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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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How many rational points does a random curve have?
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by Wei Ho
Bull. Amer. Math. Soc. 51 (2014), 27-52
Published electronically: September 30, 2013

Previous version: Original version posted September 18, 2013
Corrected version: Current version corrects publisher's error in rendering author's corrections.


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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Bibliographic 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, 2013
  • © 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