Constructing Elliptic Curves over $\mathbb{Q}(T)$ with Moderate Rank

Details Constructing Elliptic Curves over $\mathbb{Q}(T)$ with Moderate Rank Constructing Elliptic Curves over $\mathbb{Q}(T)$ with Moderate Rank

2004-06-28

arxiv.org/abs/math/0406579v1

Journal of Number Theory, Volume 123, Issue 2, April 2007, Pages 388--402

Mathematics

Number Theory

11 pages

Scientific paper

10.1016/j.jnt.2006.07.002

We give several new constructions for moderate rank elliptic curves over $\mathbb{Q}(T)$. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over $\mathbb{Q}$ using polynomials of degree two in $T$. While our method generates linearly independent points, we are able to show the rank is exactly 6 \emph{without} having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.

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