Quotients of the Hermitian curve from subgroups of $$\mathrm{PGU}(3,q)$$ without fixed points or triangles

Abstract

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

“…The following classification of subgroups of PGU(3, q) goes back to Mitchell [31] and Hartley [23]; see also [33].…”

“…• the genera g(H q /G) for the subgroups G ≤ PGU(3, q) stabilizing an F q 2 -rational point of H q are characterized in [2, Theorem 1.1]; • the genera g(H q /G) for the subgroups G ≤ PGU(3, q) stabilizing a self-polar triangle of PG(2, q 2 ) are characterized in [8, Section 3]; • the genera g(H q /G) for the subgroups G ≤ PGU(3, q) stabilizing a Frobenius-invariant triangle in H q (F q 6 ) \ H q (F q 2 ) are characterized in [6, Proposition 4.2]; • the genera g(H q /G) for the subgroups G ≤ PGU(3, q) which do not stabilize any point or triangle are characterized in [33].…”

“…In particular, F q 2 -maximal curves can be obtained as Galois F q 2 -subcovers of an F q 2 -maximal curve X , that is, as quotient curves X /G where G is a finite automorphism group of X . Most of the known maximal curves are Galois covered by the Hermitian curve; see for instance [18,6,19,34,32,33] and the references therein.…”

“…• G fixes neither points nor triangles in PG(2, q 6 ); see [33]. From the results already obtained in the literature (see [8,33] and the references therein), in order to obtain the complete list of genera of quotients H q /G, G ≤ Aut(H q ), only the following cases still have to be analyzed:…”

Classification of all Galois subcovers of the Skabelund maximal curves

“…To the best of our knowledge the genera given in these tables are new. We have checked that these values are not contained in and cannot be obtained using results from [1,2,3,5,6,7,8,9,10,11,12,14,15,22,25,26,27,29,30,32,40,41,42,43]. The paper is organized as follows: in section two, we classify Galois subcovers of Sq , while in section three, we achieve this for Rq .…”