Refinements of strong multiplicity one for \(\mathrm{GL}(2)\) (Q2079521)

Let us denote by \(F\) a number field and let \(\mathbb{A}_F\) stand for its ring of adeles. Let \(\pi_1\) and \(\pi_2\) stand for distinct cuspidal automorphic representations of \(\mathrm{GL}_2(\mathbb{A}_F)\). We denote by \(\lambda_{\pi_i}(v)\) the trace of the Langlands conjugacy class of \(\pi_i\) at unramified prime \(v\). For a real number \(\alpha\), we let \(S_{\alpha}\) denote the set of all \(v\) unramified for both \(\pi_1\) and \(\pi_2\) such that \(\lambda_{\pi_1}(v) \neq e^{i \alpha} \lambda_{\pi_2}(v)\). In the paper under the review, the author obtains a refinement of the known lower bounds of the lower Dirichlet density of \(S_{\alpha}\). In particular, it is shown that the lower Dirichlet density of \(S_{\alpha}\) is at least \(\frac{1}{16}\). This generalizes [\textit{N. Walji}, Trans. Am. Math. Soc. 366, No. 9, 4987--5007 (2014; Zbl 1305.11044)].