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.''
Citation
D. R. Heath-Brown. "The Average Analytic Rank of Elliptic Curves." Duke Math. J. 122 (3) 591 - 623, 15 April 2004. https://doi.org/10.1215/S0012-7094-04-12235-3
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