## Duke Mathematical Journal

### The Average Analytic Rank of Elliptic Curves

D. R. Heath-Brown

#### 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 $\mathbb{Q}$ 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

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

Dates
First available in Project Euclid: 22 April 2004

http://projecteuclid.org/euclid.dmj/1082665288

Digital Object Identifier
doi:10.1215/S0012-7094-04-12235-3

Mathematical Reviews number (MathSciNet)
MR2057019

Zentralblatt MATH identifier
02133077

#### Citation

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. http://projecteuclid.org/euclid.dmj/1082665288.

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