Some results in the Kummer theory of commutative 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 density 17/24. This result has later been interpreted as a statement about the properties of 2 as an element of the multiplicative group of non-zero rational numbers. This point of view eventually led to the development of the so-called Kummer theory of commutative algebraic groups. After a general introduction to the subject, which has connections to many classical problems in number theory, I will discuss some more recent results in which the multiplicative group is replaced by an abelian variety, with a particular focus on the case of elliptic curves.

Based on joint work with Antonella Perucca and Sebastiano Tronto.