Hyperelliptic Curves with Extra Involutions

Gutiérrez, Joaquín M.; Shaska, Tony

doi:10.1112/s1461157000000917

Cited by 49 publications

(96 citation statements)

References 25 publications

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“…For any given positive integer h there are at most The proof is similar to the previous lemma. In Table 2 for weighted moduli height h ≤ 4 we display how many possible tuples are there in WP 3 (1,2,3,5) (Q) and then in the next column we display how many of this tuples give us isomorphism classes of genus two curves. And this results agree with Lemma 3.…”

Section: Rational Points In the Weighted Moduli Spacementioning

confidence: 99%

The weighted moduli space of binary sextics

Beshaj¹,

Guest²

2019

Algebraic Curves and Their Applications

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We use the weighted moduli height as defined in [8] to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equationwhere H(f ) is the minimal naive height of the curve as defined in [9]. Based on the weighted moduli height h we create a database of genus two curves defined over Q with small h and show that for small such height (h < 5) about 30% of points are fine moduli points.

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Section: Rational Points In the Weighted Moduli Spacementioning

confidence: 99%

The weighted moduli space of binary sextics

Beshaj¹,

Guest²

2019

Algebraic Curves and Their Applications

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“…In [12] we intend to study generalizations of such results to all hyperelliptic curves. The dihedral invariants used here are generalized to all hyperelliptic curves in [8].…”

Section: Further Directionsmentioning

confidence: 99%

“…It seems as everything should follow smoothly as in the case of genus 3, other than the fact that computations will be more difficult. The dihedral invariants of genus g ≥ 2 hyperelliptic curves are defined in [8]. A basis for the space of holomorphic differentials is known how to be constructed.…”

Section: Introductionmentioning

confidence: 99%

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

Shaska

Shor

2016

Communications in Algebra

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“…. , t 6 . For these problems and other computational aspects of genus 3 hyperelliptic curves see [21].…”

Section: 22mentioning

confidence: 99%

“…. , t 6 invariants and other problems on genus 3 hyperelliptic curves as described in [5,6,[13][14][15][16][17][18][20][21][22] among others. …”

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Some Remarks on the Hyperelliptic Moduli of Genus 3

Shaska

2014

Communications in Algebra

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Abstract. In 1967, Shioda [23] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in [23] are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field k, char k = 2, 3, 5, 7. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants. IntroductionLet k be an algebraically closed field. A binary form of degree d is a homogeneous polynomial f (X, Y ) of degree d in two variables over k. Let V d be the k-vector space of binary forms of degree d. The group GL 2 (k) of invertible 2 × 2 matrices over k acts on V d by coordinate change. Many problems in algebra involve properties of binary forms which are invariant under these coordinate changes. In particular, any hyperelliptic genus g curve over k has a projective equation of the form Z 2 Y 2g = f (X, Y ), where f is a binary form of degree d = 2g + 2 and non-zero discriminant. Two such curves are isomorphic if and only if the corresponding binary forms are conjugate under GL 2 (k). Therefore the moduli space H g of hyperelliptic genus g curves is the affine variety whose coordinate ring is the ring of GL 2 (k)-invariants in the coordinate ring of the set of elements of V d with non-zero discriminant. It is well known that the moduli spaces H g of hyperelliptic curves of genus g, g = 4, are all rational varieties, i.e. isomorphic to a purely transcendental extension field k(t 1 , . . . , t r ); see Igusa [9], Katsylo [10].Generators for this and similar invariant rings in lower degree were constructed by Clebsch, Bolza and others in the last century using complicated symbolic calculations. For the case of sextics, Igusa [9] extended this to algebraically closed fields of any characteristic using difficult techniques of algebraic geometry. For a modern treatment of the degree six case see [11].The case of binary octavics has been first studied during the 19th century by von Gall [24] and Alagna [1,2]. Shioda in his thesis [23] determined the structure of the ring of invariants R 8 , which turns out to be generated by nine SL(2, k)-invariants J 2 , · · · , J 10 satisfying five algebraic relations. He computed explicitly these five syzygies, and determined the corresponding syzygy-sequence and therefore the structure of the ring R 8 ; see Shioda [23].This paper started as a project to implement an algorithm which determines if two genus 3 hyperelliptic curves are isomorphic over C. According to Shioda [23, Thm. 5]; two genus 3 hyperelliptic curves are isomorphic if and only if the corresponding 9-tuples (J 2 , . . . , J 10 ) are equivalent, satisfying five syzygies R i (J 2 , . . . , J 10 ) = 0, for i = 1, . . . , 5 and non-zero discriminant ∆ = 0. While trying to implement the syzygies R i (J 2 , . . . , J 10 ) = 0, for i = 1, . . . , 5 w...

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Hyperelliptic Curves with Extra Involutions

Cited by 49 publications

References 25 publications

The weighted moduli space of binary sextics

The weighted moduli space of binary sextics

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

Some Remarks on the Hyperelliptic Moduli of Genus 3

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