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

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.