Some Ree and Suzuki curves are not Galois covered by the Hermitian curve
Giovanni Zini · IRIS UNIMORE (University of Modena and Reggio Emilia) · 2017
The Deligne–Lusztig curves associated to the algebraic groups of type A 2 2 , B 2 2 , and G 2 2 are classical examples of maximal curves over finite fields. The Hermitian curve H q is maximal over F q 2 , for any prime power q , the Suzuki curve S q is maximal over F q 4 , for q = 2 2 h + 1 , h ≥ 1 , and the Ree curve R q is maximal over F q 6 , for q = 3 2 h + 1 , h ≥ 0 . In this paper we show that S 8 is not Galois covered by H 64 . We also prove an unpublished result due to Rains and Zieve stating that R 3 is not Galois covered by H 27 . Furthermore, we determine the spectrum of genera of Galois subcovers of H 27 , and we point out that some Galois subcovers of R 3 are not Galois subcovers of H 27 .