Journal of the Ramanujan Mathematical Society

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A Positive Proportion of Elliptic Curves Over Q Have Rank One


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1 Department of Mathematics, Princeton University, Princeton, 08544, United States
     

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We prove that, when all elliptic curves over Q are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are both strictly positive.
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  • A Positive Proportion of Elliptic Curves Over Q Have Rank One

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Authors

Manjul Bhargava
Department of Mathematics, Princeton University, Princeton, 08544, United States
Christopher Skinner
Department of Mathematics, Princeton University, Princeton, 08544, United States

Abstract


We prove that, when all elliptic curves over Q are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are both strictly positive.