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 -Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over ordered by height. We describe databases of elliptic curves over ordered by height in which we compute ranks and -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.
Databases of Elliptic Curves Ordered by Height]Databases of Elliptic Curves Ordered by Height and Distributions of Selmer Groups and Ranks Balakrishnan, Ho, Kaplan, Spicer, Stein, Weigandt]Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein and James Weigandt \@definecounterconj\@definecounterdefn\@definecounterhypothesis\@definecounterremark\@definecounternote\@definecounterobservation\@definecounterproblem\@definecounterquestion\@definecounteralgorithm\@definecounterexample
1 Introduction and Statement of Main Results
Over the past several decades, tables of elliptic curves defined over have been very useful in number-theoretic research. A natural ordering on elliptic curves is given by their conductor. Some of the earliest tables were those in Antwerp IV [Antwerp], which include all elliptic curves of conductor at most . In [Cremona-book], Cremona describes algorithms to list all elliptic curves of given conductor and collect arithmetic data for these curves; these algorithms have now produced an exhaustive list of curves of conductor at most