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

Scott Arms·Steven J. Miller·Alvaro Lozano-Robledo
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Abstract

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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Scott Arms, Steven J. Miller, Alvaro Lozano-Robledo. 2004-06-28. Constructing Elliptic Curves over $\mathbb{Q}(T)$ with Moderate Rank. https://doi.org/10.1016/j.jnt.2006.07.002

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