Algebraic Independence of Special Points on Shimura Varieties

TL;DR

This paper proves algebraic independence of images of special points on Shimura varieties under certain correspondences, unifying and extending previous results in unlikely intersections, multiplicative independence, and applications to abelian varieties.

Contribution

It establishes a general algebraic independence result for special points on Shimura varieties, encompassing previous specific cases and applications.

Findings

Proves algebraic independence of special points' images outside a proper Zariski closed subset.

Generalizes multiplicative independence of differences of singular moduli.

Provides applications to the structure of special points in relation to finite-rank subgroups.

Abstract

Given a correspondence $V$ between a connected Shimura variety $S$, a commutative connected algebraic group $G$, and $n \in \mathbb{N}$, we prove that the $V$-images of any $n$ special points on $S$ outside a proper Zariski closed subset are algebraically independent. Our result unifies previous unlikely intersection results on multiplicative independence and linear independence. We prove multiplicative independence of differences of singular moduli, generalizing previous results by Pila-Tsimerman, and Aslanlyan-Eterovi\'c-Fowler. We also give an application to abelian varieties by proving that the special points of $S$ whose $V$-images lie in a finite-rank subgroup of $T$ are contained in a finite union of proper special subvarieties of $S$, only dependent on the rank of the subgroup. In this way, our result is a generalization of the works of Pila-Tsimerman and Buium-Poonen.