i (x) are uniquely determined by the conditions above. For 1 ≤ i ≤ k, letn i , n i and a i be the integers: and a i = a ī n i such thatn i | q − 1. Note that n i can be greater than q − 1. It is clear that We note that if a i = 0, then n i = 1. Next we define the positive integers m i for i = 1, 2, . . . k as follows: As gcd

Main result
In this section, we derive the necessary and sufficient conditions for the existence of a rational place of E over P 0 . We also obtain the exact number of such rational places. We need a technical lemma before giving the main result.
Then S k is not empty if and only if Moreover, if S k is not empty, then | S k | =n 1n1 · · ·n k .
Let u l , i l , j l be the integers such that α −1 . . , k. We obtain the following equalities: Thus, we are looking for the number of solutions (x 1 , x 2 , . . . , x 2k−1 ) ∈ Z 2k−1 q−1 and the conditions of solvability of the following system of equations over Z q−1 : . . .
N k1 r k 1 +s k1 x 1 + · · · +N kk−1 r k k +s k1 x k−1 + r k k x k + r l k−1 x 2k−1 = N k1û1 r s k1 +u 1 + · · · +N kk−1ûk−1 r s kk−1 +u k−1 +û k r u k The system above has a solution if and only if r-adic valuation of coefficients on the left hand side is not greater than the r-adic valuation of coefficients on the right hand side. We may assume that l 1 ≤ l 2 ≤ · · · ≤ l k−1 and k k ≤ k k−1 ≤ · · · ≤ k 1 . Moreover, we have that l k−1 ≤ k k . Thus the coefficient of x k+1 has the minimum r-adic valuation, and so we can divide the coefficients of the system by r l1 . Thus we can solve the following system modulo r a−l1 and multiply its number of solutions by r (2k−1)l1 to find | U k |.
Given x 1 and x 2 , the variable x k+1 can be uniquely found. Next we consider the new system for given x 1 , x 2 and x k+1 . The coefficient of x k+2 has the minimum r-adic valuation, and so we can divide the coefficients of the system by r l1−l2 . Thus we can solve the remaining system modulo r a−l2 and multiply its number of solutions by r (2k−3)(l2−l1) to find | U k |. Then given x 3 , the variable x k+2 can be uniquely found.
If we continue in this way, the number of solutions is Therefore we have that | S k | =n 1n1 · · ·n k .
Theorem 3.2. Under the notation as in Section 2, let E = F q (x, y 1 , y 2 , . . . , y k ) be the algebraic function field with y n1 Assume that the full constant field of E is F q and [E : F q (x)] = n 1 n 2 · · · n k . Assume that q − 1 = r a for some prime number r. Moreover assume thatn i | (q − 1) for Then there exist either no or exactlyn 1n2 · · ·n k m 2 m 3 · · · m k rational places of E over P 0 . Furthermore, there exists a rational place of E over P 0 if and only if both of the following conditions hold: C1: f i (u) is ann i -power in F * q , for i = 1, 2, . . . , k, C2: the system of inequalities in (3.1) in the above lemma is satisfied.
Step 1. Let E i be the intermediate field with Let P i be an arbitrary place of E i over P i−1 . The ramification index e(P i |P i−1 ) is 1. Therefore there are either no or exactlyn i rational places of E i over P 0 . Moreover P i is a rational place of E i if and only if the evaluation f i (u) of f i (x) at P i−1 is an n i -power in F * q . Hence from here till the end of the proof we assume that condition C1 in the hypothesis of the theorem holds. Let K i be the intermediate field with The ramification index of an arbitrary place P i+1 of K i over P i is n i m i . In particular P i+1 |P i is a total ramification, P i+1 is the unique place of K i over P i , and P i is a rational place of K i . Step Let T i be the set of rational places of E i over P 0 . We note that |T i | = n 1n2 · · ·n i m 2 m 3 · · · m i−1 . Recall also that α 1 , α 2 , . . . , α i are the chosen elements of F * q with αn 1 1 = f 1 (u), αn 2 2 = f 2 (u), . . . , αn i i = f 2 (i). Let P i+2 be an arbitrary place in T i . By the evaluations . . , C i be the subgroups of F * q with |C j | =n j for j = 1, 2, . . . , i. Therefore we obtain that the map is a m 2 m 3 · · · m i−1 -to-1 map between the set T i and the Cartesian product group C 1 × C 2 × · · · × C i . Let T i be the subset of T i consisting of the places P i+2 ∈ T i such that there exists a rational place of F i over P i+2 .
Then ν Pi+2 (t i ) = 1. In particular t i is a local parameter of E i for all places in T i .
We also get ν Pi+2 Hence T i is exactly the subset of T i consisting of the places P i+2 ∈ T i such that We also have the following Let ϕ(P i+2 ) = (c 1 , c 2 , . . . , c i ). Then by definition of ϕ we get Ev Pi+2 Let P i+3 be an arbitrary place F i over P i+2 . The extension F i /E i is Galois, the ramification e(P i+2 |P i+2 ) and the inertia f (P i+2 |P i+2 ) indices are 1 and hence there are exactly m i rational places of F i over P i+2 . Therefore there are exactly m 2 m 3 · · · m i−1 m i | S i | rational places of F i over P 0 .
Step 3. Let L i be the field defined as Let P i+4 be an arbitrary place of L i over P i+3 . We obtain that Therefore the ramification index e(P i+4 |P i+3 ) is n i m i , P i+4 |P i+3 is a total ramification; and P i+4 is a rational place of L i , which is also the unique place of L i over P i+3 . Hence, for i = k we know that | S k | =n 1n2 · · ·n k by Lemma ??. Thus | T k | = n 1n2 · · ·n k m 2 m 3 · · · m k−1 . Therefore, we proved the theorem.

Remark 3.
We can obtain the analog of Theorem 3.2 for the place P ∞ . We do not state it explicitly here as it can be easily derived as we did in [10, Theorem 3].

Examples
Example 2. Let E = F 17 (x, y 1 , y 2 , y 3 ) be the function field over F 17 given by the following equations: The genus of E is g(E) = 5 and N (E) = 48. This number is a new entry in the table [13]. By using Theorem 3.2, we obtain that there are 8 rational places over each of the places P x−1 , P x−5 , P x−7 , P x−10 , P x−12 and P x−16 of rational function field.
Example 3. Let E = F 17 (x, y 1 , y 2 , y 3 ) be the function field over F 17 given by the following equations: The genus of E is g(E) = 7 and we obtain that N (E) = 60 by using Theorem 3.2. This number is a new entry in the table [13]. We note that m 2 = 2 over P ∞ .
Example 4. Let E = F 17 (x, y 1 , y 2 , y 3 ) be the function field over F 17 given by the following equations: The genus of E is g(E) = 9 and we obtain that N (E) = 72 by using Theorem 3.2. This number is a new entry in the table [13].
Example 5. Let E = F 17 (x, y 1 , y 2 , y 3 ) be the function field over F 17 given by the following equations: = 2x 4 + 3x 3 + 4x 2 + 5x + 2 The genus of E is g(E) = 11 and we obtain that N (E) = 76 by using Theorem 3.2. This number is a new entry in the table [13].
Example 6. Let E = F 17 (x, y 1 , y 2 , y 3 ) be the function field over F 17 given by the following equations: = 16x 4 + x 3 + 3x 2 + 3x + 3 The genus of E is g(E) = 13 and we obtain that N (E) = 84 by using Theorem 3.2. This number is a new entry in the table [13]. We note that m 2 = 2 over P ∞ and P x−7 .
Example 7. Let E = F 17 (x, y 1 , y 2 , y 3 ) be the function field over F 17 given by the following equations: = 15x 4 + 2x 3 + 2x 2 + 14x + 16 The genus of E is g(E) = 15 and we obtain that N (E) = 84 by using Theorem 3.2. This number is a new entry in the table [13].
Example 8. Let E = F 5 (x, y 1 , y 2 , y 3 ) be the function field over F 5 given by the following equations: The genus of E is g(E) = 33 and N (E) = 64. This number is the best value known in the table [13]. We obtain that there are 64 rational places over the place P x of the rational function field, by using Theorem 3.2.