FoCM 2014 conference

Workshop A3 - Computational Number Theory

December 12, 17:00 ~ 17:40 - Room A21

## Prime densities for $GL_1$ and $GL_2$

### Peter Stevenhagen

### Universiteit Leiden, Netherlands - psh@math.leidenuniv.nl

If we fix a rational number $x$, Artin's basic question "for how many primes $p$ does $x \bmod p$ generate the multiplicative group of non-zero integers modulo $p$?" leads to Artin's conjecture on primitive roots, and the associated prime density depends in a somewhat non-trivial way on $x$. A conceptual way to compute such densities is given by the character sum method that I developed with Moree and Lenstra, and that exploits Galois representations coming from the multiplicative group.

Artin-type questions also exist in an elliptic setting, as do the associated Galois representations. I will explain how our character sum method extends to this case.

Joint work with Pieter Moree (Max Planck Institut Bonn) and Hendrik Lenstra (Universiteit Leiden).