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How many rational points does a random curve have?


Author: Wei Ho
Journal: Bull. Amer. Math. Soc. 51 (2014), 27-52
MSC (2010): Primary 11G05, 14H52; Secondary 11G30, 14H25
DOI: https://doi.org/10.1090/S0273-0979-2013-01433-2
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.
MathSciNet review: 3119821
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Abstract: 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
Email: who@math.columbia.edu

DOI: https://doi.org/10.1090/S0273-0979-2013-01433-2
Received by editor(s): May 23, 2013
Published electronically: September 30, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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