Uniform Mordell--Lang conjecture for semiabelian varieties

Abstract

We prove the uniform Mordell--Lang conjecture for semiabelian varieties, that is, for a subvariety $X$ of a semiabelian variety, its intersection with a finite-rank subgroup of rational points, can be covered by at most $c$ cosets contained in $X$, where $c$ depends only on the dimension of the semiabelian variety, the degree of $X$, and the rank of the subgroup. Our proof is new even in the abelian case previously proven by Gao--Ge--K\"uhne.