Some cases of the Zilber-Pink conjecture for curves in $\mathcal{A}_g$

TL;DR

This paper extends height bounds for 1-parameter families of abelian varieties, leading to new unconditional results related to the Zilber-Pink conjecture for curves in the moduli space of abelian varieties.

Contribution

It generalizes André's height bounds without the need for multiplicative reduction, enabling new Zilber-Pink-type results for curves in $\\mathcal{A}_g$.

Findings

Extended height bounds for abelian families.

Unconditional Zilber-Pink-type results for curves.

No assumption of multiplicative reduction needed.

Abstract

Following our work in \cite{papas2022height}, we extend the height bounds established by Y. Andr\'e in his seminal research monograph \cite{andre1989g} for $1$-parameter families of abelian varieties defined over number fields. In our exposition we no longer assume that the family acquires completely multiplicative reduction at some point, as in Andr\'e's original result. As a corollary of these height bounds, we obtain unconditional results of Zilber-Pink-type for curves in $\mathcal{A}_g$, building upon recent results of C. Daw and M. Orr.