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Appendix: A refinement of the strong multiplicity one theorem for \(GL(2)\). (English) Zbl 0823.11021

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 H. Jacquet and 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 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 S. Gelbart and H. Jacquet [Ann. Sci. École Norm. Supér., IV. Sér. 11, 471-542 (1978; Zbl 0406.10022)].

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F12 Automorphic forms, one variable
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References:

[1] Gelbart, S., Jacquet, H.: A relation between automorphic forms on GL(2) and GL(3). Ann. Sci. Ec. Norm. Super., IV. Sér.11, 471-542 (1978) · Zbl 0406.10022
[2] Jacquet, H.: Automorphic forms on GL(2). Part II. (Lect. Notes Math., vol. 278) Berlin Heidelberg New York: Springer (1972) · Zbl 0243.12005
[3] Jacquet, H., Shalika, J.: On Euler products and a classification of automorphic representations, I and II. Am. J. Math.103, 499-558 and 777-815 (1981) · Zbl 0473.12008 · doi:10.2307/2374103
[4] Lang, S.: Algebraic number theory. Reading, M.A.: Addison-Wesley 1970 · Zbl 0211.38404
[5] Moreno, C.J.: Analytic proof of the strong multiplicity one theorem. Am. J. Math.147, 163-206 (1985) · Zbl 0564.10035 · doi:10.2307/2374461
[6] Murty, V.K.: Lacunarity of modular forms. J. Indian Math. Soc., New. Ser.52, 127-146 (1987) · Zbl 0673.10020
[7] Serre, J.-P.: Quelques applications du théorème de densité de Chebotarev. Publ. Math., Inst. Hautes Étud. Sci.54, 123-201 (1981) · Zbl 0496.12011 · doi:10.1007/BF02698692
[8] Shahidi, F.: On certain L-functions. Am. J. Math.103 (no. 2), 297-355 (1980) · Zbl 0467.12013 · doi:10.2307/2374219
[9] Shahidi, F.: On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. Math.127, 547-584 (1988) · Zbl 0654.10029 · doi:10.2307/2007005
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