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

Charles F. Schwartz

Journal de théorie des nombres de Bordeaux (1994)

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

Abstract

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Néron showed that an elliptic surface with rank 8 , and with base B = P 1 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, Y 2 = X 3 + A X 2 + B X + C (with deg ( A ) ≤ 2 , deg ( B ) ≤ 4 , and deg ( C ) ≤ 6 ) a basis ( X 1 , Y 1 ) , ⋯ , ( X 8 , Y 8 ) can be found with X i and Y i polynomial of degree ≤ 2 , ≤ 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 ≤ 2 , ≤ 3 , respectively.

How to cite

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Schwartz, 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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  5. [5] L.J. Mordell, Diophantine Equations, Academic Press, London (1969). Zbl0188.34503MR249355
  6. [6] A. Néron, Les propriétés du rang des courbes algibriques dans les corps de degré de transcendance fini, Centre National de la Recherche Scientifique, (1950), 65-69. Zbl0040.16001MR41477
  7. [7) A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proc. Int. Congress, Amsterdam, III (1954), 481-488. Zbl0074.15901MR87210
  8. [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. [9] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan24 (1972), 20-59. Zbl0226.14013MR429918
  10. [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. [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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