The average analytic rank of elliptic curves

Journal article

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. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld. 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".

Authors: Heath-Brown, D

• Publication status: Published
• Journal: DUKE MATHEMATICAL JOURNAL
• Volume: 122
• Issue: 3
• Pages: 591-623
• Publication date: 2004-04-15
• DOI: 10.1215/S0012-7094-04-12235-3
• ISSN: 0012-7094
• Language: English