# Davide Lombardo

University of Pisa

### Reductions of points of infinite order on algebraic groups

Linfoot Number Theory Seminar
27th September 2017, 11:00 am – 12:00 pm
Howard House, 4th Floor Seminar Room

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.