Hyperelliptic curves of genus three over finite fields of even characteristic

Nart, Enric; Sadornil, Daniel

doi:10.1016/j.ffa.2003.08.002

Cited by 15 publications

(21 citation statements)

References 7 publications

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“…We recall some results on hyperelliptic curves over characteristic 2 fields. For another but equivalent approach to classify such hyperelliptic curves see [6].…”

Section: How We Computed All Isomorphism Classes: the Hyperelliptic Casementioning

confidence: 99%

A census of all genus 4 curves over the field with 2 elements

Xarles

2020

Preprint

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“…We recall some results on hyperelliptic curves over characteristic 2 fields. For another but equivalent approach to classify such hyperelliptic curves see [6].…”

Section: How We Computed All Isomorphism Classes: the Hyperelliptic Casementioning

confidence: 99%

A census of all genus 4 curves over the field with 2 elements

Xarles

2020

Preprint

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“…Remark 5.6. In [15], for k = F 2 a , the authors calculate the number of kisomorphism classes of hyperelliptic curves of genus 3 with a given Newton polygon.…”

Section: Hyperelliptic Curves When P =mentioning

confidence: 99%

THE p-TORSION OF CURVES WITH LARGE p-RANK

Pries

2009

Int. J. Number Theory

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Communicated by xxxConsider the moduli space of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g − 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g − 2 and f = g − 3. Likewise, we show that a generic hyperelliptic curve with p-rank g − 2 has a-number 1 when p ≥ 3. We also show that the locus of curves with p-rank g − 2 and a-number 2 is non-empty with codimension 3 in Mg when p ≥ 5. We include some other results when f = g − 3. The proofs are by induction on g while fixing g − f . They use computations about certain components of the boundary of Mg .

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“…An earlier nonexistence result of this type is due to Ekedahl [1987], who proved that a curve X of genus g > p( p−1)/2 in characteristic p > 0 cannot be superspecial and thus a X < g. There are also other recent results about Newton polygons of hyperelliptic (that is, Artin-Schreier) curves in characteristic 2, including several nonexistence results [Blache 2012;Scholten and Zhu 2002]. In addition, there are closed formulas for the number of hyperelliptic curves of genus 3 with given 2-rank over each finite field of characteristic 2 [Nart and Sadornil 2004].…”

Section: Introductionmentioning

confidence: 98%

Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

Elkin

Pries

2013

Algebra Number Theory

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Suppose X is a hyperelliptic curve of genus g defined over an algebraically closed field k of characteristic p = 2. We prove that the de Rham cohomology of X decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the 2-torsion group scheme J X [2] of the Jacobian of X in terms of the Ekedahl-Oort type. The interesting feature is that J X [2] depends only on some discrete invariants of X , namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes that occur as the 2-torsion group schemes of Jacobians of hyperelliptic k-curves of arbitrary genus, showing that only relatively few of the possible group schemes actually do occur. Theorem 1.3. Let X be a hyperelliptic k-curve with affine equation y 2 − y = f (x) defined over an algebraically closed field of characteristic 2 as described in Notation 1.1. Then the 2-torsion group scheme of the Jacobian variety of X isand the a-number of X is a X = (g + 1 − #{α ∈ B | d α ≡ 1 mod 4})/2.

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Hyperelliptic curves of genus three over finite fields of even characteristic

Cited by 15 publications

References 7 publications

A census of all genus 4 curves over the field with 2 elements

A census of all genus 4 curves over the field with 2 elements

THE p-TORSION OF CURVES WITH LARGE p-RANK

Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

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