Silverman’s recurrence conjecture over number fields

Determine whether, for every group scheme over the integers whose generic fiber is an irreducible commutative algebraic group of dimension at least two with no unipotent part, and every point with Zariski-dense cyclic orbit in the generic fiber, the denominator equality D_{nP}=D_P holds for infinitely many positive integers n over number fields.

Background

Silverman’s conjecture asks whether a Zariski-dense point on a sufficiently high-dimensional commutative algebraic group returns to the identity with exactly the same local intersection multiplicities as the original point for infinitely many multiplication indices. The paper explains that this conjecture generalizes the normalized Ailon--Rudnick phenomenon and is logically incompatible with a universal higher-dimensional primitive-divisor theorem.

The paper constructs broad families of geometrically nonsplit semiabelian varieties and dense points satisfying the conjectured return behavior, including examples over the rationals and in every dimension at least two. These constructions establish instances of the conjecture but do not settle it for all semiabelian varieties and dense points over number fields.