Motivic & Arithmetic probability of a semistable elliptic surface with a Weierstrass torsion section

Abstract

We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by the bounded height of discriminant $\Delta(X)$. We consider the motivic probabilities over any basefield $K$ with $\text{char}(K) \neq 2,3$ of picking a nonsingular semistable elliptic surface over $\mathbb{P}^{1}$ with two marked Weierstrass sections at $\infty$ and $0$ such that marked Weierstrass section at $0$ is 2-torsion or 3-torsion. In the end, we formulate an analogous heuristics on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for the ratio of the semistable elliptic curves with a marked rational 2-torsion or 3-torsion Weierstrass point at $0$ out of all semistable elliptic curves with a marked rational Weierstrass points at $0$ over $\mathbb{Q}$ by the bounded height of discriminant $\Delta$ through the global fields analogy.