The large Galois orbits conjecture under multiplicative degeneration

TL;DR

This paper proves the large Galois orbits conjecture for certain curves in the moduli space of abelian varieties with multiplicative degeneration, advancing the understanding of unlikely intersections in algebraic geometry.

Contribution

It establishes the PEL type large Galois orbits conjecture for Hodge generic curves with multiplicative degeneration, completing the proof of Zilber-Pink in specific cases.

Findings

Proves the large Galois orbits conjecture for Hodge generic curves in $\\mathcal{A}_g$ with multiplicative degeneration.

Completes the proof of Zilber-Pink conjecture in $\\mathcal{A}_2$ for such curves.

Deduces new cases of Zilber-Pink in higher dimensions ($g \geq 3$).

Abstract

We establish the PEL type large Galois orbits conjecture for Hodge generic curves in $\mathcal{A}_g$ possessing multiplicative degeneration. Combined with our earlier works, this concludes the proof of the Zilber-Pink conjecture in $\mathcal{A}_2$ for such curves. We also deduce several new cases of Zilber-Pink in $\mathcal{A}_g$ for $g\geq 3$. Our proof uses Andr\'e's G-functions method, using formal and rigid uniformisation of semiabelian schemes to interpret the $p$-adic evaluations of the period G-functions.