Average Ranks of Elliptic Curves: Tension between Data and Conjecture

By B. Bektemirov, Barry Mazur, William A. Stein, M. Watkins. 2007

47 citations recorded by the source corpus.

Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control—thanks to ideas of Kolyvagin [Kol88]—the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today. We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures; and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

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

View Paper: https://www.semanticscholar.org/paper/f7f26b9aa5e63063c82062d166886d5b44c03c2f

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Citations in the corpus

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, background): “Computations have also not been very helpful in this regard; see [1] for a nice survey.”
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, background): “The rank distribution conjecture, due to Goldfeld [30] and Katz–Sarnak [34] (see also [3] for a beautiful survey, where it is termed the “minimalist conjecture”), states that the average rank of elliptic curves should be 1/2, with 50% of curves having rank 0 and 50% having rank 1.”