Ranks of elliptic curves

By Karl Rubin, Alice Silverberg. 2002

89 citations recorded by the source corpus.

This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and SwinnertonDyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.

Subfields: Algebra and Number Theory, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/de1a21f7fc3f64f08a819a926152620bf9c7ab78

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In Algebra and Number Theory

In Geometry and Topology

In Information Systems

Citations in the corpus

On the rank of elliptic curves (2008, methodology): “Information of the Birch and Swinnerton-Dyes conjecture can be found in [1].”
The average analytic rank of elliptic curves (2003, background): “The article by Rubin and Silverberg [8] gives an excellent survey of this topic.”
Quadratic families of elliptic curves and unirationality of degree 1 conic bundles (2014, background): “t ∈ K. Our aim is to show that there are many values t ∈ K for which the corresponding elliptic curve Et has rank ≥ 1. Conjecturally this should hold for a positive proportion of them; see the survey [RS02]. We prove rank ≥ 1 for about the square root of all t ∈ K, listed by height. We are only inte”
Three lectures on elliptic surfaces and curves of high rank (2007, unknown): “At this point we have a guess for a (usually very singular) model for the moduli curve, and if we can’t or won’t find a smooth model directly then we can ask Magma (or someday Sage) for it.”
Average Ranks of Elliptic Curves: Tension between Data and Conjecture (2007, background): “Similar surveys to this are those of Rubin and Silverberg [ RS02 ] and Kowalski [KowA], the latter of which also relates random matrix theory to the theory of elliptic curves and then discusses questions related to the inability of reconciling experimental data with theoretical asymptotics, particul”