A complete characterization of Galois subfields of the generalized Giulietti–Korchmáros function field

Communications in Algebra

Zini

2018

We investigate the genera of quotient curves H q /G of the F q 2 -maximal Hermitian curve H q , where G is contained in the maximal subgroup M q ≤ Aut(H q ) fixing a pole-polar pair (P, ℓ) with respect to the unitary polarity associated with H q . To this aim, a geometric and group-theoretical description of M q is given. The genera of some other quotients H q /G with G ≤ M q are also computed. Thus we obtain new values in the spectrum of genera of F q 2 -maximal curves. A plane model for H q /G is obtained when G is cyclic of order p · d, with d a divisor of q + 1.

“…2 6 G = Sym(3) = α ⋊ β , α of type (B2), β of type (A). 1 8 G = C 4 × C 2 = α × β , α of type (A), β of type (A). 0 7 G = C 7 = σ , σ of type (C).…”

Section: Further Resultsmentioning

confidence: 99%

On the spectrum of genera of quotients of the Hermitian curve

Communications in Algebra

Zini

2018

“…By Remark 3.2, H = M ∩ PSU(3, q) M and [M : H] = 3. Arguing as in the proof of Proposition 3.8 it can be shown that M/Ker(ϕ) is isomorphic to a subgroup of PΓL (2,9), where ϕ is the action by conjugation of M on H; H commutes elementwise with an element of order 3 in M \ H; H is contained either in M 2 or in M 3 or in M 4 as described in Theorem 3.1, a contradiction to Theorem 2.1.…”

Section: 5mentioning

confidence: 92%

“…Also, A 6 M since A 6 is characteristic in H. Hence, M/Ker(ϕ) is isomorphic to an automorphism group of A 6 , where Ker(ϕ) is the kernel of the action by conjugation of M on A 6 . As A 6 ∼ = PSL(2, 9), M/Ker(ϕ) is isomorphic to a subgroup of PΓL (2,9). Since the largest subgroup of P ΓL(2, 9) has order 720 and the order of M/Ker(ϕ) is at least |H| = 720, we have |Ker(ϕ)| = 3.…”

Section: 4mentioning

confidence: 99%

See 1 more Smart Citation

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

Zini

2019

J Algebr Comb

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“…We have indicated a genus with boldface if it gives a new addition to the genus spectrum of

F_{q^{2 n}}

‐maximal curves. To check whether or not a genus gave a new addition, we compared our results with the genera of maximal function fields given in .

q = 4

and

n = 5

{32, 156, 302, 1506, 1532}

q = 4

and

n = 7

{212, 842, 1056, 4206, 24572}

q = 5

and

n = 5

{6242, 12484, 18724}

q = 7

and

n = 5

{243, 485, 969, 1941, 4563, 9125, 18249, 36501, 50403, 100805, 201609}

. …”

Section: Determination Of the Full Automorphism Group Of Xn N⩾5mentioning

confidence: 99%

A new family of maximal curves

Beelen

2018

J. London Math. Soc.

Self Cite

In this article we construct for any prime power q and odd n 5, a new F q 2n -maximal curve Xn. Like the Garcia-Güneri-Stichtenoth maximal curves, our curves generalize the Giulietti-Korchmáros maximal curve, though in a different way. We compute the full automorphism group of Xn, yielding that it has precisely q(q 2 − 1)(q n + 1) automorphisms. Further, we show that unless q = 2, the curve Xn is not a Galois subcover of the Hermitian curve. Finally, up to our knowledge, we find new values of the genus spectrum of F q 2n -maximal curves, by considering some Galois subcovers of Xn.