Height moduli of elliptic surfaces: Motivic height zeta rationality and Kudla-Millson modularity of Mordell-Weil rank jumps

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 $n$, constructed via the height-moduli framework of Bejleri-Park-Satriano applied to $\overline{\mathcal{M}}_{1,1}\simeq\mathcal{P}(4,6)$. The Shioda-Tate formula $\rho(S)=T(S)+\mathrm{rk}(E/K)$ decomposes the Picard rank of the associated elliptic surface into the trivial lattice rank, which is local (determined by Kodaira fiber types), and the Mordell-Weil rank, which is global. The motivic height zeta function weighted by the trivial lattice rank is rational in $s=t^{1/12}$ in the dimensionally completed Grothendieck ring, via a combination of exact Euler products on the isotrivial loci $j\equiv 0, 1728$ and a motivic discriminant stabilization adapting Vakil-Wood to $\Delta=4a_4^3+27a_6^2$; over $k=\mathbb{C}$, this yields bidegree-wise Hodge number stabilization. The Kudla-Millson theta correspondence shows that the distribution of new Mordell-Weil sections by canonical height is governed by a modular form of weight $6n-2$ for $\mathrm{SL}_2(\mathbb{Z})$. Combining Shepherd-Barron's diagonalization of the Gauss-Manin connection with Kodaira-Spencer transversality, we establish unconditionally that at every Faltings height $n\ge 3$ and for every $1 \le r \le \lfloor(10n-2)/(n-1)\rfloor$, there exist infinitely many stable elliptic surfaces with Mordell-Weil rank $\mathrm{rk}(E/K) \ge r$, and that infinitely many canonical heights $\hat{h}(P)=d$ are realized by Mordell-Weil sections.