Zilber-Pink in a product of modular curves assuming multiplicative degeneration

TL;DR

This paper proves a special case of the Zilber-Pink conjecture for certain curves in modular product spaces, using a novel height bounding technique involving G-functions at non-archimedean places.

Contribution

It extends previous results by removing the asymmetry condition, employing a new approach based on relations between G-functions evaluations at multiple non-archimedean places.

Findings

Proves Zilber-Pink conjecture for specific curves in $Y(1)^n$

Introduces a height bound method using G-functions at non-archimedean places

Achieves results beyond previous asymmetry restrictions

Abstract

We prove the Zilber--Pink conjecture for curves in $Y(1)^n$ whose Zariski closure in $(\mathbb{P}^1)^n$ passes through the point $(\infty, \ldots, \infty)$, going beyond the asymmetry condition of Habegger and Pila. Our proof is based on a height bound following Andr\'e's G-functions method. The principal novelty is that we exploit relations between evaluations of G-functions at unboundedly many non-archimedean places.