Hecke orbits and the Mordell-Lang conjecture in distinguished categories
Fabrizio Barroero, Gabriel Andreas Dill
TL;DR
The paper develops a general framework of $\Sigma$-orbits inside distinguished categories to formulate Mordell-Lang-type statements in settings including semiabelian varieties and connected pure Shimura varieties. It proves an analogue of Aslanyan–Daw (Theorem AD1) in this broad context and derives unconditional Mordell-Lang-type results for fibered Legendre families and abelian-type Shimura varieties by leveraging recent work on hybrid Hecke orbits and Zilber–Pink. The approach unifies various notions of Hecke orbits and provides tools to identify finite sets of special/subvarieties (the $\Sigma$-optimal subvarieties) controlling unlikely intersections. These results yield new unconditional statements in several contexts and extend the classical Mordell-Lang paradigm to a wide categorical setting, including explicit applications to Legendre schemes and their fibered powers. The appendix further establishes a Zilber–Pink result for subvarieties of $\mathcal{A}_g$ not definable over $\bar{\mathbb{Q}}$.
Abstract
Inspired by recent work of Aslanyan and Daw, we introduce the notion of $Σ$-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 $Σ$-orbit is a subgroup of finite rank. We show that our $Σ$-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.