Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

Lercier, Reynald; Ritzenthaler, Christophe; Rovetta, Florent Ulpat; Sijsling, Jeroen

doi:10.1112/s146115701400031x

Cited by 21 publications

(40 citation statements)

References 27 publications

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“…We study the existence of representative families over k for the different strata, i.e families of curves C → S whose points are in natural bijection with the subvarieties. R. Lercier, et al in [10] explicitly constructed such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. We also refer to the second author's thesis [11,Ch.…”

Section: 2mentioning

confidence: 99%

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A note on the stratification by automorphisms of smooth plane curves of genus 6

Badr¹,

García²

2020

Colloq. Math.

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In this note, we give a so-called representative classification for the strata by automorphism group of smooth k-plane curves of genus 6, where k is a fixed separable closure of a field k of characteristic p = 0 or p > 13. We start with a classification already obtained in [3] and we mimic the techniques in [10] and [11].Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non 0-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it.We explicitly describe representative families for all strata, except for the stratum with automorphism group Z/5Z. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of Lüroth's theorem in dimension 2.

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Section: 2mentioning

confidence: 99%

“…(Cases with G ≇ Z/5Z) Clearly, the zero dimensional strata families are representative over k, since each represents a single point in the (coarse) moduli space M 6 . For the rest of cases, except for the case with G ≃ Z/5Z, we will use the same techniques used in [10] and [11]. We give a detailed example with the case G ≃ D 10 .…”

Section: 2mentioning

confidence: 99%

A note on the stratification by automorphisms of smooth plane curves of genus 6

Badr¹,

García²

2020

Colloq. Math.

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“…In other words, the problem is that given a plane quartic curve C with non-trivial automorphism group and defined over a number field k, its representative in Henn classification is not necessarily defined also over k. In [11] and [12], variations of Henn classification are given in order to get families with this property. In [12,Section 2], it is explained how to compute, given a non-hyperelliptic curve of genus 3, a representative in the modified Henn classification.…”

Section: Henn Classification Of Plane Quartic Curves With Non-trivialmentioning

confidence: 99%

“…[11]). The following families make Henn classification a representative classification of plane quartic curves in the sense that it represents each geometric point of the moduli space over non algebraically closed fields.…”

Section: Henn Classificationmentioning

confidence: 99%

Twists of non-hyperelliptic curves of genus 3

García

2018

Int. J. Number Theory

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“…Together with six additional invariants J 9 , J 12 , J 15 , J 18 , I 21 , J 21 studied by Ohno [28] they generate the full ring of invariants of ternary quartic forms, as conjectured by Shioda in [30,Appendix] and proved by Ohno in an unpublished preprint [28], and later verified by Elsenhans in the published paper [11]. These 13 invariants are collectively known as the Dixmier-Ohno invariants and have been studied by many authors [11,15,24,25]. Algorithms to compute the Dixmier-Ohno invariants of a given ternary quartic are described in [11,15,25], and Magma [4] implementations of these algorithms are available [11,15,31].…”

Section: Introductionmentioning

confidence: 96%

A database of nonhyperelliptic genus-3 curves over ℚ

Sutherland¹

2019

Open Book Series

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We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant. of this database: an enumeration of all smooth plane quartic curves with coefficients of absolute value at most B c := 9, with the aim of obtaining a set of unique Q-isomorphism class representatives for all such curves that have absolute discriminant at most B ∆ := 10 7 .Even after accounting for obvious symmetries, this involves more than 10 17.5 possible curve equations and requires a massively distributed computation to complete in a reasonable amount of time. Efficiently computing the discriminants of these equations is a non-trivial task, much more so than in the hyperelliptic case, and much of this article is devoted to an explanation of how this was done. Many of the techniques that we use can be generalized to other enumeration problems and may be of independent interest, both from an algorithmic perspective, and as an example of how cloud computing can be effectively applied to a research problem in number theory. A list of the curves that were found (more than 80 thousand) is available on the author's website [33].Remark 1.1. The informed reader will know that not every genus 3 curve over Q falls into the category of smooth plane quartics f (x, y, z) = 0 or curves with a hyperelliptic model y 2 + h(x) y = f (x). The other possibility is a degree-2 cover of a pointless conic; see [18] for a discussion of such curves and algorithms to efficiently compute their L-functions. We plan to conduct a separate search for curves of this form that will also become part of the genus 3 database in the LMFDB.1.1. Acknowledgments. The author is grateful to Nils Bruin, Armand Brumer, John Cremona, Tim Dokchitser, Jeroen Sijsling, Michael Stoll, and John Voight for their insight and helpful comments, and to the anonymous referees for their careful reading and suggestions for improvement. THE DISCRIMINANT OF A SMOOTH PLANE CURVELet C[x] d denote the space of ternary forms of degree d ≥ 1, as homogeneous polynomials in the variables x := (x 0 , x 1 , x 2 ). It is a C-vector space of dimension n d := d+2 2 equipped with a standard monomial basis

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Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

Cited by 21 publications

References 27 publications

A note on the stratification by automorphisms of smooth plane curves of genus 6

A note on the stratification by automorphisms of smooth plane curves of genus 6

Twists of non-hyperelliptic curves of genus 3

A database of nonhyperelliptic genus-3 curves over ℚ

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