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

Jun-Yong Park

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.

Keywords: motivic height zeta functions, elliptic surfaces, Grothendieck ring of stacks and Euler products.

AMS Subject Classification: Primary 14J27, 14D23, 14G10.