Families of elliptic curves ordered by conductor
Ananth N. Shankar, Arul Shankar, Xiaoheng Wang

TL;DR
This paper investigates specific families of elliptic curves ordered by conductor, providing asymptotic counts and analyzing the average size of their 2-Selmer groups, which relates to their rank distribution.
Contribution
It determines asymptotics for two special families of elliptic curves ordered by conductor and computes the average size of their 2-Selmer groups, revealing insights into their rank distribution.
Findings
Asymptotic formulas for the families ordered by conductor
Average size of 2-Selmer groups is 3 for the first family
Average rank of these elliptic curves is bounded by 1.5
Abstract
In this article, we study the family of elliptic curves , having good reduction at and , and whose -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves such that the ratio is squarefree; and second, the set of elliptic curves such that is bounded by a small power of . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is . This implies that the average rank of these elliptic curves is finite, and bounded by…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Limits and Structures in Graph Theory
