On the Genus of a Maximal Curve

Previous results on genera g of Fq2 -maximal curves are improved: (1) Either g ≤ ⌊(q − q + 4)/6⌋ , or g = ⌊(q − 1)/4⌋ , or g = q(q − 1)/2 . (2) The hypothesis on the existence of a particular Weierstrass point in [2] is proved. (3) For q ≡ 1 (mod 3), q ≥ 13, no Fq2 -maximal curve of genus (q−1)(q−2)/3 exists. (4) For q ≡ 2 (mod 3), q ≥ 11, the non-singular Fq2 -model of the plane curve of equation y + y = x is the unique Fq2 -maximal curve of genus g = (q − 1)(q − 2)/6. (5) Assume dim(DX ) = 5, and char(Fq2 ) ≥ 5. For q ≡ 1 (mod 4), q ≥ 17, the Fermat curve of equation x + y + 1 = 0 is the unique Fq2 -maximal curve of genus g = (q − 1)(q − 3)/8. For q ≡ 3 (mod 4), q ≥ 19, there are exactly two Fq2 -maximal curves of genus g = (q− 1)(q− 3)/8, namely the above Fermat curve and the non-singular Fq2 -model of the plane curve of equation y + y = x. The above results provide some new evidences on maximal curves in connection with Castelnuovo’s bound and Halphen’s theorem, especially with extremal curves; see for instance the conjecture stated in Introduction.