Heuristic for counting multiply marked stable genus-one curves

For $2\leq m\leq5$, let $\mathcal{Z}_{1,m,\mathbb{Q}}(\mathcal{B})$ count $m$-marked $(m-1)$-stable genus-one curves over $\mathbb{Z}$ ordered by $0<\operatorname{ht}(\Delta_1)\leq\mathcal{B}$. Heuristic for multiply marked genus-one curves. The expected asymptotic-tightness estimates are

$$\begin{aligned} \mathcal{Z}_{1,2,\mathbb{Q}}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac34}+b\mathcal{B}^{\frac12}+c,\\ \mathcal{Z}_{1,3,\mathbb{Q}}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac23}+b\mathcal{B}^{\frac13}+c,\\ \mathcal{Z}_{1,4,\mathbb{Q}}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac{7}{12}}+b\mathcal{B}^{\frac13}+c,\\ \mathcal{Z}_{1,5,\mathbb{Q}}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac12}+b. \end{aligned}$$

These are heuristics derived from the global-fields analogy and the appendix’s finite-field counting estimates; the source does not establish or refute them.