## Distinguishing finite group characters and refined local-global phenomena

### Volume 179 / 2017

Acta Arithmetica 179 (2017), 277-300
MSC: Primary 11F80, 20C15; Secondary 20C33.
DOI: 10.4064/aa170120-1-5
Published online: 11 July 2017

#### Abstract

Serre obtained a sharp bound on how often two irreducible degree $n$ complex characters of a finite group can agree, which tells us how many local factors determine an Artin $L$-function. We consider the more delicate question of finding a sharp bound when these objects are primitive, and answer this question for $n=2,3$. This provides some insight on refined strong multiplicity one phenomena for automorphic representations of $\operatorname{GL}(n)$. For general $n$, we also answer the character question for the families $\operatorname{PSL}(2,q)$ and $\operatorname{SL}(2,q)$.