A group law on smooth real quartics having at least 3 real branches

Johan Huisman

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 1, page 249-256
  • ISSN: 1246-7405

Abstract

top
Let C be a smooth real quartic curve in ℙ 2 . Suppose that C has at least 3 real branches B 1 , B 2 , B 3 . Let B = B 1 × B 2 × B 3 and let O ∈ B . Let τ O be the map from B into the neutral component Jac ( C ) ( ℝ ) 0 of the set of real points of the jacobian of C , defined by letting τ O ( P ) be the divisor class of the divisor ∑ P i - O i . Then, τ O is a bijection. We show that this allows an explicit geometric description of the group law on Jac ( C ) ( ℝ ) 0 . It generalizes the classical geometric description of the group law on the neutral component of the set of real points of the jacobian of a cubic curve. If the quartic curve is defined over a real number field then one gets a geometric description of a subgroup of its Mordell-Weil group of index a divisor of 8 .

How to cite

top

Huisman, Johan. "A group law on smooth real quartics having at least $3$ real branches." Journal de théorie des nombres de Bordeaux 14.1 (2002): 249-256. <http://eudml.org/doc/248897>.

@article{Huisman2002,
abstract = {Let $C$ be a smooth real quartic curve in $\mathbb \{P\}^2$. Suppose that $C$ has at least $3$ real branches $B_1, B_2 , B_3$. Let $B = B_1 \times B_2 \times B_3$ and let $O \in B$. Let $\tau _O$ be the map from $B$ into the neutral component Jac$(C)(\mathbb \{R\})^0$ of the set of real points of the jacobian of $C$, defined by letting $\tau _O (P)$ be the divisor class of the divisor $\sum P_i - O_i$. Then, $\tau _O$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$(C)(\mathbb \{R\})^0$. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of the jacobian of a cubic curve. If the quartic curve is defined over a real number field then one gets a geometric description of a subgroup of its Mordell-Weil group of index a divisor of $8$.},
author = {Huisman, Johan},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {real algebraic curve; Jacobian; Mordell-Weil group; real hyperelliptic curves},
language = {eng},
number = {1},
pages = {249-256},
publisher = {Université Bordeaux I},
title = {A group law on smooth real quartics having at least $3$ real branches},
url = {http://eudml.org/doc/248897},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Huisman, Johan
TI - A group law on smooth real quartics having at least $3$ real branches
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 249
EP - 256
AB - Let $C$ be a smooth real quartic curve in $\mathbb {P}^2$. Suppose that $C$ has at least $3$ real branches $B_1, B_2 , B_3$. Let $B = B_1 \times B_2 \times B_3$ and let $O \in B$. Let $\tau _O$ be the map from $B$ into the neutral component Jac$(C)(\mathbb {R})^0$ of the set of real points of the jacobian of $C$, defined by letting $\tau _O (P)$ be the divisor class of the divisor $\sum P_i - O_i$. Then, $\tau _O$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$(C)(\mathbb {R})^0$. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of the jacobian of a cubic curve. If the quartic curve is defined over a real number field then one gets a geometric description of a subgroup of its Mordell-Weil group of index a divisor of $8$.
LA - eng
KW - real algebraic curve; Jacobian; Mordell-Weil group; real hyperelliptic curves
UR - http://eudml.org/doc/248897
ER -

References

top
  1. [1] C. Ciliberto, C. Pedrini, Real abelian varieties and real algebraic curves. Lectures in real geometry (Madrid, 1994), 167-256, de Gruyter Exp. Math.23, de Gruyter, Berlin, 1996. Zbl0895.14013MR1440212
  2. [2] G. Fichou, Loi de groupe sur la composante neutre de la jacobienne d'une courbe réelle de genre 2 ayant beaucoup de composantes connexes réelles. Manuscripta Math.104 (2001), 459-466. Zbl1033.11031MR1836107
  3. [3] G. Fichou, Loi de groupe sur la composante neutre de la jacobienne d'une courbe réelle hyperelliptique ayant beaucoup de composantes connexes réelles, (submitted). 
  4. [4] G. Fichou, J. Huisman, A geometric description of the neutral component of the Jacobian of a real plane curve having many pseudo-lines, (submitted). Zbl1033.14018
  5. [5] A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann.10 (1876), 189-198. MR1509883JFM08.0317.04
  6. [6] J. Huisman, Nonspecial divisors on real algebraic curves and embeddings into real projective spaces. Ann. Math. Pura Appl., to appear. Zbl1072.14072
  7. [7] J. Huisman, On the neutral component of the Jacobian of a real algebraic curve having many components. Indag. Math.12 (2001), 73-81. Zbl1014.14027MR1908140
  8. [8] J.W. Milnor, Topology from the differentiable viewpoint. Univ. Press, Virginia, 1965 Zbl0136.20402MR226651
  9. [9] J.H. Silverman, The arithmetic of elliptic curves. Grad. Texts in Math.106, Springer-Verlag, 1986. Zbl0585.14026MR817210

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.