Appendix: A refinement of the strong multiplicity one theorem for \(GL(2)\)
Let \(F\) be a global field, and let \(S\) be a set of finite places of \(F\) of Dirichlet density strictly less than \(1/8\). Let \(\pi= \otimes_ v \pi_ v\), \(\pi'= \otimes_ v \pi'_ v\), be two irreducible unitary cuspidal automorphic representations of \( GL(2, \mathbb{A}_ F)\) of \(GL (2, A_ F)\). The author shows that if \(\pi_ v\simeq \pi'_ v\) for all finite \(v\) not in \(S\), then \(\pi \simeq \pi'\). This improves the standard strong multiplicity one theorem of \textit{H. Jacquet} and \textit{J. Shalika} [Am. J. Math. 103, 499-558 (1981; Zbl 0473.12008) and 777-815 (1981; Zbl 0491.10020)] where \(S\) is assumed to be finite. A refinement in a different direction was given by \textit{C. J. Moreno} [Am. J. Math. 107, 163-206 (1985; Zbl 0564.10035)]. The result is established by showing that the partial Rankin-Selberg \(L\)- function \(L (\widetilde {\pi} \times \pi', s)\) has a pole at \(s=1\). The proof of this combines analytic methods with the properties of the (self- dual) adjoint square representation \(\text{Ad} (\pi)\) on \(GL (3, \mathbb{A}_ F)\) established on \(GL (3, A_ F)\) established by \textit{S. Gelbart} and \textit{H. Jacquet} [Ann. Sci. École Norm. Supér., IV. Sér. 11, 471-542 (1978; Zbl 0406.10022)].
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