Hecke orbits and the Mordell-Lang conjecture in distinguished categories

Abstract

Inspired by recent work of Aslanyan and Daw, we introduce the notion of $\Sigma$-orbits in the general framework of distinguished categories. In the setting of connected Shimura varieties, this concept contains many instances of (generalized) Hecke orbits from the literature. In the setting of semiabelian varieties, a $\Sigma$-orbit is a subgroup of finite rank. We show that our $\Sigma$-orbits have useful functorial properties and we use them to formulate two general statements of Mordell-Lang type (one of them implying the other one). We prove an analogue of a recent theorem of Aslanyan and Daw in this general setting, which we apply to deduce an unconditional result about unlikely intersections in a fibered power of the Legendre family. In an appendix, we prove an unconditional Zilber-Pink result for subvarieties of $\mathcal{A}_g$ that cannot be defined over the algebraic numbers.