The complete list of genera of quotients of the Fq2-maximal Hermitian curve for q ≡ 1 (mod 4)

On subfields of the second generalization of the GK maximal function field

“…Another maximal subgroup of PGU (3, q), which we will denote by M ℓ , arises by considering the stabilizer of a chord ℓ. A full description of the subgroups of this group was given in [6] for even q and in [23] for q ≡ 1 (mod 4). Many genera of maximal function fields have been obtained in this way, adding to the understanding of the genus spectrum of maximal curves.…”

“…The complete list of subgroups of M ℓ up to isomorphism is known for p = 2 and q ≡ 1 (mod q), see [6] and [23] respectively. Since later, we will need these groups for a case by case analysis, we list them in the following two lemmas.…”

“…However, in the cases in whichL is tame the genus g L can be computed directly from gL without having to compute N . The values of gL can be found in [23] and will not be reproduced here. We proceed by computing N in the remaining, non-tame cases.…”

“…Proof. The genus gL is computed in [23,Proposition 3.4]. According to its proof we can describe the short-orbits structure ofL in its natural action on H q (F q 2 ).…”

“…SL(2, 5) acts semiregularly onŌ 2 . From the proof of [23,Proposition 3,4],L contains 12 cyclic subgroups of order 5 which are of type (A), that is, that fix q + 1 distinct points inŌ 2 . This implies that O 2 contains a set of 12(q + 1) points on whichL acts with stabilizer of order 5, and hence with orbits of length 24|L Z | and acts with long orbits elsewhere.…”