[Paper Review] On average sizes of Selmer groups and ranks in families of elliptic curves having marked points

Manjul Bhargava, Wei Ho|arXiv (Cornell University)|Jul 7, 2022

Analytic Number Theory Research4 citations

TL;DR

This paper establishes uniform upper bounds on the average sizes of 2- and 3-Selmer groups in families of elliptic curves with marked rational points, including those with marked points of order 2 or 3, and proves these bounds hold even when restricting to minimal curves or large subfamilies defined by congruence conditions. The key result confirms that the average rank of such curves is bounded, supporting the Poonen–Rains heuristics and extending earlier work on the average rank of elliptic curves.

ABSTRACT

We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2 imes 2 imes 2 imes 2$ and $3 imes 3 imes 3$ matrices that we studied in a previous paper. We also deduce that $100\%$ of genus one curves of the form $y^2 = Ax^4 + Bx^2 z^2 + Cz^4$ with $A, B, C \in \mathbb{Z}$, when ordered by $\max\{|B|^2,|AC|\}$, fail the Hasse principle. Other forthcoming applications include proofs that a positive proportion of integers are (respectively, are not) the sum of two rational cubes, and a positive proportion of genus one curves in $\mathbb{P}^1 imes \mathbb{P}^1$ over $\mathbb{Q}$ fail the Hasse principle.

Motivation & Objective

• To determine the average size of Selmer groups in families of elliptic curves with marked rational points, including those of order 2 or 3.
• To establish that these average sizes remain bounded even when restricting to minimal curves or large subfamilies defined by congruence conditions.
• To confirm that the average rank of such curves is bounded, supporting the Poonen–Rains heuristics for families with marked structures.
• To extend the understanding of average ranks and Selmer group sizes beyond the full family of elliptic curves to structured subfamilies with rational torsion or marked points.

Proposed method

• The authors define height functions for each family of elliptic curves, including $F_0$, $F_1$, $F_1(2)$, $F_1(3)$, and $F_2$, to order curves by size.
• They use lattice point counting in bounded regions of $\mathbb{R}^{n_j}$ to estimate the number of curves of bounded height in each family.
• They define 'large' subfamilies as those that contain all curves with $p^2 \nmid \Delta_{\text{red}}(E)$ for all but finitely many primes $p$, ensuring density preservation.
• They apply the Weil bound and modular reduction techniques to estimate the density of curves where marked points are dependent modulo $p$, particularly in $F_2$, to show independence is generic.
• They use the fact that the $p$-rank of the $p$-Selmer group bounds the algebraic rank to derive upper bounds on average rank.
• They apply the Chinese remainder theorem to combine local densities across large sets of primes to show that dependence of marked points is rare asymptotically.

Experimental results

Research questions

• RQ1 What is the average size of the 2-Selmer group in the family $F_0$ of all elliptic curves over $\mathbb{Q}$, ordered by height?
• RQ2 How do the average sizes of 2- and 3-Selmer groups change in families with marked rational points, such as $F_1$, $F_1(2)$, $F_1(3)$, and $F_2$?
• RQ3 Are the average Selmer group sizes preserved when restricting to minimal curves or large subfamilies defined by congruence conditions?
• RQ4 What is the limsup of the average rank in these families, and is it bounded?
• RQ5 Are the two marked points on a positive proportion of curves in $F_2$ independent in the Mordell-Weil group?

Key findings

• The average size of the 2-Selmer group in $F_0$ is exactly 3, and the average size of the 3-Selmer group is exactly 4.
• The average size of the 2-Selmer group in $F_1$ is at most 6, and the average size of the 3-Selmer group is exactly 12.
• The average size of the 3-Selmer group in $F_1(2)$ is exactly 4, and the average size of the 2-Selmer group in $F_1(3)$ is at most 3.
• The average size of the 2-Selmer group in $F_2$ is at most 12, confirming that the average rank in this family is bounded.
• The limsup of the average rank in $F_0$, $F_1$, $F_1(2)$, $F_1(3)$, and $F_2$ is at most $7/6$, $13/6$, $7/6$, $3/2$, and $7/2$, respectively.
• Asymptotically 100% of curves in each family have trivial rational torsion, and the marked points on $F_1$ and $F_2$ are independent with probability 1.