Constructing one-parameter families of elliptic curves with moderate rank

2004/06/28 by Scott Arms, Steven J. Miller, Álvaro Lozano‐Robledo +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #Combinatorics #Cryptography and Residue Arithmetic #Degree (music) #Discrete mathematics #Division polynomials #Elliptic curve #Hessian form of an elliptic curve #Mathematical analysis #Mathematics #Pure mathematics #Quarter period #Rank (graph theory) #Rational point #Schoof's algorithm #Supersingular elliptic curve #math.AG #math.NT #msc:11G05 #msc:11G20

paper · pdf · doi:10.1016/j.jnt.2006.07.002

published as Journal of Number Theory, Volume 123, Issue 2, April 2007, Pages 388--402 · 11 pages

arxiv created 2004/06/28 · openalex publication_date 2006/08/11 · arxiv updated 2010/11/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We give several new constructions for moderate rank elliptic curves over ℚ(T). In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over 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 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.

Citations

• One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries

Cited by

• Constructing families of moderate-rank elliptic curves over number fields