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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 [email protected]
Abstract. A general type of ray class elds of global function elds 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 diculty. 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 jF q of genus g(X ) = g de ned over the nite eld F q is usually denoted by Nq (g). In the early eighties, Serre [20{22] has written down formulas for Nq (1) and Nq (2). Since the precise value of Nq (g) is quite dicult 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 eld 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 eld extensions, to the curve X jF q we associate the global function eld K = F q (X ). Its genus gK equals g(X ), and coverings of X correspond to eld extensions of K , the degree of the covering being the degree of the extension. By construction, F q is the full constant eld, i.e. is algebraically closed in K , and we express this instance by writing K jF q . A place of K , by which we mean the maximal ideal p in some discrete valuation ring of K , with (residue eld) degree d = deg p, corresponds to (a Galois
conjugacy class of) d points in X (F qd ), and each point on X having F qd as its minimal eld of de nition over F q lies in such a conjugacy class. In particular K jF q has NK = jX (F q )j rational places, i.e. places of degree 1. Throughout this paper, K jF q is a global function eld, and all algebraic extensions of K are assumed to lie in some xed algebraic closure K of K .
2 Ray Class Fields We x a non-empty set S of places of K , and denote the greatest common divisor of the degrees of its elements by d := gcdfdeg p j p 2 S g. Let m be an S -cycle, i.e. an e ective divisor of K with support away from S . We consider the S -ray class eld mod m, denoted KSm , which is de ned as the largest abelian extension LjK 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), KSo is also known as the S -Hilbert class eld (cf. Rosen [18]). We recall that the Galois group G(KSo jK ) is isomorphic to the (ideal) class group C`(OS ) of the Dedekind ring OS consisting of all functions with poles only in S . Since S is non-empty, by class eld theory KSm is a nite (algebraic) extension of K . In fact, using C ebotarev's Density Theorem, any nite abelian LjK is seen to be equal to some KSm . Here for m we can take the conductor of LjK , and S can always be chosen nite. Furthermore, the ray class elds 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 eld of KSm has degree d over F q , thus KSmjF qd . (b) If S T and m n, then KSm KTn . fm;ng , where the minimum is taken coecientwise. (c) KSm \ KTn = KSmin [T In terms of the ray class elds of K , we can write down the genus for any abelian extension of K . Theorem 1. Let S , d, m = Pp mp p be as above, and L an intermediate eld of KSm jK . Then the genera gK and gL of K and L satisfy mp XX ? d (gL ? 1) = [L : K ] gK ? 1 + deg2 m ? 21 [L \ KSm?np : K ] deg p : p n=1 This formula can be proved either by applying Mobius inversion to the Conductor Discriminant Product Formula, as done by Cohen et al. [2] in the number eld case, or by using Hilbert's Di erent Formula, the Hasse-Arf Theorem and the connection between upper rami cation groups and higher unit groups known from local class eld theory (see [1]). We observe that computation of the genus of ray class eld 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 rami cation at only one place p of K jF q (outside S ), i.e. to m = mp with m 2 N 0 . Recall that, by de nition, the residue eld F p of p satis es [F p : F q ] = deg p. We determine the degrees [KSmp : K ] in three steps. First of all, from what has been said about the Hilbert class eld, [KSo : K ] equals the S -class number hS := jC`(OS )j. Its computation is connected with the problem of nding generators for the group OS of S -units, which in turn are needed for the other two steps. Indeed, by class eld theory, the Galois group G(KSp jKSo ) is isomorphic to the cokernel of the canonical group homomorphism OS ! F p . Since F q OS , deg p it follows that [KSp : KSo ] divides q q?1?1 . Similarly, the cokernel of OS \ (1 + p) ! (1 + p)=(1 + pm ) is isomorphic to G(KSmpjKSp ). According to the following theorem, its order can be determined for all m 2 N simultaneously. Let p be the characteristic of K , and de ne daep := minfpl j a pl ; l 2 N 0 g for any real a > 0. Theorem 2. There are s := jS j ? 1 positive integers n1; : : : ; ns depending only on S and p such that
KSmp : KSp = q(m?1) deg p
s l Y
i=1
mm ni p
for all m 2 N . The proof given in [1, p. 43] provides an algorithm for the computation of the numbers n1 ; : : : ; ns . Since the order of these numbers is irrelevant, the behaviour of the degrees can be summarized in the polynomial
S;p :=
s X i=1
tni ;
which is uniquely determined by S and p. Unfortunately we have no explicit formula for 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 S;p can be given explicitly. Theorem 3. Let K be the rational function eld over the prime eld F p , S a non-empty set of rational places of K and p aPrational place of K not occurring in S ; thus 0 s := jS j ? 1 < p. Then S;p = sn=1 tn .
In particular KS(s+1)p = K , and for m s + 2 it follows that KSmp has exactly
NKSmp = 1 + [KSmp : 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 2 f5; 7g and di erent values of s and m. The results are displayed in the tables 1 and 2.
Table 1. Ray class elds over F 5 s m
1 2 3 4 1 2 3 4 1 2 3 4 3 4 5-6 7 4 5-6 7 8 5-6 7 8 9 [KSmp : K ] 5 5 5 5 25 25 25 25 125 125 125 125 gKSmp 2 4 6 10 22 34 56 70 172 284 356 420 NKSmp 11 16 21 26 51 76 101 126 251 376 501 626
Table 2. Ray class elds over F 7 s m
1 2 3 4 5 6 1 2 3 4 5 6 3 4 5 6 7-8 9 4 5 6 7-8 9 10 [KSmp : K ] 7 7 7 7 7 7 49 49 49 49 49 49 gKSmp 3 6 9 12 15 21 45 69 93 117 162 189 NKSmp 15 22 29 36 43 50 99 148 197 246 295 344
Now let q = pe with e 2 N be an arbitrary power of the characteristic p again. Take Q := f1; : : : ; q ? 1g 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 jF p ) acts on Q. Clearly, two elements n; n0 2 Q lie in the same G-orbit Gn = Gn0 i n0 pl n mod q ? 1 for some l 2 N 0 . For n 2 N we de ne (
en := jGnj if n 2 Q and n = min Gn; 0 otherwise. The following has been proved by Lauter [11].
Theorem 4. Let K be the rational function eld over F q , p a rational place
of K , and S the set consisting of the other q rational places. Then S;p = P n n2N en t .
Let us set r := pq or ppq according to whether q is a square or not. By investigating the numbers en , we see that KS(r+1)p = K . Furthermore we obtain two elds, namely KS(r+2)p and KS(2r+2)p of degree r and rq over K if q is a square, and p ? 1 elds KS(r+i+1)p with 1 i < p of degree qi 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 de ning 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 := pq 2 N and let y; z 2 K satisfy yr + y = xr+1 and zq ? z = xp2r (xq ? x). Then KS(r+2)p = K (y) and KqS(2r+2)p = K (y; z). (b) For r := pq 2 N let y1; : : : ; yp?1 2 K satisfy yi ? yi = xir=p(xq ? x). Then KS(r+i+1)p = K (y1 ; : : : ; yi ) for i 2 f1; : : : ; p ? 1g.
5 Tables Now we use ray class eld extensions of small genus ground elds K jF 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 Nq (g). The upper bound is taken from [6]. The lower bound is attained by a eld LjF q of genus gL = g with precisely that many rational places, and is set in boldface if our example actually improves the lower bound known before. The eld L satis es KS(m?1)p ( L KSmp (thus has conductor mp) with m 2 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 [KSmp : 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 ?1 ? d mX np gL = 1 + [L : K ] gK ? 1 + md ? 2 2 n=0 [KS : K ] : Since the inertia degree of p in L is hS =hS[fpg , L has (
NL [L : K ] jS j + hS if hS = hS[fpg and d = 1 0 otherwise rational places. Here equality holds i 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 eld L occurring in the tables 3{5 is given in [1].
Table 3. Function elds over F 2 with many rational places g N2 (g) [L : K ] S m d gK
6 10 10 7 10 2 8 11 2 9 12 4 10 13 4 12 14{15 7 14 15{16 15 15 17 8 16 17{18 2 17 17{18 16 19 20 4 22 21{22 4
j
1 5 5 3 3 2 1 2 8 1 5 5
j
21 26 10 1 42 41 16 14 71 14 1 51 26 12 1
1 1 2 0 2 0 0 0 5 0 1 2
g N2 (g) [L : K ] jS j m d gK 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 27 1
Table 4. Function elds over F 3 with many rational places g N3 (g) [L : K ] S m d gK
5 12{13 7 16 9 19 10 19{21 14 24{26 15 28 16 27{29 17 24{30 19 28{32 22 30{36 24 31{38 30 37{46
3 8 3 9 3 9 9 6 3 3 3 9
j
j
4 22 2 14 6 51 2 51 8 52 3 61 3 32 4 51 9 12 1 10 3 5 10 14 1 4 81
1 0 2 0 2 0 0 2 3 3 4 1
g N3 (g) [L : K ] jS j m d gK 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
Table 5. Function elds over F 4 with many rational places g N4 (g) [L : K ] S m d gK
4 15 2 5 17{18 4 6 20 4 8 21{24 2 9 26 8 10 27{28 12 11 26{30 2 12 29{31 4 13 33 8 14 32{35 8 19 37{43 4 20 40{45 8 21 41{47 4 22 41{48 4 23 45{50 4
j
j
7 61 4 61 5 23 10 6 1 3 31 2 21 13 4 3 7 81 4 61 4 23 9 10 1 5 33 10 8 1 10 10 1 11 10 1
1 0 0 3 1 1 3 1 0 0 2 0 3 3 3
g N4 (g) [L : K ] jS j m d gK 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
References 1. R. Auer. Ray class elds of global function elds with many rational places. Dissertation 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 nite elds of characteristic 2 with many rational points. C. R. Acad. Sci. Paris 317 (1993) serie I, 593{597. 4. G. van der Geer, M. van der Vlugt. How to construct curves over nite elds 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 nite elds 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. Kluners, M. Pohst, K. Roegner, M. Schornig, K. Wildanger. KANT V4. J. Symb. Comp. 24/3 (1997) 267{283. 10. K. Lauter. Deligne Lusztig curves as ray class elds. Manuscripta Math. 98/1 (1999) 87{96. 11. K. Lauter. A formula for constructing curves over nite elds with many rational points. J. Number Theory 74/1 (1999) 56{72. 12. H. Niederreiter, C. P. Xing. Explicit global function elds over the binary eld with many rational places. Acta Arith. 75/4 (1996) 383{396.
13. H. Niederreiter, C. P. Xing. Cyclotomic function elds, Hilbert class elds, and global function elds with many rational places. Acta Arith. 79/1 (1997) 59{76. 14. H. Niederreiter, C. P. Xing. Algebraic curves over nite elds with many rational points. Number theory (Eger 1996) 423{443, de Gruyter, Berlin, 1998. 15. H. Niederreiter, C. P. Xing. Global function elds with many rational places over the ternary eld. Acta Arith. 83/1 (1998) 65{86. 16. H. Niederreiter, C. P. Xing. A general method of constructing global function elds with many rational places. Algorithmic Number Theory (Portland 1998), Lect. Notes in Comp. Sci. 1423 555{566, Springer, Berlin, 1998. 17. M. Perret. Tours rami ees in nies de corps de classes. J. Number Theory 38 (1991) 300{322. 18. M. Rosen. The Hilbert class eld in function elds. 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 algebrique sur un corps ni. C. R. Acad. Sci. Paris 296 (1983) serie I, 397{402. 21. J.-P. Serre. Nombres de points des courbes algebrique sur F q . Seminaire de Theorie des Nombres de Bordeaux 22 (1982/83). 22. J.-P. Serre. Resume des cours de 1983{1984. Annuaire du College 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.
Abstract. A general type of ray class elds of global function elds 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 diculty. 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 jF q of genus g(X ) = g de ned over the nite eld F q is usually denoted by Nq (g). In the early eighties, Serre [20{22] has written down formulas for Nq (1) and Nq (2). Since the precise value of Nq (g) is quite dicult 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 eld 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 eld extensions, to the curve X jF q we associate the global function eld K = F q (X ). Its genus gK equals g(X ), and coverings of X correspond to eld extensions of K , the degree of the covering being the degree of the extension. By construction, F q is the full constant eld, i.e. is algebraically closed in K , and we express this instance by writing K jF q . A place of K , by which we mean the maximal ideal p in some discrete valuation ring of K , with (residue eld) degree d = deg p, corresponds to (a Galois
conjugacy class of) d points in X (F qd ), and each point on X having F qd as its minimal eld of de nition over F q lies in such a conjugacy class. In particular K jF q has NK = jX (F q )j rational places, i.e. places of degree 1. Throughout this paper, K jF q is a global function eld, and all algebraic extensions of K are assumed to lie in some xed algebraic closure K of K .
2 Ray Class Fields We x a non-empty set S of places of K , and denote the greatest common divisor of the degrees of its elements by d := gcdfdeg p j p 2 S g. Let m be an S -cycle, i.e. an e ective divisor of K with support away from S . We consider the S -ray class eld mod m, denoted KSm , which is de ned as the largest abelian extension LjK 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), KSo is also known as the S -Hilbert class eld (cf. Rosen [18]). We recall that the Galois group G(KSo jK ) is isomorphic to the (ideal) class group C`(OS ) of the Dedekind ring OS consisting of all functions with poles only in S . Since S is non-empty, by class eld theory KSm is a nite (algebraic) extension of K . In fact, using C ebotarev's Density Theorem, any nite abelian LjK is seen to be equal to some KSm . Here for m we can take the conductor of LjK , and S can always be chosen nite. Furthermore, the ray class elds 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 eld of KSm has degree d over F q , thus KSmjF qd . (b) If S T and m n, then KSm KTn . fm;ng , where the minimum is taken coecientwise. (c) KSm \ KTn = KSmin [T In terms of the ray class elds of K , we can write down the genus for any abelian extension of K . Theorem 1. Let S , d, m = Pp mp p be as above, and L an intermediate eld of KSm jK . Then the genera gK and gL of K and L satisfy mp XX ? d (gL ? 1) = [L : K ] gK ? 1 + deg2 m ? 21 [L \ KSm?np : K ] deg p : p n=1 This formula can be proved either by applying Mobius inversion to the Conductor Discriminant Product Formula, as done by Cohen et al. [2] in the number eld case, or by using Hilbert's Di erent Formula, the Hasse-Arf Theorem and the connection between upper rami cation groups and higher unit groups known from local class eld theory (see [1]). We observe that computation of the genus of ray class eld 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 rami cation at only one place p of K jF q (outside S ), i.e. to m = mp with m 2 N 0 . Recall that, by de nition, the residue eld F p of p satis es [F p : F q ] = deg p. We determine the degrees [KSmp : K ] in three steps. First of all, from what has been said about the Hilbert class eld, [KSo : K ] equals the S -class number hS := jC`(OS )j. Its computation is connected with the problem of nding generators for the group OS of S -units, which in turn are needed for the other two steps. Indeed, by class eld theory, the Galois group G(KSp jKSo ) is isomorphic to the cokernel of the canonical group homomorphism OS ! F p . Since F q OS , deg p it follows that [KSp : KSo ] divides q q?1?1 . Similarly, the cokernel of OS \ (1 + p) ! (1 + p)=(1 + pm ) is isomorphic to G(KSmpjKSp ). According to the following theorem, its order can be determined for all m 2 N simultaneously. Let p be the characteristic of K , and de ne daep := minfpl j a pl ; l 2 N 0 g for any real a > 0. Theorem 2. There are s := jS j ? 1 positive integers n1; : : : ; ns depending only on S and p such that
KSmp : KSp = q(m?1) deg p
s l Y
i=1
mm ni p
for all m 2 N . The proof given in [1, p. 43] provides an algorithm for the computation of the numbers n1 ; : : : ; ns . Since the order of these numbers is irrelevant, the behaviour of the degrees can be summarized in the polynomial
S;p :=
s X i=1
tni ;
which is uniquely determined by S and p. Unfortunately we have no explicit formula for 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 S;p can be given explicitly. Theorem 3. Let K be the rational function eld over the prime eld F p , S a non-empty set of rational places of K and p aPrational place of K not occurring in S ; thus 0 s := jS j ? 1 < p. Then S;p = sn=1 tn .
In particular KS(s+1)p = K , and for m s + 2 it follows that KSmp has exactly
NKSmp = 1 + [KSmp : 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 2 f5; 7g and di erent values of s and m. The results are displayed in the tables 1 and 2.
Table 1. Ray class elds over F 5 s m
1 2 3 4 1 2 3 4 1 2 3 4 3 4 5-6 7 4 5-6 7 8 5-6 7 8 9 [KSmp : K ] 5 5 5 5 25 25 25 25 125 125 125 125 gKSmp 2 4 6 10 22 34 56 70 172 284 356 420 NKSmp 11 16 21 26 51 76 101 126 251 376 501 626
Table 2. Ray class elds over F 7 s m
1 2 3 4 5 6 1 2 3 4 5 6 3 4 5 6 7-8 9 4 5 6 7-8 9 10 [KSmp : K ] 7 7 7 7 7 7 49 49 49 49 49 49 gKSmp 3 6 9 12 15 21 45 69 93 117 162 189 NKSmp 15 22 29 36 43 50 99 148 197 246 295 344
Now let q = pe with e 2 N be an arbitrary power of the characteristic p again. Take Q := f1; : : : ; q ? 1g 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 jF p ) acts on Q. Clearly, two elements n; n0 2 Q lie in the same G-orbit Gn = Gn0 i n0 pl n mod q ? 1 for some l 2 N 0 . For n 2 N we de ne (
en := jGnj if n 2 Q and n = min Gn; 0 otherwise. The following has been proved by Lauter [11].
Theorem 4. Let K be the rational function eld over F q , p a rational place
of K , and S the set consisting of the other q rational places. Then S;p = P n n2N en t .
Let us set r := pq or ppq according to whether q is a square or not. By investigating the numbers en , we see that KS(r+1)p = K . Furthermore we obtain two elds, namely KS(r+2)p and KS(2r+2)p of degree r and rq over K if q is a square, and p ? 1 elds KS(r+i+1)p with 1 i < p of degree qi 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 de ning 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 := pq 2 N and let y; z 2 K satisfy yr + y = xr+1 and zq ? z = xp2r (xq ? x). Then KS(r+2)p = K (y) and KqS(2r+2)p = K (y; z). (b) For r := pq 2 N let y1; : : : ; yp?1 2 K satisfy yi ? yi = xir=p(xq ? x). Then KS(r+i+1)p = K (y1 ; : : : ; yi ) for i 2 f1; : : : ; p ? 1g.
5 Tables Now we use ray class eld extensions of small genus ground elds K jF 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 Nq (g). The upper bound is taken from [6]. The lower bound is attained by a eld LjF q of genus gL = g with precisely that many rational places, and is set in boldface if our example actually improves the lower bound known before. The eld L satis es KS(m?1)p ( L KSmp (thus has conductor mp) with m 2 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 [KSmp : 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 ?1 ? d mX np gL = 1 + [L : K ] gK ? 1 + md ? 2 2 n=0 [KS : K ] : Since the inertia degree of p in L is hS =hS[fpg , L has (
NL [L : K ] jS j + hS if hS = hS[fpg and d = 1 0 otherwise rational places. Here equality holds i 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 eld L occurring in the tables 3{5 is given in [1].
Table 3. Function elds over F 2 with many rational places g N2 (g) [L : K ] S m d gK
6 10 10 7 10 2 8 11 2 9 12 4 10 13 4 12 14{15 7 14 15{16 15 15 17 8 16 17{18 2 17 17{18 16 19 20 4 22 21{22 4
j
1 5 5 3 3 2 1 2 8 1 5 5
j
21 26 10 1 42 41 16 14 71 14 1 51 26 12 1
1 1 2 0 2 0 0 0 5 0 1 2
g N2 (g) [L : K ] jS j m d gK 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 27 1
Table 4. Function elds over F 3 with many rational places g N3 (g) [L : K ] S m d gK
5 12{13 7 16 9 19 10 19{21 14 24{26 15 28 16 27{29 17 24{30 19 28{32 22 30{36 24 31{38 30 37{46
3 8 3 9 3 9 9 6 3 3 3 9
j
j
4 22 2 14 6 51 2 51 8 52 3 61 3 32 4 51 9 12 1 10 3 5 10 14 1 4 81
1 0 2 0 2 0 0 2 3 3 4 1
g N3 (g) [L : K ] jS j m d gK 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
Table 5. Function elds over F 4 with many rational places g N4 (g) [L : K ] S m d gK
4 15 2 5 17{18 4 6 20 4 8 21{24 2 9 26 8 10 27{28 12 11 26{30 2 12 29{31 4 13 33 8 14 32{35 8 19 37{43 4 20 40{45 8 21 41{47 4 22 41{48 4 23 45{50 4
j
j
7 61 4 61 5 23 10 6 1 3 31 2 21 13 4 3 7 81 4 61 4 23 9 10 1 5 33 10 8 1 10 10 1 11 10 1
1 0 0 3 1 1 3 1 0 0 2 0 3 3 3
g N4 (g) [L : K ] jS j m d gK 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
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