Abstract for the talk on 10.06.2021 (15:15 h)Leipzig seminar on Algebra, Algebraic Geometry and Algebraic Topology
Davide Lombardo (Università di Pisa)
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.