On the analytic strong multiplicity one for SL(2)

With the results on GL(3) and the adjoint lift, a restricted version of analytic strong multiplicity one is proved which holds for SL(2), the restriction being that the representations are required to be generic with respect to the same additive character. In this case there exists a positive constant c depending on e > 0 and the number field F only, such that p can be decided completely by its local components p(v) with norm N(v) < c.C(p )(6+e), where C(p) is the analytic conductor of p with respect to the adjoint lift. A similar argument also allows us to simplify the proof of the restricted version of strong multiplicity one for SL(2) originally given in [J. Chai and Q. Zhang, A strong multiplicity one theorem for SL2, Pacific J. Math. 285(2) (2016) 345-374].