Néron showed that an elliptic surface with rank , and with base , and geometric genus , may be obtained by blowing up 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 points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the points ; we observe that, relative to the Weierstrass form of the equation, (with , and a basis can be found with and polynomial of degree , 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 and polynomial of degree , respectively.
@article{JTNB_1994__6_1_1_0, author = {Charles F. Schwartz}, title = {An elliptic surface of {Mordell-Weil} rank $8$ over the rational numbers}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {1--8}, publisher = {Universit\'e Bordeaux I}, volume = {6}, number = {1}, year = {1994}, doi = {10.5802/jtnb.102}, mrnumber = {1305284}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.102/} }
TY - JOUR TI - An elliptic surface of Mordell-Weil rank $8$ over the rational numbers JO - Journal de Théorie des Nombres de Bordeaux PY - 1994 DA - 1994/// SP - 1 EP - 8 VL - 6 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.102/ UR - https://www.ams.org/mathscinet-getitem?mr=1305284 UR - https://doi.org/10.5802/jtnb.102 DO - 10.5802/jtnb.102 LA - en ID - JTNB_1994__6_1_1_0 ER -
Charles F. Schwartz. An elliptic surface of Mordell-Weil rank $8$ over the rational numbers. Journal de Théorie des Nombres de Bordeaux, Volume 6 (1994) no. 1, pp. 1-8. doi : 10.5802/jtnb.102. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.102/
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