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

Let $k$ be a perfect field with $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)\mathrm{\ne }2,3$, 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 ${\bar{\mathcal{M}}}_{1,1}\simeq \mathcal{P}(4,6)$. The Shioda-Tate formula $\rho (S)=T(S)+\mathrm{r}\mathrm{k}(E\mathrm{/}K)$ 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 $s=t^{1\mathrm{/}12}$ 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 $\mathrm{\Delta }=4a_4^3+27a_6^2$; 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 ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. 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 $1\le r\le \lfloor (10n-2)\mathrm{/}(n-1)\rfloor$, there exist infinitely many stable elliptic surfaces with Mordell-Weil rank $\mathrm{r}\mathrm{k}(E\mathrm{/}K)\ge r$, and that infinitely many canonical heights $\widehat{h}(P)=d$ are realized by Mordell-Weil sections.