Genus 3 curves with many involutions and application to maximal curves in characteristic 2

Let k = Fq be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C2 × C2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an ArtinSchreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H , and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then characterize when a triple (E1, E2, E3) of elliptic curves admits an ArtinSchreier cover by a genus 3 curve, and we apply this result to the construction of maximal curves. As a consequence, when q is nonsquare and m := ⌊2√q⌋ ≡ 1, 5, 7 (mod 8), we obtain that Nq(3) = 1 + q + 3m. We also show that this occurs for an infinite number of values of q nonsquare. Let C be a smooth, absolutely irreducible, projective curve of genus g > 0 over a finite field k = Fq. The question to determine the maximal number of points Nq(g) of such a curve C is a tantalizing one. Curves such that #C(k) = Nq(g) are called maximal curves. Serre-Weil bound shows that Nq(g) ≤ 1 + q + gm, where m = ⌊2 √ q⌋. However, no general formula is known for the value of Nq(g) (so far not even for infinitely many values of q) when g > 2 is fixed and q is not a square. For g = 3, because of the so-called Serre’s twisting factor (or Serre’s obstruction, see [LR08], [LRZ08]), the best general result is that for a given q, either q + 1+ 3m−Nq(3) ≤ 3 or Mq(3)− (q + 1− 3m) ≤ 3, where Mq(3) is the minimum number of points [Lau02]. Although this obstruction is now better understood and can be computed in some cases [Rit09], we are still not able to find Nq(3) for a general q. However, when q is a square, Nq(3) is known for infinitely many values; see [Ibu93] when the characteristic is odd, and [NR08] for the characteristic 2 case, where Nq(3) is determined for all square q. ∗The authors acknowledge support from the project MTM2006-11391 from the Spanish MEC