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arXiv · 1602.01894

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

Jennifer S. Balakrishnan·Wei Ho·Nathan Kaplan·Simon Spicer·William Stein·James Weigandt
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Abstract

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ ordered by height in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon observed in these databases is that the average rank eventually decreases as height increases.

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Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein, James Weigandt. 2016-02-05. Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks. https://doi.org/10.1112/s1461157016000152

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