The Average Analytic Rank of Elliptic Curves

2004/04/15 by D. R. Heath‐Brown · 7 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.1215/s0012-7094-04-12235-3

openalex publication_date 2004/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

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.''

Citations

• The average rank of elliptic curves I
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• On the low-lying zeros of Hasse–Weil L-functions for elliptic curves
• ON THE NON-VANISHING OF ELLIPTIC CURVE L-FUNCTIONS AT THE CENTRAL POINT
• Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
• One level density of low-lying zeros of quadratic Hecke L-functions of Imaginary Quadratic Number Fields
• Birch and Swinnerton-Dyer conjecture [wikipedia]
• Elliptic curve [wikipedia]
• Rank of an elliptic curve [wikipedia]