TY - JOUR
T1 - Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
AU - Balakrishnan, Jennifer S.
AU - Ho, Wei
AU - Kaplan, Nathan
AU - Spicer, Simon
AU - Stein, William
AU - Weigandt, James
N1 - Publisher Copyright:
© The Author(s) 2016.
PY - 2016
Y1 - 2016
N2 - 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 Q, ordered by height. We describe databases of elliptic curves over 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.
AB - 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 Q, ordered by height. We describe databases of elliptic curves over 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.
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U2 - 10.1112/S1461157016000152
DO - 10.1112/S1461157016000152
M3 - Article
AN - SCOPUS:84984555015
SN - 1461-1570
VL - 19
SP - 351
EP - 370
JO - LMS Journal of Computation and Mathematics
JF - LMS Journal of Computation and Mathematics
IS - A
ER -