“Primitive root problems for algebraic groups”
Martedì 25 Febbraio 2025, ore 17:00 - Aula 2BC30 - Francesco Campagna (Maitre de Conference at Clermont Auvergne)
Abstract
In 1927, Artin proposed a conjectural density for the set of primes p for which a given integer g is a primitive root mod p. The problem studied by Artin has an inherently geometric nature as it concerns the multiplicative group scheme over the integers, and thus admits various generalizations to other algebraic groups. Already in the case, first studied by Lang and Trotter in 1977, where the multiplicative group is replaced by an elliptic curve over the rationals, one can observe new phenomena that do not occur in Artin’s original problem.
In this talk, I will trace the history of Artin’s conjecture and explain the heuristic behind it. I will then compare what is known for Artin’s original problem to more recent developments in the context of elliptic curves.
