Géométrie du groupe des points de Weierstrass d'une quartique lisse

Girard, Martine

doi:10.1006/jnth.2001.2732

Cited by 4 publications

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“…Suppose that this divisor is in the kernel of the isogeny from the Jacobian to the product of the three elliptic curves E 1 × E 2 × E 3 and compute its images on each elliptic factor. We obtain that n 1 = n 2 = n 4 = n 5 = n 6 = 0, 2m 3 If one of the m i is odd, then twice the divisor is also in the kernel of the isogeny. As before we show that none of the possible divisors with only even coefficients can occur as they would either correspond to a bitangent through two hyperflexes, or give rise to a degree 2 map to the projective line or translate into the existence of a conic tangent to the curve at P 1 or P 2 and two of the three points P 5 , P 7 , and P 8 , which is ruled out by the fifth relation of Proposition 5.2.…”

Section: 4mentioning

confidence: 84%

“…For completeness, we recall the previous results obtained in [3] for the generic quartic and for the two irreducible strata M 1 and M 2 : …”

Section: Elliptic Factors Of the Jacobianmentioning

confidence: 98%

“…Using specialization of curves in these families, we obtain bounds both on the rank and on the torsion part of the group generated by the Weierstrass points for a generic quartic in these strata. In particular, in [3], we showed that for a generic quartic, it is a free abelian group of rank at least 11. In in Section 6, we pursue the study carried out in [4], [3] and improve some of the bounds previously obtained.…”

mentioning

confidence: 99%

“…In particular, in [3], we showed that for a generic quartic, it is a free abelian group of rank at least 11. In in Section 6, we pursue the study carried out in [4], [3] and improve some of the bounds previously obtained.…”

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confidence: 99%

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The group of Weierstrass points of a plane quartic with at least eight hyperflexes

Girard

2006

Math. Comp.

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Abstract. The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space M 3 of curves of genus 3.The Weierstrass points of an algebraic curve, which can be defined over some extension of the base field, form a distinguished set of points of the curve having the property of being geometrically intrinsic. Curves of genus 0 or 1 have no Weierstrass points, and for hyperelliptic curves, the Weierstrass points can easily be characterized as the ramification points under the hyperelliptic involution, which generate the 2-torsion subgroup of the Jacobian. For nonhyperelliptic curves of genus 3, which admit a plane quartic model, the structure of the Weierstrass subgroup, i.e., the subgroup of the Jacobian generated by the images of the Weierstrass points under the Abel-Jacobi map, cannot be characterized so easily. This structure is known for curves with many automorphisms ([10], [12], [13]), in which cases the groups are finite. As with Weierstrass points, the Weierstrass subgroup is a geometric invariant of the curve, which we study in this paper. More precisely, we determine the structure of this group for all curves having either eight (Theorems 4.1 and 5.1) or nine hyperflexes (Theorem 3.1), i.e., points at which the tangent line to the curve meets the curve with multiplicity four. These are the first nontrivial cases of interest since there are no curves with either ten or eleven hyperflexes, and the two possible cases of curves with twelve hyperflexes have already been treated ([7], [13]). We show:

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Section: 4mentioning

confidence: 84%

“…For completeness, we recall the previous results obtained in [3] for the generic quartic and for the two irreducible strata M 1 and M 2 : …”

Section: Elliptic Factors Of the Jacobianmentioning

confidence: 98%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The group of Weierstrass points of a plane quartic with at least eight hyperflexes

Girard

2006

Math. Comp.

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“…For some particular curves with large automorphisms groups (for instance, Fermat curves [18]), these groups have been found to be torsion. The first author provided the first examples where this group has positive rank ( [7], [8]) and obtained a lower bound of 11 on the rank of the generic genus 3 curve. The motivation of this paper was to bridge the gap between this bound and the expected bound of 23 -meaning that there are no relations between the Weierstrass points on the generic genus 3 curve.…”

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The Weierstrass Subgroup of a Curve Has Maximal Rank

Girard¹,

Kohel²,

Ritzenthaler³

2006

Bull. London Math. Soc.

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We show that the Weierstrass points of the generic curve of genus g over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian.The Weierstrass points are a set of distinguished points on curves, which are geometrically intrinsic. In particular, the group these points generate in the Jacobian is a geometric invariant of the curve. A natural question is to determine the structure of this group. For some particular curves with large automorphisms groups (for instance, Fermat curves [18]), these groups have been found to be torsion. The first author provided the first examples where this group has positive rank ([7], [8]) and obtained a lower bound of 11 on the rank of the generic genus 3 curve. The motivation of this paper was to bridge the gap between this bound and the expected bound of 23 -meaning that there are no relations between the Weierstrass points on the generic genus 3 curve. The result we obtain is valid for generic curves of any genus. More precisely, let the Weierstrass subgroup of a curve C be the group generated by the Weierstrass points in the Jacobian of the curve C. We show that Theorem 1. The Weierstrass subgroup of the generic curve of genus g ≥ 3 is isomorphic to Z g(g 2 −1)−1 .As a consequence of this theorem, we deduce the following corollaries.Corollary 2. For any number field K, the group generated by the Weierstrass points of a curve over K in its Jacobian is isomorphic to Z g(g 2 −1)−1 , outside of a set of curves whose moduli lie in a thin set in M g (K).Corollary 3. For each g ≤ 13, the curves of genus g over Q for which the group generated by the Weierstrass points in its Jacobian is isomorphic to Z g(g 2 −1)−1 , determine a Zariski dense set of moduli in M g .

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An extraordinary origami curve

Herrlich

Schmithüsen

2008

Mathematische Nachrichten

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We study the origami W defined by the quaternion group of order 8 and its Teichmüller curve C(W ) in the moduli space M3. We prove that W has Veech group SL2(Z), determine the equation of the family over C(W ) and find several further properties. As main result we obtain infinitely many origami curves in M3 that intersect C(W ). We present a combinatorial description of these origamis.Origami curves are certain special Teichmüller curves in some moduli space of curves. They are obtained from an unramified covering of a once punctured torus, see Section 1.1 for a precise definition.First examples of Teichmüller curves were already given by Veech in [V]. In recent years they have attracted a lot of attention, partly because of their relation to rational billiards, see e.g. [McM] and references therein; a survey of examples defining primitive Teichmüller curves can be found in [HuSc]. Another topic of interest is the action of the absolute Galois group of the rationals on them as discussed in [Lo] and [M1].In some respects Teichmüller curves arising via origamis are more accessible than general ones. Precisely for them, the Veech group (as defined in Section 1.1) is a subgroup of SL 2 (Z), cf. [GJ]. It can be determined for each origami explicitly, cf. [S]. Nevertheless, it is still difficult to approach the question how their Teichmüller curves are located in the moduli space. There are only a few origami curves for which explicit equations have been found so far, cf. [H] and [M1].In this note we present an extraordinary origami curve in genus 3. It is the smallest nontrivial example of a normal origami having as Veech group the full group SL 2 (Z), see Prop. 2. The Jacobian of the associated family of curves has a two-dimensional fixed part, see Proposition 7. M. Möller has observed that this implies that our origami curve is also a Shimura curve; recently he proved that it is the only algebraic curve in a moduli space of curves of genus at least 2 which is at the same time a Teichmüller curve and a Shimura curve, cf.[M2]. An explicit equation for the associated family of curves is given in Proposition 5. This family of curves has been studied by several authors, and some of the results of Section 1 have been known previously, cf.[Gi], [Gu2], [KK]; nevertheless the use of the "origami" structure allows for new proofs and makes the exposition considerably more elementary.A very remarkable property of this origami curve, and the main new result of this paper, is the fact that it intersects infinitely many other origami curves, see Theorem 1 in Section 3. To our knowledge this is the first example of origami curves that intersect in moduli space.The decomposition of the Jacobian gives a second map onto an elliptic curve; if this map is ramified only over torsion points it can be made into an origami. In Section 2 we develop explicit formulas for these maps to determine when this particular type of ramification occurs.We give a combinatorial description of the infinitely many origamis that intersect our ori...

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Géométrie du groupe des points de Weierstrass d'une quartique lisse

Cited by 4 publications

References 9 publications

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The Weierstrass Subgroup of a Curve Has Maximal Rank

An extraordinary origami curve

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