Rationality of the trivial lattice rank weighted motivic height zeta function for elliptic surfaces

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.