Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point

Source abstract

Let $E_{A,B}\colon y^2 +Axy + By = x^3$ be an elliptic curve over $\mathbb{Q}$. The $3$-torsion point $(0,0)$ induces a $3$-isogeny $φ\colon E_{A,B}\to E_{A,B}'$. Assuming $E_{A,B}$ has good reduction at $3$, we construct an explicit $(m+t)\times m$ matrix $M_{A,B}'$ over $\mathbb{F}_3$, whose kernel encodes the dual isogeny Selmer group $\operatorname{Sel}_{\widehatφ}(E_{A,B}')$ modulo the image of the torsion point $(0,0)$. Here, $m = ω(B) - 1$, and $t$ encodes the \emph{global Selmer ratio} $3^{t-2}$. We compute $M_{A,B}'$ in various regimes for billions of elliptic curves $E_{A,B}$. Based on our data, and motivated by prevalence of random linear algebraic models throughout number theory, we conjecture that, for fixed $m$ and $t$, the matrices $M_{A,B}'$ become uniformly distributed, and we formulate a corresponding conjecture for the distribution of $\operatorname{Sel}_φ(E_{A,B})$. Our model predicts that, for fixed $t$, the average size of $\operatorname{Sel}_φ(E_{A,B})$ is $1 + 3^t$.