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

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.

11 pages

References in corpus (3)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• Variation in the number of points on elliptic curves and applications to excess rank
• Primitive root producing quadratics

Cited by in corpus (3)

• Effective equidistribution and the Sato-Tate law for families of elliptic curves
• An identity for sums of polylogarithm functions
• Constructing families of moderate-rank elliptic curves over number fields