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

TL;DR

The paper proves the Zilber--Pink conjecture for irreducible curves in $Y(1)^n$ whose Zariski closure in $(\mathbb{P}^1)^n$ passes through $(\infty,\ldots,\infty)$, removing the previous asymmetry restriction. It develops height bounds via André's G-functions method, leveraging new global relations that combine archimedean and non-archimedean period data arising from two elliptic-family frameworks: the $1/j$ and Tate families. By constructing G-functions from locally invariant periods and establishing non-trivial global relations, the authors derive a lower bound for Galois orbits and an effective height bound, which feed into a Pila–Zannier strategy to obtain ZP for lines in $Y(1)^n$ and related cases. The results extend the scope of Zilber--Pink in modular settings and showcase the power of integrating non-archimedean period relations with G-function techniques, suggesting new avenues for effective unlikely intersections in moduli spaces of abelian varieties.

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 André's G-functions method. The principal novelty is that we exploit relations between evaluations of G-functions at unboundedly many non-archimedean places.

Theorems & Definitions (72)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• proof
• Theorem 1.6
• Conjecture 1.7
• Theorem 1.8
• Lemma 2.1