Arithmetic Moduli of Elliptic Surfaces

By considering arithmetic algebraic geometry of orbifold curves on modular curve, we formulate the moduli stack of minimal elliptic surfaces with section over any base field $K$ with $\mathrm{char}(K) \neq 2,3$. Inspired by the classical work of [Tate] which allows us to determine the Kodaira-Néron types of fibers over global fields, we establish Tate's correspondence between the moduli stacks $\mathrm{Rat}_{n}^{\gamma}(C_g, \overline{\mathcal{M}}_{1,1})$ of rational maps with vanishing constraints $\gamma$ and $\mathrm{Hom}^{\Gamma}_n(\mathcal{C}_g, \overline{\mathcal{M}}_{1,1})$ of morphisms with cyclotomic twistings $\Gamma$. Consequently, we acquire the exact arithmetic invariants of the moduli for each Kodaira-Néron types which naturally render new sharp enumerations with the main leading term of order $\mathcal{B}^{\frac{5}{6}}$ and secondary & tertiary order terms $\mathcal{B}^{\frac{1}{2}} ~\&~ \mathcal{B}^{\frac{1}{3}}$ on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting elliptic curves over $\mathbb{P}_{\mathbb{F}_q}^{1}$ with additive reductions ordered by bounded height of discriminant $\Delta$. The emergence of non-constant lower order terms due to unstable elliptic curves is in stark contrast with counting semistable (i.e., strictly multiplicative reductions) elliptic curves. In the end, we formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for counting elliptic curves over $\mathbb{Q}$ through global fields analogy. This is a joint work in progress with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).