An elliptic surface of Mordell-Weil rank $8$ over the rational numbers

@article{Schwartz1994,
abstract = {Néron showed that an elliptic surface with rank $8$, and with base $B = P_1 \mathbb \{Q\}$, and geometric genus $=0$, may be obtained by blowing up $9$ points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the $9$ points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the $9$ points ; we observe that, relative to the Weierstrass form of the equation,\begin\{equation*\} Y^2 = X^3 + AX^2 + BX + C \end\{equation*\}(with $\deg (A) \le 2, \deg (B) \le 4$, and $\deg (C) \le 6)$ a basis $\left\lbrace (X_1,Y_1), \dots , (X_8, Y_8)\right\rbrace $ can be found with $X_i$ and $Y_i$ polynomial of degree $\le 2, \le 3$, respectively. One explicit example is computed, showing that for almost every elliptic surface given by a Weierstrass equation of the above form, a basis may be found with $X_i$ and $Y_i$ polynomial of degree $\le 2,\; \le 3$, respectively.},
author = {Schwartz, Charles F.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {elliptic surface with rank 8; parameterizations of the coefficients of the Weierstrass equations; Mordell-Weil groups},
language = {eng},
number = {1},
pages = {1-8},
publisher = {Université Bordeaux I},
title = {An elliptic surface of Mordell-Weil rank $8$ over the rational numbers},
url = {http://eudml.org/doc/247532},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Schwartz, Charles F.
TI - An elliptic surface of Mordell-Weil rank $8$ over the rational numbers
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 1
SP - 1
EP - 8
AB - Néron showed that an elliptic surface with rank $8$, and with base $B = P_1 \mathbb {Q}$, and geometric genus $=0$, may be obtained by blowing up $9$ points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the $9$ points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the $9$ points ; we observe that, relative to the Weierstrass form of the equation,\begin{equation*} Y^2 = X^3 + AX^2 + BX + C \end{equation*}(with $\deg (A) \le 2, \deg (B) \le 4$, and $\deg (C) \le 6)$ a basis $\left\lbrace (X_1,Y_1), \dots , (X_8, Y_8)\right\rbrace $ can be found with $X_i$ and $Y_i$ polynomial of degree $\le 2, \le 3$, respectively. One explicit example is computed, showing that for almost every elliptic surface given by a Weierstrass equation of the above form, a basis may be found with $X_i$ and $Y_i$ polynomial of degree $\le 2,\; \le 3$, respectively.
LA - eng
KW - elliptic surface with rank 8; parameterizations of the coefficients of the Weierstrass equations; Mordell-Weil groups
UR - http://eudml.org/doc/247532
ER -