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

Abstract

Following Park-Schmitt, let ${\mathcal{W}}_{\min ,n}$ denote the moduli stack of minimal Weierstrass fibrations of Faltings height $n\ge 2$ over an unparameterized ${\mathbb{P}}_{\mathbb{C}}^1$. A very general member has Mordell-Weil rank zero. For every integer $r$ satisfying $1\le r\le \lfloor \frac{10n-2}{n-1}\rfloor$, we prove that the locus of simply-branched elliptic surfaces with only $I_1$ singular fibres and Mordell-Weil rank at least $r$ is analytically dense in ${\mathcal{W}}_{\min ,n}$. The locus of such surfaces with rank exactly one is also analytically dense. More precisely, within the simply-branched all-$I_1$ 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.

30 pages. New analytic density results, including exact rank one, and local comparison of section moduli with Hodge loci; transversality statements revised, realization proofs rewritten, motivic and modularity sections removed, paper retitled. Comments welcome