Davide Lombardo

Reductions of points of infinite order on algebraic groups

Over 50 years ago, Hasse proved that the set of prime numbers dividing at least one integer of the form \(2^n+1\) has natural density \(17/24\). One can interpret this result as a statement about the rational point (of infinite order) 2 in \(G_m(Q)\), and this point of view leads to the following general question: fix an algebraic group A over a number field K, a point α ∈ A(K) of infinite order, and a prime l. For "how many" places p of K does l divide the order of α mod p? I will describe a general framework to tackle this and similar questions, and provide an answer when A is the product of an abelian variety and a torus. If time permits I will also discuss an unexpected consequence of the result: the density of such places p is a rational number whose denominator is (essentially) independent of the underlying algebraic group and of the choice of α. This is joint work with Antonella Perucca.