GAAG seminar - Global to local obstructions for primitivity

An integer is called a primitive root modulo a prime number p when its reduction generates the multiplicative group of the field ofp elements.
Similarly, a point on an elliptic curve over Q is called a primitive point modulo p when its reduction generates the point group of the elliptic curve modulo p.
Proving that a given integers is a primitive root modulo infinitely many primes can only be done under the assumption of Riemann hypotheses, 
and proving primitivity of a given point modulo infinitely many primes is currently beyond our technical means.

Contrary to this, proving that primitivity occurs only modulo finitely many primes tends to be easy, as it is implied by global properties of a “finite" kind.
I will discuss such global properties, which can be of a non-trivial group theoretical nature.

This lecture is based on joint work with Francesco Campagna, Francesco Pappalardi and Nathan Jones.