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

Abstract

The second generalized GK function fields Kn are a recently found family of maximal function fields over the finite field with q 2n elements, where q is a prime power and n ≥ 1 an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of Kn. In case gcd(q + 1, n) = 1 and either q is even or q ≡ 1 (mod 4), we find a complete list of Galois subfields of Kn. Our construction adds several previously unknown genera to the genus spectrum of maximal curves.

New examples of maximal curves with low genus

“…Proof. To prove the statement it is sufficient to observe that, if f (X) = X 12 − aX 10 − 33X 8 + 2aX 6 − 33X 4 − aX 2 + 1 then f (X) has multiple roots if and only if a 2 + 108 = 0 since the resultant of f (X) and its derivative f ′ (X) is equal to (a 2 + 108) 8 . Now the claim follows from [74, Proposition 3.…”

“…The problem was solved in [35], where examples of F q 2 -maximal curves, with q = p 3h > 8, p a prime, non-covered by H q were constructed. Applying Kleiman-Serre covering result to these curves and to their generalizations [30], [7] and [71] provided further examples of maximal curves; see [1,8,15,22,38,39]. Other recent constructions can be found in [42,[78][79][80][81].…”

Classification of all Galois subcovers of the Skabelund maximal curves

“…A preliminary study in [5] already revealed that new genera of maximal curves can be obtained by considering Galois subcovers of K n . A more detailed study of Galois subcovers of the second generalization of the GK function field was provided in [6].…”

“…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 .…”

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 .…”