The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1

By Manjul Bhargava, A. Shankar. 2013

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In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new lower bound on the equidistribution of root numbers of elliptic curves; we prove that there is a family of elliptic curves over $\mathbb{Q}$ having density at least $55\%$ for which the root number is equidistributed.

Subfields: Algebra and Number Theory, Geometry and Topology, History and Philosophy of Science

View Paper: https://www.semanticscholar.org/paper/e4c9a6fd50d8a991b76b544f6a8c412554e128c4

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In Algebra and Number Theory

  1. Manjul Bhargava
  2. Melanie Matchett Wood
  3. B. Gross
  4. A. Shankar
  5. Kiran S. Kedlaya
  6. Bjorn Poonen
  7. Jordan S. Ellenberg
  8. L. Pierce

In Geometry and Topology

  1. Manjul Bhargava
  2. Melanie Matchett Wood
  3. Bjorn Poonen
  4. B. Gross
  5. A. Shankar
  6. Xiaoheng Wang
  7. Jordan S. Ellenberg
  8. Alexei Oblomkov

In History and Philosophy of Science

  1. Manjul Bhargava

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