On strong multiplicity one for automorphic representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $\pi$ be a unitary, cuspidal, automorphic representation of $G{L}_n({\text{A}}_K)$. Let $S$ be a set of finite places of $K$, such that the sum ${\sum }_{v\in S}N{v}^{-2\mathrm{/}(n^2+1)}$ is convergent. Then $\pi$ is uniquely determined by the collection of the local components ${\pi_v\mid v\not\in S, ~v ~$\text{finite}$}$of$\pi$. Combining this theorem with base change, it is possible to consider sets $S$ of positive density, having appropriate splitting behavior with respect to solvable extensions of $K$, and where $\pi$ is determined upto twisting by a character of the Galois group of $L$ over $K$.