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

Xarles, Xavier

doi:10.48550/arxiv.2007.07822

Cited by 5 publications

(13 citation statements)

References 5 publications

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“…Over F 2 , we found a unique curve of genus 4 and gonality 5 up to isomorphism. This agrees with the recent calculations of Xarles [27]. Over F 3 , we found 145 isomorphism classes of genus 4 curves with gonality 5.…”

Section: Now We Turn Tosupporting

confidence: 92%

Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points

Faber,

Grantham

2020

Preprint

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We extend the computations from our previous paper [5] to determine the maximum number of rational points on a curve over F 3 and F 4 with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus g has gonality at most g+3 2 . Second, a curve of gonality γ and large genus over F q has γ(q + 1) rational points.

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Section: Now We Turn Tosupporting

confidence: 92%

Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points

Faber,

Grantham

2020

Preprint

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“…• The positivity condition: the number of degree-i places on C must be nonnegative for all i ≥ 1. • Data from LMFDB (genus ≤ 3), [34] Proof. From Lemma 5.7, for g = 0, .…”

Section: Exhaustion Over Weil Polynomialsmentioning

confidence: 99%

“…To complete the proofs, we identify candidates for C with a given zeta function using data from LMFDB [19], which includes a table of genus-4 curves by Xarles [34], plus a similar table for g = 5 computed by Dragutinović [4]. We then make a computation of abelian extensions of function fields in Magma.…”

Section: Introductionmentioning

confidence: 99%

The relative class number one problem for function fields, I

Kedlaya¹

2022

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We classify finite extensions of function fields (of curves over finite fields) with the same class number, except when both curves have base field F 2 and genus > 1. In this case, we show that the zeta functions of the curves form one of 208 explicit pairs, and find 64 examples of cyclic extensions; the list is provably complete except if the base curve has genus 6 or 7 or if the extension is noncyclic. These cases will be resolved in subsequent papers.

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“…In [Xar20], Xarles gave the approach to compute all (smooth) curves of genus 4 over F 2 up to isomorphism. The given algorithm for determining the hyperelliptic curves over F 2 can be generalized to higher genera, and here, we will use it to obtain the set Hyp 5 (F 2 ).…”

Section: Hyperelliptic Curvesmentioning

confidence: 99%

“…In [Xar20], Xarles determined all curves of genus 4 defined over F 2 . His approach for computing the hyperelliptic curves is a universal one.…”

Section: Introductionmentioning

confidence: 99%

Computing binary curves of genus five

Dragutinović¹

2022

Preprint

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Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in P 4 . We present and explain algorithms we used to determine, up to isomorphism over F 2 , all genus 5 curves defined over F 2 , and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over F 2 of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over F 2 weighted by the size of their F 2 -automorphism groups.

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

Abstract: We explain how we computed equations for all genus 4 curves defined of the field F2 up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.

Cited by 5 publications

References 5 publications

Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points

Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points

The relative class number one problem for function fields, I

Computing binary curves of genus five

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