The average analytic rank of elliptic curves

By D. R. Heath-Brown. 2003

85 citations recorded by the source corpus.

All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over Q is at most 2, thereby improving a result of Brumer [2]. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld [3]. A third result concerns the density of curves with analytic rank at least R, and shows that the proportion of such curves decreases faster than exponentially as R grows. The proofs depend on an analogue of Weil’s “explicit formula”.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e779644d0c9534e87c16b51a723cac204f7bd926

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Low-lying zeros of families of elliptic curves (2004, methodology): “ his doctoral thesis [Mil], has independently proved density theorems for various families ofelliptic curves (alongwith otherthings), but more restricted by the support of φb. Brumer [B], Heath-Brown [H-B], Michel [Mic], Silverman [Si3], and others have proved results on the average rank of certain ”
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, methodology): “Still assuming the generalized Riemann hypothesis and the Birch–Swinnerton-Dyer conjectures, this constant was subsequently improved to 2 by Heath-Brown [26] and to 25/14 ∼ 1.”
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, methodology): “We thank Noam Elkies, Tom Fisher, Dorian Goldfeld, Dick Gross, Roger Heath-Brown, Harald Helfgott, Wei Ho, Henryk Iwaniec, Barry Mazur, Bjorn Poonen, Peter Sarnak, Michael Stoll, Alice Silverberg, Christopher Skinner, Jacob Tsimerman, Jerry Wang, and Kevin Wilson for helpful conversations.”