2000/08/26 by Gabor Korchmaros, Fernando Torres
Mathematics · #math.AG #msc:14G #msc:11G
published as Math. Ann. 323(3) (2002), 589--608 · 23 pages, LaTex2e, ps file available at http://www.ime.unicamp.br/~ftorres
arxiv created 2000/08/26 · arxiv updated 2009/11/30
Previous results on genera g of Fq2-maximal curves are improved: (1) Either g≤ (q2-q+4)/6, or g=\lfloor(q-1)2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \citeat is proved; (3) For q≡ 1\pmod3, q≥ 13, no Fq2-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q≡ 2\pmod3, q≥ 11, the non-singular Fq2-model of the plane curve of equation yq+y=x(q+1)/3 is the unique Fq2-maximal curve of genus g=(q-1)(q-2)/6; (5) Assume dim(\cD_\cX)=5, and char(\fq)≥ 5. For q≡ 1\pmod4, q≥ 17, the Fermat curve of equation x(q+1)/2+y(q+1)/2+1=0 is the unique Fq2-maximal curve of genus g=(q-1)(q-3)/8. For q≡ 3\pmod4, 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 yq+y=x(q+1)/4. 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.