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Points de Weierstrass et jacobienne de courbes algebriques de genre 3

Abstract : The domain of this thesis is the geometry of algebraic curves and of their Jacobians (in zero characteristic). More precisely, the object of this thesis is the study of the group generated by the Weierstrass points in the Jacobian for some smooth plane curves of genus three. We determine this group for certain families of curves of genus three. We proceed in two steps. We use first the geometry of the curve and of its Jacobian to reduce the number of generators. We will see that it is not possible to reduce this number more. In order to prove this, we use several techniques: in the second part, we carry out an explicit descent using an isogeny; in the third part, we use reduction modulo a prime of good reduction. When we are dealing with families, this kind of geometric restrictions is valid for the whole family. On the other hand, the techniques we use in the second step only yield the result for a particular curve. In each case, we use a specialisation theorem to conclude. Moreover, we compute this group for the only plane quartic, apart for Fermat's quartic, possessing the minimal number of Weierstrass points, that is twelve; also in this case, the geometry of the Jacobian is useful to compute this group. These computations enable us to give an estimation on the rank of this group and on the torsion part for a generic quartic, depending on the number of hyperflexes (that is points where the tangent meets the curve with multiplicity four).
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Contributor : Martine Girard <>
Submitted on : Tuesday, February 26, 2002 - 12:24:28 PM
Last modification on : Friday, March 27, 2020 - 2:59:26 AM
Long-term archiving on: : Friday, April 2, 2010 - 7:52:00 PM


  • HAL Id : tel-00001137, version 1



Martine Girard. Points de Weierstrass et jacobienne de courbes algebriques de genre 3. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2000. Français. ⟨tel-00001137⟩



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