Rational points on cubic surfaces and AG codes from the Norm-Trace curve Matteo Bonini ∗1, Massimiliano Sala2, and Lara Vicino3 1School of Mathematics and Statistics, University College Dublin, Dublin, Ireland e-mail: [email protected] 2Department of Mathematics, University of Trento, Trento, Italy e-mail: [email protected] 3Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark e-mail: [email protected] Abstract In this paper we give a complete characterization of the intersections between the Norm-Trace curve 3 2 over Fq3 and the curves of the form y = ax + bx + cx + d, generalizing a previous result by Bonini and Sala, providing more detailed information about the weight spectrum of one-point AG codes arising from such curve. We also derive, with explicit computations, some general bounds for the number of rational points on a cubic surface defined over Fq. Keywords: Norm-trace curve - AG Code - Weight spectrum - Cubic Surfaces MSC Codes: 14G50 - 11T71 - 94B27 Many of the best performing algebraic codes are known to be Algebraic Geometry (AG) codes, which arise from algebraic varieties over finite fields. Among them, the most studied codes are the ones arising from algebraic curves, that were introduced by Goppa in the ’80s; see [18, 19] for a detailed description. Let be an algebraic curve defined over the finite field with q elements Fq. The parameters of codes arising fromX strictly depend on some geometrical properties of the curve. In general, curves with many X Fq-rational places with respect to their genus give rise to long AG codes with good parameters. For this arXiv:2102.05478v1 [math.AG] 10 Feb 2021 reason, maximal curves, that is, curves attaining the Hasse-Weil upper bound, have been widely investigated in the literature; see [2, 6, 30, 35, 37, 39]. In general, the determination of the weight spectrum of a code C (i.e. the set of the possible weights of C) is a very difficult task, see for instance [25]. ∗The author was supported by the Irish Research Council, grant n. GOIPD/2020/597. 1 For AG codes, it is possible to derive information about their weight spectrum by the study of the intersection of the base curve and low degree curves, as done in [3,5,12,28,29]. The Norm-Trace curves areX a natural generalization of the celebrated Hermitian curve to any extension field Fqr , and their codes have been widely studied; see [3, 8, 16, 17, 31]. In this paper, we focus on the intersection between the Norm-Trace curve over Fq3 and curves of the form y = Ax3 + Bx2 + Cx + D, giving the proof of a result which corrects a previous conjecture stated in [8]. In addition to this, we partially deduce the weight spectrum of its one-point codes in the place at the infinity. In order to obtain these results, we translate the problem of finding the planar intersection between the cubic Norm-Trace curve and the above mentioned curves into that of counting the number of Fq-rational points of certain cubic surfaces, which is a well-knwon topic in algebraic geometry in positive characteristic, see [11, 27, 38]. For this reason, in section 2 we start by presenting general results on the number of Fq- rational points on cubic surfaces defined over Fq. These results end up constituting a partial generalization of a classical result by Weil and we obtain them exploiting classical results on cubic surfaces over finite fields. In sections 3, 4 and 5 we analyze cubic surfaces arising from the intersection between the Norm-Trace 3 2 curve and curves of the form y = Ax + Bx + Cx + D over Fq3 , proving Theorem 3.3. Finally, in section 6, we use these results to investigate the weight spectrum of a certain family of one-point codes arising from the cubic Norm-Trace curve. 1 Preliminaries h Let q = p , where p is a prime and h> 0 an integer, and denote with Fq the finite field with q elements. Fqr Fqr F r We recall that the norm NFq and the trace TFq , where r is a positive integer, are functions from q to Fq such that r − F r q 1 r−1 r−2 q q−1 q +q +···+q+1 NFq (x)= x = x and F r−1 r−2 qr q q q TF (x)= x + x + + x + x. q ··· When q and r can be derived unequivocally from the context, we will omit the subscripts. 1.1 The Norm-Trace curve 2 The Norm–Trace curve r is the curve defined over the affine plane A (Fqr ) by the equation N N(x)=T(y). (1) If r = 2 the curve r is smooth, while if r 3 it can be easily seen that r has a singular point which is N ≥ 2r−1 N the point at the infinity P∞. It is then well-known that r has q affine places and a single place at the N infinity; in fact, there is exactly one place centered at each affine point of r (these are all smooth points) and the point at the infinity is either smooth (in the case r = 2) and henceN center of exactly one place, or it 2 is singular and center of only one branch of the curve (if r 3), so that r has a unique place at the infinity also in this case. ≥ N If r = 2, r coincides with the Hermitian curve, and this is the only case in which r is smooth, since, N N as noted above, for r 3 it has a singularity in P∞. ≥ Moreover it is known that its Weierstrass semigroup in the place centered at P∞ is generated by r r−1 q −1 q , q−1 , see [17]. Also, the automorphism group is determined by the following result. D E r−1 r Theorem 1.1 ( [7]). The automorphism group of r Aut( r) has order q (q 1) and is a semidirect product G ⋊ C, where N N − G = (x, y) (x, y + a) T(a)=0 { 7→ | } qr −1 q−1 ∗ C = (x, y) (bx,b y) b F r . { 7→ | ∈ q } Our main aim is the study of the planar intersection (i.e. the intersection counted without multiplicity) 3 2 between r and the cubic curves of the form y = ax + bx + cx + d, where a,b,c,d Fqr and a = 0, in the case Nr = 3. The case r = 2 and a = b = 0 was investigated in [4] and the case r∈= 2 and a =6 0 was completely investigated in [14, 28]. On the other hand, the case r = 3 and a = 0 was investigated in [8]. In [8], the authors claim the following result. 3 Theorem 1.2. The number of planar intersections between 3 and cubic curves of the form y = ax + 2 2 N bx + cx + d, where a,b,c,d Fq3 , a = 0, is bounded by q +7q +1, when the surface defined over Fq3 by the equation ∈ 6 3 q 3 q2 3 2 q 2 q2 2 q q2 X0X1X2 = AX0 + A X1 + A X2 + BX0 + B X1 + B X2 + CX0 + C X1 + C X2 + E is irreducible, for A,B,C,D,E F 3 . ∈ q Still, a proof of this result is not given in the paper. The aim of this paper is to give a detailed proof of this bound in any possible case, showing that this is not correct when such surface is a cone over an elliptic curve. 1.2 Algebraic Geometry codes In this section we recall some basic facts on AG codes. For a detailed discussion we refer to [36]. Let be a projective curve over the finite field Fq and consider the function field Fq( ) of rational X X functions defined over Fq. Denote with (Fq) the set of the Fq-rationals points of . A divisor D on can be seen as a finite sum n P ,X where the n s are integers. For a functionX f F ( ), (f) denotesX P ∈X (Fq ) P P q F ∈ X q q the divisor associated toPf. A divisor D is q-rational if it coincides with its image P ∈X (F ) nP P under F q the Frobenius map over q. P F n Given an q-rational divisor D = i=1 niPi on , its support is defined as supp(D)= Pi : ni =0 . X F { 6 } The Riemann-Roch space associatedP with D is the q-vector space L (D)= f Fq( ) (f)+ D 0 0 . { ∈ X | ≥ }∪{ } 3 It is known that L (D) is an Fq-vector space of finite dimension ℓ(D). The exact dimension of this space can be computed using Riemann-Roch theorem. n F F Consider now the divisor D = i=1 Pi where all the Pi’s are q-rational. Let G be another q-rational divisor on such that supp(G) suppP (D)= . Consider the evaluation map X ∩ ∅ n eD : L (G) F −→ q f eD(f) = (f(P ),...,f(Pn)). 7−→ 1 The map eD is Fq-linear and it is injective if n> deg(G). The AG code CL (D, G), also called functional code, associated with the divisors D and G is defined as n CL (D, G) := eD(L (G)) = (f(P ),...,f(Pn)) f L (G) F . Such a code is an [n,ℓ(G) ℓ(G D), d]q { 1 | ∈ }⊆ q − − code, where d d¯= n deg(G) and d¯ is the so-called designed minimum distance (of such code).