On Subfields of the Hermitian Function Field
Arnaldo Garcia, Henning Stichtenoth, Chaoping Xing · Compositio Mathematica · 2000
The Hermitian function field H = K ( x , y ) is defined by the equation y q + y = x q +1 ( q being a power of the characteristic of K ). Over K = ${\mirrored F}$ q 2 it is a maximal function field; i.e. the number N ( H ) of ${\mirrored F}$ q 2 -rational places attains the Hasse–Weil upper bound N ( H )= q 2 +1+2 g ( H )· q . All subfields K [subnE ] E ⊆ H are also maximal. In this paper we construct a large number of nonrational subfields E ⊆ H , by considering the fixed fields H $^{\mathcal G}$ under certain groups ${\mathcal G}$ of automorphisms of H / K . Thus we obtain many integers g [ges ]0 that occur as the genus of some maximal function field over ${\mirrored F}$ q 2 .