On Subfields of the Hermitian Function Field

Abstract

The Hermitian function field H = K(x, y) is defined by the equation yq + y = xq+1 (q being a power of the characteristic of K). Over K = double-struck F signq2 it is a maximal function field; i.e. the number N(H) of double-struck F signq2-rational places attains the Hasse-Weil upper bound N(H) = q2+1+2g(H)·q. All subfields K (subset of with not equal to) E ⊆ H are also maximal. In this paper we construct a large number of nonrational subfields E ⊆ H, by considering the fixed fields Hscript g sign under certain groups script g sign of automorphisms of H/K. Thus we obtain many integers g ≥ 0 that occur as the genus of some maximal function field over double-struck F signq2.