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High rank elliptic surfaces

Meijer, M. (1999) High rank elliptic surfaces. Master's Thesis / Essay, Mathematics.

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Abstract

Let E/Q(t) be an elliptic curve given by the equation Y2 = X3 + g(t6) where g is a polynomial. In the case that g is a quadratic polynomial with simple zeroes ≠ 0, we show that rankE(Q(t)) is either 16 or 18. Moreover we prove that rank 18 really occurs and we give 18 independent points in certain explicit examples. For the subgroup E(Q(t)) we give the theoretical upper bound 9. However, using quadratic polynomials g, we never found more than 4 independent points in E(Q(t)). Finally we study the case where deg(g) = 3. Ve show that rankE(Q(t)) ≥ 24 occurs, by providing an example where one can write down 24 independent points. We finish by studying a case where the Mordell-Weil rank over Q(t) is at least 5.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 15 Feb 2018 07:28
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/8622

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