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Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. (English) Zbl 1276.14088
This paper contains a thorough treatment of geometric, arithmetic and algorithmic aspects of hyperelliptic curves of small genus with special emphasis on genus \(3\) curves.
Shioda’s computation of the graded ring \(\mathcal{I}_8\) of invariants of binary octics is reviewed and shown to be valid for any algebraically closed field of positive characteristic \(p>7\). The ring is generated by \(9\) fundamental invariants \(J_2,\dots,J_{10}\) which satisfy \(5\) relations. This yields a representation of the coarse moduli space of genus \(3\) hyperelliptic curves as a projective variety defined by the five Shioda relations on a weighted projective space of dimension \(9\) whose points are of the form \((J_2: J_3:\dots : J_{10})\) and the coordinates have weights \(2,3,\dots,10\). This description of the moduli space facilitates the enumeration of rational points in the strata determined by the automorphism group, over a finite field. The paper contains an exhaustive description of these strata and their characterization in terms of equations defined by the invariants, and it corrects several mistakes that occur in previous monographs on this topic.
The construction of curves with prefixed values of the fundamental invariants is also tackled. The authors are inspired by the general method of J.-F. Mestre [Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 313–334 (1991; Zbl 0752.14027)], based on the computation of the eight points of intersection of a non-singular conic \(\mathcal{Q}\) with a degree \(4\) curve \(\mathcal{H}\) whose coefficients are invariants. The construction depends on the stratum of the moduli space to which the point \((J_2: J_3:\dots : J_{10})\) belongs. Algorithms for the computation of the coefficients of \(\mathcal{H}\) as polynomials in the Shioda invariants are developed and specific tricks are used in the cases where the conic \(\mathcal{Q}\) is singular.
Finally, several arithmetic questions are addressed, again by working specifically on the different strata of the moduli space. Among others, the following problems are solved for each stratum: Is the field of moduli automatically a field of definition? Can the curve be always hyperelliptically defined over the field of moduli? Can one construct a model over the field of moduli when there is no obstruction?
A Magma code containing the various algorithms to check the computational assertions, calculate invariants and construct curves with prescribed invariants is available on the web page of the authors.

MSC:
14Q05 Computational aspects of algebraic curves
13A50 Actions of groups on commutative rings; invariant theory
14H10 Families, moduli of curves (algebraic)
14H37 Automorphisms of curves
Software:
Magma
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