Francesco Campagna

At a number theory conference in Banff (1988) Erdős raised the following question: Let a,b be two integers with the property that for all positive integers n the set of prime numbers dividing a^n-1 is equal to the set of prime numbers dividing b^n-1. Is then a=b? In collaboration with Gabriel Dill we study the modular analogue of this problem, where the values of the polynomials X^n-1 at integers are replaced by the values of Hilbert class polynomials at elements belonging to various Dedekind domains of interest. Our investigations are motivated by unlikely intersections results of different authors, as well as by some philosophical analogies between the realms of multiplicative groups, abelian and Shimura varieties. I will try to explain all this in the talk.