Rationality of the trivial lattice rank weighted motivic height zeta function for elliptic surfaces
Jun-Yong Park
TL;DR
This work studies a trivariate motivic height zeta function for elliptic surfaces, weighting by the trivial lattice and Mordell--Weil data. It develops a local-to-global framework via twisted maps and inertia data to obtain a finite Euler product, proving rationality for the trivial-lattice specialization after inverting the Lefschetz class $\mathbb{L}$ and providing an explicit formula. By contrast, it conjectures irrationality for the Mordell--Weil and Néron--Severi refinements, arguing that global lattice jumps are not captured by the local Kodaira stratification and hence do not admit a finite motivic Euler product. The results highlight a locality-to-globality dichotomy in motivic height zetas for elliptic surfaces and clarify which refinements admit finite descriptions.
Abstract
Let $k$ be a perfect field with $\mathrm{char}(k)\neq 2,3$, set $K=k(t)$, and let $\mathcal{W}_n^{\min}$ be the moduli stack of minimal elliptic curves over $K$ of Faltings height $n$ from the height-moduli framework of Bejleri-Park-Satriano applied to $\overline{\mathcal{M}}_{1,1}\simeq \mathcal{P}(4,6)$. For $[E]\in \mathcal{W}_n^{\min}$, let $S \to \mathbb{P}^1_{k}$ be the associated elliptic surface with section. Motivated by the Shioda-Tate formula, we consider the trivariate motivic height zeta function \[ \mathcal{Z}(u,v;t):= \sum_{n\ge0}\Bigl(\sum_{[E]\in \mathcal{W}_n^{\min}} u^{T(S)}v^{\mathrm{rk}(E/K)}\Bigr)t^n \in K_0(\mathrm{Stck}_k)[u,v][[t]] \] which refines the height series by weighting each height stratum with the trivial lattice rank $T(S)$ and the Mordell--Weil rank $\mathrm{rk}(E/K)$. We prove rationality for the trivial lattice specialization $Z_{\mathrm{Triv}}(u;t)=\mathcal{Z}(u,1;t)$ by giving an explicit finite Euler product. We conjecture irrationality for the Néron-Severi $Z_{\mathrm{NS}}(w;t)=\mathcal{Z}(w,w;t)$ and the Mordell-Weil $Z_{\mathrm{MW}}(v;t)=\mathcal{Z}(1,v;t)$ specializations.