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

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Mathematics Elliptic curve Rank (graph theory) Algebraic number Supersingular elliptic curve Order (exchange) Conductor Pure mathematics Database Combinatorics Geometry Mathematical analysis Computer science Finance Economics

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

YOU? · · 2016 · Open Access · · DOI: https://doi.org/10.1112/s1461157016000152 · OA: W2255367803

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and 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 that we observe in our database is that the average rank eventually decreases as height increases.