How many rational points does a random curve have?

Ho, Wei

doi:10.1090/S0273-0979-2013-01433-2

How many rational points does a random curve have?
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by Wei Ho

Bull. Amer. Math. Soc. 51 (2014), 27-52

DOI: https://doi.org/10.1090/S0273-0979-2013-01433-2

Published electronically: September 30, 2013

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Previous version: Original version posted September 18, 2013
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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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