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

Journal de théorie des nombres de Bordeaux (2002)

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

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topHuisman, 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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- [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] A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann.10 (1876), 189-198. MR1509883JFM08.0317.04
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- [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] J.W. Milnor, Topology from the differentiable viewpoint. Univ. Press, Virginia, 1965 Zbl0136.20402MR226651
- [9] J.H. Silverman, The arithmetic of elliptic curves. Grad. Texts in Math.106, Springer-Verlag, 1986. Zbl0585.14026MR817210

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