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

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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

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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

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  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
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