Number of points on a family of curves over a finite field

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1 arxiv: v1 [math.nt] 10 Oct 2016 Number of points on a family of curves over a finite field Thiéyacine Top Abstract In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field F q with many rational points. The results obtained improve the known bounds for a few pairs (q,g). Key words. Curves over finite fields, jacobian varieties. AMS subject classification. 11G20, 14G15, 14H45. 1 Introduction Let C be a (projective, non-singular and geometrically irreducible) curve of genus g defined over the finite field F q with q elements. We denote by N q (g) the maximum number of rational points on a curve of genus g over F q, namely N q (g) = max{ C(F q ) : C is a curve over F q of genus g}. In the last years, due mainly to applications in Coding Theory and Cryptography (see e.g.[8]), there has been considerable interest in computing N q (g). It is a classical result that N q (0) = q+1. Deuring and Waterhouse[10], and Serre [6] computed N q (1) andn q (2) respectively. Serre also computed N q (3) for q < 25 and Top [7] extended these computations to q < 100. For g 3 no such general formula is known. However, Serre [6] building on earlier results by Hasse and Weil proved the following upper-bound: N q (g) q +1+g 2 q, Université Blaise Pascal, Clermont-Ferrand, France ( thieyacine.top@math.univ-bpclermont.fr) 1

2 2 T. Top where x denotes the floor part of a real number x. This bound is now called Hasse-Weil-Serre bound. Some partial results for some specific pairs (q, g) are recorded on the website in the form N q (g) [c 1,c 2 ] or [...,c 2 ], where c 2 is the bound given by Hasse-Weil-Serre or by "explicit formulas", or by more intricate arguments. c 1 is the bound given by the existence of a curve C with C(F q ) = c 1,... when c 1 c 2 / 2. Our work consists in studying geometric families of curves and studying among them those which have many points over F q. The curves we are considering are certain fibre products of hyperelliptic curves, we prove a formula for the genus and a formula for the number of rational points of such curves that depend are the polynomial f i s and the hyperelliptic they define. In particular by building curves of given genus explicitly, we have the following result: Let k be an integer 1 and q 2 be a prime power. Let I be a non-empty subset {1,2,...,k} and denote by I the set of all non-empty subsets of I. Let f 1,f 2,...,f k be polynomials of respective degrees d i s, with coefficients in the finite fields F q such that the polynomial f 1 f 2... f k is separable. We define the polynomial f I (x) = i I f i (x) and denote by C I the hyperelliptic curve of equation y 2 = f I (x). Let A I = q +1 C I (F q ). Consider now the fibre product C of which an open affine subset is given by y1 2 = f 1 (x). y 2 k = f k (x) Let N = C(F q ) where C(F q ) is the number of F q -rational points on C. Theorem 1.1. In this notation: The genus g of C is givin by:, g = 2 k 2 (d d k 4)+1+δ k,

3 Many points 3 with δ k = 2 k 2 (resp. δ k = 0) if one of the d i s is odd (resp. all of the d i s are even). There are 2 k 1 (resp. 2 k ) points at infinity if one of the d i s is odd (resp. all of the d i s are even). The number of F q -rational points of C is given by: N = q +1 I I A I. The proof of the above theorem follows from the lemma 2.1. and the lemma 3.3. for the computation of g and C(F q ) respectively. In a second part of the paper, we use Theorem1.1 on specific examples in order to improve a current lower bounds for N q (g) for some values of (q,g). The results we obtain are listed in Table 1. g q N q (g) N g q q (g) New enter Old New enter Old [...,53] [...,156] [...,60] [...,165] [...,67] [...,172] [...,80] [136,180] [...,84] [...,193] [...,96] [...,72] [...,102] [232,300] [...,106] [376,460] [...,113] [60,78] [...,124] [80,92] [...,133] [102,114] [...,137] [132,150] [...,148] [72,100] [...,152] [42,55] Table 1: The specified interval (Old) is that given by the website as of Avril Geometry of curves Let k be a positive integer. Consider now the fibre product C = C f1...f k of which an open affine subset is given by y1 2 = f 1 (x). y 2 k = f k (x),

4 4 T. Top where the f i s are coprime polynomials over F q of respective degrees d i s. Lemma 2.1. Suppose the polynomial f(x) := i f i(x) is separable, then the affine curve is smooth. The genus of the complete curve is g = 2 k 2 (d d k 4)+1+δ k, with δ k = 2 k 2 (resp. δ k = 0) if one of the d i s is odd (resp. all of the d i s are even). There are 2 k 1 (resp. 2 k ) points at infinity if one of the d i s is odd (resp. all of the d i s are even). Proof. Smoothness follows from the jacobian criterion applied to the matrix f 1(x) 2y 1 0 f 2 (x) 0 2y f k (x) 0 2y k If none of the y i s is zero, there exists a minor equal to 2 k 1 y 1...y i 1 f i (x)y i+1...y k 0; If for some i, y i = 0, then f i (x) = 0 and therefore f i (x) 0 and, for j i, we have f j (x) 0 and there exists a minor equal to 2 k 1 y 1...y i 1 f i(x)y i+1...y k 0. The group G = {±1} k = (Z/2Z) k acts in an obvious way on C by [ε](x,y 1,...,y k ) = (x,ε 1 y 1,...,ε k y k ) and is the Galois group of the covering given by φ : C P 1 (x,y 1,...,y k ) x. The group G acts transitively on the set C, of points at infinity; the inertia group is cyclic therefore is trivial or reduced to Z/2Z. If one of the d i s is odd (resp. all d i s are even), there is ramification above P 1 hence we obtain C = 2 k 1 (resp. = 2 k ). The Riemann-Hurwitz formula applied to the morphism φ gives the genus, observing that φ is branched at every point x = α i with f i (α i ) = 0, i.e (x,y 1,...,y k ) = (α i,± f 1 (α i ),...,0,...,± f k (α i )), and possibly above.

5 Many points 5 3 Computation the number of points We keep the same notation as before. Definition 3.1. Let I be a non-empty subset of {1,...,k}, we define: 1. the polynomial f I (x) = i I f i(x) and we denote by d I its degree; 2. the smooth projective curve C I whose affine model is given by v 2 = f I (u); 3. we denote by g I the genus of C I ; 4. the morphism φ I : C C I given by φ I (x,y 1,...,y k ) = (x,y I ) (where y I = i I y i). We denote by I the set of non-empty subsets of {1,...,k}. Lemma 3.2. The morphism is a separable isogeny. Ψ : I I Jac(C I) Jac(C) (C I ) I I I I φ I (C I). Proof. We first verify that I I g I = g; indeed, if we putd I = i I d i = degf I then g I = d I 1 2, which we may write d I 1 ε I with ε 2 I = 0 or 1. So g I = ( k d I 1 ε I = 1 d i N i 2 k +1 ) ε I 2 2 I I I I i=1 I where N i is the number of I s containing i, i.e N i = 2 k 1. If all d i are even, all the ε I = 1 and 1 2 I (1+ε I) = 2 k 1 and the formula follows. If at least one of d i is odd, denote by M the number of I with d I odd, then 1 2 I (1+ε I) = 2 k 1 M ; we conclude by using M = 2 2k 1. The abelian varieties I I Jac(C I) andjac(c) have the same dimension. Furthermore, if η j = u j 1 du/v is a regular differential form on C I (for 1 j g I ) then ω I,j := φ I (η j) = x j 1 dx/y I is regular on C and these forms are linearly independent. Indeed an equality of type: g I λ I,j ω I,j = 0, I j=1 implies P I (x)y I c = 0, I

6 6 T. Top where I c := [1,k]\I ; which implies the P I s are zero. The differential of Ψ is an isomorphism, which proves that Ψ is a separable isogeny. Remark. We can deduce from the previous calculation, when k 2, that the curve C is not hyperelliptic. In fact the canonical morphism P (ω I,j (P)) may be written P (1,x,x 2,...,y 1,...,y k,...) thus is generically of degree 1, therefore an isomorphism (if C were hyperelliptic, it would be a morphism of degree 2). Lemma 3.3. Let f 1,...,f k be polynomials with coefficients in the finite field F q such that the product f 1...f k is separable, and let C be the associated curve. Let A I = q +1 C I (F q ) then, C(F q ) = q +1 I I A I. Proof. The abelian varieties Jac(C) and I Jac(C I) are F q -isogenous therefore have the same number of points on F q m. If then and if then and thus therefore C(F q m) = q m +1 (β m βm 2g ) Jac(C)(F q m) = 1 i 2g (1 β m i ) C I (F q m) = q m +1 ((α (I) 1 ) m +...+(α (I) 2g ) m ) Jac(C)(F q m) = I Jac(C I )(F q m) = 1 i 2g Jac(C I )(F q m) = I (1 (α (I) i ) m ) (1 (α (I) j ) m ), C I (F q m) = q m +1 ((α (I) 1 )m +...+(α (I) 2g )m ) = q m +1 I and, in particular, for m = 1, we get C(F q ) = q +1 I 1 j g I α (I) j = q +1 I I A I. j 1 j g I (α (I) j ) m,

7 Many points 7 4 Numerical examples Examples of genus-g curves y 2 1 = f 1, y 2 2 = f 2 over F q having many points. The meaning of the quantities A 1, A 2, A 3, and N is explained in the text. The following examples improve some results listed on the website and were computed using the computer algebra package Magma. q = 17 f 1 = x 4 +x 3 +16x 2 +15x+1 A 1 = 8 N = 48 [...,53] f 2 = x 4 +13x 3 +16x A 2 = 6 A 3 = 16 q = 19 f 1 = x 4 +x 3 +18x 2 +13x+14 A 1 = 8 N = 52 [...,60] f 2 = x 4 +4x 3 +18x 2 +7x+12 A 2 = 8 A 3 = 16 q = 23 f 1 = x 4 +19x 3 +7 A 1 = 9 N = 60 [...,67] f 2 = x 3 +x+11 A 2 = 9 A 3 = 18 q = 29 f 1 = x 4 +x 3 +28x 2 +28x+18 A 1 = 10 N = 72 [...,80] f 2 = x 4 +27x 3 +27x 2 +28x+15 A 2 = 10 A 3 = 22 q = 31 f 1 = x 4 +x 3 +30x 2 +30x+10 A 1 = 11 N = 76 [...,84] f 2 = x 4 +30x 3 +5x 2 +19x+14 A 2 = 10 A 3 = 23 q = 37 f 1 = x 4 +x 3 +35x 2 +32x+1 A 1 = 12 N = 88 [...,96] f 2 = x 4 +11x 3 +29x 2 +13x+29 A 2 = 12 A 3 = 26 q = 41 f 1 = x 4 +x 3 +40x 2 +40x+36 A 1 = 12 N = 94 [...,102] f 2 = x 4 +23x 3 +10x 2 +20x+36 A 2 = 11 A 3 = 29 q = 43 f 1 = x 4 +x 3 +41x 2 +34x+1 A 1 = 13 N = 100 [...,106] f 2 = x 4 +12x 3 +17x 2 +41x+38 A 2 = 13 A 3 = 30 q = 47 f 1 = x 4 +25x 3 +x 2 +2x+31 A 1 = 12 N = 102 [...,113] f 2 = x 3 +x+38 A 2 = 13 A 3 = 29 q = 53 f 1 (x) = x 4 +x 3 +52x 2 +47x+1 A 1 = 14 N = 120 [...,124] f 2 (x) = x 4 +16x 3 +36x 2 +18x+46 A 2 = 14 A 3 = 38

8 8 T. Top q = 59 f 1 (x) = x 4 +x 3 +54x 2 +6x+1 A 1 = 15 N = 124 [...,133] f 2 (x) = x 4 +13x 3 +22x 2 +3x+9 A 2 = 15 A 3 = 34 q = 61 f 1 (x) = x 4 +x 3 +58x 2 +18x+1 A 1 = 15 N = 126 [...,137] f 2 (x) = x 4 +27x 3 +5x 2 +47x+28 A 2 = 14 A 3 = 35 q = 67 f 1 (x) = x 4 +x 3 +66x 2 +57x+1 A 1 = 16 N = 136 [...,148] f 2 (x) = x 4 +2x 3 +49x 2 +24x+1 A 2 = 16 A 3 = 36 For q = 71 f 1 (x) = x 4 +x 3 +x 2 +44x+1 A 1 = 16 N = 144 [...,152] f 2 (x) = x 4 +9x 3 +8x 2 +21x+64 A 2 = 16 A 3 = 40 For q = 73 f 1 (x) = x 4 +x 3 +66x 2 +57x+1 A 1 = 17 N = 148 [...,156] f 2 (x) = x 4 +2x 3 +49x 2 +24x+1 A 2 = 17 A 3 = 40 For q = 79 f 1 (x) = x 4 +x 3 +3x 2 +7x+1 A 1 = 16 N = 156 [...,165] f 2 (x) = x 4 +4x 3 +5x 2 +24x+68 A 2 = 16 A 3 = 36 For q = 83 f 1 (x) = x 4 +x 3 +x 2 +5x+1 A 1 = 18 N = 162 [...,172] f 2 (x) = x 4 +72x 3 +54x 2 +29x+36 A 2 = 16 A 3 = 44 For q = 89 f 1 (x) = x 4 +x 3 +3x 2 +7x+1 A 1 = 16 N = 168 [136,180] f 2 (x) = x 4 +4x 3 +5x 2 +24x+68 A 2 = 16 A 3 = 36 q = 97 f 1 (x) = x 4 +8x 3 +3x 2 +23x+1 A 1 = 19 N = 180 [...,193] f 2 (x) = x 4 +9x 3 +63x 2 +28x+91 A 2 = 19 A 3 = 44 q = 5 2 f 1 (x) = x 4 +x 2 +rx A 1 = 9 N = 68 [...,72] f 2 (x) = x 4 2rx 3 +rx+2 A 2 = 9 with r 2 +r +1 = 0 A 3 = 24 q = 13 2 f 1 (x) = (x 2)(x 2 2) A 1 = 26 N = 295 [232,300] f 2 (x) = x 4 4x 2 +1 A 2 = 21 A 3 = 78 q = 17 2 f 1 (x) = (x+2)(x+10)(x 2 +6) A 1 = 33 N = 454 [376,460] f 2 (x) = (x 2 +10)(x 2 +6x+3) A 2 = 29 A 3 = 102 Table 2: Examples of genus-5.

9 Many points 9 q = 23 f 1 (x) = x 5 +12x 4 +19x 3 +x+2 A 1 = 13 N = 66 [60,78] f 2 (x) = x 3 +12x 2 +18x+4 A 2 = 9 A 3 = 20 q = 31 f 1 (x) = x 5 +6x 4 +4x 3 +4x 2 +17x+16 A 1 = 19 N = 84 [80,92] f 2 (x) = (x) 3 +13x 2 +15x+13 A 2 = 11 A 3 = 22 q = 41 f 1 (x) = x 5 +31x 4 +11x 3 +14x 2 +35x+40 A 1 = 23 N = 104 [102,114] f 2 (x) = x 3 +23x 2 +5x+17 A 2 = 12 A 3 = 27 q = 59 f 1 (x) = x 5 +57x 4 +17x A 1 = 25 N = 134 [132,150] f 2 (x) = x 3 +2x+22 A 2 = 15 A 3 = 34 Table 3: Examples of genus-6. q = 29 f 1 (x) = x 6 +9x 5 +3x 4 +1 A 1 = 46 N = 80 [72,100] f 2 (x) = x 4 +13x 3 +28x 2 +12x+1 A 2 = 40 A 3 = 54 Table 4: Examples of genus-7. Acknowledgements We thank Professor Marc Hindry (Paris 7), the mathematics lab of the University Blaise Pascal in particular Nicolas Billerey and Marusia Rebelledo. References [1] Everett W. Howe. New bounds on the maximum number of points on genus-4 curves over small finite fields, Arithmetic, Geometry, Cryptography and Coding Theory (Y. Aubry, C. Ritzenthaler, and A. Zykin, eds.), Contempory Mathematics 574,in American Mathematical Sociaty,Providence, RI, 2012, pp [2] Yasutaka Ihara. Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3): (1982), 1981.

10 10 T. Top [3] Tetsuo Kodama, Jaap Top, and Tadashi Washio. Maximal hyperelliptic curves of genus three. Finite Fields Appl., 15(3): , [4] Stephen Meagher and Jaap Top. Twists of genus three curves over finite fields. Finite Fields and their Appl., 16: , [5] Jean-Pierre Serre. Nombres de points des courbes algébriques sur F q. In Seminar on number theory, (Talence, 1982/1983), pages Exp. No. 22, 8. Univ. Bordeaux I, Talence, [6] Jean-Pierre Serre. Rational points on curves over finite fields. Unpublished notes by F.Q. Gouvêa of lectures at Harvard [7] Jaap Top. Curves of genus 3 over small finite fields. Indag. Math. (N.S.), 14(2): , [8] Michael Tsfasman, Serge Vladut, and Dmitry Nogin. Algebraic geometric codes: basic notions, volume 139 of Mathematical Survey and Monographs. American Mathematical Society, Providence, RI, [9] Gerard van der Geer and Marcel van der Vlugt. Tables of curves with many points. Math. Comp., 69(230): , [10] William C. Waterhouse. Abelian varieties over finite fields. Ann. Sci. Ecole Norm. Sup. (4), 2: , 1969.