Towards a characterization of subfields of the Deligne–Lusztig function fields

In this paper, we give a characterization of subgroups contained in the decomposition group A ( P ∞ ) of a rational place P ∞ by means of a necessary and sufficient condition for each of the three types of function fields of Deligne–Lusztig curves. In particular, we translate the problems on the genera of subfields of the Deligne–Lusztig function fields to the combinatorial problems concerning some specific vector spaces and their dimensions. This allows us to determine the genera set consisting of all the genera of the fixed fields of subgroups of the decomposition group A ( P ∞ ) for the Hermitian function field over F q where q is a power of an odd prime. Promising results pertaining to the genera of subfields of the other types of Deligne–Lusztig function fields are provided as well. Indeed, it turns out that we improve many previous results given by Garcia–Stichtenoth–Xing, Giulietti–Korchmáros–Torres and Çakçak–Özbudak on the subfields of function fields of Deligne–Lusztig curves.