On strong multiplicity one for automorphic representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let π be a unitary, cuspidal, automorphic representation of GLn(AK). Let S be a set of finite places of K, such that the sum ∑v S Nv−2/(n2+1) is convergent. Then π is uniquely determined by the collection of the local components {πvv S, v finite} of π. Combining this theorem with base change, it is possible to consider sets S of positive density, having appropriate splitting behavior with respect to a solvable extension L of K, and where π is determined up to twisting by a character of the Galois group of L over K.