Height-ordered distribution of dual 3-isogeny Selmer groups

Establish that, for each fixed integer t and the family of elliptic curves E over Q with a rational point of order 3 and global Selmer ratio c(\phi,E)=3^{t-2}, the limiting proportion of curves of height at most X whose dual isogeny Selmer group \Sel_{\widehat\phi}(E') has dimension d+1 equals the probability \mathbb{P}_t(d) that a random (m+t)\times m matrix over \mathbb{F}_3 has nullity d, for every integer d\geq\max(0,-t).

Background

The paper studies elliptic curves over \mathbb{Q} admitting a rational point of order 3, represented by normalized models y^2+Axy+By=x^3. The point (0,0) induces a 3-isogeny \phi:E\to E'. The authors control for the global Selmer ratio c(\phi,E)=3^{t-2}, because variation in this ratio can itself force large Selmer groups and obscure any random-matrix behavior.

For curves with good reduction at 3 and without additional 3-power isogenies, an explicit cubic-residue matrix M'{A,B} has dimensions (m+t)\times m, where m=\omega(B)-1, and its nullity equals \dim{\mathbb{F}3}\Sel{\widehat\phi}(E')-1. The conjecture asserts that, after ordering curves by height and fixing the Selmer-ratio parameter t, the resulting limiting nullity distribution agrees with the limiting nullity distribution of random matrices over \mathbb{F}_3. The paper does not prove this height-ordered assertion; it only provides computational evidence and explains that a sufficiently uniform form of the componentwise conjecture would imply it on the matrix-defined subfamily.