Some Cases of the Zilber–Pink Conjecture for Curves in 𝒜g

Abstract

Abstract Following our work in [ 30], we extend the height bounds established by Y. André in his seminal research monograph [ 1] 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 André’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.