The average rank of elliptic curves

By D. R. Heath-Brown. 2004

122 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/5e1bc285922b11a80966dd704ca9125e9240c6e8

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Citations in the corpus

A note on S(t) and the zeros of the Riemann zeta‐function (2005, background): “It is interesting that Brumer [1] obtained the same constant 1/2 for a similar bound for the rank of an elliptic curve in terms of the conductor.”
The average analytic rank of elliptic curves (2003, background): “Our first result concerns the average over all such curves, and sharpens an estimate of Brumer [2].”
ALMOST ALL ELLIPTIC CURVES ARE SERRE CURVES (2006, unknown): “Since (c.f. [1]), this implies (2).”
One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries (2003, result): “erate support. Converting to our language, for small support the 1- level densities for many families have been shown equal to the Katz-Sarnak predictions: all elliptic curves (Brumer and Heath-Brown [Br], [BHB5], support less than 2 3 ); twists of a given curve (support less than 1); oneparameter f”
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, methodology): “In [9], Brumer showed that the generalized Riemann hypothesis and the Birch–Swinnerton-Dyer conjectures together imply that the average rank of all elliptic curves, when ordered by their heights, is finite and is in fact bounded”
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, background): “(Conditional on GRH and BSD, a finite upper bound of 2.3 on the average rank was demonstrated by Brumer [13]; improved conditional upper bounds of 2.0 and 1.79 were given by Heath-Brown [32] and more recently by Young [42], respectively.)”