Duke Mathematical Journal

The Average Analytic Rank of Elliptic Curves

D. R. Heath-Brown

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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 $\mathbb{Q}$ 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.''

Article information

Duke Math. J., Volume 122, Number 3 (2004), 591-623.

First available in Project Euclid: 22 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}


Heath-Brown, D. R. The Average Analytic Rank of Elliptic Curves. Duke Math. J. 122 (2004), no. 3, 591--623. doi:10.1215/S0012-7094-04-12235-3. https://projecteuclid.org/euclid.dmj/1082665288

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