New maximal curves as ray class fields over Deligne-Lusztig curves

Abstract

Abstract. We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchmáros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the GiuliettiKorchmáros curve is equal to the above-mentioned ray class field extension of the Hermitian curve.

On some Galois covers of the Suzuki and Ree curves

“…Skabelund [25,Sec. 3] introduced the F q 4 -maximal curveS q defined over F q with affine equationsS q :…”

“…The automorphism group Aut(H q ) ofH q is defined over F q 3 and has a normal subgroup of index d := gcd (3, m) which is isomorphic to SU(3, q 0 ) × C m/d , where SU(3, q 0 ) is the special unitary group over F q and C m/d is a cyclic group of order m/d. Analogously, Skabelund [25] constructed Galois covers of the Suzuki and Ree curves as follows. Let q 0 = 2 s with s ≥ 1 and q = 2q 2 0 = 2 2s+1 .…”

“…Also in [25] the automorphism groups S(q) of S q and R(q) of R q were lifted to subgroups of the full automorphism groups Aut(S q ) and Aut(R q ) ofS q andR q , respectively. We show that the lifted groups are actually the full automorphism groups ofS q andR q .…”

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

“…The automorphism group R(q) := Aut(R q ) is isomorphic to the simple Ree group 2 G 2 (q). We state some other properties of R(q); see [40] and [43].…”

“…The following constructions are due to D. Skabelund [43]. LetS q be the curve defined over F q by the affine equationsS q :…”