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

Journal de théorie des nombres de Bordeaux (1994)

- Volume: 6, Issue: 1, page 1-8
- ISSN: 1246-7405

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topSchwartz, Charles F.. "An elliptic surface of Mordell-Weil rank $8$ over the rational numbers." Journal de théorie des nombres de Bordeaux 6.1 (1994): 1-8. <http://eudml.org/doc/247532>.

@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 -

## References

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- [7) A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proc. Int. Congress, Amsterdam, III (1954), 481-488. Zbl0074.15901MR87210
- [8] C.F. Schwartz, A Mordell-Weil group of rank 8, and a subgmup of finite index, Nagoya Math. J.93 (1984), 19-26. Zbl0504.14031MR738915
- [9] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan24 (1972), 20-59. Zbl0226.14013MR429918
- [10] T. Shioda, An infinite family of elliptic curves over Q with large rank via Néron's method, Invent. Math.106 (1991), 109-119. Zbl0766.14024MR1123376
- [11] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli40 (1991), 83-99. Zbl0757.14011MR1104782

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