Quotients of the Hermitian curve from subgroups of $$\mathrm{PGU}(3,q)$$ PGU ( 3 , q ) without fixed points or triangl
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Quotients of the Hermitian curve from subgroups of PGU(3, q) without fixed points or triangles Maria Montanucci1 · Giovanni Zini2 Received: 10 April 2018 / Accepted: 23 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we deal with the problem of classifying the genera of quotient curves Hq /G, where Hq is the Fq 2 -maximal Hermitian curve and G is an automorphism group of Hq . The groups G considered in the literature fix either a point or a triangle in the plane PG(2, q 6 ). In this paper, we give a complete list of genera of quotients Hq /G, when G ≤ Aut(Hq ) ∼ = PGU(3, q) does not leave invariant any point or triangle in the plane. Also, the classification of subgroups G of PGU(3, q) satisfying this property is given up to isomorphism. Keywords Hermitian curve · Unitary groups · Quotient curves · Maximal curves Mathematics Subject Classification 11G20
1 Introduction Let Fq be a finite field of order q and X be a projective, irreducible, non-singular algebraic curve of genus g defined over Fq . The problems of determining the maximum number of points over Fq that X can have and finding examples of algebraic curves X with many rational points have been important, not only from the theoretic perspective, but also for applications in Coding Theory; see, for instance, [11,32,33]. The Hasse–
This research was partially supported by Ministry for Education, University and Research of Italy (MIUR) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).
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Giovanni Zini [email protected] Maria Montanucci [email protected]
1
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Asmussens Allé, Building 303B, 2800 Kongens Lyngby, Denmark
2
Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, Milano, Italy
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Journal of Algebraic Combinatorics
Weil theorem provides an upper bound for the number of rational points |X (Fq )| √ that a curve X defined over Fq can have, namely |X (Fq )| ≤ q + 1 + 2g q. If √ |X (Fq )| = q + 1 + 2g q, then the curve X is said to be Fq -maximal. Clearly, X can be Fq -maximal only if either g is zero or q is a square. A natural question in this context is: Over a finite field Fq 2 of square cardinality, which nonnegative integers g can be realized as the genera of maximal curves over Fq 2 ? A leading example of a maximal curve is the Hermitian curve Hq over Fq 2 , where q is a power of a prime p. It is defined as the non-singular plane curve admitting one of the following birational equivalent plane models: X q+1 + Y q+1 + Z q+1 = 0 and X q Z + X Z q = Y q+1 . For fixed q, the curve Hq has the largest genus g(Hq ) = q(q − 1)/2 that an Fq 2 -maximal curve can have; see [12] and references therein. The full automorphism group Aut(Hq ) is isomorphic to PGU(3, q), the group of projectivities of PG(2, q 2 ) commuting with the unitary polarity associated with Hq . The automorphism group Aut(Hq ) is extremely la
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