On subfields of the Hermitian function field

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 = F-q2 it is a ma ximal function field; i.e. the number N(H) of F-q2-rational places attains the Hasse-Weil upper bound N(H)=q(2)+1+2g(H).q. All subfields K not subset of or equal to E subset of or equal to H are also maximal. In this paper we construct a large number of nonrational subfields E subset of or equal to H, by considering the fixed fields H-g under certain groups g of automorphi sms of H/K. Thus we obtain many integers g greater than or equal to 0 that occur as the genus of some maximal function field over F-q2.