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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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How many rational points does a random curve have?
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by Wei Ho PDF
Bull. Amer. Math. Soc. 51 (2014), 27-52 Request permission


A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over ${\mathbb {Q}}$, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank $0$ and rank $1$, with higher ranks being negligible. We will describe these conjectures and discuss some results on bounds for average rank, highlighting recent work of Bhargava and Shankar.
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Additional Information
  • Wei Ho
  • Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
  • MR Author ID: 770878
  • Email:
  • Received by editor(s): May 23, 2013
  • Published electronically: September 30, 3013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 51 (2014), 27-52
  • MSC (2010): Primary 11G05, 14H52; Secondary 11G30, 14H25
  • DOI:
  • MathSciNet review: 3119821