Analytically dense Mordell-Weil rank jumps on elliptic surfaces via transverse realization

Abstract

Following Park-Schmitt, let $\mathcal{W}_{\mathrm{min},n}$ denote the moduli stack of minimal Weierstrass fibrations of Faltings height $n\ge2$ over an unparameterized $\mathbb{P}^1_{\mathbb{C}}$. A very general member has Mordell-Weil rank zero. For every integer $r$ satisfying $1\le r\le\left\lfloor\frac{10n-2}{n-1}\right\rfloor$, we prove that the locus of simply-branched elliptic surfaces with only ${\mathop{\rm I}}_1$ singular fibres and Mordell-Weil rank at least $r$ is analytically dense in $\mathcal{W}_{\mathrm{min},n}$. The locus of such surfaces with rank exactly one is also analytically dense. More precisely, within the simply-branched all-${\mathop{\rm I}}_1$ locus, every point is an analytic limit of pairwise non-isomorphic Jacobian elliptic surfaces carrying $r$ independent sections whose canonical heights all tend to infinity. These sections arise from individually primitive integral $(1,1)$ classes imposing $r(n-1)$ independent period conditions. After base change to a marked deformation chart, the deformation germ of each constructed surface with its ordered sections is identified with the corresponding smooth Hodge-locus germ. The proof combines Shepherd-Barron's infinitesimal period calculation with a block Vandermonde construction, lattice approximation, and the holomorphic implicit function theorem.